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- Permanent Link:
- http://ufdc.ufl.edu/UFE0050230/00001
## Material Information- Title:
- The Effects of Environmental Variation, Density, Reproduction and Size-Selective Fishing Mortality on Fish Life History Traits
- Creator:
- Matthias, Bryan G
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2016
- Language:
- english
- Physical Description:
- 1 online resource (182 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Fisheries and Aquatic Sciences
Forest Resources and Conservation - Committee Chair:
- AHRENS,ROBERT
- Committee Co-Chair:
- ALLEN,MICHEAL S
- Committee Members:
- LINDBERG,WILLIAM J
ST MARY,COLETTE MARIE WALTER,JOHN - Graduation Date:
- 8/6/2016
## Subjects- Subjects / Keywords:
- Female animals ( jstor )
Fish ( jstor ) Fisheries ( jstor ) Fishing ( jstor ) Mating behavior ( jstor ) Modeling ( jstor ) Mortality ( jstor ) Mortality rates ( jstor ) Population growth ( jstor ) Population growth rate ( jstor ) Forest Resources and Conservation -- Dissertations, Academic -- UF evolution -- fisheries -- growth -- lifehistory -- reproduction Greater Orlando ( local ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Fisheries and Aquatic Sciences thesis, Ph.D.
## Notes- Abstract:
- Accurate estimates of life history traits including growth, survival, and reproductive schedules are necessary components of many fisheries population and assessment models. Commonly we assume these traits are temporally static; however life history traits can change as a result of biotic, abiotic, and anthropogenic factors. The purpose of my dissertation was to estimate and investigate the impacts of temporal changes in life history traits as they relate to the development of fisheries regulations. My first objective was to estimate the impacts of density and environmental variation on lifetime growth patterns of Black Crappie (BC). My second objective was to compare the von Bertalanffy model to multiphasic growth models accounting for reproductive schedules of Gag, a protogynous hermaphrodite, and to determine the impacts of using different growth models on fisheries reference points using a yield-per-recruit framework. My final objective was to construct a simulation to predict the evolutionary responses of sequentially hermaphroditic species to exploitation and to determine how mechanisms controlling sexual transition influence these responses. Results from BC showed that density and temperature had the largest impacts on growth in length, but water level and Chlorophyll A also influenced BC growth. These results suggest changes in growth from density and environmental conditions can influence the effectiveness of size-based regulations, especially as environmental variation is expected to increases as a result of climate change. Comparing growth models for Gag indicate that growth patterns change at maturation and transition and using the von Bertalanffy model to describe Gag growth can lead to biased management recommendations. Finally, I found that mechanisms controlling transition had large impacts on the evolutionary trajectories of the timing of maturation and transition, which led to differences in sustainability of hermaphroditic populations to exploitation. Populations in which transition was static or under genetic control were much more likely to suffer from sperm limitation and recruitment failure than populations with social control of transition. Managing fisheries into the future with prospects of climate change and long-term impacts of fishing requires an understanding of how life history traits change through time and how these changes influence the management of exploited resources. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2016.
- Local:
- Adviser: AHRENS,ROBERT.
- Local:
- Co-adviser: ALLEN,MICHEAL S.
- Statement of Responsibility:
- by Bryan G Matthias.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Bryan G Matthias. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Classification:
- LD1780 2016 ( lcc )
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PAGE 1 THE EFFECTS OF ENVIR ONMENTAL VARIATION, DENSITY, REPRODUCTIO N AND SIZE SELECTIVE FISHING MO RTALITY ON FISH LIFE HISTORY TRAITS By BRYAN GLEN MATTHIAS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2016 PAGE 2 2016 Bryan Glen Matthias PAGE 3 To my wife and family PAGE 4 4 ACKNOWLEDGMENTS I would like to thank the National Marine Fisheries Service Recruitment, Training, and Research Program, grant number NA11NMF4550121, for funding and support that made this work possible. I would also like to thank my supervisory committee for their help and guidance throughout this process. I am particularly grateful for the support and guidance from Rob Ahrens and Mike Allen throughout my time here at the University of Florida. I thank my fellow colleagues and students for their insight and help and would specifically like to thank Zach Siders, Ed Camp, and Dan Gwinn. Finally, I would like to thank my wife for her unwavering support. PAGE 5 5 TABL E OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 7 LIST OF FIGURES .......................................................................................................... 8 ABSTRACT ................................................................................................................... 11 CHAPTER 1 INTRODUCTION AND JUSTIFICATION ................................................................ 13 2 DECOUPLING THE EFFECTS OF DENSITY AND ENVIRONMENTAL VARIABILITY ON FISH GROWTH ......................................................................... 20 Introduction ............................................................................................................. 20 Methods .................................................................................................................. 23 Results .................................................................................................................... 27 Discussion .............................................................................................................. 29 3 COMPARISON OF GR OWTH MODELS FOR SEQUENTIAL HERMAPHRODITES BY CO NSIDERING MULTI PHASIC GROWTH ................... 48 Introduction ............................................................................................................. 48 Methods .................................................................................................................. 51 Growth Models ................................................................................................. 52 Yield Per Recruit Models .................................................................................. 53 Results .................................................................................................................... 56 Discussion .............................................................................................................. 58 4 FISHERY INDUCED CHANGES TO LIFE HISTORY TRAITS IN SEQUENTIAL HERMAPHRODITES DUE T O SIZE SELECTIVE FISHING .................................. 77 Introduction ............................................................................................................. 77 Methods .................................................................................................................. 80 Recruitment ...................................................................................................... 82 Survival ............................................................................................................. 83 Maturation ........................................................................................................ 85 Transition .......................................................................................................... 85 Growth .............................................................................................................. 86 Gamete Production .......................................................................................... 87 Reproducti on .................................................................................................... 88 Inheritance ........................................................................................................ 89 Model Initialization and Analysis ....................................................................... 91 PAGE 6 6 Results .................................................................................................................... 92 Evolutionary R esponses to F ishing .................................................................. 92 Reference P oint T rends .................................................................................... 93 Impacts of the M ale S ize A dvantage P ........................................... 95 Discussion .............................................................................................................. 96 5 CONCLUSIONS AND MANAGEMENT IMPLICATIONS ...................................... 110 APPENDIX A CHAPTER 1 R CODE ........................................................................................... 117 Jags Code for von Bertalanffy Growth Model Incorporating Variation in Environmental Conditions and Density .............................................................. 117 B CHAPTER 2 R CODE ........................................................................................... 119 von Bertalanffy Model ........................................................................................... 119 Jags Code for von Bertalanffy Growth Model ................................................. 119 Yield per recruit Code for von Bertalanffy ...................................................... 119 Bi phasic Model .................................................................................................... 122 Jags Code for Bi phasic Growth Model .......................................................... 122 Yield per recruit Code for Bi phasic Model ..................................................... 123 Tri phasic Model ................................................................................................... 125 Jags Code for Tri phasic Growth Model ......................................................... 125 Yield per recruit Code for Bi phasic Model ..................................................... 126 C CHAPTER 3 R CODE ........................................................................................... 131 Function for Starting Values .................................................................................. 131 Static Timing of Transition .................................................................................... 132 Genetic Control of Transition ................................................................................ 140 Social Control of Transition for Well mixed Populations ....................................... 148 Social Control of Transition for Segregated Populations ....................................... 156 LIST OF REFERENCES ............................................................................................. 165 BIOGRAPHICAL SKETCH .......................................................................................... 182 PAGE 7 7 LIST OF TABLES Table page 2 1 List of equations used in the growth model and likelihood along with priors used in estimating Black Crappie growth. ........................................................... 39 2 2 Parameter estimates and 95% credible intervals (CI; in parentheses) from the von Bertalanffy model and the lengthweight relationship along with their associated fixed effects regression coefficients representing density and environmental effects. ........................................................................................ 41 3 1 List of equations used in describing lengthand maturity at age for Gag. .......... 64 3 2 List of equati ons used in the yieldper recruit analysis for Gag. ......................... 66 3 3 Mean parameter estimates and 95% credible intervals (in parentheses) f rom each of the growth models. The tri phasic Lester estimates of g and k are broken into female specific (within the tri phasic Lester column) and male specific estimates (in the male estimates column). Note that and k from the bi and tri phasic Lester growth models were derived from the estimates of h and g .......................................................................................... 67 3 4 Fishing mortality rates resulting in maximum yield per recruit ( FMAX) and spawning stock biomasses of 0.35 of unfished for combined sexes ( F35T), females only ( F35F) and males only ( F35M). The 95% credible intervals for each estimate are in parentheses. The von Bertalanffy and bi phasic Lester models assume sizebased, sex independent natural mortality rates equal to the Brody growth coefficient ( k). The tri phasic Lester model with variable M assumes size based natural mortality changes after sex change and equals the sex specific k whereas the tri phasic Lester model with constant M assumes natural mortality is equal to the femalespecific estimate of k for all ages. ................................................................................................................... 68 3 5 Fishing mortality rates resulting in maximum yieldper recruit (FMAX) and spawning stock biomasses of 0.35 of unfished for combined sexes ( F35T), females only ( F35F) and males only ( F35M). The 95% credible intervals for each estimate are in parentheses. All models assume a sizebased natural mortality rate, with natural mortality constant across all growth models (i.e., M = 0.1342 yr 1). .................................................................................................... 69 4 1 Parameter values used in the individual based model. ..................................... 104 PAGE 8 8 LIST OF FIGURES Figure page 2 1 Environmental factors and indices of abundance from 1998 through 2013 on Lochloosa Lake. ................................................................................................. 42 2 2 Workflow diagram outlining the structure and components used to determine the impacts of density and environmental variation on Black Crappie growth in le ngth and weight. ........................................................................................... 43 2 3 Predicted effects on growth in length and weight of Lochloosa Lake Black Crappie from 1998 to 2013. ................................................................................ 44 2 4 Cohort specific estimates of L ,c and coefficient of variation ( cvc) for Black Crappie from the 1998 2013 cohorts. ................................................................. 46 2 5 Predicted cohort specific von Bertalanffy growth curves for Black Crappie from the 1998 to 2013 cohorts. ........................................................................... 47 3 1 Observed and predicted length, maturity, and sex changeat age of Gag using the von Bertalanffy, bi phasic Lester, and tri phasic Lester models. ......... 70 3 2 Residual plots of the predicted lengthat age for each of the growth models. .... 72 3 3 Yield per recruit, total and sex specific spawning stock biomass per recruit over a range of instantaneous fishing mortality rates and growth models, assuming natural mortality rate linked with k. ..................................................... 73 3 4 Yield per recruit, total and sex specific spawning stock biomass per recruit over a range of instantaneous fishing mortality rates and growth models, assuming a natural mortality rate equal to 0.1342 yr1. ....................................... 75 4 1 Mean length at maturity for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. ....................................................................... 105 4 2 Mean length at transition for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 wi th fishing starting in year 200. ....................................................................... 106 4 3 Sex ratio for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. .......................................................................................... 107 4 4 Spawning potential ratio for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. ....................................................................... 1 08 PAGE 9 9 4 5 Fertilization rate and probability of recruitment failure for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. ............ 109 PAGE 10 10 LIST OF ABBREVIATIONS BC Black Crappie Chl a Chlorophyll A concentration CPUE Catch per unit effort DIC Deviance information criterion Eq Equation (s) F Instantaneous fishing mortality rate F 35F Fishing mortality rate resulting in 35% of unfished female spawning stock biomass F 35M Fishing mortality rate resulting in 35% of unfished male spawning stock biomass F 35T Fishing mortality rate resulting in 35% of unfished total spawning stock biomass F MAX Fishing mortality rate resulting in maximum yield per recruit FWC Florida Fish and Wildlife Conservation Commission NMFS National Marine Fisheries Service SPR Spewing potential ratio SSB /R Spawning stock biomass per recruit SSB F /R Female spawning stock biomass per recruit SSB M /R Male spawning stock biomass per recruit SSB T /R Total spawning stock biomass per recruit VB von Bertalanffy WC White Crappie YPR Yield per recruit PAGE 11 11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE EFFECTS OF ENVIR ONMENTAL VARIATION, DENSI TY, REPRODUCTION AND SIZE SELECTIVE FISHING MO RTALITY ON FISH LIFE HISTORY TRAITS By Bryan Glen Matthias August 2016 Chair: Robert Ahrens Major: Fisheries and Aquatic Sciences Accurate estimates of life history traits including growth, survival and reproductive schedules are necessary components of many fisheries population and assessment models. Commonly we assume these traits are temporally static; however life history traits can change as a result of bi otic, abiotic, and anthropogenic factors. The purpose of my dissertation was to estimate and investigate the impacts of temporal changes in life history traits as they relate to the development of fisheries regulations My first objective was to esti mate the impacts of density and environmental variation on lifetime growth patterns of Black Crappie (BC). My second objective was to compare the von Bertalanffy model to multiphasic growth models account ing for reproductive schedules of Gag a protogynous hermaphrodite, and to determine the impacts of using different growth models on fisheries reference points using a yieldper recruit framework. My final objective was to construct a simulation to predict the evolutionary responses of sequentially hermaphroditic species to exploitation and to determ ine how mechanisms controlling sexual transition influence these responses Results from BC showed that density and temperature had the largest impacts on growth in length, but water level and Chlorophyll A also influenced BC growth. These results suggest PAGE 12 12 changes in growth from density and environmental conditions can influence the effectiveness of size based regulations especially as environmental variation is expected to increases as a result of climate change. Comparing growth models for Gag indicate that growth patterns change at maturation and transition and using the von Bertalanffy model to describe Gag growth can lead to biased management recommendations Finally, I found that mechanisms controlling transition had large impacts on the evolutionary t rajectories of the timing of maturation and transition, which l e d to differences in sustainability of hermaphroditic populations to exploitation. Populations in which transition was static or under genetic control were much more likely to suffer from sperm limitation and recruitment failure than populations with social control of transition. Managing fisheries into the future with prospects of climate change and long term impacts of fishing requires an understanding of how life history traits change through time and how these changes influence the management of exploited resources. PAGE 13 13 CHAPTER 1 INTRODUCTION AND JUST IFICATION The management of exploited species often relies on population models to develop management strategies and regulations (Ricker 1975; Jennings et al. 2001; Radomski et al. 2001) Accurate estimates of life history traits including mortality, growth, and reproductive schedules are necessary components of many of these models. Estimates of these traits are used to derive survival, vulnerability, and fecundity schedules (Beverton and Holt 1957; Ricker 1975; Lorenzen 2000; Walters and Martell 2004) and are often used to estimate stock productivity. Commonly, we assume estimates of these life history traits remain constant or stationary through time (PFMC 2008; Tuck 2009; Whitten et al. 2013) and/or over the lifetime of an individual (but see Gulland 1987; Lorenzen 2000; Lester et al. 2004) Systematic change in these life history characteristic s over time that are not captured in underlying population models could lead to ineffective management recommendations and inappropriate regulations (reviewed in Lorenzen 2016) One of the current challenges in the management of fisheries is accounting for and integrating the impacts of climate change and the impacts that past exploitation into our understanding of current and future stock productivity. Changes in stock productivity can influence biological and ma nagement reference points used to manage exploited fish stocks (see Rijnsdorp et al. 2009; Heino et al. 2013 for review ). Predicting how future stock productivity changes requires an understandi ng of how environmental conditions and exploitation influence mortality, growth, and reproduction (Rijnsdorp et al. 2009; Heino et al. 2013; Morrongiello et al. 2014) It is well supported that environmental conditions (e.g., temperature, water PAGE 14 14 clarity, water level etc.) density, and anthropogenic impacts (e.g., exploitation, ecosystem changes, habitat alterations etc. ) influence natural mortality, growth, and reproductive schedules of fish ( e.g., Fry 1947; J rgensen 1990; Glover et al. 2013) but are commonly ignored and in population models Additionally, e stimates of these traits are highly dependent on the conditions prior to sampling (Lorenzen 2016) and making predictions using these temporal snapshots can be problematic if the underlying conditions are dif ferent or changing through time. How individual life history traits respond to changes in biotic, abiotic, and anthropogenic factors can have large impacts on population productivity and the effectiveness of management regulations. Predicting how populations respond to changing environmental conditions or exploitation requires an understanding of how biotic, abiotic, and anthropogenic factors influence life history traits across various time scales. Short term changes can result from environmental variation (e.g., Gardner 1981; Gaeta et al. 2014; Kazyak et al. 2014) changes in density (e.g., Beverton and Holt 1957; Lorenzen 1996a; Vincenzi et al. 2012) or from failure to account for changes in mortality, growth, reproductive investment, etc. happening over the lifetime o f an individual (e.g., Pauly 1980; St. Mary 1996; Lester et al. 2004) Long term processes can result from long term trends in biotic or abiotic factors, such as climate change (Rijnsdorp et al. 2009; Neuheimer and Grnkjaer 2012; Morrongiello et al. 2014) or long term changes in density dependent proc esses as a result of exploitation (Rijnsdorp and Van Beek 1991; Law 2000; Conover and Munch 2002) Additional long term trends can stem from the selective processes of fisheries that result in evolutionary changes in life history traits, including growth and the timing of maturation or sexual transition (Jrgensen 1990; Rijnsdorp PAGE 15 15 1993; Enberg et al. 2012; Fenberg and Roy 2012) Failure to acco unt for these processes can result in inaccurate representations mortality, growth, and reproductive patterns, which could lead to negative management outcomes as they relate to current management objectives and influence the sustainability of the populati on and fishery (Conover and Munch 2002; Eikeset et al. 2013) Many studies have assessed and developed methods to account for some sources of variation in life history traits. T he impacts of variation in environmental conditions (Vllestad and Olsen 2008; Davidson et al. 2010; Kazy ak et al. 2014) and density (Walters and Post 1993; Post et al. 1999; Lorenzen and Enberg 2002) on fish growth have been assessed. Many population and assessment models incorporate size dependent survival schedules. I n recent years t he development and use of growth models that account for the cost of reproduction has been increasing (see Minte Vera et al. 2015) Finally, e vidence for fisheries induced evolution or change in life history traits has been linked with the selective properties of fisheries (Law 2000; Heino and God 2002; Cassoff et al. 2007; Mollet et al. 2007) W ith the exception of sizebased mortality, many population models still assume temporal invariance of many life history traits and understanding the impacts of this assumption is needed. The effectiveness of many regul ations is influenced by changes in life history traits that arise through variation in environmental conditions and density. Much work has been conducted to address how biotic and abiotic factors impact fish growth, but many of these methods are difficult to transfer to population and assessment models. For instance, many models rely on repeatedly observing individuals through time (e.g., Haugen et al. 2007; Vllestad and Olsen 2008; Davidson et al. 2010; Kazyak et al. PAGE 16 16 2014) or by back calculation techniques (e.g., Maceina and Shireman 1982; Maceina et al. 1991; Maciena 1992; Morrongiello et al. 2011, 2014; Urbach et al. 2012; Glover et al. 2013) Often these studies utilize methods to assess the growth increment that do not rely on an underlying growth function, such as the von Bertalanffy function, making them difficult to incorporate into population models. With concerns of increased environmental variation associated with climate change, it is becoming even more important to develop and utilize methods that assess the impacts of biotic and abiotic variation on fisheries management. In addition to variation in biotic and abiotic factors, many morphological and physiological changes happen over the lifetime of an individual and underst anding how these changes influence management can be challenging. These changes can occur as a result of differences in predation risk with size (Pauly 1980; Gulland 1987; Lorenzen 1996a) ontogenetic shifts (Beverton and Holt 1957; Walters and Martell 2004; Einum et al. 2006) along with changes in reproductive investment that happen over the lifetime of an individual (Charnov 1993; Charnov et al. 2001; Lester et al. 2004; MinteVera et al. 2015) Some of these patterns are commonly incorporated into population models, specifically size dependent natural mortality patter n s (Gulland 1987; Lorenzen 1996a, 2000) or the impacts of size and age on maturation (see Dieckmann and Heino 2007) O thers are often ignored, especially with the frequent use of the von Bertalanffy function to describe the growth patterns of fish. This model assumes all individuals follow a single growth pattern for their entire life. This assumption is often violated at young ages (Beverton and Holt 1957; Ricker 1975; Walters and Martell 2004) and as a result of changes in reproductive investment associated with maturation (Charnov 1993; Day PAGE 17 17 and Taylor 1997; Charnov et al. 2001; Lester et al. 2004) For fish with additional physiological changes happening after maturation, such as sexual transition, growth models incorporating multiple changes in reproductive investment are needed. For sequential hermaphroditic species, ignoring changes in reproductive investment and the subsequent impacts on growth patterns associated with maturation and transition could result i n biased management recommendations. Long term changes in the mean life history traits can cause large impacts on the sustainability of fisheries estimates of stock productivity, and predictions of stock recovery (Law 2000; Conover and Munch 2002; Eikeset et al. 2013) These changes can occur as a result of fishery dependent and fishery independent processes. If these processes result in the preferential reproductive success of individuals that display certain heritable phenotypic traits (e.g., early maturation, high aggression, etc.), then selection will lead to an evolutionary change within the population (Law 2000; Heino and God 2002; Kuparinen and Meril 2007) In recent years the impacts of fishery dependent changes in life history traits, specifically changes in maturation and growth trends, have been extensively studied. T hese trends are usually assessed on dioecious (separate sex) species changes and have largely ignor ed the evolutionary consequences of fishing on sequential hermaphrodites (Hamilton et al. 2007; Sattar et al. 2008; Collins and McBride 2011; Fenberg and Roy 2012; Mariani et al. 2013) By ignoring the evolutionary consequences of sizeselective fishing on sequential hermaphroditic species, managers run the risk of reducing the long term sustainability of the population and fishery yi elds. These consequences can arise from misidentifying populationlevel responses to fishing by assuming static maturation and sexual PAGE 18 18 transition as opposed to changes in the timing of maturation or sexual transition. Therefore, understanding how hermaphroditic populations respond to exploitation can provide valuable insight into possible management strategies of hermaphroditic species. T o improve the management of exploited resources, managers and researchers need to consider models that account for changes in life history traits of individuals and populations. F or my dissertation I sought to develop methods to both explore the impacts of variation in life history traits and allow for easy incorporation into population models. The overall objective of my dis sertation was to estimate and investigate the impacts of temporal changes in life history traits as they relate to the development of fisheries regulations aimed at achieving common fisheries management objectives Because variation in life history traits can impact many different aspects of fishery management, I chose to focus my dissertation on three separate life history components using two case studies and a simulation My second chapter focused on estimating the impacts of variation in density and env ironmental conditions on the lifetime growth patterns of a short lived mesopredator. Black Crappie Pomoxis nigromaculatus were chosen because they are known to have highly variable growth (see Maceina and Shireman 1982; Miller et al. 1990; Allen et al. 1998) making the species a good candidate to estimate the impacts of environmental conditions and density on growth. For the third chapter, I sought to develop and assess the management implications of using models that accounting for changes in growth patterns following maturation and sexual transition of a sequential hermaphrodite. In this study Gulf of Mexico Gag were chosen because t he timing of maturation and sexual transition has remained relatively stable since the 1970s (SEDAR 2014) and there was PAGE 19 19 a large data set on length, maturity and transitionat age data, making this population an ideal example to compare growth models The fourth chapter focused on predicting the evolutionary and population level consequences of sizeselective harvest on sequential hermaphrodites. Finally, I discuss the overall implications of this research on fisheries management. PAGE 20 20 CHAPTER 2 DECOUPLING THE EFFECTS OF DENSITY AND ENVIRONMENTAL VARIABILITY ON FISH GROWTH Introduction Management of fish populations often relies on estimates of growth to develop size and harvest limits (Ricker 1975; Jennings et al. 2001; Radomski et al. 2001) by providing information to derive vulnerability, fecundity, and survival schedules for population and assessment models (Ricker 1975; Lorenzen 2000; Walters and Martell 2004) It is common to assume that estimates from these growth models remain constant through time (PFMC 2008; Tuck 2009; Whitten et al. 2013) even though changes in growth rates can be density dependent or vary with environmental variation (see Beverton and Holt 1957; Lorenzen and Enberg 2002; LobnCervi 2010 and references therein). Detecting variation in growth caused by changes in density and fluctuating environments can be confounded due to concurrent changes in both density and the environment (Barbraud and Weimerskirch 2003; Haugen et al. 2007; Vllestad and Olsen 2008; Davidson et al. 2010) With growing concerns related to the impacts of climate and ecosystem changes on exploited wild populations, assessing the combined impacts of density dependent and density independent effects on fish growth rates is becom img increasingly important to understand the dynamics of exploited populations (Vllestad and Olsen 2008) and the effectiveness of management options. Density dependence arises from resource competition and generally affects mortality and/or growth, but the magnitude of these effects depends on the life stage and competit ive asymmetries among individuals (e.g., juvenile vs. adult; for review see Vincenzi et al. 2012) During the early stag es of life, density dependent mortality is the primary mechanism of population regulation via impacts on numerical abundance (Elliott PAGE 21 21 1994; Milner et al. 2003; Vincenzi et al. 2012) For larger juveniles and adults, d ensity dependent growth regulates population biomass and reproductive output as a result of interactions with fecundity schedules and the timing of maturation (Lorenzen 2008) Because growth depends on food availabili ty, r eductions in per capita food availability cause an inverse relationship between density and mean growth rates as a result of fewer prey per predator or reduced capture efficiency (e.g., Wa lters and Post 1993; Post et al. 1999; Lorenzen and Enberg 2002) Environmental factors, such as temperature (see Fry 1947; De Staso and Rahel 1994; Kazyak et al. 2014) water clarity (see Gardner 1981; Craig and Babaluk 1989) and water level (see Gaeta et al. 2014) have also been found to influence growth. One of the mos t important and well studied environmental factors influencing growth is temperature because it influences consumption, metabolism, and behavior of fish (Fry 1947; De Staso and Rahel 1994; Kazyak et al. 2014) W ater clarity also plays a major role in fish growth due to its influence on f oraging rates and thus condition and growth of many visual predators (Gardner 1981; Craig and Babaluk 1989) Effects of fluctuating water level on growth have received much less attention than other environmental factors, but is of increasing importance due to predicted increase of droughts in response to climate change (Lake 2011; Romm 2011) Reductions in lake levels often decrease the amount of available littoral habitat and result in the loss of important structures such as fallen trees (Ficke et al. 2007; Lake 2011; Gaeta et al. 2014) Water level fluctuations also change the foraging arena by concentrating or dispersing predators and prey int o smaller or larger areas effectively changing density if total abundance remains relatively stable. Because populations are often experiencing PAGE 22 22 changes in density at the same time the environment is changing, understanding the combined e ffects of both density dependence and environmental variation on fish growth is critical for managing fisheries in the face of climate change and other anthropogenic influences. Understanding factors that affect growth in fish populations is critical for management, particularly in a changing climate. Changes in growth rates may diminish the effectiveness of sizebased r egulations and such regulations could allow periodic overfishing if growth rates are changing through time in response to changes in density and environmental conditions. Our objective was to determine the impacts of density and environmental variation on lifetime growth using Black Crappie Pomoxis nigromaculatus (hereafter referred to as BC) in a north central Florida lake as a case study Because fish growth is highly plastic and growth variation can manifest in changes to both length and weight, ignoring changes in one could result in the misspecification of the impacts of either density of environmental variation on overall fish growth. Therefore I chose to assess both using a set of nonlinear models. Growth variation in length was assessed using the von Bertalanffy growth model incorporating hierarchical mixed effects to estimate time and cohort specific growth parameters. Variation in mean weight was assessed using a nonlinear Bayesian fixed effects model. Black Crappie provide a unique study species due to their relatively short life span (<10 years in Lochloosa Lake) and highly variable recruitment allowing large fluctuations in population size over short time periods. Additionally, north central Florida has experienced several droughts and hurricanes over the past 15 years, resulting in a high PAGE 23 23 level of environmental variation and allowing us to assess the impacts of fluctuating water levels on BC growth. Methods Black Crappie length, weight, and age data were obtained from Florida Fish and Wildlife Co nservation Commission (FWC) annual trawl ( October or November 19982013) and recreational catch sampling ( January through April 20062013) from Lochloosa Lake in north central Florida (see Tuten et al. 2008, 2010 for description of trawl and trawling methods) Only trawl data were used to assess variation in the lengthweight relationship because fish weights were not obtained from recreational catch samples. A ges of 0 and 1 year old fish from the trawl surveys were estimated from a length distribution combined with age verification All other fish, along with the age 0 and 1 subsample, were aged using either whole or sectioned otoliths following Florida FWC protocols outlined in (Tuten et al. 2008, 2010) Brief ly, two independent readers examined whole otoliths and if three or more annuli were found, one of the otoliths was sectioned. Fractional ages were used in the analysis and all fish were assumed to have a birthdate of March 1st. N o fish captured in the 1998 2000 and 2004 trawl surveys were aged; therefore 0 and 1 year old fish were obtained only from the length distribution during these years Recreational catches were sampled from discarded carcasses obtained at fish camps and boat ramps at Lochloosa Lake (for detailed methods see (Wilson et al. 2015) Water quality data ( i.e., Chloro phyll A concentrations, water level, and water temperature) were obtained from the St. Johns Water Management District (2014) Water quality surveys were usually completed monthly or every other month from 1997 throug h present ( Figure 2 1 ). PAGE 24 24 For the analysis, environmental data were aggregated at the monthly level and missing data points were averaged over the two closest sample dates. Year specific estimates of relative density were assumed t o be proportional to the trawl catch per unit effort (CPUE) of all fish greater than or equal to age1 during the specified year ( Figure 2 1 ). Relative cohort density was assumed to be proportional to the age0 trawl CPUE, and all years had an estimate of age0 CPUE ( Figure 2 1 ). All CPUE and environmental variables used in the model were standardized to have a mean of zero and a standard deviation of one for the benefit of model convergence and stability. Environmental effects assessed were relative density during the sampling year and cohort specific relative density along with mean chlorophyll A concentrations (hereafter referred to as Chla), water level, and temperature. Table 2 1 contains a list of all equations, priors, and likelihood functions us ed in the analysis, and the workflow diagram ( Figure 2 2 ) shows the links between the data, effects, and model components used in the analyses. An incremental von Bertalanffy growth function was fit ted to the combined lengthat ag e data from both gears using a Bayesian hierarchical mixed effects model to estimate impacts of density and environmental variation on length (skeletal growth; Eq 2 1 2 2, 2 3 and 2 20; Table 2 1 ). The incremental form of the v on Bertalanffy growth function was used because it can be easily modified to account for changes in growth during time t (where t is the length of time since the cohort was last observed). This can be accomplished by multiplying the predicted growth incr ement ( i.e., the expected increase in length over t ) by a lognormal regression ( t) ) representing conditions present during t ( Eq 2 1 and 2 3; Table 2 1 ). The von Bertalanffy parameters k and to PAGE 25 25 had uninformative priors and values for L,c and cvc were estimated using cohort specific effects ( Eq 2 6 to 2 13: Table 2 1 ). The model incorporated both cohort specific and timespecific effects ( Figure 2 2 ) Cohort specific effects can be defined as factors that affect a specific cohort, such as within cohort competition, whereas timespecific effects are factors that affect all cohorts over t such as the mean water level from time t to time t+t O nly rando m effects for cohort specific L,c and cvc were used with the von Bertalanffy model due to non convergence when adding cohort specific fixed effects ( Eq 2 1, 2 2 and 2 6 to 2 13; Table 2 1 ). Random effect priors were informed via uninformative hyper parameters representing the overall mean ( and ) and precision ( and ) for each set of random effects. Time effects were incorporated into the incremental growth model using a lognormal regression acting on the growth increment over t ( Eq 2 1 and 2 3; Table 2 1 ). Regression coefficient priors were uninformative ( Eq 2 18; Table 21 ) and t ime effects assessed were relative density during the sampling year along with mean Chla, water level, and temperature between sampling events at time t and time t+t An agespecific gear effect was implemented into the likelihood component due to the selectivity of recreational anglers Recreational anglers tend to harvest BC larger than 20 cm and select for older ages (ages 36; Miranda and Dorr 2000; Wilson et al. 2015) For harvest oriented fisheries the size selective nature results in the removal of the largest individuals in the younger age classes ( i.e., ages 23; Conover and Munch 2 002; Hamilton et al. 2007; Conover et al. 2009) and can result in biased estimates of mean length at age for those younger age classes (Miranda et al. 1987) Therefore I applied a recreational angling specific gear effect to represent the bias associated with PAGE 26 26 the preferential removal of the largest individuals in the youngest age class es ( Eq 2 14 and 2 20; Table 21 ). I assumed t hat the recreational angling specific bias was greater at younger ages and approached zero at older ages ( i.e., both gears appeared to have unbiased estimates of mean length in oldest age classes [ages 710]) and I arbitrarily chose age2 to represent this agespecific bias. An uninformative prior was used to inform the gear effect gi and a positive estimated value of gi would confirm my hypothesis that anglers select larger fish at younger ages. A negative estimated value would indicate that anglers select smaller fish at younger ages and an estimated gi with 95% credible intervals containing zero would indicate similar bias between the gears. Variation in mean weight ( i.e., allometric growth) was assessed using a modified lengthweight relationship ( Eq 2 4, 2 5, and 2 21; Table 21 ). The model incorporated changes only in the shape parameter describing fish body length ( e.g., b ) using a fixed effects regression ( Eq 2 4 and 2 5; Table 21 ) The a parameter was assumed to remain constant over time as it converts lengthb to the appropriate unit of weight (e.g., millimeters to grams, centimeters to kilograms, etc.) and I used an uninformative prior on the log of a ( Eq 2 4 and 2 15; Table 21 ) Because weight data w ere not obtained from recreation al catches, Chla, water level and temperature were averaged over the year prior to the trawl sample. Additionally, the regression coefficient for cohort density converged in the model and therefore was included as a fixed effect in the model ( Eq 2 5; Table 21 ). All regression coefficient priors were used in the model were uninformative ( Eq 2 19; Table 21 ). A single coefficient of variation with an uninformative prior was used to describe the uncertai nty around the weight at length relationship ( Eq 2 17 and 2 21; Table 21 ). PAGE 27 27 Models were run concurrently in program R version 3.1.3 using runjags version 3.3.0 (Denwood 2013; R Development Core Team 2013) Bayesian models were run using four chains and for one million iterations with a thinning rate of fifty. Convergence was verified using Gelman and Rubin diagnostics (Gelman and Rubin 1992; Brooks and Gelman 1998) Results A total of 6,195 Black Crappie were sampled with 4,156 (67%) from FWC trawl surveys from 19982013 and 2,039 (33%) from recreational catch samples from 2006 2012. The trawl survey s elect ed primarily for young ( less than age3 ), but did catch larger and older fish as well. Recreational anglers select ed for larger, older fish ( greater than age3 ). C hl a concentrations were quite variable between 19982013 ( Fig ure 2 1 ) rang ing from 0 273 mg/m3 and were, on average, higher from 1998 2002 with another peak between 20082009 ( Figure 21 ). Water level was also dynamic, ranging from a maximum depth of 1.23.6 m and fluctuated throughout this time period ( Figure 21 ). Water temperature seasonally fluctuated, with maximum summer water temperatures remaining relatively consistent each year between 28 and 32C and minimum winter water temperatures reaching between 7 and 18C ( Figure 21 ). Trawl CPUE for age 1+ BC were highest between 2005 and 2008, indicating years of high relative densities, and appeared to be low in 2000, 2002, 2010, and 2011, indicating years of low relative densities ( Figure 21 ). Age 0 trawl CPUE varied through time with an exceptionally large recruiting class in 2006 and relatively large cohorts in 2004, 2005, and 2010 ( Figure 21 ). Populationlevel a verage von Bertalanffy parameter estimates were 33 c m for 0.40 yr1 for k, 0.27 yr for t0, and 0.13 for ( Table 2 2 ). A nglers caught larger fish PAGE 28 28 than the trawl at younger ages and the model was able to account for this bias by estimating a positive, age specific gear effect for angler caught fish ( Table 22 ; Figure 2 3 A ). All fixed effects acting on the growth increment (in length) had 95% credible intervals that did not contain zero, with age 1+ trawl CPUE and temperature showing negative relationships and both Chla and water level showing positive effects on growth ( Table 22 ; Figure 23 B E ). Predicted impacts on growth (arbitrarily represented as length at age1 in Figure 23 ) showed that age1+ trawl CPUE and water temperature had the largest impacts on growth in length (predicted lengths at age1 from about 11 cm to 16 cm over the range of observed values), while Chla and water level had less of an impact (predicted lengths at age1 from about 12 cm to 14 cm; Figure 23 B E). The weight length parameters were also influenced by density and environmental conditions ( Table 22 ). Age 0 trawl CPUE, representing cohort specific relative density, had a positive effect on weight at length, while age1+ trawl CPUE, representing year specific relative density, and C hl a both had negative effects on weight at age1 ( Table 2 2 ). The impacts of both age0 and age1+ trawl CPUE effects were relatively small compared to the effects of Chl a on allometric growth in weight of an average age1 fish (26 27 g and 2528 g vs. 22 32 g respectively; Table 22 ; Figure 23 F J). Water level and temperature had 95% credible intervals containing zero, indicating no effect on allometric growth ( Figure 2 3 P Q). Cohort specific asymptotic length ( ) and growth variation ( cvc) indicate d large differences in the growth potential and variation around the mean growth of each cohort from 1998 to 2013 ( Figure 2 4 ). Cohort specific , ranged from 28 t o 3 8 cm for the PAGE 29 29 2000 and 2012 cohorts respectively. Estimates of cohort specific cvc ranged from 0.09 to 0.23 for the 2001 and 2011 cohorts respectively. The poster ior distributions of showed higher than average growth potential for the 2001, 2007, 2008 and 2012 cohorts and lower than average growth potential for the 2000 and 2010 cohorts. For cvc, the posterior distributions showed that the 20102011 cohorts had high variation around mean growth and only the 2007 cohort had lower than average grow th variation. Growth curves varied substantially among cohorts ( Figure 2 5 ) and generally the 95% credible intervals for the cohort specific mean relationship did not contain the overall mean growth curve. The model was able to capture cohort specific growth trajectories for all cohorts, regardless of sample size or age distribution ( Figure 25 ). T he growth of older (age 2+) individuals for certain cohorts was sometimes underestimate d ( e.g., 2004, 2009, and 2010 cohorts). Discussion Variation in density dependent and independent factors caused substantial impacts on BC growth in both length and weight, and the impacts generally had a greater effect on growth in length than on the allometric growth parameter An environmental factor had the greatest impact on allometric growth compared to density related factors, similar to Haugen et al. (2007) Vllestad and Olsen (2008) and Davidson et al. (2010) but this trend was not evident with growth in length. As observed in many other studies (e.g., Beverton and Holt 1957; Walters and Post 1993; Post et al. 1999; Lorenzen and Enberg 2002; Sass et al. 2004; Casini et al. 2014) I identified decreased growth in both length and weight associated with increases in year specific density T emperature had the greatest environmental effect on BC growth in length, but PAGE 30 30 I found no impact of temperature on the lengthweight relationship. O ur study also showed that higher water level s and Chl a concentrations led to increases in growth in length. For growth in weight, no evident relationship was found with water level, but decreased growth in weight w as associated with high Chl a concentrations Because of the high plasticity in fish growth, understanding the relationships between density, environmental conditions, and growth in exploited fish populations is necessary for managing fish populations in highly dynamic ecosystems and under changing climates. Even though BC often experience highly variable growth patterns, relatively few studies have assessed temporal trends in density dependent growth of BC. Schramm et al. (1985) and Miller et al. (1990) noted temporal changes in BC growth before, during and after a largescale experimental commercial fishery from 1976 to 1987 for BC and Lepomis spp. but these studies did not quantify the relationship between density and growth. Growth of White Crappie (hereafter referred to as WC) P. annularis incr eased after stocking of a predator in impoundments (Gabelhouse 1984; Boxrucker 1987, 2002; Galinat et al. 2002) but these s tudies failed to assess the density growth relationship. Allen et al. (1998) found a quadratic density growth relationship across Florida BC populations with the lowest growth rates at intermediate densities. Pope et al. (2004) found a linear relationship between density and growth in age 02, but not age 3+ W C in reservoirs across Texas. Guy and Willis (1995) sampled BC across South Dakota reservoirs and lakes and found a negative relationship between log (CPUE) and growth determined via back calculation methods and condition. The studies by Guy and Willis (1995), Allen et al. (1998), and Pope et al. (2004) assessed the density growth relationship across water bodies, but did not evaluate temporal trends in growth. Unlike PAGE 31 31 the previously mentioned st udies, this study not only provides evidence of temporal density dependent growth in a BC population, but also quantified the density growth relationship. Similar to many studies (e.g., Beverton and Holt 1957; Walters and Post 1993; Post et al. 1999; Lorenzen and Enberg 2002; Sass et al. 2004; Casini et al. 2014) higher relative densities w ithin a year were associated with decreased fish growth in both length and weight. I also estimated slightly fatter fish associated with cohorts of higher relative density, contrary to expect ations This suggests that there were other factors associated wi th cohort specific relative densities that I did not account for, such as 1) changes in the functional form of density dependence ( e.g., density dependent mortality verses growth) through the lifetime of each cohort ; 2) trawl CPUE was not an appropriate index of cohort strength; and/or 3) incorrect form of the density growth relationship. Future studies should seek to understand how these different hypotheses and model structures can influence density dependent effect s in growth and will provide valuable guidance to detect ing density dependent impacts to growth in exploited fish populations Density dependent changes in growth result from changes in per capita prey availability that leads to an inverse relationship bet ween density and growth or survival In this study I focused solely on density dependent growth in this analysis T he magnitude of density effects depends on competitive asymmetries between individuals of different sizes or stages ( e.g., juvenile vs. adult ) and on differences in resource use across life stage (e.g., ontogeny; Walters and Post 1993; Post et al. 1999; Lorenzen and Enberg 2002; Einum et al. 2006; Vi ncenzi et al. 2012) Juvenile fish appear most PAGE 32 32 sensitive to competition from densities within their cohort, but as fish transition into adulthood, competition among cohorts is expected to be of greater consequence (Vincenzi et al. 2012) As competition transitions from intr a to int e r cohort competition, the per capita rate influenced by density shifts from mortality (or mortality growth interactions via size based mortality) on young fish to growth on adults (e.g., biomass accumulation and egg production; Walters and Post 1993; Post et al. 1999; Lorenzen and Enberg 2002; Einum et al. 2006; Vincenzi et al. 2012) Because g rowth observations for fish younger than six months old were lacking and I was interested in the effects of density dependent growth, ignoring the role of density dependent mortality likely did not impact the results except through its influence on adult density Many fishery assessment models often use CPUE as an index of abundance (see Campbell 2015) This assumes a direct relationship between catch and abundance ( e.g., constant catchability) and these trends can often be misleading due to changes in catchability (Clark 1985; Harley et al. 2001; Bishop 2006; Ye and Dennis 2009; Hangsl eben et al. 2013) C atchability is often influenced by environmental factors such as water clarity, temperature, and habitat types (Kirkland 1965; Simpson 1978; Gillialand 1987; Danzmann et al. 1991; Hangsleben et al. 2013) Additionally, fish density can influence the capture efficiency of bottom trawls and can result in either hyperdepletion or hyperstability depending on which species is targeted (God et al. 1999; ODriscoll et al. 2002; Hoffman et al. 2009; Kotwicki et al. 2013, 2014) Changes in environmental conditions and relative density that were observed on Lochloosa Lake between 1998 and 2013 likely introduced bias in the estimates of relative density used in this study but the magnitude and direction of these effects on trawl CPUE remain PAGE 33 33 largely unexplored. Therefore, further work is needed to address the applicability of using trawl CPUE as an index of abundance for Lochloosa Lake BC Density dependent relationships have been described using many different mathematical relationships with linear and negative power functions being two of the most common forms (Beverton and Holt 1957; Lorenzen 1996b; Lorenzen and Enberg 2002; Grant and Imre 2005; LobnCe rvi 2007, 2010) For this study I assumed a lognormal relationship between trawl CPUE h owever the model essentially predicted a linear trend between BC trawl CPUE and growth. This implies that growth will change systematically over a range of populati on densities, similar to the trends predicted by Beverton and Holt (19 57) Lorenzen (1996b) and Lorenzen and Enberg (2002) If the density dependent effects on growth were only evident at low population sizes, a negative power function would have been more appropriate (Grant and Imre 2005; LobnCervi 2007) In this study, I identified both positive and negative impacts of C hl a on BC growth. Growth in length increased and growth in weight decreased with increasing C hl a concentrations. Because C hl a is correlated with increased turbidity and decreased water clarity, I focus on their combined effects on fish growth even though I did not directly test the effects of turbidity or water clarity on BC growth. Numerous studies have assessed the impacts of water clarity on fish growth and foraging efficiency. For many species including BC, decreased water clarity has been found to decrease the activity of visual predators foraging efficiency, reactive distances, and feeding rat es (Moore and Moore 1976; Vinyard and OBrien 1976; Ellison 1984; Miner and Stein 1993) It follows then that increased water clarity was associated with higher growth PAGE 34 34 rates and increased condition of many species (Sto ne and Modde 1982; Craig and Babaluk 1989; Hoxmeier et al. 2009) similar to what was observed between BC growth in weight and C hl a Hydrilla Hydrilla verticillata an invasive submersed aquatic plan present in Lochloosa Lake also interacts with C hl a turbidity and water clarity (Canfield et al. 1983, 1984) Hydrilla and its associated epiphyton competes with planktonic algae for nutrients, decr eases wind based resuspension of particles and nutrients, increases planktonic algal sedimentation due to reduced water turbulence, and reduces overall planktonic Chl a concentrations (Canfield et al. 1983, 1984) Generally Hydrilla affords increased invertebrate abundance while decreasing the accessibility of prey (Maceina a nd Shireman 1982; Maceina et al. 1991) leading to decreas es in predator activity and foraging success due to high stem densities (Crowder and Cooper 1979; Savino and Stein 1982) Because juvenile BC rely upon zooplankton and openwater insect larvae (Keast 1968; Edwards et al. 1982) and Threadfin Shad Dorosoma petenense (a pelagic planktivore) are a major component of adult BC diets (Keast 1968) it is not surprising that numerous studies have shown increased BC growth after Hydrilla removal (Maceina and Shireman 1982; Maceina et al. 1991) Therefore it is likely that the positive relationship between C hl a and growth in length observed in this study is at least partially impacted by the presence of Hydrilla in Lochloosa Lake, but the extent of which is unknown. Impacts of water temperature are one of the most studied environmental factors because temperature is a major factor determining whether or not growth will happen (Allen 1941) The relationship between growth rate and temperature should be domeshaped with the highst growth rates at an optimal temperature and decreasing growth PAGE 35 35 rates as temperatures deviate from optimal ( Parker 1974) I ncreases in temperature up to the optimal correlate with increased catabolism (proportional to k from the von Bertalanffy equation) and greater food intake (e.g., Wingfield 1940; Haugen et al. 2007; Vllestad and Olsen 2008; Davidson et al. 2010; Morrongiello et al. 2011, 2014) At temperatures above optimal catabolism should continue to increase (higher k) but we expet to see decreased in growth due to lower consumption rates (Hayward and Arnold 1996; Hale 1999; Bajer 2005; Kazyak et al. 2014) I n WC m aximum consumption was found to be around 24C with a sharp decline in consumption around 27C (Hayward and Arnold 1996; Bajer 2005) Additionally in Crappies Pomoxis spp., temperat ures above 2628C have been found to increase mortality and decrease growth rates if sufficient prey is not available (Ellison 1984; Hale 1999; Michaletz et al. 2012) Assuming thermal limits between BC and WC are similar due to the spatial overlap in their range and habitat, BC in Lochloosa Lake experienced increased stress for about four months per year (e.g., temperatures exceeding 26C) and temperatures above 30C almost yearly between 20042012. The model identified a relationship between temperature and growth in lengt h but not weight, suggesting adequate prey was available when temperatures exceeded 26C to prevent starvation. Of further consequence, this model predicted the best growth when temperatures were the coldest I t was possible that BC experienced the fastes t growth during low temperatures (16 C average), but this relationship was a result of assuming a lognormal relationship between temperature and growth when a parabolic model would have been more appropriate. I was not able to address this issue due to the lack of data during the hottest months of each year (i.e. there were no observations from May through PAGE 36 36 September ) and attempts to fit a parabolic temperaturegrowth relationship did not converge. Effects of fluctuating wa ter level in lakes on growth have received much less attention than other environmental factors, but is of increasing importance due to predicted increase of dr o ughts due to climate change (Lake 2011; Romm 2011) Reductions in lake levels often decrease the amount of littoral habitat available (Ficke et al. 2007; Lake 2011) due to the loss of important structures such as fallen trees (Gaeta et al. 2014) Consistent with studies on Largemouth Bass Micropterus salmoides (Gaeta et al. 2014) and Golden P erch Macquaria ambigua (Morrongiello et al. 2011) reductions of growth in length were associated with decreased water levels. For Lochloosa Lake BC, I found no significant effects on growth in weight as a result of water levels This suggests that BC were still able to attain adequate food resources to maintain weight, but not enough to add additional resources to increasing length. With the prospects of increased environmental variability along with increased duration of droughts and wet periods (IPCC 2015) understanding the impacts of fluctuating water levels in lakes on fish growth will be needed for additional species and other life history traits, such as mortality Studies assessing the impacts of environmental variation often use methods such as comparing growth over large spatial areas ( e.g., Donald et al. 1980; Guy and Willis 1995; Allen et al. 1998; McInerny and Cross 1999; Hoxmeier et al. 2009; Gertseva et al. 2010; Casini et al. 2014) or growth increment analyses to assess temporal changes in growth. Growth increments can be obtained via mark recapture methods ( e.g., Haugen et al. 2007; Vllestad and Olsen 2008; Davidson et al. 2010; Kazyak et PAGE 37 37 al. 2014) or b y using back calculation methods ( e.g., Maceina and Shireman 1982; Maceina et al. 1991; Maciena 1992; Morrongiello et al. 2011, 2014; Urbach et al. 2012; Glover et al. 2013) Back calculation methods suffer from a range of problems, such as the Lees phenomenon (Lee 1912) and assume density and environmental factors influence structure growth ( e.g., scale, otholith, spine, etc.) in similar ways that somatic growth is influenced. M ethods used to assess temporal variation in growth generally do not assume an underlying growth curve ( e.g., the von Bertalanffy growth function) making it difficult to translate these results into assessment or agestructure population models. I used repeated measures of independent lengthat age observations to test for changes in growth using the von Bertalanffy growth function, which means that it can be directly incorporated into assessment methods or agestructured models. In addition, many populations already have a time series of lengthat age data and constructing models similar to the one developed in this manuscript will be relatively easy if there is adequate environmental data. Understanding temporal variation in fish growth rates caused by changes in density or the environment will be needed to manage fisheries as environments become more variable. By ignoring variation in fish growth and assuming static lifetime growth could lead to changes in the efficacy of management strategies, especially sizebased regulations (e.g., length limit s). As this and other studies show (Barbraud and Weimerskirch 2003; Haugen et al. 2007; Vllestad and Olsen 2008; Davidson et al. 2010) the combined effects of density dependence and environmental variation can cause substantial impacts on fish growth. W ith the prospects of increased environmental variation associated with climate change (IPCC 2015) ignoring t emporal PAGE 38 38 variation in fish growth may no longer be an option and understanding how changes in fish growth can impact the effi cacy of size based regulations is needed. PAGE 39 39 Table 21 List of equations used in the growth model and likelihood along with priors used in estimating Black Crappie growth. Component Equation Growth Equations 2 1 Predicted length L at age a for an individual in cohort c at time t where , = ? , + rk , ( 1 rF exp ( rF G P ) ) exp ( ) 2 2 Predicted length L at age 0 for an individual in cohort c = ( 1 exp ( ) ) 2 3 Time effect t and time + = W of an individual from cohort c at time t = , c and time specific t shape parameter b describing fish body shape = + + Growth parameter priors 2 6 Cohort specific ~ ~ ( 0 10 ) 2 8 Precision hyper prior for random cohort effect on ~ ( 1 1 ) 2 9 Brody growth coefficient ~ ( 1 1 ) 2 10 Theoretical age at which length is zero ~ ( 0 10 ) 2 11 Cohort specific cv exp ( ) ~ ( ) 2 12 Mean coefficient of variation over all cohorts ~ ( 0 10 ) 2 13 Precision hyper prior for random cohort effect on ~ ( 1 1 ) 2 14 Gear effect for fish obtained via trawl (0) or recreational sample (1) = 0 ~ ( 0 10 ) 2 15 Length weight scaling coefficient ~ exp ( 0 10 ) ~ ( 0 10 ) 2 17 Coefficient of variation around the length weight relationship ~ ( 1 1 ) PAGE 40 40 Table 21. Continued Component Equation Regression coefficient priors 2 18 Regression coefficient i for fixed effects on growth in length ~ ( 0 10 ) 2 19 Regression coefficient i for fixed effects on lengthweight relationship ~ ( 0 10 ) Likelihoods 2 20 Likelihood of observed length L at age a from individual i of cohort c during time t , ~ , exp ( ) , W from individual i of cohort c during time t , ~ , PAGE 41 4 1 Table 22. Parameter estimates and 95% credible intervals (CI; in parentheses) from the von Bertalanffy model and the lengthweight relationship along with their associated fixed effects regression coefficients representing density and environmental effects. von Ber talanffy Length Weight Component Est. (95% CI) Component Est. (95% CI) Regression Coefficients Age 1+ Trawl CPUE 0.132 ( 0.146, 0.118) Age 0 Trawl CPUE 0.002 (4.6*10 4 0.003) C hl a 0.046 (0.025, 0.066) Age 1+ Trawl CPUE 0.004 ( 0.006, 0.002) Water Level 0.063 (0.038, 0.088) C hl a 0.019 ( 0.030, 0.007) Tempera t ure 0.152 ( 0.249, 0.056) Water Level 0.007 ( 0.021, 0.006) Tempera t ure 0.011 ( 0.007, 0.029) PAGE 42 42 Figure 2 1. Environmental factors and indices of abundance from 1998 through 2013 on Lochloosa Lake. A) Chlorophyll A concentration obtained from the St. Johns Water Management District (2014) B) Water level obtained from the St. Johns Wat er Management District (2014) C) W ater temperature obtained from the St. Johns Water Management District (2014) D) T rawl catch per unit effort (CPUE) for ages greater than 1 representing yearly density f rom annual FWC trawl surveys E) T rawl CPUE for age 0 representing cohort specific estima tes of relative density f rom annual FWC trawl surveys PAGE 43 43 Figure 22. Workflow diagram outlining the structure and components used to determine the impacts of density and environmental variation on Black Crappie growth in length and weight. PAGE 44 44 Figure 23. Predicted effects on growth in length and weight of Lochloosa Lake Black Crappie from 1998 to 2013. A) Angler specific gear effect and me an length at age. B) Effect of a g e 1+ trawl catch per unit effort (CPUE) on predicted length at age1. C) Effect of Chlorophyll A (Chl a) on predicted length at age1 D) Effect of water level on predicted length at age1. E ) Effect of temperature on predicted length at age1. F) Effect o f age 0 trawl CPUE on predicted weight at age1. G) Effect of age 1+ trawl CPUE on predicted weight at age1. H) Effect of Chl a on predicted weight at age1. I) Effect of water level on predicted weight at age1. J) Effect of temperature on predicted weig ht at age 1. Predicted length and weight at age1 over the ranges of effects (standardize between 2 and 2) while holding all other effects at the mean value ( i.e., zero). Light grey shading indicates credible intervals that do not PAGE 45 45 contain zero and dark gr ey shading indicates credible intervals that contain zero. PAGE 46 46 Figure 24 Cohort specific estimates of L ,c and coefficient of variation ( cvc) for Black Crappie from the 1998 2013 cohorts A) Cohort specific estimates of L ,c. B) Cohort specific estimates of cvc. Black points represent the mean estimates and black lines represent the 95% credible intervals The vertical dashed lines represent the overall mean estimates and the grey shaded regions are the 95% credible intervals for the overall mean. PAGE 47 47 Figure 25 Predicted cohort specific von Bertalanffy growth curves for Black Crappie from the 1998 to 2013 cohorts. The dashed black lines represent the cohort specific growth curves with 95% credible intervals for the mean ( dark gr ey ) and observations ( light grey ) f or Black Crappie caught by trawling ( yellow circles) and recreational anglers ( blue triangles). The overall m ean von Bertalanffy growth curve s are shown with solid black line s. Note that the 95% credible intervals for the mean lengthat age are difficult to see for some cohorts. PAGE 48 48 CHAPTER 3 COMPARISON OF GROWTH MODELS FOR SEQUENTIAL HERMAPHRODITES BY CONSIDERING MULTI PHASIC GROWTH Introduction The von Bertalanffy growth mo del (hereafter referred to as VB; von Bertalanffy 1938) has been extensively used to describe growth of fish and other taxa that display indeterminate growth (Ricker 1975; Lester et al. 2004) One of the main advantages of the VB model is its strong biological and empirical support (Beverton and Holt 1957; Chen et al. 1992; Lester et al 2004) However this model has been criticized because it seems unlikely that one growth curve should be able to represent the complex physiological changes happening throughout the life of an organism (Day and Tay lor The VB model is relatively inflexible as it considers only decreasing incremental growth throughout the life of the organism, which may not hold true for very young fish (e.g. larval and early juvenile growth phases; Beverton and Holt 1957; Ricker 1975; Walters and Martell 2004) More recently the model has been criticized because it does not account for changes in energ y allocation to reproduction after a fish reaches maturity (Charnov 1993; Charnov et al. 2001; Lester et al. 2004) This suggests that using the VB model can lead to misspecification of management reference points and could result in over or under exploitation because the VB model ignores potential changes in growth patterns before and after maturation. Accurately estimating life history parameters, such as growth and reproductive schedules, are crucial to management because these traits are often used to develop size based regulations (e.g. size l imits, harvest slots, etc.), to set harvest limits, and are major components in fisheries assessment models (Ricker 1975; Jennings et al. 2001; PAGE 49 49 Radomski et al. 2001; Walters and Martell 2004) Commonly, fish growth is estimated independently of reproductive schedules (e.g. length or age at maturation), where growth is assumed to follow the VB model and estimates of age/length at mat uration are obtained using a logistic regression. This process assumes that age and length interact to influence the timing of maturation via: 1) the relationship between age and length or 2) directly accounting for the growth variation by estimating both age and length regression coefficients. This ignores any interaction between maturity and growth, even though it has been shown that both the timing of maturation and the amount of energy allocated towards reproduction may influence lifetime growth (Charnov 1993; Charnov et al. 2001; Lester et al. 2004) There are numerous examples of growth models that incorporate the infl uence of maturity on growth (e.g., Brody 1945; Lester et al. 2004) and assume that fish grow according to multiple phases throughout their life. One such example, developed by Lester et al. (2004) assumes a period(s) of linear growth prior to maturation ( i.e., no reproductive investment) and growth following the VB model after maturation. Because this and similar models incorporate the age at maturation as a parameter, changes in the timing of or biased estimates of maturity can have large impacts on the subsequent growth curves. Thus, fish experiencing physiological or behavioral changes after maturity, such as sexual transition, should experience an additional growth phase once individuals change sex This is because the energetic costs of producing eggs are markedly higher than those of producing sperm (Asher et al. 2008) Therefore in protogynous hermaphrodites (individuals initially mature as female) females that have transition ed to male will have additional energy resources to devote to either growth or to mate PAGE 50 50 acquisition (St. Mary 1994; Chu and Lee 2012; Cogalniceanu et al. 2013) Additionally males often suffer from high energetic costs and increased mortality due to increased levels of aggression and resource defense, reduced time foraging, or fasting during mating season (e.g., Neuhaus and Pelletier 2001; Hoffman et al. 2008; Georgiev et al. 2014) T hese changes associated with sex change are often ignored when estimating growth because many authors use the VB model to describe growth rates (for examples see: Buxton 1992; Garratt et al. 1993; Alonzo and Mangel 2004; Alonzo et al. 2008; Cossington et al. 2010; Linde et al. 2011; Fenberg and Roy 2012) I t is important to consider models other than the VB to desc ribe the complex changes happening throughout the life of a sex changing fish a s a result of the physiological and behavioral changes likely associated with sex change. Several authors have modified the VB model to explain the sexual size dimorphism observ ed in many sex changing species (Garratt et al. 1993; Adams and Williams 2001; Munday et al. 2004; Linde et al. 2011) but have largely ignored the physiological and behavioral changes also associated with changing sex. Gar ratt et al. (1993) developed a bi phasic VB model that described accelerated growth after transition ( i.e., a growth spurt) but did not incorporate changes associated with maturation. Several authors back calculated lengthat age estimates to compare the growth rates of fish that had changed sex to those that were still the primary sex, but did not assess potential changes in growth due to maturation or sex change (e.g., Adams and Williams 2001; Munday et al. 2004; Linde et al. 2011) I sought to expand on these studies by modifying the Lester et al. (2004) growth model to account for an additional growth phase associated with sex change. Our primary objective was to determine if accounting for just maturity ( i.e., the bi phasic PAGE 51 51 model developed by Lester et al. 2004) or accounting for growth transitions associated with maturity and sex change ( i.e., tri phasic Lester) would more accurately describe the growth patterns of a protogynous hermaphroditic fish than the standard VB model Our second objective was to compare management reference points from each model using a yield per recruit frame work to determine the implications of using each growth model. Methods I used the Gulf of Mexico Gag Mycteroperca microlepis as a case study for this analysis. Gag is a long lived (max imum observed age 31 years) protogynous hermaphrodite. Gag are targeted in both commercial and recreational fisheries. Gag length (fork length in mm), age, and histology data were obtained from fisheries dependent and independent samples between 19792012 from the National Marine Fisheries Service (NMFS) used in the 2013 Gag stock assessment (SEDAR 2014) Gag length, maturity and sex changeat age data were fit using a Bayesian hierarchical framework to predict growth using VB, bi and tri phasic Lester models, and timing of maturation and sex change assuming logistic functions. Growth models were run in program R version 3.1.3 using runjags version 3.3.0 (Denwood 2013; R Development Core Team 2013) and yieldper recruit models were run in program R version 3.1.3. All equations for the growth models and yield per recruit equations are presented in Tables 3 1 and 3 2 respectively. The data came from multiple fishery dependent sources (~31,700) and fishery independent surveys (~1,500). Because of the sel ectivity of the fishery dependent sources, growth was modeled in the recent Gag assessment using a truncated normal distribution with a constant standard deviation to account for minimum length limits in commercial and recreational fisheries (developed by McGarvey and Fowler 2002; PAGE 52 52 SEDAR 2014) Additionally, they used a modified VB model that assumed a period of linear growth from age 0 (fixed at 10 cm) to age1 (SEDAR 2014) I chose to use the traditional formulation of the VB model incorporating an agespecific standard deviation in order to reduce model complexity and avoid the assumption of a constant standard deviation. Because there were samples of small, young fish from fishery independent surveys (~1,750 less than 500 mm and almost 300 less than age1), data were aggregated without any consideration for sample si zes within each gear type following the recommendations of Wilson et al. (2015) As shown in Wilson et al. (2015) when there are samples of small, young fish, this method helps account for some of the effects of gear selectivity on growth parameter estimation. Growth Models Mean lengthat age from the VB model was estimated using the standard formulation of the von Bertalanffy growth equation; where is the average maximum attainable length, k is the Brody growth coefficient scaling size to catabolism and t0 is the theoretical lengthat age 0 if the fish always grew according to the VB model ( Eq 3 1; Table 3 1 ) For the bi phasic Lester model, m ean length at age was estimated using the growth model developed by Lester et al. (2004) ; where h is the prereproductive growth rate, t1 is the age intercept for the pre reproductive growth phase, and T is age at maturity ( Eq 3 2; Table 31 ) The Lester et al. (2004) formulation also estimates reproductive investment ( g ), which is used to estimate k and in the post maturation growth phase ( Eq 3 2; Table 31 ). The tri phasic Lester model is identical to the bi phasic Lester, except there is an additional growth phase after transition to male at age and sex specific estimates of gs, ks, and ( Eq 3 3; Table 31 ). I assumed length- PAGE 53 53 at age was normally distributed with a constant coefficient of variation ( Eq 3 4; Table 31 ). Age based maturity ma,i and sex ual transition Da,i for individual i was described using Bernoulli trial s with agespecific probabilities of being mature or male ( Eq 3 7 and 3 8; Table 31 ). Age specific probabilities of being mature or male were estimated with a logistic function where T and are the age at which 50% of the of individuals are mature or male and the sigma terms ( and ) represent the sl ope of the logistic function for being mature ( i.e., female) or male, respectively ( Eq 3 5 and 3 6; Table 31 ). These equations were the same for all growth models. The VB, bi and tri phasic Lester m odel fits were compared using deviance information criterion ( D IC ; Spiegelhalter et al. 2002) where ( | ) is the value of the likelihood at the mean parameter values, ( | ) is the value of the likelihood function for draw n of the joint posterior distribution, and N is the number of samples in the posterior distribution ( Eq 3 9 and 3 10; Table 31 ) Bayesian models were run using seven chains, each generating 1,500 samples of the posterior distribution using a thinning rate of 100. Convergence was verified using Gelman and Rubin diagnostics (Gelman and Rubin 1992; Brooks and G elman 1998) All estimated parameters from the growth models and logistic regressions had uninformative priors. Yield Per Recruit Models I used an age structured y ield per recruit models to calculate the fishing mortality rates that would result in maximum yieldper recruit ( FMAX) and spawning stock biomass being 35% of unfished condition using total biomass ( F35T) and sex specific biomasses ( F35F, F35M for female and male respectively). The models incorporated agespecific PAGE 54 54 growth in length and weight, and survival schedules. Growth equations used, described in the previous section, were the VB, bi phasic Lester and tri phasic Lester ( Eq 3 1 to 3 3; Table 31 ). For this model I chose to make the simplifying assumption that the timing of both maturation and sex change remained constant over exploitation rates, sim ilar to many assessment models including Gag (SEDAR 2014; see Provost and Jensen 2015) Survivorship schedules ( la) were used to calculate the probability of a recruit surviving to each age using Lorenzen (2000) size based natural mortality ( Ma), fishing mortality ( F ) and agespecific vulnerability to harvest ( Va; Eq 3 1 1 ; Table 3 2 ) Unfished conditions were determined by setting F to zero when calculating the survivorship schedule ( Eq 3 1 1; Table 32 ) Uncertainty in overall natural mortality rate ( Mbase) was assessed using two methods; 1) Mbase was assumed to be equal to the k estimated from the appropriate growth model ( Walters and Martell 2004) ; and 2) Mbase was held constant over all growth models and set to 0.1342 yr1, the estimate of natural mortality used in the 2013 stock assessment ( Eq 3 1 2 and 3 1 3; Table 32 ; SEDAR 2014) These two scenar ios represent likely scenarios for assessment models, where an estimate of natural mortality is obtained using surrogate information obtained from the growth model (Charnov 1993; Jensen 1996; Walters and Martell 2004) and when there is an independent estimate of natural mortality ( i.e., via methods developed by Hoenig 1983; SEDAR 2014) W hen Mbase was set equal to k, I assumed two scenarios f or the tri phasic Lester model where; 1) Mbase was sex specific and changed with sexual transition and 2) Mbase was constant post maturation and was set to the female estimate of k ( Eq 3 1 3; Table 32 ) Juvenile mortality rates were held constant and set to equal PAGE 55 55 the female estimate for k from the appropriate growth model. Age specific vulnerability was simulated using a logistic model where Llim is the minimum length l imit in the Gulf of Mexico recreational fishery (56 cm; Eq 3 1 4; Table 32 ). T hese models produced yield per recruit and spawning stock biomass per recruit Because Gag are sequential hermaphrodites, I chose to assess the impact s of fishing mortality rate for each sex separately and for combined sexes. Generally assessment models are concerned with either female biomass or total biomass (Brooks et al. 2008) effectively ignor ing male reproductive contribution. Therefore I assessed male spawning stock biomass in order to highlight the sex specific impacts of exploitation on the malephase because overharvest of males could result in sperm limitation if male biomass falls too low (Coleman et al. 1996; Alonzo and Mangel 2004) Weight at age was calculated using the lengthweight equation ( Eq 3 1 5; Table 3 2 ) and was used to calculate adult female and male spawning stock biomass per recruit ( SSBF/R and SSBM /R respectively ; Eq 3 1 6 and 3 1 7; Table 32 ). Parameter values for the lengthweight equation were obtained from SEDAR (2014) Total spawning stock biomass ( SSBT/R ) was calculated as the sum of SSBF/R and SSBM/R for a given fishing mortality rate. T he probabilities associated with being a female ( Pfem,a) or male ( Pmale,a) at age were modeled using a double logistic for females ( i.e., the probability of being mature minus the probability of being male) and a single logistic for males ( Eq 3 1 8 and 3 1 9; Table 32 ). Y ield per recruit ( Y PR ) was summed over all ages for each fishing mortality rate ( Eq 3 2 0; Table 32 ). For each growth model, the yield per recruit model was repeated N= 10,500 times and parameter values for the PAGE 56 56 growth models and probabilities of being female or male were drawn from the joint posterior distributions. Results The tri phasic Lester model performed the best, using DIC, when describing the lengthat age and re productive schedules of Gag Additionally, both the bi and tri phasic Lester models performed better than the VB model D IC values of 110, 0, and 722 respectively ). A ll of the models predicted similar growth patterns up to the age at transition, about ag e 12 ( Figure 3 1 A ). All models overestimated the lengthat age of individuals between the ages of 1 and 2, presumably due to selectivity, but appeared to fit the samples less than age1 and greater than age 2 ( Figure 3 2 ). After about age 12, the models diverged, with the bi phasic Lester model slightly overestimating and the VB model slightly underestimating the length of the oldest fish relativ e to the tri phasic model ( Figure 31 A and Figure 32 ) C omparing the agebased maturity estimates the VB model ( i.e., growth and reproductive schedules estimated independently) had a higher estimate of ageat 50% maturity than the other models ( T = 3.5 vs. 2.6 and 2.7 years for the bi and tri phasic Lester models respectively; Table 3 3 ; Figure 31 ). For the timing of sex change, all models produced similar estimates and fits, with the ageat 50% sex change around 12. 6 or 12. 7 years; Table 33 ; Figure 31 ). The tri D IC, indicating that growth decreases after females transition to males. This model suggests that fish allocated more resources to reproduction after sex change than before ( gs=f = 0.28 yr1 ( 0.270.29) vs. gs=m = 0.32 yr1 ( 0.310.33)). Further, estimates from the tri phasic Lester model indicated higher metabolic rates ( k) and lower for males compared with females PAGE 57 57 ( Table 33 ). This model also appeared to perform best at estimating the length of the oldest fish given the available data (ages 2030; Figure 31 A and Figure 32 ). Assuming metabolic rates are proportional to the base natural mortality, t he FMAX estimates for each model ranged from 0. 210.3 3 yr1, with the estimates using the tri phasic Lester model resulting in the lowest and the VB being the hi ghest values ( Table 3 4 ; Figure 3 3 ). Fishing mortality rates resulting in total spawning stock biomasses ( F35T) were quite variable, ranging from 0.15 to 0.18 yr1, with the VB model resulting in the highest estimate and both tri phasic Lester models being the lowest ( Table 34 ; Figure 33 ). Female specific SSB/R F35F estimates from the bi phasic Lester model were lowest ( F35F = 0.23 yr1) and estimates from the VB and both tri phasic Lester were similar ( F35F = 0.34 yr1; Table 34 ; Figure 33 ). F ishing mortality rates result ing in the male spawning stock biomasses of 0.35 of unfished were similar between all models ( F35M = 0.070.08 yr1; Table 34 ; Figure 33 ). With equal values for the base mortality rate ( Mbase = 0.1342 yr1), t he FMAX estimates for each model ranged from 0.30 to 0.40 yr1, w ith the estimates using the bi phasic Lester model resulting in the highest and the VB being the lowest values ( Table 3 5 ; Figure 3 4 ). Fishing mortality rates resulting in total spawning stock biomasses were the lowest for the VB model ( F35T = 0.18 yr1) and were the highest for the bi phasic Lester model ( F35T = 0.26 yr1; Table 35 ; Figure 34 ). Female specific SSB /R F35F estimates from the bi phasic Lester model were lowest ( F35F = 0.32 yr1) and highest from the tri phasic Lester model ( F35F = 0.44 yr1; Table 35 ; Figure 34 ). Estimates of fishing mortality rates result ing in the male spawning stock biomasses of PAGE 58 58 0.35 of unfished were similar between all models ( F35M = 0.08 yr1; Table 35 ; Figure 34 ). Discussion G rowth models incorporating changes in growth rates at maturation and sex change were found to be statistically better at describing Gag growth than the VB model Following the growth models through a simple YPR analysis revealed very different estimates of FMAX and fishing mortality rates that resulted in the spawning stock biomasses falling below 35% of the unfished conditions. Along with the work of Charnov, Lester, and associates (Charnov 1993; Charnov et al. 2001; Lester et al. 2004) this analysis showed that it is important to consider multi phasic growth models for species with complex life histor y traits such as sex change when predicting growth patterns and developing management strategies. Our study supports the arguments of Charnov (1993), Day and Taylor (1997), ov et al. (2001), and Lester et al. (2004) that a single growth curve is n o t suited to describe the lifetime growth of fish. I t is important to note that the predicted lengthat age for fish below approximately age 1213 were almost visually identical between all of the models. It was also striking how similar the VB and the tri phasic Lester model predictions were for all ages, even for the oldest ages (2030 years). This suggests the VB model can be used to predict the growth of sex changing fish when reproductive status is not available. However it is important to avoid using the VB model without consideration of other models if life history traits are linked to growth characteristics The tri phasic model predicted higher energy allocation towards reproduction in the male phase than the female phase. T his does not imply that males are allocating PAGE 59 59 energy towards increasing gonad tissue growth. The increase in reproductive investment is likely associated with increased aggression and changes in behavior because more aggressive males will have a greater chance at defending mating opportunities mating sites and territories (Tecot et al. 2013; Georgiev et al. 2014) T his usually comes at the cost of riskier behaviors, higher metabolic rates ( k) and increased mortality from decreased foraging time or direct malemale conflict (e.g., Neuhaus and Pelletier 2001; Hoffman et al. 2008; Georgiev et al. 2014) An increase in mortality after transitioning to males is also supported by the empiri cal relationship between natural mortality and k (e.g. M = 1.65 k, M = 1.5 k or M = k; Charnov 1993; Jensen 1996; Walters and Martell 2004) as well as the invariant relationship between reproductive effort and M derived using the VB model (Charnov 2008) F urther work is needed to determine if there are differences in sex specific natural mortality rates for Gag. Currently, models used to predict the growth of sex changing fish ignore gender by using the VB model or estimate separate growth curves for fish that changed sex and those that had not (e.g., Garratt et al. 1993; Adams and Williams 2001; Munday et al. 2004; Linde et al. 2011) O ften maturation or gender data are not available so using a single model to describe fish growth may be unavoidable. Using models such as those developed by Garratt et al. (1993), Adams and Williams (2001), Munday et al. (2004), or Linde et al. (2011) would exclude agelength pairs where there is no information on the reproductive state. In the case of Gag this would result in the loss of over 97% of the total data. By excluding this much data, I would introduce additional parameter uncertainty and potential biases if reproductive state information were not missing at random Using growth models such as the tri phasic Lester model would PAGE 60 60 avoid some of these issues and unlike the VB models provide a way to describ e sexspecific changes in growth when at least some gender and maturity data exist Other studies assessing changes in growth associated with sex change have found that males experience a gr owth spurt surrounding the timing of transition (e.g., Garratt et al. 1993; Walker and McCormick 2004, 2009; Walker et al. 2007; Munday et al. 2009; McCormick et al. 2010) These studies mainly used daily otolith increment analysis to measure the otolith growth rates before and after sex change for small, short lived sex changing species (e.g., Walker and McCormick 2004, 2009; Walker et al. 2007; Munday et al. 2009; M cCormick et al. 2010) It is expected that this growth spurt allow s newly transitioned males to gain an additional size advantage over large females within the harem and further allow s them to suppress the growth of the largest females (McCormick et al. 2010) Garratt et al. (1993) fit a sex specif ic biphasic VB growth model to a protogynous hermaphrodite Chrysoblephus puniceus to describe the male biased sexual size dimorphism by assuming a male growth spurt. This model assumed females followed the typical VB model and fish that transitioned to male were fit with the bi phasic VB model Unlike the aforementioned studies, Adams and Williams (2001), Munday et al. (2004), and Linde et al. (2011) examined the growth of protogynous hermaphrodites Coral Trout Plectropomus maculatus and Rainbow Wrasse Coris julis and did not find a growth spurt at sex change for either of these species. Similar to these studies, the tri phasic Lester model does not provide evidence for a transitional growth spurt in Gag because the estimates of the ageat sex change we re almost identical for all models and because the predicted growth curves were very similar Additionally it is believed that Gag have a lek mating system not haremic PAGE 61 61 suggesting that a lack of a male growth spurt could provide evidence against haremic m ating systems I did not directly test for transitional growth spurt and further work is needed to address the possibility of a growth spurt in Gag Management of fish populations often relies on estimates of growth and reproductive schedules to set harves t regulations. Ageand length structured population models are often constructed using information from the VB model, and surrogate information on agespecific fecundity, vulnerability, and natural mortality that are dependent on the VB model ( e.g., using the relationships by Jensen 1996; Lorenzen 1996a, 2000; Gwinn et al. 2010) When I assumed Mbase = k, the VB models generally produced the highest estimates for each reference point. Conversely, when using an estimate of natural mortality that was not based on information from the growth model, the VB model generally produced the lowest reference point estimates. Because of the differences in reference points based on my assumptions, obtaining accurate estimates of both growth and natural mortality are necessary to inform management decisions and preform stock assessments, especially when natural mortalit y is based on information from the growth model T he major focus to improve growth estimates has been to assess ways to improve the VB fit to account for nonrepresentative sampling (see Gwinn et al. 2010; Wilson et al. 2015) T hese studies have focused on the effects of sampling bias on parameter estimates, not on the impacts of using models other than the VB. By using models other than the VB, my results showed that common management reference points ( FMAX, F35T, F35F, and F35M) were highly dependent on the assumptions of how fish grow throughout their lives. Further, using the incorrect growth model will affect PAGE 62 62 biological reference points associated with spawning stock biomass or spawning potential ratio. In this study, I assessed the sensitivity of equilibrium reference points to different growth models. F urther research will be needed to assess the impacts of switching from the VB to tri phasic Lester mo dels when using more advanced assessment models. Management reference points often rely solely on mature female biomass for dioecious species and are focused either on total mature biomass or just female biomass for hermaphroditic species (Brooks et al. 2008) S imilar to Alonzo et al. (2008) my results show that relatively low fishing mortality rates decreased male spawning stock biomass to 35% of the unfished condition, and therefore can become severely depleted at fishing mortality rates associated with FMAX, F35T and F35F. T he extent and implications of severely reduced male biomass depends on fertilization rates, which are often unknown for many species (Brooks et al. 2008) and plasticity in the timing of sex change. Brooks et al. (2008) suggested using femalespecific reference points if fertilization rates are high, but at low fert ilization rates, male reference points performed better when the timing of sex change is static. Because I used a per recruit framework and did not incorporate stock recruitment relationships, egg fertilization, or variation in the timing of sex change, I did not assess the implications of using sex specific or total spawning stock biomass reference points and further work is still needed to determine the appropriate reference points for sex changing fish. In addition to differences in management and biological reference points between the VB and tri phasic Lester models, the assumptions being made about the interactions between growth and maturity can have large impacts on management PAGE 63 63 outcomes. It is well su pported that life history traits, such as growth and reproductive traits can change due to size selective fishing mortality (Jrgensen 1990; Rijnsdorp 1993; Grift et al. 2003; Olsen et al. 2004, 2005; Sattar et al. 2008) Because the VB model does not explicitly ac count for reproduction, changes in the timing of maturation and/or sex change must be accounted for by assuming changes in one or more parameters (e.g. k or ) Unlike the VB model, changes in growth can solely be accounted for by assuming changes in the timing of maturity or sex change when using the tri phasic Lester model. For instance, a decrease in the age at maturity will also decrease the mean sizeat age of mature fish because they spend less time as a juvenile experiencing high growth rates. This will also influence the timing of sex change due to the decreased sizeat age and interactions between fertility, mortality and the population structure (Warner 1988; Iwasa 1991; Munday et al. 2006) Therefore, in populations that have experienced changes in the timing of maturation or sex change, it is even more important to consider models other than the VB to account for the impacts of reproduction on fish growth. PAGE 64 64 Table 31. List of equations used in describing lengthand maturity at age for Gag. Description Equation Components and Priors Growth Models 3 1 von Bertalanffy predicted length (mm) at age a = 1 exp ( ) ~ ( 0 10 ) ~ ( 10 10 ) ~ ( 0 10 ) 3 2 bi phasic Lester predicted length (mm) at age a = ( ) + ( 1 ) < = ln 1 + 3 = uD C ? ~ ( 0 10) ~ ( 0 10) ~ ( 10 10 ) 3 3 tri phasic Lester predicted length (mm) at age a = ) EB = rQ . ? A? f f8 f-+ ( 1 ) < + @ ( 1 rF A ? f f8 f. ) EB = > = ln 1 + 3 = uD C ? ~ ( 0 10) ~ ( 0 10 ) ~ ( 10 10 ) 3 4 Likelihood of observed length L at age a for individual i ~ ( ) ~ ( 10 10 ) Reproductive M odels 3 5 Probability of being mature at age a = ( 1 + exp ( ( ) ) ~ ( 10 10 ) ~ ( 10 10 ) 3 6 Probability of being male at age a = ( 1 + exp ( ( ) ) ~ ( 10 10 ) ~ ( 10 10 ) 3 7 Likelihood of age a individual i being mature ~ PAGE 65 65 Table 31. Continued Description Equation Components and Priors 3 8 Likelihood of age a individual i being male ~ Model Comparison 3 9 Deviance Information Criterion = 2 2 ln ( | ) = 2 ln ( | ) PAGE 66 66 Table 32. List of equations used in the yieldper recruit analysis for Gag. Description Equation Components 3 1 1 Survivorship at age a = 1 = exp ( ) a = ln 1 + ( exp ( ) 1 ) a = ln 1 + ( exp ( ) 1 ) , = Sex specific Base Natural Mortality Rate 3 1 4 Vulnerability to harvest at age a = ( 1 + exp ( ( ) ( ) ) ) = 1 17 10 = = 3 1 8 Probability of being female at age a = ( 1 + exp ( ( ) ) / ) 1 + exp ( ) / a = ( 1 + exp ( ( ) ) / ) 3 2 0 Yield per recruit ( Y PR ) for a given F = ( 1 exp ( ) ) PAGE 67 67 Table 3 3. Mean parameter estimates and 95% credible intervals (in parentheses) from each of the growth models. The tri phasic Lester estimates of g and k are broken into femalespecific (within the triphasic Lester column) and malespecific estimates (in the male estimates column). Note that and k from the bi and tri phasic Lester growth models were derived from the estimates of h and g Parameter v on Bertalanffy Bi phasic Lester Tri phasic Lester Male Est. h (mm) 139.3 (137.7, 141.0) 134.2 (131.6, 136.4) g (yr 1 ) 0.30 (0.30, 0.31) 0.28 (0.27, 0.29) 0.32 (0.31, 0.33) k (yr 1 ) 0.14 (0.1 4 0.1 4 ) 0.10 (0. 09 0.1 0 ) 0.0 9 (0. 09 0. 09 ) 0.1 0 (0.1 0 0.1 0 ) t 0 or t 1 (yr) 1.0 5 ( 1. 08 1. 02 ) 1. 08 ( 1.1 2 1. 05 ) 1.1 7 ( 1. 22 1.1 3 ) cv 0.11 (0.1 1 0.1 1 ) 0.1 1 (0.1 0 0.1 1 ) 0.1 1 (0.1 0 0. 11 ) T (yr) 3.4 5 ( 3.20 3.65 ) 2. 60 ( 2.54 2.66 ) 2. 70 ( 2.62 2.82 ) (yr) 12. 60 (1 2 00 1 3 31 ) 12. 61 (1 2.01 1 3.32 ) 12.7 0 (1 2 .1 0 1 3 43 ) 0.5 3 (0. 44 0. 66 ) 0.8 6 (0. 75 0. 99 ) 0. 81 (0. 70 0. 94 ) 1.8 4 (1. 58 2.15 ) 1.8 4 (1. 59 2.14 ) 1.8 6 (1. 61 2 .1 8 ) PAGE 68 68 Table 3 4. Fishing mortality rates resulting in maximum yield per recruit ( FMAX) and spawning stock biomasses of 0.35 of unfished for combined sexes ( F35T), females only ( F35F) and males only ( F35M). The 95% credible intervals for each estimate are in parentheses. The von Bertalanffy and bi phasic Lester models assume sizebased, sex independent natural mortality rates equal to the Brody growth coefficient ( k). The tri phasic Lester model with variable M assumes size based natural mortality changes after sex change and equals the sex specific k whereas the tri phasic Lester model with constant M assumes natural mortality is equal to the femalespecific estimate of k for all ages. Tri phasic Lester (Variable M ) Tri phasic Lester (Constant M ) Metric von Bertalanffy Bi phasic Lester F MAX (yr 1 ) 0.3 26 ( 0.318 0. 333 ) 0.2 45 (0. 237 0. 253 ) 0. 206 (0. 200 0. 211 ) 0. 206 (0. 200 0. 211 ) F 35T (yr 1 ) 0. 184 (0. 181 0. 188 ) 0. 176 (0.1 70 0.1 82 ) 0.1 45 (0.1 42 0.1 48 ) 0.1 45 (0.1 42 0.1 48 ) F 35F (yr 1 ) 0.336 (0. 309, 0.354 ) 0.231 (0.218, 0.243 ) 0.337 (0.301, 0.344 ) 0.337 (0.301, 0.344 ) F 35M (yr 1 ) 0.076 (0.071, 0.078 ) 0.076 (0.071, 0.081 ) 0.071 (0.067, 0.071 ) 0.071 (0.067, 0.071 ) PAGE 69 69 Table 3 5. Fishing mortality rates resulting in maximum yieldper recruit (FMAX) and spawning stock biomasses of 0.35 of unfished for combined sexes ( F35T), females only ( F35F) and males only ( F35M). The 95% credible intervals for each estimate are in parentheses. All models assume a sizebased natural mortality rate, with natural mortality constant acr oss all growth models ( i.e., M = 0.1342 yr 1). Metric von Bertalanffy Bi phasic Lester Tri phasic Lester F MAX (yr 1 ) 0. 303 ( 0.302 0. 304 ) 0. 389 (0. 381 0. 394 ) 0. 394 (0. 385 0. 410 ) F 35T (yr 1 ) 0. 175 (0. 174 0. 175 ) 0. 256 (0. 248 0. 264 ) 0. 223 (0. 220 0. 227 ) F 35F (yr 1 ) 0.329 (0. 302, 0.347 ) 0.323 (0.305, 0.342 ) 0.442 (0.399, 0.454 ) F 35M (yr 1 ) 0.075 (0.071, 0.078 ) 0.081 (0.075, 0.087 ) 0.075 (0.072, 0.078 ) PAGE 70 70 Figure 31. Observed and predicted length, maturity, and sex changeat age of Gag using the von Bertalanffy, bi phasic Lester, and tri phasic Lester models. A) Observed (dots) and predicted l ength at age of Gag using the von Bertalan ffy (solid line), bi phasic Lester (long dashed line) and tri phasic Lester (short dashed line). B) Predicted m aturity at age of Gag using a logistic regression. C) Predicted s ex changeat age using a logistic regression. D) Predicted m aturity at age using a logistic regression combined wit h the bi phasic Lester E) Predicted s ex changeat age using a logistic regression for the bi phasic Lester F) Predicted m aturity at age PAGE 71 71 using a logistic regression combined with the tri phasic Lester. G) Predicted sex changeat age using a logistic regre ssion combined with the tri phasic Lester. PAGE 72 72 Figure 32. Residual plots of the predicted lengthat age for each of the growth models. A) Residuals for the von Bertalanffy model. B) Residuals for the bi phasic Lester. C) Residuals for the tri phasic Lester. The solid line represents a Lowess smoother and the dotted line is set at zero. PAGE 73 73 Figure 33. Yield per recruit, total and sex specific spawning stock biomass per recruit over a range of instantaneous fishing mortality rates and growth models, assuming natur al mortality rate linked with k. A) Yield per recruit (YPR ) model with estimates of the fishing mortality rate resulting in the maximum YPR for each of the growth models ( vertical lines ) B) Total spawning stock biomass per recruit ( SSBT/R ) with estimates of the fishing mortality rate resulting in 35% of unfished SSBT/R for each of the growth models ( vertical lines ) C) Female spawning stock biomass per recruit ( SSBF/R ) with estimates of the fishing mortality rate resulting in 35% of unfished SSBF/R for each of the growth models (vertical lines) D) Male spawning stock biomass per recruit ( SSBM/R ) with estimates of the fishing mortality rate resulting in 35% of unfished SSBM/R for each of the growth models (vertical lines) The von Bertalanffy and bi phasic Lester models (solid and dotted black lines PAGE 74 74 respectively) assume a sex independent natural mortality rate equal to the Brody growth coefficient ( k). The tri phasic Lester model with variable M (dashed black line) assumes natural mortality rate changes after sex change and equals the sex specific k whereas the tri phasic Lester with constant M (dashed grey line) assumes natural mortality is equal to the femalespecific estimate of k for the entire life. It is important to no te that both of the tri phasic Lester estimates are very similar PAGE 75 75 Figure 3 4 Yield per recruit, total and sex specific spawning stock biomass per recruit over a range of instantaneous fishing mortality rates and growth models, assuming a natur al mor tality rate equal to 0.1342 yr1. A) Yield per recruit ( YPR ) model with estimates of the fishing mortality rate resulting in the maximum YPR for each of the growth models (vertical lines). B) Total spawning stock biomass per recruit ( SSBT/R ) with estimates of the fishing mortality rate resulting in 35% of unfished SSBT/R for each of the growth models (vertical lines). C) Female spawning stock biomass per recruit ( SSBF/R ) with estimates of the fishing mortality rate resulting in 35% of unfishe d SSBF/R for each of the growth models (vertical lines). D) Male spawning stock biomass per recruit ( SSBM/R ) with estimates of the fishing mortality rate resulting in 35% of unfished SSBM/R for each of the growth PAGE 76 76 models (vertical lines). All models assume a size based natural mortality rate with a natural mortality constant across all growth models. The von Bertalanffy bi phasic Lester and tri phasic Lester models are represented by solid dotted and dashed black lines respectively PAGE 77 77 CHAPTER 4 FISHERY INDUCED CHANGES TO L IFE HISTORY TRAITS I N SEQUENTIAL HERMAPHRODITES DUE T O SIZE SELECTIVE FISHING Introduction Concerns have been mounting about fisheries induced changes to exploited fish populations. This can result from intensive harvest along with the selective practices of many fisheries and has been identified as a key driving force behind changes in the population structure and phenotypic traits of many exploited fish stocks (Rijnsdorp 1993; Grift et al. 2003; Olsen et al. 2005) These changes in population structure and life history traits can lead to long term reductions in catches (Belgrano and Fowler 2013) delayed recovery from overexploitation (Hutchings 2004, 2005; Walsh et al. 2006) and can shift management reference points (Heino et al. 2013) Further, if the selective practices of a fishery result in the preferential removal of individuals that display certain phenotypic traits (e.g., fast growth, late maturation, etc.) and the traits are heritable, fishing will lead to a genetic change within the population (Law 2000; Heino and God 2002; Kuparinen and Meril 2007) Much of the work assessing the impacts of fishing on wild populations has been conducted on dioecious spec ies, largely ignoring the impacts of exploitiaton on life history traits of sequential hermaphrodites (but see Hamilton et al. 2007; Collins and McBride 2011; Fenberg and Roy 2012; Mariani et al. 2013) Decreases in the timing of maturation and transition have been identified in m any sequentially hermaphroditic species (Hamilton et al. 2007; Collins and McBride 2011; Fenberg and Roy 2012; Mariani et al. 2013) and may be indicative of an evolutionary response to harvest if maturation and/or transition are heritable. F or other species, changes in the timing of maturation and transition have not been identified (SEDAR PAGE 78 78 2014; Provost and Jensen 2015) When the timing of maturation and/or transition does not respond to changes in the population structure, populations can experience sperm or egg limitation from the loss of terminal sex individuals (Huntsman and Schaaf 1994; Armsworth 2001; Alon zo and Mangel 2004) T he extent of these impacts depends on how the timing of transition is controlled and can have very different outcomes if the timing of transition is static or changes in response to genetic, environmental, biological, or social fac tors (Huntsman and Schaaf 1994; Alonzo and Mangel 2005) There are many different models used to predict when transition should occur. The hypothesis that a developmental process initiates transition in fish (e.g. a critical size or age) has been around for about 70 years (Liu 1944) suggesting that for some species the timing of transition is likely genetically controlled. T h e way this model is generally applied in population assessments or simulations assumes that the timing of transition is static and wont change as a population is exploited (e.g., Alonzo and Mangel 2004; Brooks et al. 2008; SEDAR 2014; Provost and Jensen 2015) W ork by Shapiro, War ner, Charnov and coauthors suggest that transition should occur when individuals can improve their reproductive fitness (e.g., the size advantage model; Ghiselin 1969) and often occurs in response to social cues, behavioral responses, or demographic changes (for examples see: Charnov 1982; Shapiro and Boulon 1982; Warner 1988) This suggests that the timing of transition is highly plastic and can change rapidly in response to selective harvest. These hypotheses predict very differ ent evolutionary responses to fishing and can have large impacts on the long term sustainability of a population and fishery. PAGE 79 79 Whether the timing of transition is static or controlled by some other processes, it is important to understand how differences i n the mechanisms controlling transition interact to drive changes in life history traits and impact population structure. Many simulations have been used to study the impacts of fishing on sequential hermaphrodites, however the majority of simulations assu me d static timing of maturation and transition (but see Huntsman and Schaaf 1994; Alonzo and Mangel 2005; Sattar et al. 2008) Simulations that assume static timing of transition have consistently found that harvest can lead to sperm limitation at high fishing mortality rates (e.g., Huntsman and Schaaf 1994; Armsworth 2001; Alonzo and Mang el 2004, 2005; Brooks et al. 2008) W hen the timing of transition was allowed to vary, simulations consistently predicted decreases in the timing of transition and often little sperm limitation (Huntsman and Schaaf 1994; Alonzo and Mangel 2005; Sattar et al. 2008) Further, when the timing of maturation was allowed to vary in response to fishing, it has consistently been found to decrease as a result of fishing (Huntsman and Schaaf 1994; Sattar et al. 2008) The results of these studies show the importance of comparing multiple mechanisms controlling transition and allowing the timing of maturation to vary to fully understand the impacts of fishing on sequential hermaphrodites. Up to this point, few of these models have assessed the potential evolutionary consequences of fishing on sequential hermaphrodites. Sattar et al. ( 2008) used an evolutionary model to predict how the timing of maturation and transition evolve in response to various levels of fishing mortality. In that study, the authors assessed the evolutionary endpoints and only used the sizeadvantage model to d escribe the timing of transition. In this study, I expanded on the work by Huntsman and Schaaf (1994), PAGE 80 80 Alonzo and Mangel (2005), and Sattar et al. (2008) by comparing how different rules of transition influence the evolutionary response to selective harvest. Further, I did this by comparing the evolutionary trajectories of life history traits (e.g., the timing of maturation and transition) and populationlevel traits through time. My primary objective was to determine how the timing of maturation and transition of a sequential hermaphrodite evolve under size selective fishing mortality. My secondary objective was to assess how mechanisms controlling the rules of transition (e.g., critical size/age or social control) impact the evolutionary trajectories and the populationlevel response to size selectivity. My final objective was to determine which mechanisms driving transition are likely the most susceptible to overfishing and fisheries induced evolution. Methods I used an individual based model representing a large, long lived protogynous hermaphrodite (e.g., Gag Mycteroperca microlepis Red Grouper Epinephelus morio, Hogfish Lachnolaimus maximus California Sheephead Semicossyphus pulcher ) to determine the impacts of sizeselective fishing mortality on the evolutionary trajectories in life history traits and temporal trends in populationlevel traits. In this model, fish were subject to sizebased natural and fishing mortality, maturation, transition, growth, and reproduction. The timing of maturation was allowed to evolve and the timing of transition was also allowed to evolve or change as a result of social conditions. I assumed a heritability of 0.2 for both of these life history traits (Mousseau and Roff 1987; Roff 1993; Weigensberg and Roff 1996; Jnasson et al. 1997; Conover and Munch 2002) Individuals were allowed to mature and transition to male in the same year, effectively bypassing female function and reproducing as a male for their entire adult life. PAGE 81 81 Juvenile growth rates and adult reproductive investment were allowed to vary between individuals, but for simplicity were not allowed to evolve over time and did not change as a function of density. A tri phasic growth model was used to describe juvenile, female, and male specific growth rates. Reproduction happened at the end of the year and determined the total number of age1 fish the fol lowing year as well as the distr ibution of genetically controlled parameters. Each year, eggs produced during the previous year recruit to the population as 1 year old fish. Parameter values used in the model are located in Table 4 1 The mechanisms controlling transition can have large im pacts on how sequentially hermaphroditic populations respond to sizeselective fishing mortality (Huntsman and Schaaf 1994; Alonzo and Mangel 2005) Therefore, transition was modeled using various assumptions on the mechanisms driving the timing of transition. I modeled two scenarios of critical size at transition and two of socially controlled transition. Critical size at transition was modeled as either static, as is commonly assumed in population models, and genet ically controlled, in which the mean size at transition was determined by inheritance. For social c ontrol of transition, the size advantage model developed by Ghiselin (1969) was used to determine which individuals transitioned to male. This model predicts that females transitioned to male when their reproductive success as a mal e became greater than if they remained female. The two social control models differed solely in the order of survival, maturation, and transition to represent species in which males and females are well mixed throughout the year (i.e., maturation and transition happen immediately prior to reproduction) or if there is a lag between the maturation/transit ion process and PAGE 82 82 reproduction (i.e., mortality occurs between maturation/transition and reproduciton). This allowed me to determine how the mechanisms controlling transition influenced 1) the evolutionary trends in the timing of maturation and transition, 2) the temporal trends in biological reference points and 3) the sustainability of a population and fishery. Recruitment Individuals ( i ) entered the population at age1 with growth parameters juvenile growth rate hi and femalespecific reproductive investment gi ,, drawn from random normal distributions. ~ ( ) ( 4 1) ~ ( 4 2) Here and are the populationlevel mean juvenile (stage si = 1) growth rate and femalespecific reproductive investment respectively and and are coefficients of variation describing the standard deviation around the mean values. The values of and were based off of Gag (Matthias et al. 2016) and resulted in a mean femalespecific asymptotic length of approximately 130 cm. The coefficients of variation for the random number generators were assigned so the population would have roughly a 10% coefficient of variation around the meanlengthat ag e in the unfished condition, similar to previous estimates for Gag (Matthias et al. 2016) Using the assigned growth parameters, I then calculated the remaining growth parameters. Reproductive investment of male Gag, estimated using a tri phasic growth model, was recently found to be about 1.2 times greater than that of females (Matthias et al. 2016) Therefore, the relationship = 1 2 ( 4 3) PAGE 83 83 was used to describe the change in reproductive investment after transition (stage si = 2) for each individual i and was held constant over time and all models. Sex specific Brody growth coefficients ki,s, and sex specific asymptotic size , for individual i was calculated using = ln 1 + 3 ( 4 4) , = 3 ( 4 5) and were derived by (Lester et al. 2004) to relate reproductive investment and juvenile growth to the von Bertalanffy parameters. Survival Survival was modeled as Bernoulli trial with the probability of surviving for individual i ( ) to year y ( ) = , 0 < < , , ( 4 6) as a function of sizebased natural mortality , a juvenile mortality component and fishing mortality F if vulnerable to harvest (i.e., = 1 ). The juvenile natural mortality component represents a c ost of being a juvenile (e.g., living in riskier habitats than adults). Without the population becomes dominated by individuals that wait to mature until because grow th rates are higher prior to maturation. Therefore was used to deter mine when an individual reaches maturation and was set so that the mean length at maturation was approximately equal to 66% of the mean female PAGE 84 84 The size based natural mortality was calculated using a modified Lorenzen mortality equation (Lorenzen 2000) = ln 1 + max 1 , 1 , ( 4 8) incorporating the length in the previous year , sex specific Brody growth coefficient , and a sex specific base natural mortality rate , T he base natural mortality rate for individual i was calculated as a function of the Brody growth coefficient, , = 1 5 (Jensen 1996) For juveniles, , was calculated using the femalespecific for individual i The traditional formulation of the Lorenzen mortality equation has been extensively used for describing the average sizebase natural mortality of a population. However, the Lorenzen model scales changes in mortality relative to and i ndividuals with the same k experience identical natural mortality patterns, regardless of their asymptotic size. To account for individual variation in asymptotic le ngth, mortality was scaled assuming size dependent mortality relative to 100 cm and , for individuals greater than 100cm. This assumes that individuals with the same k have the same sizebased mortality patterns, regardless of age or asymptotic size. Selectivity to harvest was modeled to represent a minimum length limit = 0 , < 1 ( 4 7) where the vulnerability to the harvest was zero if the length last year, Li,y 1, was less than the l ength limit and one if greater than or equal to the length limit. PAGE 85 85 Maturation The length at maturation for individual i , was modeled using a linear relationship to describe a change in the length required to mature as a function of age , = + ( 4 8) in which the intercept, , is the size component and the slope, , is the age component and were inherited from the parents. For simplicity, I initially assumed that , decreased as a function of age (i.e., was negative) and was allowed to evolve. Individuals matured when their length in year y was greater than the agespecific length required to mature , = 1 , 0 ( 4 9) where mati = 0 represents juveniles and mati = 1 represents adults. Transition Transition was modeled using various assumptions on the mechanisms driving the timing of transition. I modeled two scenarios of critical size at transition and two of socially controlled transition. For critical size at maturation, individuals transitioned from female to male when their length in year y was greater than the assigned length at transition = 2 , 1 ( 4 10) where si = 1 represents juveniles and females and si = 2 represents males. For social control, the size advantage model was used to determine which individuals transitioned to male (Ghiselin 1969) Individuals changed sex when the PAGE 86 86 number of gametes able to be fertilized as a male [i.e., min , ] became greater than gametes produced as a female (i.e., , ). = 2 min , , 1 ( 4 11) In order to prevent small females from transitioning before large females, this process was simulated for each female starting with the largest female. If a female transitioned they were removed from the female population and added to the male population. This reduced the total number of eggs produced and increased the competition for eggs by reducing the proportion of eggs fertilized by each male This was repeated for the next largest female and in order to reduce computation time, this process stopped after 100 females did not transition. Growth Fish growth followed a tri phasic model developed by (Matthias et al. 2016) and the length in year y for individual i was calculated by = 0 = 0 + = 0 , + , 1 , = 1 ( 4 12) in which growth was divided into three phases. Juveniles (i.e., mati = 0) experienced linear growth until reaching maturation. After reaching maturity (i.e., mati = 1), individuals grew according to the von Bertalanffy model and switched to a different von Bertalanffy after transition. Transition from the female to male growth phase occurred when individuals attained a critical size or when reproductive success as a male was greater than if they remained female. Length at age0 was held constant at 0 cm for all individuals. PAGE 87 87 Gamete Production Female egg production was assum ed to be proportional to weight and accounted for reproductive investment ( ) = , ( 4 13) where and b were the scalar and body shape parameters relating length to weight. Total egg production was multiplied by the reproductive investment because it is related to the gonadosomatic index (Lester et al. 2004) Using reproductive investment in the calculation of egg production accounts for the tradeo ff between the investment into reproduction and growth. Therefore, individuals with high g produced more eggs than a female of the same length with low g but those individuals with high g did not get as large (i.e., smaller ). Sperm production by male i in year y was calculated similar to female egg production, = , ( 4 14) but also accounted for the smaller gamete size using a scaling parameter c. Total sperm production of a male was arbitrarily assumed to be ten times greater than the egg production of a comparable female. This was set to account for the small size of sperm and excess sperm released during external fertilization. When there was excess sperm in year y, the proportion of eggs allocated to each male was determined by the individuals size relative to other males = , ( 4 15) where the first component is the total egg production by all n1 females in y ear y, Li,y is the length of male i in year y, describes the relationship between male size and PAGE 88 88 access to females, and the component in parentheses prevents the total number of fertilized eggs from being greater than total egg production. The parameter was set to represent the size advantage of male access to females. For example, if = 0 then there is no size advantage for males and all males have equal access to females. If > 0 then larger males have greater access to females than smaller males Values of were arbitrarily set at 1, 4, and 8 to represent various male size advantages. This means that in scenarios in which = 8, males have greater access to females (i.e., more females per male) than in scenarios when = 1. Reproduction The tot al number of recruits Ry entering the population in year y was calculated using the Beverton Holt stock recruitment function = 1 + ( 4 16) where abh is the maximum recruit survival per unit fecundity, bbh is the scaling parameter representing density dependent mortality, and is the total number of eggs fertilized by males in the previous year. The stock recruitment function was parameterized using steepness = 4 ( 1 ) ( 4 17) where H is the steepnes s parameter and represents the ratio of recruits produced at 20% of unfished stock size to the unfished stock size. Steepness was assumed to be 0.8, similar to the steepness values used and/or estimated in recent assessments for Gag, Red Grouper and Hogfis h (Cooper et al. 2013; SEDAR 2014, 2015) The scaling parameter bbh, was calculated using = ( 4 18) PAGE 89 89 where Req is the equilibrium recruitment. Equil ibrium recruitment was set so that the adult population remained above 2,000 individuals in the unfished condition. T he number of fertilized eggs was calculated via = ( 4 19) where is the total number of eggs fertilized by male j* (i.e., male fecundity), n2 is the total number of males alive in year y, and j* = ( 1 , n2) are the males alive in year y. Fecundity of individual i in year y was calculated separately juveniles, females, and males using = 0 = 0 , = 1 & = 1 min , , = 2 ( 4 20) where the top component is set to zero if juvenile, the middle is egg production if female, and bottom is the minimum of sperm production or the proportion of total eggs produced in year y allocated to individual i if male Taking the minimum of these two values accounted for the potential of sperm limitation when the male population size was low. This occurred when the total sperm production was less than eggs production. When sperm was limiting, males did not compete for eggs, there was no size advantage for males other than being able to produce more sperm, and the total num ber of fertilized eggs was then calculated as the sum of male sperm production (i.e., = ). Inheritance In order to determine the evolutionary impacts of fishing on the timing of maturation and transition (when genetically determined), both trai ts were allowed to evolve through time as a function of the reproductive population. Reproduction occurred PAGE 90 90 at the end of each year and was modeled using random mating. For each of the Ry recruits in year y, a male and female parent was randomly assigned based on the relative fecundity of the parent in year y. For a female, this was the proportion of fertilized eggs produced and for a male, was the proportion of eggs fertilized in year y. Parameters were assigned using inheritance and represented the intercept and slope of the maturation norm along with the size at transition when transition was under genetic control. These parameters were assigned using a normal random number generator ~ , ( 4 21) where is the predicted mean for offspring i given the values for the parents and heritability, and is the standard deviation. The predicted offspring mean was calculated using = , , + ( 4 22) where h2 is narrow sense heritability, and are the beta values for parents j1 and j2, and was the fecundity weighted mean in year y of the reproductive population. For static transition, was assigned the in a similar manor, except was kept constant through time and over all individuals. The standard deviation was set at 40, which was the equilibrium standard devi ation of as a result of variation in growth rates and the length required to mature (see description in the maturation section). The standard deviation was set at 14 which kept the estimated standard deviation of the lengthat transition appr oximately equal for all scenarios. These parameters were held constant through time. PAGE 91 91 Model Initialization and Analysis For the first 50 years of the simulation, approximately 23 generations, the year specific means for (e.g., ) were held constant to prevent rapid evolution while the simulation was being initialized. After 50 years, were allowed to change via inheritance. The model ran for 700 years and fishing started after year 200. This represented approximately ten generations pr ior to fishing and 25 generations after fishing started. A range of fishing mortality rates ( F = 0.0, 0.1,, 0.5) were used to represent a variety of scenarios from no fishing to exploitation rates sufficient to collapse the stock. Once fishing was initiated, fishing mortality rates were held constant for the remainder of the simulation. One hundred iterations were run for each combination of fishing mortality rate and mechanism controlling transition. Changes in the timing of maturation and transition were assessed via trends in the mean lengthat maturity and mean lengthat transition. Trends in population level traits were assessed with reference points commonly used in fisheries management, including male:female sex ratio, spawning potential ratio ( SPR ), fertilization rate, and the probability of recruitment failure. SPR was calculated as the ratio of fertilized eggs produced when unfished (i.e., year 199) to the number of fertilized eggs produced after fishing started and is an index of relative reproduction in a given year. Fertilization rate was the proportion of eggs fertilized in a given year and was used to assess sperm limitation. The probability of recruitment failure was calculated as the proportion of simulations in which no recruits were produced in a given year. PAGE 92 92 Results Evolutionary R esponses to F ishing In all scenarios, sizeselective fishing mortality caused evolutionary changes in the mean length at maturation and decreases in the timing of transition when tra nsition was under genetic and social control ( Figure 4 1 and 4 2 ). After 500 years of fishing mortality, changes in mean length at maturation and transition were greater when transition was socially contr olled (up to a 28 and 39 cm change respectively ) compared to the genetically determined critical size at transition models (up to a 12 and 5 cm change respectively; Figure 41 and 4 2 ). Additionally, the mechanism controlling transition had large impacts on the evolutionary trajectories in the timing of maturation and trends in the mean size at transition. For all mechanisms controlling transition higher fishing mortality rates resulted i n greater evolutionary responses to fishing and changes in the timing of transition ( Figure 41 and 4 2 ). It is important to note that the evolutionary trends in the timing of maturation were primarily a response to changes in , the intercept of the maturation norm and not the slope, as the mean of did not change as a result of selection used in this model. Therefore changes in the mean length at transition represent changes in M ean l ength at maturation increased as a result of fishing ( Figure 41 A F ) when transition was held static and genetically determined. The increases in the timing of maturation were generally greater when the timing of transition was s tatic ( Figure 41 A F ). Additionally at high fishing mortality rates ( F = 0.4 and 0.5) and when males had greater size advantages ( = 4 and 8), the evolutionary trends in the timing of maturation were highly variable and unstable as a result of low population sizes and these populations often crashed prior to year 700 ( Figure 41 A F ). When the timing of PAGE 93 93 matu ration was socially controlled, the mean length at maturation decreased thought time as a result of fishing. These decreases were generally greater for populations well mixed throughout the year ( Figure 41 G L ). Trends in the mean length at transition were also highly dependent on mechanisms controlling transition. When the timing of transition was allowed to evolve or change as a result of social conditions, the mean length at transition decreased after fishing s tarted in year 200, but the trends were very different between genetic and social control of transition ( Figure 4 2 D L ). When under genetic control, the mean length at transition slowly and consistently decreased through time up to 5 cm after 500 years of fishing ( Figure 42 D F ). For social control models we see reductions in the mean leng th at transition up to 39 cm for well mixed populations and 35 cm for populations with a lag between maturation/trans ition and reproduciton ( Figure 42 G L ). T he model also predict ed rapid declines in the mean length at transition from 5 14 cm during the first 10 years of fishing to compensate for the initial loss of large males from fishing ( Figure 4 2 G L ). Reference P oint T rends In all scenarios, the mechanisms controlling transition had large impacts on the trends in reference points. After the onset of fishing, changes in sex ratios were identified for all mechanisms controlling transition, but the sex ratio trends were very different between critical size and social control of transition ( Figure 4 3 ). Decreases in SPR were evident for all mechanisms controlling transition and in all case s higher fishing mortality rates lead to lower SPR values ( Figure 4 4 ). Finally, reduced fertilization rates and recruitment failure were only found when transition was modeled assuming static or genetically controlled size at tra nsition ( Figure 4 5 ). PAGE 94 94 Trends in the sex ratios were similar between static and genetic control of transition. After fishing started in year 200, sex ratios decreased rapidly within 10 years and higher fishing mortality rates lead to lower sex ratios ( Figure 43 A F ). After year 215, the sex ratios generally increased through time and as a result of changes in the timing of maturation and, for genetic control, decreases in the mean length at transition ( Figure 43 A F ). A t fishing mortality rates above 0.3 with high male size advantages ( = 4 and 8) the sex ratios were unstable as a result of low population sizes and a high proportion of simulations resulted in extirpation. For s ocial control models, sex ratios were generally higher after the onset of fishing than in the unfished condition ( Figure 4 3 G L ). W hen the male size advantage was high ( = 4 and 8), sex ratios initially decreased for five years after fishing started followed by a rapid increase over the next 10 years ( Figure 43 G L ). The initial drops in sex ratios were the result of losing large males in the first few years of the fishery. After these large males died they were replaced by many small males, which lead to increases in the sex ratios over the next few years. After sex ratios peaked around year 220, they decreased through time as a result of changes in the timing of both maturation and transition ( Figure 43 G L ). The largest decreases in SPR occurred within 10 year s after fishing started when transition was static or under genetic control ( Figure 44 A F ). These initial decreases in SPR reflect reduced egg production when sperm was not limiting and low fertilization rates when sperm was limiting ( Figure s 4 4 and 4 5 A F ). After year 210, trends in SPR continued to decrease through time with low male size advantages ( = 1; Figure 44 A F ). These decreases were associated with reduced egg production due to increases in PAGE 95 95 the timing of maturation. At higher male size advantages and higher exploitation rates the SPR either remained stable or increased after year 220 ( Figure 44 A F ). The increases in SPR were associated with decreases in the timing of transition combined with low fertilization rates when transition was genetically controlled. Similar to static and genetic control of transition, fishing caused large decreases in SPR within 10 years after the onset of fishing ( Figure 44 G L ). Because there was no sperm limitation with social control of transition, the initial reductions in SPR reflect decreases in egg production from both increased mortality and the loss of females through transition. After 20 years of fishing, SPR consistently increased through time when transition was under social control ( Figure 44 G L ). These increases were the result of higher total egg production from individuals maturing earlier. Reduced fertilization rates and recruitment failure (i.e., no recruits produced in a given year) were only evident in the critical size at transition scenarios. In general, higher fishing mortality rates resulted in reduced fertilization rates and higher probabilities of recruitment failure ( Figure 45 ). Additionally, higher values had lower fertil ization rates and higher probabilities of recruitment failure at a given fishing mortality rate ( Figure 4 5 ). This was likely associated with fewer males prior to the start of fishing and longer time between maturation and transit ion because the length required to transition was larger at higher values. Impacts of the M ale S ize A dvantage P arameter Trends in maturation and transition were influenced by the male size advantage parameter On average, lower values had greater changes in the timing of maturation and transition than higher values for a given fishing mortality rate ( Figure s 4 1 and 4 2 ). In the unfished condition higher values were also associated with higher PAGE 96 96 mean lengths at transition ( Figure 4 2 ). Additionally when = 4 and 8, the mean lengths at transition for the critical size models were about 10 cm greater than the social control models ( Figure 42 ). The male size advantage parameter also influenced the reference point trends. As expected the sex ratios were negatively correlated with indicating fewer males per female when males had large size advantages ( Figure 4 3 ). Addi tionally, sex ratios were lower for critical size compare to social control of transition models as a result of differences in the mean length at transition ( Figure s 4 2 and 4 3 ). After fishing was initia ted, had the largest impact on SPR fertilization rate, and probability of recruitment failure trends for critical size at transition models ( Figures 44 and 4 5 ). Higher values of consistently resulted in lower SPR fertilization rates, and higher probabilities of recruitment failure for a given fishing mortality rate ( Figure s 4 4 and 4 5 ). Additionally higher values were unable to sustain hig h fishing mortality rates and populations often experienced extirpation (i.e., SPR = 0 and probability of recruitment failure = 1). Discussion To fully understand the consequences of sizeselective fishing mortality on sex changing fish, it is important to consider the evolutionary impact on life history traits a s well as mechanisms that determine the timing of transition. By incorporating multiple rules of transition, very different trends in the timing of transition and subsequently evolut ionary impacts on maturation were identified When assuming critical size at transition, the model showed relatively slow changes to the timing of maturation and transition. This slow response to sizeselective fishing resulted in lower sex ratios, the pot ential for sperm limitation, and high probabilities of recruitment failure, which were PAGE 97 97 similar to predictions in other studies in which transition was held static or slow to respond (Huntsman and Schaaf 1994; Armsworth 2001 ; Alonzo and Mangel 2004, 2005; Brooks et al. 2008) Conversely when assuming social control of transition, the model predicted rapid changes in the mean length at transition, substantial decreases in the mean length at maturation, and increased sex ratios. The high plasticity in transition prevented both sperm limitation and recruitment failure when transition was socially controlled. Similar to Alonzo and Mangel (2005) these results support the need for a better understanding of transition rules to predict not only the response to sizeselective fishing, but also the impacts of different management strategies. Many studies have identified decreases in the tim ing of maturation and transition in exploited hermaphroditic populations (e.g., Hamilton et al. 2007; Collins and McBride 2011; Fenberg and Roy 2012; Mariani et al. 2013) suggesting the potential for fisheries induced evolution. Until recently, few simulation studies have assessed the evolutionary impacts of size selective fishing mortality on sequential hermaphrodites (e.g., Sattar et al. 2008) Assessing the evolutionary impacts of fishing has mainly been confined to dioecious (i.e., separate sex) species. These studies have generally identified decreases in the timing of maturation associated with sizeselective fishing mortality (e.g., Jrgensen 1990; Rijnsdorp 1993; Grift et al. 2003; Olsen et al. 2004, 2005) similar to the findings presented in this study with socially controlled transition and Sattar et al. (2008) However, maturation trends for the critical size models were very different from those produced by socially controlled transition. For both the static and genetic control of transition models, the mean length at maturation increased as a result of exploitation. The differences in maturation trends between the critical size and the PAGE 98 98 social control of transition models suggest a tradeoff between survival, growth, and the reproductive schedules. Life history theory predicts that exploitation will cause decr eases in the timing of maturation (Gross 1985; Law and Grey 1989; Rowell 1993; Roff 2002; Ernande et al. 2004) I ndividuals that mature early will be favored over late maturing individuals as a result of reduced life expectancy from exploitation (Gross 1985; Law and Grey 1989; Rijnsdorp 1993; Rowell 1993; Law 2000) These predic tions have been corroborated by many studies on both dioecious and hermaphroditic species ( e.g., Rijnsdorp 1993; Trippel 1995; Conover and Munch 2002; Olsen et al. 2004, 2005; Mollet et al. 2007; Sattar et al. 2008; Pardoe et al. 2009) and r esults presented in this study from models assuming social control of transition. R esults from models with static and genetic control of transition did not conform to the predictions from life history theory. When transition was static or genetically controlled the ability to transition was dependent on attaining a critical size Decreases in the timing of maturation increase the probability of surviving to reproduce (a fitness advantage) but also increase the time (and probability of mortality) it will take to reach the size required to transition (a fitness dis advantage if males have high reproductive success) Conversely higher length at maturation decreases the probability of surviving to maturation, but also decreases the time to reach critical size to change sex as a result of increased growth. This decrease in time required to reach transition afforded those individuals greater reproductive contributions if they survived to reproduce as males, especially when few males were in the population. Interestingly this tradeoff was not present with the social control because PAGE 99 99 individuals did not need to attain a critical size to change sex and those individuals that were able to reproduce prior to entering the fishery had higher reproductive success. Similar to other simulations (Huntsman and Schaaf 1994; Alonzo and Mangel 2005; Sattar et al. 2008) and field studies (Hamilton et al. 2007; Fenberg and Roy 2012; Mariani et al. 2013) this model predicted dec reases in the timing of transition when under genetic and social control. As expected, decreases in mean length at transition were also influenced by the magnitude of the fishing mortality rate. Similar to this model, Sattar et al. (2008) used an individual based model to assess the impacts of fishing on maturation and transition on a protogynous hermaphrodite and found greater decreases in the timing of maturation and transition at higher fishing mor tality rates. These authors assessed the evolutionary end points from sizeselective fishing mortality and did not assess trends through time. B y assessing trends through time, I was able to identify differences in maturation, transition, and sex ratios th at could be used to help differentiate between genetic and social control of transition. The rules of transition used in this model resulted in widely differing magnitudes and trends in the mean size at transition and sex ratios. T he interesting differenc es arise immediately after fishing started in year 200. Genetically controlled transition showed a consistent decrease through time, whereas socially controlled transition showed an abrupt drop in the timing of transition after the initiation of fishing as a result of changes in population structure. We also see very different sex ratio trends that vary based on the rules of transition. Sex ratio for critical size at transition always decreased, resulting in more females per male. For social control, we see the opposite trends. In PAGE 100 100 general, sex ratios increased resulting in more males per female. These results present interesting implications for the ability to detect differences in the rules of transition. For many exploited hermaphroditic species it can be difficult to determine the rules governing transition to inform management and assessment models. The trends in the mean size at transition and sex ratio predicted from this model could be used to help predict transition rules for exploited species. Due t o the similarity between static and genetic control, I expect that it will be very difficult to detect differences between these rules over short time spans (i.e., 50 to 100 years). If there are abrupt changes or spatial differences in fishing mortality id entifying whether transition is based on a critical size or social control should be much easier For example, there are large spatial differences in the timing of transition in Hogfish in which males inhabiting nearshore reefs on the Florida Coast are substantially smaller than those occupying offshore reefs (Collins and McBride 2011) These differences are thought to be a result of heavy recreational spear fishing on the nearshore reefs and as a res ult, the mean sizes at transition were almost 15 cm smaller for those reefs (i.e., 32.7 vs. 59.2 cm for nearshore and offshore reefs respectively; (Collins and McBride 2011) Along with the differences in the size at transition, Hogfish are haremic protogynous hermaphrodites and males and females inhabit the same habitat throughout the year ( Collins and McBride 2011, 2015) suggesting social control of transition with relatively low uncertainty of when to change sex. On the other han d for Gag, a lekking protogynous hermaphrodite in which females and males occupy spatially distinct habitat d uring the nonspawning season, the timing of transition has been relatively stable since the 1970s (SEDAR 2014) Additionally, the sex ratios for Gag drastically dropped from about 18% males in the 1970s to around 1% PAGE 101 101 in the 1990s (Coleman et al. 1996; McGovern et al. 1998) suggesting genetic control of transition. However, only a few transition rules were assessed in this model and other m echanisms of transition that were not addressed in this manuscript will also need to be considered for exploited species. Many transition rules have been developed and used to describe the patterns of transition for hermaphroditic species. The hypothesis t hat developmental processes (e.g., a critical size or age) initiates transition is a common assumption for many assessment and population models (e.g., Armsworth 2001; Alonzo and Mangel 2004; Brooks et al. 2008; SEDAR 2014; Provost and Jensen 2015) This assumption ignores the possibility that size at transition should correspond with maximizing reproductive fitness (Munday et al. 2006) and is influenced by the social structure of the population (Shapiro and Boulon 1982) The size advantage model, which was used in this study to describe social control of transition, predicts that individuals will change sex when they can improve their reproductive fitness (Ghiselin 1969; Warner 1975, 1988; Warner et al. 1975; Charnov 1982) and often occurs in response to social cues, behavioral responses, or demographic changes (Warner 1988) In both of the social control models, I assumed that individuals had perfect knowledge of the population. This is likely not the case for many populations and uncertainty in when to transition would have likely resulted in slower changes in timing of maturation and transition. Additional work will be needed to fully understand the evolutionary and population level effects of uncertainty for populations with socially controlled transition. Over the range of fishing mortality rates used in this study, populations with socially controlled transition were much less likely to experience recruitment failure. PAGE 102 102 Further these populations were able to persist at fishing mortality rates that extirpated populations with static and genetic control of transition. However at low fishing mortality rates, populations with critical size at transition had higher SPR values than those with social control, similar to Huntsman and Schaaf (1994) and Alonzo and Mangel (2005) Because of the rapid decrease in the mean size at transition for socially controlled transition, these populations had greater reductions in egg production than critical size populations as a result of females transitioning to male at smaller sizes. This trend was reversed once fishing mortality rates were high enough to cause sperm limitation and populations with critical size at transition were much more likely to suffer from recruitment failure. These differences between critical size and social control of transition highlight the need to determine the mechanisms controlling transition for harvested species. The basis of this project was centered on the idea that life history traits, specifically the timing of maturation and, for certain scenarios, transition were heritable. The speed of the evolutionary response depends on narrow sense heritability, h2, and the amount of variability in the lengthat maturation and transition present in the population (Zimmer and Emlen 2013) I assumed the narrow sense heritability was equal to 0.2, which is typical for life history traits (Mousseau and Roff 1987; Roff 1993; Weigensberg and Roff 1996; Jnasson et al. 1997; Conover and Munch 2002) I f narrow sense heritability or the variances associated with the maturation and transition trends (i.e., ) were either larger or smaller than those used in this model, the speed at which maturation and transition evolved would have been faster or slower, respectively. This would not have affected the overall patterns and trends, especially the PAGE 103 103 differences observed between critical size and social control of transition. Further, given the overall differences between the different rules of transition, this model provides valuable insight into potential evolutionary pathways for hermaphroditic species. Many sequentially hermaphroditic marine organisms are targeted in recreational and/or commercial fisheries and there is a lot of uncertainty around how these populations change in response to the size selective properties of many fisheries. Similar to findings of other studies (e.g., Huntsman and Schaaf 1994; Sattar et al. 2008; Fenberg and Roy 2012; Mariani et al. 2013) I showed that selectively harvesting larger sequential hermaphrodites could decrease timing of maturation and transition. If transition is static or slow to change, high fishing mortality can cause reproductive limitation resulting from the loss of terminal sex individuals, but this is likely not the case if the timing of transition changes rapidly in response to the selective properties of fisheries. Additionally by considering multiple transition rules, this model predicts very different sex ratio trends as a result of sizeselective fishing mortality. These differences for transition rules not only highlight the need for a better understanding of the mechanisms controlling transition, but also provide tools that could be used to help predict the possible transition rules for a given species. PAGE 104 104 Table 41. Parameter values used in the individual based model. Parameter Value Description (cm) 13.0 Mean juvenile growth rate 0.2 Coefficient of variation around (yr 1 ) 0.3 Mean reproductive investment 0.1 Coefficient of variation around , (cm) 80.7 Starting value for the intercept predicting age specific size required for maturation 40.0 Standard deviation for 2.0 Starting value for the slope predicting age specific size required for maturation 0.1 Standard deviation for (cm) = 1 = 4 = 8 110.5 120.5 129.5 Length at transition for static control of transition and starting values genetic control of transition for each of the male size advantage values ( 14.0 Standard deviation for 1 17 10 b 3.0 Allometric growth parameter describing fish body form c 10.0 Parameter describing sperm production of males, arbitrarily assumed to be 10 times greater than fecundity of a comparable female 0.2 Narrow sense heritability H 0.8 Steepness for the stock recruit curve M juv ( = 1 ) Static Genetic Social Social Lag 0.050 0.050 0.040 0.040 Additional juvenile mortality rate for low male size advantage ( = 1) and different mechanisms controlling transition, static, genetic, social, and social with a year lag M juv ( = 4 ) Static Genetic Social Social Lag 0.085 0.085 0.060 0.055 Same as above for moderate male size advantage ( = 4) M juv ( = 8 ) Static Genetic Social Social Lag 0.095 0.095 0.070 0.065 Same as above for high male size advantage ( PAGE 105 105 Figure 4 1. Mean length at maturity for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. A) S tatic timing of transition (gr ey) with male size advantage of 1. B) S tatic timing of transition with of 4. C) S tatic timing of transition with of 8. D) Genetic control of transition (red) with of 1. E) Genetic control of transition with of 4. F) Genetic control of transition with of 8. G) S ocial control with well mixed populations (green) with of 1 H) Social contr ol with well mixed populations with of 4. I) Social contr ol with well mixed populations with of 8. J) S ocial control with a lag between maturity/transition and reproduction ( blue) with of 1. K) Social control with a lag between maturity/transition and reproduction with of 4. L) Social control with a lag between maturity/transition and reproduction with of 8. Note lighter shades indicate higher fishing mortality rates each line represents the average of 100 iterations and the scales are different between critical size and s ocial control of transition. PAGE 106 106 Figure 4 2. Mean length at transition for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. A) Static timing of transition (grey) with male size advantage of 1. B) Static timing of transition with of 4. C) Static timing of transition with of 8. D) Genetic control of transition (red) with of 1. E) Genetic control of transition with of 4. F) Genetic control of transition with of 8. G) Social control with well mixed populations (green) with of 1. H) Social contr ol with well mixed populations with of 4. I) Social contr ol with well mixed populations with of 8. J) Social control with a lag between maturity/transition and reproduction (blue) with of 1. K) Social control with a lag between maturity/transition and reproduction with of 4. L) Social control with a lag between maturity/transition and reproduction with of 8. Note lighter shades indicate higher fishing mortality rates each line represents the average of 100 iterations and the scales are different between critical size and social control of transition. PAGE 107 107 Figure 4 3. Sex ratio for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. A) Static timing of transition (grey) with male size advantage of 1. B) Static timing of transition w ith of 4. C) Static timing of transition with of 8. D) Genetic control of transition (red) with of 1. E) Genetic control of transition with of 4. F) Genetic control of transition with of 8. G) Social control with well mixed populations (green) wi th of 1. H) Social contr ol with well mixed populations with of 4. I) Social contr ol with well mixed populations with of 8. J) Social control with a lag between maturity/transition and reproduction (blue) with of 1. K) Social control with a lag between maturity/transition and reproduction with of 4. L) Social control with a lag between maturity/transition and reproduction with of 8. Note lighter shades indicate higher fishing mortality rates each line represents the average of 100 iterations and the scales are different between critical size and social control of transition. PAGE 108 108 Figure 4 4. Spawning potential ratio for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. A) Static timing of transition (grey) with male size advantage of 1. B) Static timing of transition with of 4. C) Static timing of transition with of 8. D) Genetic control of transiti on (red) with of 1. E) Genetic control of transition with of 4. F) Genetic control of transition with of 8. G) Social control with well mixed populations (green) with of 1. H) Social control with well mixed populations with of 4. I) Social control with well mixed populations with of 8. J) Social control with a lag between maturity/transition and reproduction (blue) with of 1. K) Social control with a lag between maturity/transition and reproduction with of 4. L) Social control with a l ag between maturity/transition and reproduction with of 8. Note lighter shades indicate higher fishing mortality rates and each line represents the average of 100 iterations. PAGE 109 109 Figure 4 5. Fertilization rate and probability of recruitment failure for different mechanisms controlling transition from year 100 to 700 of the simulation over fishing mortality rates F from 0 to 0.5 with fishing starting in year 200. A) Fertilization rate for static timing of transition (grey) with male size advantage o f 1. B) Fertilization rate for static timing of transition with of 4. C) Fertilization rate for static timing of transition with of 8. D) Fertilization rate for genetic control of transition (red) with of 1. E) Fertilization rate for genetic control of transition with of 4. F) Fertilization rate for genetic control of transition with of 8. G) Probability of recruitment failure for static timing of transition with of 1. H) Probability of recruitment failure for static timing of transition with of 4. I) Probability of recruitment failure for static timing of transition with of 8. J) Probability of recruitment failure for genetic control of transition with of 1. K) Probability of recruitment failure for genetic control of transition with of 4. L) Probability of recruitment failure for genetic control of transition with of 8. Fertilization rates and probabilities of recruitment failure were one and zero respectively for scenarios with social control of transition. Note lighter shades indicate higher fishing mortality rates and each line represents the average of 100 iterations. PAGE 110 110 CHAPTER 5 CONCLUSIONS AND MANAGEMENT IMPLICATIONS Managing fisheries into the future requires an understanding of how temporal changes in life history traits influence stock productivity and how these changes can influence the management of exploited species. Ignoring changes in productivity from environmental variation, changes in density, trends in the timing of maturation and transition will lead to biased estimates of stock productivity. For Black Crappie, increases in temperature from climate change are predicted to decrease growth in length, which will lead to decreases in per capita fecundity. Long term changes in Black Crappie density are also predicted to have significant impacts on growth and could lead to stun ting at high population sizes or could result in a trophy fishery at low densities For exploited sequential hermaphroditic populations with socially controlled transition individuals maturing early had the highest reproductive success leading to decreases in the overall stock productivity as a result of decrease in mean sizeat age. When transition was s tatic or genetically determined individuals that postponed maturation until larger si zes and were able to transition had the highest reproductive success because males were often found to be limiting the population productivity under selective exploitation. Understanding how changes in stock productivity are influenced by t emporal variation and interactions between survival, growth, and reproductive patterns will be necessary as we manage fisheries into the future. Concern of the impacts of climate change on fish stocks has received much attention in the past 20 25 years. I t is well known that temperature is a major factor that influences fish growth and g enerally higher temperatures lead to increased growth rates ( e.g., Wingfield 1940; Haugen et al. 2007; Vllestad and Olsen 2008; Davidson et PAGE 111 111 al. 2010; Morrongiello et al. 2011, 2014) F or species in which water temperatures exc eed optimal growth for part of the year, such as Black Crappie in Lochloosa Lake, increases in temperature from climate change could lead to decreased growth rates (Hayward and Arnold 1996; Hale 1999; Mi chaletz et al. 2012; Kazyak et al. 2014) and increased mortality (Ellison 1984; Hale 1999; Michaletz et al. 2012) In addition to warmer mean global temperatures, we expect to see increases in environmental variability and the duration of droughts (Lake 2011; Romm 2011; IPCC 2015) which can lead to low water levels and reduced growth rates of Black Crappie and other species (Morrongiello et al. 2011; Gaeta et al. 2014) Additionally, low water levels can decrease the amount of littoral habitat, vegetation and fallen trees (Ficke et al. 2007; Lake 2011; Gaeta et al. 2014) further impacting fish populations. The prospects of higher temperatures and increased environmental variability emphasize the need to predict how populations respond to climate chang e and to understand how these changes influence the management of these populations. Density dependence is one of the major forms of population regulation (Walters and Post 1993; Post et al. 1999; Lorenzen and Enberg 2002; Einum et al. 2006; Lorenzen 2008; Vincenzi et al. 2012) Many population and assessment models only consider density dependent mortality through the use of a stock recruitment function and ignore density dependent changes in growth (e.g., Walters and Martell 2004) but short term variation and long term trends in density can have large impacts on estimates of optimal or maximum sustainable yield (see Lorenzen 2008 for review). Similar to many other studies (e.g., Beverton and Holt 1957; Walters and Post 1993; Post et al. 1999; Lorenzen and Enberg 2002; Sass et al. 2004; C asini et al. 2014) Black PAGE 112 112 Crappie growth in length and weight was negatively correlated to density, suggesting that years with high density decrease per capita egg production By ignoring density dependent growth in population models we overlook one of the compensatory responses to changes in density, which can lead to inappropriate sizebase regulations and biased estimates of population productivity. Fisheries induced evolution has been a major concern for the past 1520 years and much of this attent ion has been focused on the evolution of maturation in dioecious species (but see Hamilton et al. 2007; Sattar et al. 2008; Collins and McBride 2011; Fenberg and Roy 2012; Mariani et al. 2013) S equential hermaphroditic fish provide a unique opportunity to study fisheries induced evolution because differences in the mechanisms controlling sexual transition can have large impacts on populationlevel traits along with evolutionary responses to exploitation. Many simulation studies have been used to predict the impacts of sizeselective fishing on protogynous hermaphrodites and most assume both the timing of maturation and sexual transition are static. These models consistently predict that and fishing will lead to sperm limitation (Huntsman and Schaaf 1994; Armsworth 2001; Alonzo and Mangel 2004, 2005; Brooks et al. 2008) I found similar results when the timing of transition was static and genetically controlled, even when the timing of maturation was allowed to evolve. Additionally, the impacts of sperm limitation were highly dependent on how reproductive success scaled with male size. Populations with large male size advantage and static or genetic control of tr ansition were more susceptible to sperm limitation than those with relatively small male size advantage or when transition was socially controlled. T here are many hypotheses on when individuals should transition and allowing the timing of PAGE 113 113 transition to change as a result of social conditions leads to very different results. Similar to other studies when the timing of sexual transition was allowed to change in response to social conditions (Huntsman and Schaaf 1994; Alonzo and Mangel 2005; Sattar et al. 2008) decreases in the timing of transition mitigate the effects of sperm limitation. These differences in the expected responses to size selective harvest emphasiz e the importance of understanding mechanisms that control transition for exploited species in order to develop appropriate management strategies. Using models that accurately describe life history traits is important when constructing population and assess ment models because many traits are dependent on lengthat age. Therefore, we often link survival, reproductive, and vulnerability schedules with the growth curve via mean lengthat age to describe how the these schedules change over the lifetime of an individual (see Walters and Martell 2004) For instance, the Lorenzen mortality curve relies on length to describe sizebased mortality (Lorenzen 2000) fecundity is generally assumed to be proportional to length cubed (i.e., weight), and vulnerability is modeled using logistic equations incorporating lengthat age information. Population and assessment models rely on this information to estimate management reference points such as maximum yieldper recruit, maximum or optimum sustainable yield, and the corresponding f ishing mortality rates. As shown with Gag, changing the assumption on how the lifetime growth patterns are influenced by maturation and reproduction can result in very different management and biological reference points. Similar changes would likely be as sociated when considering the impacts of variation in environmental factors and density, as they were shown to influence the lifetime growth patterns of Black Crappie. Regardless of the reason for PAGE 114 114 changes in growth, using methods that accurately represent these changes is important to reduce bias in management recommendations and is especially important when estimates of natural mortality and reproductive schedules are unavailable. Greater accuracy in growth parameter estimates is important when natural mor tality rates or reproductive schedules are not known because we rely on surrogate methods to estimate these parameters. Many of the surrogate methods utilize information from the von Bertalanffy equation, such as the relationship between M and k (e.g., M = k, M = 1.5 k, or M = 1.65 k ; Beverton and Holt 1959; Charnov 1993; Jensen 1996; Walters and Martell 2004) and the relationships between maturation or transition and L (e.g., length at maturity = 0.66 L and length at transition = 0.79 L; Beverton and Holt 1959; Charnov 1993; Charnov and Skladttir 2000; Allsop and West 2003) Therefore developing more accurate methods for predicting and describing growth patterns is necessary. For Gag, the tri phasic growth model predicted an increase in k after females transitioned to male and, assuming M is related to k, this would suggest an increase in the natural mortality rate for males as well Additionally, the assumption of natural mortality being related to k versus m aximum age, as was assumed in the recent Gag stock assessment (SEDAR 2014) resulted in very different estimates of FMAX and fishing mortality rates that resulted in the spawning stock biomass falling below 0.35. This shows how important it is to have estimates of life history trai ts and accurate descriptions of the growth patterns when these estimates are unavailable. Long term consequences of climate change and exploitation will need to be accounted for as we manage fisheries into the future. We know that both changing PAGE 115 115 environmen ts and exploitation impact population, community, and ecosystem dynamics through effects on hydrological and oceanic regimes, primary and secondary production, nutrient availability, trophic structure, predator prey relationships, and life history traits (see Enberg et al. 2009; Rijnsdorp et al. 2009; Belgrano and Fowler 2013; Heino et al. 2013; Crozier and Hutchings 2014). As these conditions change, traits and strategies present i n a population will evolve or change to match these new conditions. These changes will result in some stocks being more productive than in current conditions due to longer growing seasons, decreased comp et ition, predator reductions, etc. (e.g., Enberg et al. 2009; Crozier and Hutchings 2014) Other stocks will lose productivity due to prolonged periods of thermal stress, higher total mortality, faster life histories (e.g., smaller sizeat age, earlier maturation, higher reproductive investment), etc. ( e.g., Enberg et al. 2009; Heino et al. 2013; Crozier and Hutchings 2014) Developing methods that can estimate changes in stock productivity and be integrated in to population and assessment models will be necessary to develop appropriate management strategies for stocks impacted by climate change and exploitation. Much work has been done to improve estimates and develop more accurate representations of life hist ory traits. There are many models, including those developed in my dissertation, that can be used to estimate or account for environmental conditions (e.g., Porch et al. 2002) density (e.g., Lorenzen 1996) reproductive schedules (e.g., Brody 1945; Lester et al. 2004; MinteVera et al. 2015) and fisheries induced evolution (e.g., Dieckmann and Heino 2007; Kuparinen and Meril 2007) For many populations, data are not available to estimate changes in life history traits. W hen the data are available we need to determine if it is appropriate to assume the life history traits are PAGE 116 116 temporally static and to understand how this variation influences fisheries management. To ensure sustainability of stocks and fisheries into the future, managers will need an understanding on how changes in biotic, abiotic, and anthropogenic factors influence population productivity through changes in mortality, growth and reproductive schedules. PAGE 117 117 APPENDIX A CHAPTER 1 R CODE Jags Code for von Bertalanffy Growth Model Incorporating Variation in Environmental Conditions and Density modelFilename = "vonB.bug" cat(" model { for(crt in 1:ncohort) { lpred[crt,1]< linf[crt]*(1 exp(k*tnot)) # length at age 0 for(a in 1:nages[crt]) { # time effect, the j[x,y] is a matrix that turns the effect on or off lt[crt,a]< beta[1]*yrden[myrs[crt,a]]+ # year density effect beta[2]*tschla[crt,a]+ # chlorophyll A effect beta[3]*tsdm[crt,a]+ # depth effect beta[4]*tstemp[crt,a] # temperature effect # calculates the growth increment obtained during age a1 to # age a times the time effect lpre d [crt,a+1]< lpred[crt,a]+ # predicted length at age a1 (linf[crt] lpred[crt,a])*(1 exp( k*(ts[crt,a]/12)))* # additional growth to age a exp(lt[crt,a]) # time effect v[crt,a]< pow(lpr ed[crt,a+1]*exp(lncv[crt]),2) # variance for cohort for(i in 1:nacrt[crt,a]) { lre[crt,a,i]< gears[crt,a,i]*LRE*(1/pow(age[crt,a],2)) lena[crt,a,i]~dnorm(lpred[crt,a+1]*exp(lre[crt,a,i]),1/v[crt,a]) # likelihood for length gear (angling) effect on lpred } } } for (i in 1:nwt) { b[i]< mub+ # intercept for l w theta[1]*crtden[cohort[wtobs[i]]]+ # cohort density effect theta[2]*yrden[year[wtobs[i]]]+ # year density effect theta[3]*btchla[wtobs[i]]+ # chla effect theta[4]*btdm[wtobs[i]]+ # depth effect theta[5]*bttemp[wtobs[i]] # temperature effect wpred[i]< exp(lna)*len[wtobs[i]]^b[i] # predicted weight PAGE 118 118 wt[wtobs[i]]~dnorm(wpred[i],1/pow(wpred[i]*wcv,2)) } # random effects priors for(crt in 1:ncohort) { linf[crt]~dno rm(mulinf,prec[1]) lncv[crt]~dnorm(mucv,prec[2]) } LRE~dnorm(0,0.0001) # RE hyperpriors prec[1]~dgamma(1,1) prec[2]~dgamma(1,1) # Fixed effects priors for(i in 1:4) { beta[i]~dnorm(0,0.00001) # prior for beta (effects on length ) } for(i in 1:5) { theta[i]~dnorm(0,0.00001) # prior for theta (effects on weight) } # VB priors mulinf~dnorm(0,10^ 4) # prior for mean linf k~dgamma(1,1) # prior for k tnot~dnorm(0,0.00001) # prior for tnot mucv~dnorm(0,0.00001) # prior for mean cv (in log space) # lengthweight priors mub~dgamma(1,1) lna~dnorm(0,0.00001) wcv~dgamma(1,1) } ", fill=TRUE, file=modelFilename) PAGE 119 119 APPENDIX B CHAPTER 2 R CODE von Bertalanffy Model Jags Code for von Bertalanffy Growth Model modelFilename = "GagLester1.bug" cat(" model { for(i in 1:n) { lpred[i]< linf*(1 exp( k*(age[i] to))) # predicted length at age v[i]< pow(lpred[i]*mucv,2) # variance at age lena[i]~dnorm(lpred[i],1/v[i]) # likelihood component propmat[i]< 1/(1+exp( (age[i] T)/sig[1])) # proportion mature propmale[i]< 1/(1+exp( (age[i] tau)/sig[2])) # proportion male matobs[i]~dbern(propmat[i]) # maturity likelihood component maleobs[i]~dbern(propmale[i]) # male likeliho od component } # priors k~dgamma(1,1) linf~dnorm(0,0.000001) to~dnorm(0,0.0001) T~dgamma(1,1) tau~dgamma(1,1) mucv~dgamma(1,1) sig[1]~dgamma(1,1) sig[2]~dgamma(1,1) } ", fill=TRUE, file=modelFilename) Yield per recruit Code for von Bertalanffy # Yield Per recruit function biphasic M=k YPRLvbfun< function(k,linf,to,T,tau,mucv,si g) { lmat< linf*(1 exp( k*(T to))) # length at maturation wmat< alpha*lmat^beta # weight at maturation ltran< linf*(1 exp( k*(tau to))) # length at tra nsition PAGE 120 120 lta< NULL len< NULL ek< matrix(NA,nrow=30,ncol=2) mat< NULL mal< NULL vul< NULL fecf< NULL fecm< NULL ek1< NULL SL< NULL la< fagef< fagem< ya< matrix(NA,nrow=30,ncol=fiter) for(a in 1:30) { # length up to min(age,transition) len[a]< linf*(1 exp( k*(a to))) # length vul[a]< 1/(1+exp((560len[a])/(560*mucv))) # vulnerability # female and male fecundity fecf[a]< alpha*len[a]^beta* (1/(1+exp((T a)/sig[1])) 1/(1+exp((taua)/sig[2]))) fecm[a]< a lpha*len[a]^delta*(1/(1+exp((taua)/sig[2]))) # Lorenzen survival SL[a]< (1+(linf/len[a])*(exp(k) 1))^( (k*Mmult)/k) M< log(SL[max(1,a1)]) # survivorship la[a,]< ifelse(a==rep(1,fiter),1,la[a1,]*exp( M fvec*vul[ifelse(a==rep(1,fiter),1,a 1)])) fagef[a,]< la[a,]*fecf[a] fagem[a,]< la[a,]*fecm[a] ya[a,]< la[a,]*vul[a]*(1 exp( fvec))*alpha*len[a]^beta } # YPR info ff1< colSums(fagef) fm1< colSums(fagem) yield1< colSums(ya) fopt1< fvec[which(colSums(ya)==max(colSums(ya)))] return(list(ff1,fm1,yield1,fopt1)) } PAGE 121 121 # Yield Per recruit function biphasic M=0.1342 YPRLMvbfun< function(k,linf,to,T,tau,mucv,sig) { lmat< linf*(1 exp( k*(T to))) wmat< alpha*lmat^beta ltran< linf*(1 exp( k*(tau to))) lta< NULL len< NULL ek < matrix(NA,nrow=30,ncol=2) mat< NULL mal< NULL vul< NULL fecf< NULL fecm< NULL ek1< NULL SL< NULL la< fagef< fagem< ya< matrix(NA,nrow=30,ncol=fiter) for(a in 1:30) { # length up to min(age,transition) len[a]< linf*(1 exp( k*(a to))) vul[a]< 1/(1+exp((560len[a])/(560*mucv))) fecf[a]< alpha*len[a]^beta* (1/(1+exp((T a)/sig[1])) 1/(1+exp((taua)/sig[2]))) fecm[a]< alpha*len[a]^delta*(1/(1+exp((taua)/sig[2]))) SL[a]< (1+(linf/len[a])*(exp(k) 1))^( 0.1342/k) # M from SEDAR 33 M< log(SL[max(1,a1)]) la[a,]< ifelse(a==rep(1,fiter),1,la[a1,]*exp( M fvec*vul[ifelse(a==rep(1,fiter),1,a1)])) fagef[a,]< la[a,]*fecf[a] fagem[a,]< la[a,]*fecm[a] ya[a,]< la [a,]*vul[a]*(1 exp( fvec))*alpha*len[a]^beta } ff1< colSums(fagef) fm1< colSums(fagem) yield1< colSums(ya) PAGE 122 122 fopt1< fvec[which(colSums(ya)==max(colSums(ya)))] return(list(ff1,fm1,yield1,fopt1)) } Bi phasic Model Jags Code for Bi phasic Growth Model modelFilename = "GagLester.bug" cat(" model { k< log(1+g/3) # k for vb growth curve after maturity linf< 3*h/g # asymptotic length at age infinity for(i in 1:n) { ek[i]< k*(age[i] T) # mature[i]< ifelse(age[i]>T,1,0) # If age>T, mature= 1, else 0 # length up to min(age,transition) lpred[i]< h* # Linear growth rate (depending on age) (min(age[i],T) t1)* # determines amount of time fish grew according to h[X] exp(ek[i]*mature[i])+ # exp( kT) if they are mature, else it equals 1 linf[1]*(1 exp(ek[i]))*mature[i] # growth after maturity if they are mature v[i]< pow(lpred[i]*mucv,2) lena[i]~dnorm(lpred[i],1/v[i]) propmat[i]< 1/(1+exp( (age[i] T)/sig[1])) propmale[i]< 1/(1+exp( (age[i] tau)/sig[2])) matobs [i]~dbern(propmat[i]) maleobs[i]~dbern(propmale[i]) } # priors g~dgamma(1,1) h~dgamma(1,1) t1~dnorm(0,0.0001) T~dgamma(1,1) tau~dgamma(1,1) mucv~dgamma(1,1) sig[1]~dgamma(1,1) PAGE 123 123 sig[2]~dgamma(1,1) } ", fill=TRUE, file=modelFilename) Yield per recruit Code for Bi phasic Model # Yield Per recruit function biphasic M=k YPR2Lfun< function(g,h,t1,T,tau,mucv,sig) { lmat< h*(T t1) # length at maturity wmat< alpha*lmat^beta # weight at maturity k< log(1+g/3) # k for vb growth after maturity linf< 3*h/g # asymptotic length at age infinity ltran< h* # Linear growth rate (depending on age) (T t1)* # determines amount of time fish grew according to h[X] exp( k[1]*(tau T))+ # exp( kT) if they are mature, else it equals 1 linf[1] *(1 exp( k[1]*(tau T))) lta< NULL len< NULL ek< matrix(NA,nrow=30,ncol=2) mat< NULL mal< NULL vul< NULL fecf< NULL fecm< NULL ek1< NULL SL< NULL la< fagef< fagem< ya< matrix(NA,nrow=30,ncol=fiter) for(a in 1:30) { ek1[a]< k*(min(a,tau) T) # k time mat[a]< ifelse(a>T,1,0) # If age>T, mature=1, else 0 mal[a]< ifelse(a>tau,1,0) # If age>tau, male=1, else 0 # length up to min(age,transition) len[a]< h* # Linear growth rate (min(a,T)t1)* # time fish grew according to h[X] exp(ek1[a]*mat[a])+ # exp( kT) if they are mature, else 1 linf[1]*(1 exp(ek1[a])) *mat[a] # growth after maturity vul[a]< 1/(1+exp((560len[a])/(560*mucv))) fecf[a]< alpha*len[a]^beta* PAGE 124 124 (1/(1+exp((T a)/sig[1])) 1/(1+exp((tau a)/sig[2]))) fecm[a]< alpha*len[a]^delta*(1/(1+exp((taua)/sig[2]))) SL[a]< (1+(linf/len[a])*(exp(k) 1))^( (k*Mmult)/k) M< log(SL[max(1,a1)]) la[a,]< ifelse(a==rep(1,fiter),1,la[a1,]*exp( M fvec*vul[ifelse(a==rep(1,fiter),1,a1)])) fagef[a,]< la[a,]*fecf[a] fagem[a,]< la[a,]*fecm[a] ya[a,]< la[a,]*vul[a]*(1 exp( fvec))*alpha*len[a]^beta } ff2< colSums(fagef) fm2< colSums(fagem) yield2< colSums(ya) fopt2< fvec[which(colSums(ya)==max(colSums(ya)))] return(list(ff2,fm2,yield2,fopt2)) } # Yield Per recruit function biphasic M=0.1342 YPR2LMfun< function(g,h,t1,T,tau,mucv,sig) { lmat< h*(T t1) wmat< alpha*lmat^beta k< log(1+g/3) # k for vb growth linf< 3*h/g # asymptotic length at age infinity ltran< h* # Linear growth rate (depending on age) (T t1)* # amount of time fish grew according to h[X] exp( k[1]*(tau T))+ # exp( kT) if they are mature, else it equals 1 linf[1]*(1 exp( k[1]*(tau T))) lta< NULL len< NULL ek< matrix(NA ,nrow=30,ncol=2) mat< NULL mal< NULL vul< NULL fecf< NULL fecm< NULL ek1< NULL SL< NULL la< fagef< fagem< ya< matrix(NA,nrow=30,ncol=fiter) PAGE 125 125 for(a in 1:30) { ek1[a]< k*(min(a,tau) T) # mat[a]< ifelse(a>T,1,0) # If age>T, mature=1, else 0 mal[a]< ifelse(a>tau,1,0) # If age>tau, male=1, else 0 # length up to min(age,transition) len[a]< h* # Linear growth rate (depending on age) (min(a,T)t1)* # amount of time fish grew according to h[X] exp(ek1[a]*mat[a]) + # exp( kT) if they are mature, else 1 linf[1]*(1 exp(ek1[a]))*mat[a] # growth after maturity vul[a]< 1/(1+exp((560len[a])/(560*mucv))) fecf[a]< alpha*len[a]^beta* (1/(1+exp((T a)/sig[1])) 1/(1+exp((taua)/sig[2]))) fecm[a]< alpha*len[a] ^delta*(1/(1+exp((taua)/sig[2]))) SL[a]< (1+(linf/len[a])*(exp(k) 1))^( 0.1342/k) # M from SEDAR 33 M< log(SL[max(1,a1)]) la[a,]< ifelse(a==rep(1,fiter),1,la[a1,]*exp( M fvec*vul[ifelse(a==rep(1,fiter),1,a1)])) fagef[a,]< la[a,]*fecf[a] fagem[a,]< la[a,]*fecm[a] ya[a,]< la[a,]*vul[a]*(1 exp( fvec))*alpha*len[a]^beta } ff2< colSums(fagef) fm2< colSums(fagem) yield2< colSums(ya) fopt2< fvec[which(colSums(ya)==max(colSums(ya)))] return(list(ff2,fm2,yield2,fopt2)) } Tri phasic Model Jags Code for Tri phasic Growth Model modelFilename = "GagLester 3 .bug" cat(" model { k< log(1+g/3) # k for vb growth curve after maturity linf< 3*h/g # asymptotic length at age infinity for(i in 1:n) PAGE 126 126 { ek[i,1]< k[1]*(min(age[i],tau) T) # exp(k) component for female ek[i,2]< k[2]*(age[i]tau) # exp(k) component for males mature[i]< ifelse(age[i]>T,1,0) # If age>T, mature=1, else 0 male[i]< ifelse(age[i]>tau,1,0) # If age>tau, male=1, else 0 # lengt h up to min(age, age at transition) lt[i]< h* # Linear growth rate (depending on age) (min(age[i],T) t1)* # amount of time fish grew according to h[X] exp(ek[i,1]*mature[i])+ # exp( kT) if they are mature, else equals 1 linf[1]*(1 exp(ek[i,1] ))*mature[i] # growth after maturity if mature lpred[i]< ifelse(male[i]==0,lt[i], # if not male, take lt lt[i]*exp(ek[i,2])+linf[2]*(1 exp(ek[i,2]))) # male growth using vb v[i]< pow(lpred[i]*mucv,2) # variance lena[i]~dnorm(lpred[i],1/ v[i]) # likelihood component for length propmat[i]< 1/(1+exp( (age[i] T)/sig[1])) # proportion mature propmale[i]< 1/(1+exp( (age[i] tau)/sig[2])) # proportion male matobs[i]~dbern(propmat[i]) # maturation likelihood component maleobs[i]~dbern(propmale[i]) # transition likelihood component } # priors g[1]~dgamma(1,1) g[2]~dgamma(1,1) h~dgamma(1,1) t1~dnorm(0,0.0001) T~dgamma(1,1) tau~dgamma(1,1) mucv~dgamma(1,1) sig[1]~dgamma(1,1) sig[2]~dgamma(1,1) } ", fill=TRUE, file=modelFilename) Yield per recruit Code for Bi phasic Model # Yield Per recruit function biphasic M=k YPR3Lfun< function(h,T,t1,g,tau,mucv,sig) { lmat< h*(T t1) # length at maturity wmat< alpha*lmat^beta # weight at maturity k< log(1+g/3) # k for vb growth after maturity linf< 3*h/g # asymptotic length at age infinity PAGE 127 127 ltran< h* # Linear growth rate (depending on age) (T t1)* # amount of time fish grew according to h[X] exp( k[1]*(tau T))+ # exp( kT) if they are mature, else it equals 1 linf[1]*(1 exp( k[1]*(tau T))) lta< NULL len< NULL ek< matrix(NA,nrow=30,ncol=2) mat< NULL mal< NULL vul< NULL fecf< NULL fecm< NULL SL< matrix(NA,nrow=30,ncol=2) SL1< mhat< matrix(NA,nrow=30,ncol=2) la< fagef< fagem< ya< matrix(NA,nrow=30,ncol=fiter) la1< fagef1< fagem1< ya1< matrix(NA,nrow=30,ncol=fiter) ek1< matrix(NA,nrow=30,ncol=2) for(a in 1:30) { ek1 [a,1]< k[1]*(min(a,tau) T) # female vbk time ek1[a,2]< k[2]*(a tau) # male vbk time mat[a ]< ifelse(a>T,1,0) # If age>T, mature=1, else 0 mal[a]< ifelse(a>tau,1,0) # If age>tau, male=1, else 0 # length up to min(age,transition) lta[a]< h* # Linear growth rate (depending on age) (min(a,T)t1)* # amount of time fish grew ac cording to h[X] exp(ek1[a,1]*mat[a])+ # exp( kT) if they are mature, else 1 linf[1]*(1 exp(ek1[a,1]))*mat[a] # growth after maturity len[a]< ifelse(mal[a]==0,lta[a], lta[a]*exp(ek1[a,2])+linf[2]*(1exp(ek1[a,2]))) vul[a]< 1/(1+exp((560len[a])/(560*mucv))) fecf[a]< alpha*len[a]^beta* (1/(1+exp((T a)/sig[1])) 1/(1+exp((taua)/sig[2]))) fecm[a]< alpha*len[a]^delta*(1/(1+exp((taua)/sig[2]))) # survival lorenzen with everything from female SL[a,]< (1+(linf[c(1,2)]/len[a])*(exp(k[c(1,2)]) 1))^ PAGE 128 128 ( (k[c(1,2)]*Mmult)/k[c(1,2)]) M< log(SL[max(1,a1),]) mhat[a,]< M[1]+fvec[c(1,50)]*vul[ifelse(a==1,1,a1)] la[a,]< ifelse(a==rep(1,fiter),1,la[a1,]*exp( M[ifelse(a<=rep(tau,fiter),1,2)] fv ec*vul[ifelse(a==rep(1,fiter),1,a1)])) fagef[a,]< la[a,]*fecf[a] fagem[a,]< la[a,]*fecm[a] ya[a,]< la[a,]*vul[a]*(1 exp( fvec))*alpha*len[a]^beta # survival lorenzen with sex specific parameters SL1[a,]< (1+(linf[c(1,1)]/len[a])*(exp(k[c(1,1)] ) 1))^ ( (k[c(1,1)]*Mmult)/k[c(1,1)]) M1< log(SL1[max(1,a1),]) la1[a,]< ifelse(a==rep(1,fiter),1,la1[a1,]* exp( M1[ifelse(a<=rep(tau,fiter),1,2)] fvec*vul[ifelse(a==rep(1,fiter),1,a1)])) fagef1[a,]< la1[a,]*fecf[a] fagem1[a,]< la1[a,]*fecm[a] ya1[a,]< la1[a,]*vul[a]*(1exp( fvec))*alpha*len[a]^beta } ff3< colSums(fagef) fm3< colSums(fagem) yield3< colSums(ya) fopt3< fvec[which(colSums(ya)==max(colSums(ya)))] ff31< colSums(fagef1) fm31< colSums(fagem1) yield31< col Sums(ya1) fopt31< fvec[which(colSums(ya1)==max(colSums(ya1)))] return(list(ff3,fm3,yield3,fopt3,ff31,fm31,yield31,fopt31)) } # Yield Per recruit function biphasic M=0.1342 YPR3LMfun< function(h,T,t1,g,tau,mucv,sig) { lmat< h*(T t1) # length at matur ity wmat< alpha*lmat^beta # weight at maturity k< log(1+g/3) # k for vb growth after maturity linf< 3*h/g # asymptotic length at age infinity ltran< h* # Linear growth rate (depending on age) (T t1)* # amount of time fish grew according to h[X] exp( k[1]*(tau T))+ # exp( kT) if they are mature, else 1 linf[1]*(1 exp( k[1]*(tau T))) PAGE 129 129 lta< NULL len< NULL ek< matrix(NA,nrow=30,ncol=2) mat< NULL mal< NULL vul< NULL fecf< NULL fecm< NULL SL< matrix(NA,nrow=30,ncol=2) SL1< mhat< matrix(NA,nrow=30,ncol=2) la< fagef< fagem< ya< matrix(NA,nrow=30,ncol=fiter) la1< fagef1< fagem1< ya1< matrix(NA,nrow=30,ncol=fiter) ek1< matrix(NA,nrow=30,ncol=2) for(a in 1:30) { ek1 [a,1]< k[1]*(min(a,tau) T) # female k time ek1[a,2]< k[2]*(a tau) # male k time mat[a]< ifelse(a>T,1,0) # If age>T, mature=1, else 0 mal[a]< ifelse(a>tau,1,0) # If age>tau, male=1, else 0 # length up to min(age,transition) lta[ a]< h* # Linear growth rate (min(a,T)t1)* # time fish grew according to h[X] exp(ek1[a,1]*mat[a])+ # exp( kT) if they are mature, else 1 linf[1]*(1 exp(ek1[a,1]))*mat[a] # growth after maturity len[a]< ifelse(mal[a]==0,lta[a], lta[a]*exp(ek1[a,2])+linf[2] *(1 exp(ek1[a,2]))) vul[a]< 1/(1+exp((560len[a])/(560*mucv))) fecf[a]< alpha*len[a]^beta* (1/(1+exp((T a)/sig[1])) 1/(1+exp((taua)/sig[2]))) fecm[a]< alpha*len[a]^delta*(1/(1+exp((taua)/sig[2]))) # lorenzen with female components SL[a,] < (1+(linf[c(1,2)]/len[a])*(exp(k[c(1,2)]) 1))^( 0.1342/k[c(1,2)]) M< log(SL[max(1,a1),]) mhat[a,]< M[1]+fvec[c(1,50)]*vul[ifelse(a==1,1,a1)] la[a,]< ifelse(a==rep(1,fiter),1,la[a1,]*exp( M[ifelse(a<=rep(tau,fiter),1,2)] PAGE 130 130 fvec*vul[ifelse(a==rep(1,fiter),1,a1)])) fagef[a,]< la[a,]*fecf[a] fagem[a,]< la[a,]*fecm[a] ya[a,]< la[a,]*vul[a]*(1 exp( fvec))*alpha*len[a]^beta # lorenzen with sex specific components SL1[a,]< (1+(linf[c(1,1)]/len[a])*(exp(k[c(1,1)]) 1))^ ( (0.1342)/k[c(1,1)]) # M=K from SEDAR 33 M1< log(SL1[max(1,a1),]) la1[a,]< ifelse(a==rep(1,fiter),1,la1[a1,]* exp( M1[ifelse(a<=rep(tau,fiter),1,2)] fvec*vul[ifelse(a==rep(1,fiter),1,a1)])) fagef1[a,]< la1[a,]*fecf[a] fagem1[a,]< la1[ a,]*fecm[a] ya1[a,]< la1[a,]*vul[a]*(1exp( fvec))*alpha*len[a]^beta } ff3< colSums(fagef) fm3< colSums(fagem) yield3< colSums(ya) fopt3< fvec[which(colSums(ya)==max(colSums(ya)))] ff31< colSums(fagef1) fm31< colSums(fagem1) yield31< colSums(y a1) fopt31< fvec[which(colSums(ya1)==max(colSums(ya1)))] return(list(ff3,fm3,yield3,fopt3,ff31,fm31,yield31,fopt31)) } PAGE 131 131 APPENDIX C CHAPTER 3 R CODE Function for Starting Values funinits< function(sratio='high') { # assign sex ratio here lst < list(R=5000, # max recruits npyr=100, # number of burnin years nyr=600, # number of years Fyr=200, # year prior to start of fishing (e.g. fishing starts in Fyr+1) alw=1.17*10^ 8, # a for lengthweight blw=3, # scalar for relating fem ale length to fecundity delta=1, # parameter relating male length to reproductive success # (if set at blw, then it is a ssuming male RS scales linearly # with weight) cmale=10, # parameter relating relative fecundity of males to size, # 1= same as female, 2=twice as much RS as a female of the # same size) h2=0.2, # heritability b1start=807, # starting value for beta 1 (intercept) betsd1=40, # stdev around beta1 (intercept on pmrn) b2start=2, # starting value for slope betsd2=0.1, # stdev around beta2 (slope on pmrn) betsd3=14, # stdev around beta3 (timing of transition) lim=800, # length limit Mlorlen=1000, # reference size for Lorenzen (any fish above this size has # no size based mortality) mk=1.5, # relationship between M and k, M=mk*k mug=0.3, # mean reproductive investment mugcv=0.1, # cv around mug muh=130, # mean juvenile growth rate muhcv=0.15, # cv around muh juvM=c(0.06,0.055,0.085,0.085), # juvenile mortality rate, order is for # social control well mixed, social control lag # static, and genetic control of transition if(sratio=='high') # params for high delta { lst$delta< 8 # delta lst$b3start< 1295 # critical size at transition lst$juvM< c(0.07,0.065,0.095,0.095) # juvenile mortality rates }else{ if(sratio=='med') # params for medium delta { lst$delta< 4 # delta PAGE 132 132 lst$b3start< 1205 # critical size at transition lst$juvM< c(0.06,0.055,0.085,0.085) # juvenile mortality rates }else{ # params for low delta lst$delta< 1 # delta lst$b3start< 1105 # critical size at transition lst$juvM< c(0.04,0.04,0.05,0.05) # juvenile mortality rates } } return(lst) } Static Timing of Transition FunLt< function(f,inits) # ltran static { # extract starting parameters R< inits$R npyr< inits$npyr nyr< inits$nyr+npyr+1 Fyr< inits$Fyr alw< inits$alw blw< inits$blw delta< inits$delta cmale< inits$cmale h2< inits$h2 b1start< inits$b1start betsd1< inits$betsd1 b2start< inits$b2start betsd2< inits$betsd2 b3start< inits$b3start betsd3< inits$betsd3 lim< inits$lim Mlorlen< inits$Mlorlen mk< inits$mk mug< inits$mug mugsd< mug*inits$mugcv muh< inits$muh muhsd< muh*inits$muhcv juvM< inits$juvM[3] # stockrecruit steepness< 0.8 # stockrecruit steepness bha< 4*steepness/(1steepness) # Maximum recruit survival bhb< bha/R # scalar for maximum number of recruits # create storage parameters pop< matrix(NA,nrow=nyr,ncol=R) PAGE 133 133 len< matrix(NA,nrow=nyr,ncol=R) mat< matrix(NA,nrow=nyr,ncol=R) male< matrix(NA,nrow=nyr,ncol=R) k< list() g< list() linf< list() M< list() k[[1]]< array(NA,c(nyr,R)) # female vbk for each individual in each year k[[2]]< array(NA,c(nyr,R)) # male vbk for each individual in each year g[[1]]< array(NA,c(nyr,R)) # female reproductive investment for each # individual in each year g[[2]]< array(NA,c(nyr,R)) # male reproductive investment for each # individual in each year linf[[1]]< array(NA,c(nyr,R)) # female lin f for each individual in each year linf[[2]]< array(NA,c(nyr,R)) # male linf for each individual in each year h< array(NA,c(nyr,R)) # juvenile growth rate for each individual in # each year M[[1]]< array(NA,c(nyr,R)) # female natural mortality for each individual in # each year M[[2]]< array(NA,c(nyr,R)) # male natural mortality for each individual in # each year beta1< array(NA,c(nyr,R)) # intercept for the maturation norm for each # individual in each year beta2< array(NA,c(nyr,R)) # slope of the maturation norm for each # individual in each year beta3< array(NA,c(nyr,R)) # length required to transition for each # individual in each year mub1< vector(length=nyr) # mean intercept (beta1) for each year mub2< vector(length=nyr) # mean slope (beta2) for each year mub3< vector(length=nyr) # mean timing of transition (beta3) for each # year newmales< rep(0,length=nyr) # number of new in each year matlen < rep(NA,length=nyr) # mean length of maturing individuals sdmatlen< rep(NA,length=nyr) # sd of length of maturing individuals Ltran< array(NA,c(nyr,R)) # lengthat transition matrix mulmale< rep(0,length=nyr) # mean length of all males muNmale< rep(0 ,length=nyr) # mean length of new males sdNmale< rep(0,length=nyr) # standard deviation of new male lengths N< array(0,c(nyr,3)) # total numbers, number s of females, and # number s of male per year eggs< vector(length=nyr) # eggs produced in each year feggs< vector(length=nyr) # eggs fertilized in each year Rp1< NULL # recruits for the next year (plus 1) sdb1< NULL # estimated standard deviation for beta1 in year y PAGE 134 134 sdb2< NULL # same for beta2 sdb3< NULL # same for beta3 # start simulation yr=1 pop[yr,]< 1 # set recruits in year as alive g [[1]][yr,]< rnorm(R,mug,mugsd) # assign reproductive investment female g[[2]][yr,]< g[[1]][yr,]*1.2 # reproductive investment male k[[1]][yr,]< log(1+g[[1]][yr,]/3) # k for females k[[2]][yr,]< log(1+g[[2]][yr,]/3) # k males h[yr,]< rnorm(R,muh,muhsd) # juvenile growth rate linf[[1]][yr,]< 3*h[yr,]/g[[1]][yr,] # linf females linf[[2]][yr,]< 3*h[yr,]/g[[2]][yr,] # linf males M[[1]][yr,]< mk*k[[1]][yr,] # natural mortality M[[2]][yr,]< mk*k[[2]][yr,] # natural mortality len[yr,]< h[yr,] # length at age 1 mat[yr,]< 0 # set initial maturities as immature male[yr,]< 0 # set initial males as not males # individual relationship for timing of maturation mub1[yr]< b1start # mean intercept for maturation norm mub2[yr]< b2start # mean slope for maturation norm mub3[yr]< b3start # mean length required to transition beta1[yr,]< rnorm(R,b1start,betsd1) # random intercept for each individual beta2[yr,]< r norm(R,b2start,betsd2) # random slope for each individual beta3[yr ,]< rnorm(R,b3start,betsd3) # random length for transition # add for next year mub1[yr+1]< b1start # mean intercept for maturation norm mub2[yr+1]< b2start # mean slope for maturation norm m ub3[yr+1]< b3start # mean size required for transition beta1[yr+1,]< rnorm(R,b1start,betsd1) # random intercept for each individual beta2[yr+1,]< r norm(R,b2start,betsd2) # random slope for each individual beta3[yr+1,]< rnorm(R, b3start,betsd3) # random length for transition Rp1[yr]< R # recruits for next year for(yr in 2:(nyr 1)) { # set up recruits PAGE 135 135 g [[1]][yr,]< rnorm(R,mug,mugsd) # reproductive investment female g[[2]][yr,]< g[[1]][yr,]*1.2 # reproductive investment male k[[1]][yr,]< log(1+g[[1]][yr,]/3) # k females k[[ 2]][yr,]< log(1+g[[2]][yr,]/3) # k males h[yr,]< rnorm(R,muh,muhsd) # juvenile growth rate linf[[1]][yr,]< 3*h[yr,]/g[[1]][yr,] # linf females linf[[2]][yr,]< 3*h[yr,]/g[[2]][yr,] # linf males M[[1]][y r,]< mk*k[[1]][yr,] # female natural mortality M[[2]][yr,]< mk*k[[2]][yr,] # male natural mortality # deal with all other fish alive take< which(pop==1) # take fish that are alive # survive takejf< which(male[take]==0) # juveniles and females tak em< which(male[take]==1) # males Mlor< vector(length=length(take)) # lorenzen mortality Mlor[takejf]< log((1+pmax(1,Mlorlen/len[take[takejf]])* (exp(k[[1]][take[takejf]]) 1))^ (M[[1]][take[takejf]]/k[[1]][take[takejf]])) # lorenzen for females if(length(takem)>0) { Mlor[takem]< log((1+pmax(1,Mlorlen/len[take[takem]])* (exp(k[[2]][take[takem]]) 1))^ (M[[2]][take[takem]]/k[[2]][take[takem]])) # lorenzen for males } if(yr<=Fyr) # no fishing at beginning, starts after 50 yrs { surv< rbinom(length(take),1,exp( MlorjuvM*(1 mat[take]))) # survival unfished condition }else{ surv< rbinom(length(take),1, exp( Mlor juvM*(1 mat[ take]) F[yr]*(len[take]>=lim))) # survival fished condition } pop[take]< surv # update survivors take2< which(pop==1) # take the survivors # mature take3< which(pop==1&mat==0) # alive and immature PAGE 136 136 ages< yr (take31)%%nyr # ages of individuals that are alive takemat< which(len[take3]>=(beta1[take3] beta2[take3]*ages)) # individuals that are larger than the inherited length at maturation mat[take3[takemat]]< 1 # set to mature matlen[yr]< mean(len[take3[takemat]]) # mean length of newly matured individs sdmatlen[yr]< sd(len[take3[takemat]]) # st dev of newly matured individs # transition take4< which(pop==1&male==0) # take individuals not male tran< which(len[take4]>=beta3[take4]) # new males male[take4[tran]]=1 # set to mature mulmale[yr]< mean(len[which(male==1&pop==1)]) # mean length of males muNmale[yr]< mean(len[take4[tran]]) # mean length of new males sdNmale[yr]< sd(len[take4[tran]]) # stdev of new males # grow take2jf< which(male[take2]==0) # females and juveniles take2m< which(male[take2]==1) # males len[take2[take2jf]]< pmin(linf[[1]][take2[take2jf]], # max length len[take2[take2jf]]+ # prior length ifelse(mat[take2[take2jf]]==0,h[take2[take2jf]], # if juvenile then linear (linf[[1]][take2[take2jf]] len[take2[take2jf]])* # if female than vb (1 exp( k[[1]][take2[take2jf]])))) # last part of vb if(length(take2m)>0) { len[take2[take2m]] < pmin(linf[[1]][take2[take2m]], # max length len[take2[take2m]]+ # prior length ifelse(len[take2[take2m]]>=linf[[2]][take2[take2m]], 0, # if larger than male linf, zero growth (linf[[2]][take2[take2m]] len[take2[take2m]])* # otherwise they grow (1 exp ( k[[2]][take2[take2m]])))) } take4< which(pop==1) ages< yr (take41)%%nyr # ages of individuals that are alive mu5[yr]< mean(len[take4[which(ages==5)]]) # mean length @ age 5 mu10[yr]< mean(len[ta ke4[which(ages==10)]]) # mean length @ age 10 mu15[yr]< mean(len[take4[which(ages==15)]]) # mean length @ age 15 # reproduce if(yr<=50) { # if year is less than 50, hold betas and R constant PAGE 137 137 Rp1[yr]< R # recruits next year # individual relationship for timing of maturation # beta values for next year's age 1 beta1[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b1start,betsd1) beta2[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b2start,betsd2) beta3[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b3start,betsd3) # m ean beta values and stdevs associated with them mub1[yr+1]< mean(beta1[yr+1,1:Rp1[yr]]) mub2[yr+1]< mean(beta2[yr+1,1:Rp1[yr]]) mub3[yr+1]< mean(beta3[yr+1,1:Rp1[yr]]) sdb1[yr+1]< sd(beta1[yr+1,1:Rp1[yr]]) sdb2[yr+1]< sd(beta2[yr+1,1:Rp1[yr] ]) sdb3[yr+1]< sd(beta3[yr+1,1:Rp1[yr]]) }else{ # if year is greater than 50, reproduction happens takefem< which(pop==1&mat==1&male==0) # take females takemale< which(pop==1&male==1) # take males if(length(takefem)==0|length(takem ale)==0) { # if no males or females, no reproduction Rp1[yr]< 0 mub1[yr+1]< NA mub2[yr+1]< NA mub3[yr+1]< NA sdb1[yr+1]< NA sdb2[yr+1]< NA sdb3[yr+1]< NA }else{ betwtf< (g[[1]][takefem]*alw*len[takefem]^blw) # fec based weighting eggs[yr]< sum(betwtf) # total egg production bmt1< cmale*(g[[2]][takemale]*alw*len[takemale]^blw) # total number of 'eggs' that male can fertilize (sperm production) bmt2< sum(betwtf)*alw*len[takemale]^del ta/s um(alw*len[takemale]^delta) # number 'eggs' the male s have access to betwtm< pmin(bmt1,bmt2) # contribution of each male feggs[yr]< min(eggs[yr],sum(betwtm)) # total fertilized eggs in year Rp1[yr]< ifelse(yr<=50,1,round(bha*feggs[yr ]/(1+bhb*feggs[yr]))) # recruits for next year # the following calculates the mean betas weighted on fecundity bet1f< sum(beta1[takefem]*betwtf/sum(betwtf)) PAGE 138 138 bet1m< sum(beta1[takemale]*betwtm/sum(betwtm)) bet2f< sum(beta2[takefem]*bet wtf/sum(betwtf)) bet2m< sum(beta2[takemale]*betwtm/sum(betwtm)) bet3f< sum(beta3[takefem]*betwtf/sum(betwtf)) bet3m< sum(beta3[takemale]*betwtm/sum(betwtm)) # average the male and female values b1< mean(c(bet1f,bet1m)) # response, new beta1 b2< mean(c(bet2f,bet2m)) # response, new beta2 b3< mean(c(bet3f,bet3m)) # response, new beta2 # assigns the parents to recrui ts if there is only 1 parent, # allocate all to that parent parent1< ifelse(length(takefem)==rep(1,round(Rp1[yr])), rep(takefem,round(Rp1[yr])), sample(takefem,round(Rp1[yr]), replace=TRUE,prob=betwtf/sum(betwtf))) parent2< ifelse(length(takemale)==rep(1,round(Rp1[yr])), rep(takemale,round(Rp1[yr])), sample(takemale,round(Rp1[yr]), replace=TRUE,prob=betwtm/sum(betwtm))) # takes the mean and standard deviation of the parental values for beta1 pb1< rowMeans(cbind(beta1[parent1],beta1[parent2])) pb1std< apply(cbind(beta1[parent1],beta1[parent2]),1,FUN=sd) # calculates new beta1 for offspring using heritability mb1< h2*(pb1b1)+b1 # takes the mean and standard deviation of the parental values for beta2 pb2< rowMeans(cbind(beta2 [parent1],beta2[parent2])) pb2std< apply(cbind(beta2[parent1],beta2[parent2]),1,FUN=sd) # calculates new beta2 for offspring using heritability mb2< h2*(pb2b2)+b2 # takes the mean and standard deviation of the parental values for beta2 pb3< rowMeans(cbind(beta3[parent1],beta3[parent2])) pb3std< apply(cbind(beta3[parent1],beta3[parent2]),1,FUN=sd) # calculates new beta2 for offspring using heritability mb3< h2*(pb3b3)+b3 # be ta values for next year's age 1 beta1[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb1,betsd1)#pb1std) beta2[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb2,betsd2)#pb2std) PAGE 139 139 beta3[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b3start,betsd3)#pb2std) # calculate mean beta values and stdevs, used for checking mub1[yr+1]< mean(beta1[yr+1,1:Rp1[yr]]) mub2[yr+1]< mean(beta2[yr+1,1:Rp1[yr]]) mub3[yr+1]< mean(beta3[yr+1,1:Rp1[yr]]) sdb1[yr+1]< sd(beta1[yr+1,1:Rp1[yr]]) sdb2[yr+1]< sd(beta2[yr+1,1:Rp1[yr]]) sdb3[yr+1]< sd(beta3[yr+1,1:Rp1[yr]]) } } # add age 1 fish to the population pop[yr,]< ifelse(yr<=rep(50,R),1,c(rep(1,Rp1[yr 1]),rep(0,R Rp1[yr 1]))) # set recruits in year as alive # do this at the end so you don't have to run over age 1's in simulation len[yr,]< h[yr,] # length at age 1 mat[yr,]< 0 # set initial maturities as immature male[yr,]< 0 # ... with males # total pop size, number of females, and number of males in year N[yr,1]< sum(pop,na.rm=TRUE) N[yr,2]< sum(pop[which(mat==1&male==0)]) N[yr,3]< sum(pop[which(male==1)]) # if pop size is zero, then fill in and break loop if(sum(pop,na.rm=TRUE)==0) { mub1[(yr+1):nyr]< NA mub2[(yr+1):nyr]< NA mub3[(yr+1):nyr]< NA newmales[(yr+1):nyr]< NA matlen[(yr+1):nyr]< NA mulmal e[(yr+1):nyr]< NA muNmale[(yr+1):nyr]< NA N[(yr+1):nyr,]< 0 eggs[(yr+1):nyr]< 0 feggs[(yr+1):nyr]< 0 mu5[(yr+1):nyr]< NA mu10[(yr+1):nyr]< NA mu15[(yr+1):nyr]< NA break } } PAGE 140 140 # return values of interest return(list(betas=cbind(mub1[(npyr+1):(nyr 1)], mub2[(npyr+1):(nyr1)]), newmales=newmales[(npyr+1):(nyr 1)], matlen=matlen[(npyr+1):(nyr 1)],mulmale=mulmale[(npyr+1):(nyr 1)], muNmale=muNmale[(npyr+1):(nyr 1)], N=N[(npyr+1):(nyr1),],eggs=eggs[(npyr+1):(nyr 1)], feggs=feggs[(npyr+1):(nyr 1)], SR=N[(npyr+1):(nyr 1),3]/(N[(npyr+1):(nyr1),2]+ N[(npyr+1):(nyr 1),3]),mu5=mu5[(npyr+1):(nyr 1)], mu10=mu10[(npyr+1):(nyr 1)],mu15=mu15[(npyr+1):(nyr 1)])) rm(list (pop,len,mat)) } Genetic Control of Transition FunLtG< function(f,inits) # ltran genetic { # extract starting parameters R< inits$R npyr< inits$npyr nyr< inits$nyr+npyr+1 Fyr< inits$Fyr alw< inits$alw blw< inits$blw delta< inits$delta cmale< inits $cmale h2< inits$h2 b1start< inits$b1start betsd1< inits$betsd1 b2start< inits$b2start betsd2< inits$betsd2 b3start< inits$b3start betsd3< inits$betsd3 lim< inits$lim Mlorlen< inits$Mlorlen mk< inits$mk mug< inits$mug mugsd< mug*inits$mugcv muh< inits$muh muhsd< muh*inits$muhcv juvM< inits$juvM[3] # stockrecruit steepness< 0.8 # stockrecruit steepness bha< 4*steepness/(1steepness) # Maximum recruit survival bhb< bha/R # scalar for maximum number of recruits PAGE 141 141 # create storage parameters pop< matrix(NA,nrow=nyr,ncol=R) len< matrix(NA,nrow=nyr,ncol=R) mat< matrix(NA,nrow=nyr,ncol=R) male< matrix(NA,nrow=nyr,ncol=R) k< list() g< list() linf< list() M< list() k[[1]]< array(NA,c(nyr,R)) # female vbk for each individual in each year k[[2]]< array(NA,c(nyr,R)) # male vbk for each individual in each year g[[1]]< array(NA,c(nyr,R)) # female reproductive investment for each # individual in each year g[[2]]< array(NA,c(nyr,R)) # male reproductive investment for each # individual in each year linf[[1]]< array(NA,c(nyr,R)) # female linf for each individual in each year linf[[2]]< array(NA,c(nyr,R)) # male linf for each individual in each year h< array(NA,c(nyr,R)) # juvenile growth rate for each individual in # each year M[[1]]< array(NA,c(nyr,R)) # female natural mortality for each individual in # each year M[[2]]< array(NA,c(nyr,R)) # male natural mortality for each individual in # each year beta1< array(NA,c(nyr,R)) # intercept for the maturation norm for each # individual in each year beta2< array(NA,c(nyr,R)) # slope of the maturation norm for each # individual in each year beta3< array(NA,c(nyr,R)) # length required to transition for each # individual in each year mub1< vector(length=nyr) # mean intercept (beta1) for each year mub2< vector(length=nyr) # mean slope (beta2) for each year mub3< vector(length=nyr) # mean timing of transition (beta3) for each # year newmales< rep(0,length=nyr) # number of new in each year matlen< rep(NA,length=nyr) # mean length of maturing individuals sdmatlen< rep(NA,length=nyr) # sd of length of maturing individuals Ltran< array(NA,c(nyr,R)) # lengthat transition matrix mulmale< rep(0,length=nyr) # mean length of all males muNmale< rep(0,length=nyr) # mean length of new males sdNmale< rep(0,length=nyr) # standard deviation of new male lengths N< array(0,c(nyr,3)) # total numbers, number s of females, and # number s of male per year eggs< vector(length=nyr) # eggs produced in each year feggs< vector(length=nyr) # eggs fertilized in each year PAGE 142 142 Rp1< NULL # recruits for the next year (plus 1) sdb1< NULL # estimated standard deviation for beta1 in year y sdb2< NULL # same for beta2 sdb3< NULL # same for beta3 # start simulation yr=1 pop[yr,]< 1 # set recruits in year as alive g [[1]][yr,]< rnorm(R,mug,mugsd) # assign reproductive investment female g[[ 2]][yr,]< g[[1]][yr,]*1.2 # reproductive investment male k[[1]][yr,]< log(1+g[[1]][yr,]/3) # k for females k[[2]][yr,]< log(1+g[[2]][yr,]/3) # k males h[yr,]< rnorm(R,muh,muhsd) # juvenile growth rate linf[[1]][yr,]< 3*h[yr,]/g[[1]][yr,] # l inf females linf[[2]][yr,]< 3*h[yr,]/g[[2]][yr,] # linf males M[[1]][yr,]< mk*k[[1]][yr,] # natural mortality M[[2]][yr,]< mk*k[[2]][yr,] # natural mortality len[yr,]< h[yr,] # length at age 1 mat[yr,]< 0 # set initial maturities as immature male[yr,]< 0 # set initial males as not males # individual relationship for timing of maturation mub1[yr]< b1start # mean intercept for maturation norm mub2[yr]< b2start # mean slope for maturation norm mub3[yr]< b3start # mean len gth required to transition beta1[yr,]< rnorm(R,b1start,betsd1) # random intercept for each individual beta2[yr,]< r norm(R,b2start,betsd2) # random slope for each individual beta3[yr ,]< rnorm(R,b3start,betsd3) # random length for transition # add for next year mub1[yr+1]< b1start # mean intercept for maturation norm mub2[yr+1]< b2start # mean slope for maturation norm m ub3[yr+1]< b3start # mean size required for transition beta1[yr+1,]< rnorm(R,b1start,betsd1) # random intercept for each indiv idual beta2[yr+1,]< r norm(R,b2start,betsd2) # random slope for each individual beta3[yr+1,]< rnorm(R,b3start,betsd3) # random length for transition Rp1[yr]< R # recruits for next year for(yr in 2:(nyr 1)) PAGE 143 143 { # set up recruits g [[1]][yr,]< rnorm (R,mug,mugsd) # reproductive investment female g[[2]][yr,]< g[[1]][yr,]*1.2 # reproductive investment male k[[1]][yr,]< log(1+g[[1]][yr,]/3) # k females k[[ 2]][yr,]< log(1+g[[2]][yr,]/3) # k males h[yr,]< rnorm(R,muh,muhsd) # juvenile growth rate linf[[1]][yr,]< 3*h[yr,]/g[[1]][yr,] # linf females linf[[2]][yr,]< 3*h[yr,]/g[[2]][yr,] # linf males M[[1]][y r,]< mk*k[[1]][yr,] # female natural mortality M[[2]][yr,]< mk*k[[2]][yr,] # male natural mortality # deal with all other fish alive take< which(pop==1) # take fish that are alive # survive takejf< which(male[take]==0) # juveniles and females takem< which(male[take]==1) # males Mlor< vector(length=length(take)) # lorenzen mortality Mlor[takejf]< log((1+pmax(1,Mlorlen/len[take[takejf]])* (exp(k[[1]][take[takejf]]) 1))^ (M[[1]][take[takejf]]/k[[1]][take[takejf]])) # lorenzen for females if(length(takem)>0) { Mlor[takem]< log((1+pmax(1,Mlorlen/len[take[takem]])* (exp(k[[2]][take[takem]]) 1))^ (M[[2]][take[takem]]/k[[2]][take[takem]])) # lorenzen for males } if(yr<=Fyr) # no fishing at beginning, starts after 50 yrs { surv< rbinom(length(take),1,exp( MlorjuvM*(1 mat[take]))) # survived }else{ surv< rbinom(length(take),1,exp( MlorjuvM*(1 mat[take]) F[yr]*(len[take]>=lim))) # those that survived from fishing } pop[take]< surv # update survivors take2< which(pop==1) # take the survivors # mature PAGE 144 144 take3< which(pop==1&mat==0) ages< y r (take31)%%nyr # ages of individuals that are alive takemat< which(len[take3]>=(beta1[take3] beta2[take3]*ages)) # individuals that are larger than the inherited length at maturation mat[take3[takemat]]< 1 # set to mature matlen[yr]< mean(len[take3[takemat]]) # mean length of newly matured individs sdmatlen[yr]< sd(len[take3[takemat]]) # stdev of newly matured individuals # transition take4< which(pop==1&male==0) tran< which(len[take4]>=beta3[take4]) male[take4[tran]]=1 mulmale[yr]< mean(len[which(male==1&pop==1)]) muNmale[yr]< mean(len[take4[tran]]) sdNmale[yr]< sd(len[take4[tran]]) # grow take2jf< which(male[take2]==0) # females and juveniles take2m< which(male[take2]==1) # males len[take2[take2jf]]< pmin(linf[[1]][take2[take2jf]], # max length len[take2[take2jf]]+ # prior length ifelse(mat[take2[take2jf]]==0,h[take2[take2jf]], # if juvenile then linear (linf[[1]][take2[take2jf]] len[take2[take2jf]])* # if female than vb (1 exp( k[[1]][take2[take2jf]])))) # last part of vb if(length(take2m)>0) { len[take2[take2m]]< pmin(linf[[1]][take2[take2m]], # max length len[take2[take2m]]+ # prior length ifelse(len[take2[take2m]]>=linf[[2]][take2[take2m]], 0, # if larger than male linf, zero growth (linf[[2]][take2[take2m]] len[take2[take2m]])* # otherwise they grow (1 exp ( k[[2]][take2[take2m]])))) } take4< which(pop==1) ages< yr (take41)%%nyr # ages of individuals that are al ive mu5[yr]< mean(len[take4[which(ages==5)]]) # mean length at age 5 mu10[yr]< mean(len[take4[which(ages==10)]]) # mean length at age 10 mu15[yr]< mean(len[take4[which(ages==15)]]) # mean length at age 15 # reproduce if(yr<=50) { PAGE 145 145 Rp1[yr]< R # individual relationship for timing of maturation # beta values f or next year's age 1 beta1[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b1start,betsd1) beta2[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b2start,betsd2) beta3[yr+1,1:Rp1[yr]]< rnorm(Rp1[ yr],b3start,betsd3) mub1[yr+1]< mean(beta1[yr+1,1:Rp1[yr]]) mub2[yr+1]< mean(beta2[yr+1,1:Rp1[yr]]) mub3[yr+1]< mean(beta3[yr+1,1:Rp1[yr]]) sdb1[yr+1]< sd(beta1[yr+1,1:Rp1[yr]]) sdb2[yr+1]< sd(beta2[yr+1,1:Rp1[yr]]) sdb3[yr+1]< sd(bet a3[yr+1,1:Rp1[yr]]) }else{ takefem< which(pop==1&mat==1&male==0) takemale< which(pop==1&male==1) if(length(takefem)==0|length(takemale)==0) { Rp1[yr]< 0 mub1[yr+1]< NA mub2[yr+1]< NA mub3[yr+1]< NA sdb1[yr+1]< NA sdb2[yr+1]< NA sdb3[yr+1]< NA }else{ betwtf< (g[[1]][takefem]*alw*len[takefem]^blw) # fec based weight eggs[yr]< sum(betwtf) bmt1< cmale*(g[[2]][takemale]*alw*len[takemale]^blw) # total number of 'eggs' male can fertilize bmt2< sum(betwtf)*alw*len[takemale]^delta/sum(alw*len[takemale]^delta # number 'eggs' male has access to betwtm< pmin(bmt1,bmt2) # contribution of that male feggs[yr]< min(eggs[yr],sum(betwtm)) # fertilized eggs Rp1[yr]< ifelse(yr< =50,1,round(bha*feggs[yr]/(1+bhb*feggs[yr]))) # recruits for next year bet1f< sum(beta1[takefem]*betwtf/sum(betwtf)) bet1m< sum(beta1[takemale]*betwtm/sum(betwtm)) bet2f< sum(beta2[takefem]*betwtf/sum(betwtf)) bet2m< sum(beta2[t akemale]*betwtm/sum(betwtm)) PAGE 146 1 46 bet3f< sum(beta3[takefem]*betwtf/sum(betwtf)) bet3m< sum(beta3[takemale]*betwtm/sum(betwtm)) # mean beta of the parents b1< mean(c(bet1f,bet1m)) b2< mean(c(bet2f,bet2m)) b3< mean(c(bet3f,bet3m)) # assigns the parents to recrui ts parent1< ifelse(length(takefem)==rep(1,round(Rp1[yr])), rep(takefem,round(Rp1[yr])), sample(takefem,round(Rp1[yr]), replace=TRUE,prob=betwtf/sum(betwtf))) parent2< ifelse(length(takemale)==rep(1,round(Rp1[yr])), rep(takemale,round(Rp1[yr])), sample(takemale,round(Rp1[yr]), replace=TRUE,prob=betwtm/sum(betwtm))) # takes the mean and standard deviation of the parental values for beta1 pb1< r owMeans(cbind(beta1[parent1],beta1[parent2])) pb1std< apply(cbind(beta1[parent1],beta1[parent2]),1,FUN=sd) # calculates new beta1 for offspring using heritability mb1< h2*(pb1b1)+b1 # takes the mean and standard deviation of the parental values for beta2 pb2< rowMeans(cbind(beta2[parent1],beta2[parent2])) pb2std< apply(cbind(beta2[parent1],beta2[parent2]),1,FUN=sd) # calculates new beta2 for offspring using heritability mb2< h2*(pb2b2)+b2 # takes the mean a nd standard deviation of the parental values for beta2 pb3< rowMeans(cbind(beta3[parent1],beta3[parent2])) pb3std< apply(cbind(beta3[parent1],beta3[parent2]),1,FUN=sd) # calculates new beta2 for offspring using heritability mb3< h2*(pb3b3)+b3 # beta values for next year's age 1 beta1[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb1,betsd1)#pb1std) beta2[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb2,betsd2)#pb2std) beta3[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb3,betsd3)#pb2std) mub1[yr+ 1]< mean(beta1[yr+1,1:Rp1[yr]]) mub2[yr+1]< mean(beta2[yr+1,1:Rp1[yr]]) PAGE 147 147 mub3[yr+1]< mean(beta3[yr+1,1:Rp1[yr]]) sdb1[yr+1]< sd(beta1[yr+1,1:Rp1[yr]]) sdb2[yr+1]< sd(beta2[yr+1,1:Rp1[yr]]) sdb3[yr+1]< sd(beta3[yr+1,1:Rp1[yr]]) } } # add age 1 fish to the population pop[yr,]< ifelse(yr<=rep(50,R),1,c(rep(1,Rp1[yr 1]),rep(0,R Rp1[yr 1]))) # set recruits in year as alive # do this at the end so you don't have to run over age 1's in simulation len[yr,]< h[yr,] # length at age 1 mat[yr,]< 0 # set initial maturities as immature male[yr,]< 0 # ... with males N[yr,1]< sum(pop,na.rm=TRUE) N[yr,2]< sum(pop[which(mat==1&male==0)]) N[yr,3]< sum(pop[which(male==1)]) if(sum(pop,na.rm=TRUE)==0) { mub1[(yr+1):nyr] < NA mub2[(yr+1):nyr]< NA mub3[(yr+1):nyr]< NA newmales[(yr+1):nyr]< NA matlen[(yr+1):nyr]< NA mulmale[(yr+1):nyr]< NA muNmale[(yr+1):nyr]< NA N[(yr+1):nyr,]< 0 eggs[(yr+1):nyr]< 0 feggs[(yr+1):nyr]< 0 mu5[(yr+1):nyr]< NA m u10[(yr+1):nyr]< NA mu15[(yr+1):nyr]< NA break } } #mean(sdNmale[50:700]) return(list(betas=cbind(mub1[(npyr+1):(nyr 1)], mub2[(npyr+1):(nyr1)]), newmales=newmales[(npyr+1):(nyr 1)], matlen=matlen[(npyr+1):(nyr 1)], mulmale=mulmale[(npyr+1):(nyr 1)], muNmale=muNmale[(npyr+1):(nyr 1)], N=N[(npyr+1):(nyr1),],eggs=eggs[(npyr+1):(nyr 1)], PAGE 148 148 feggs=feggs[(npyr+1):(nyr 1)], SR=N[(npyr+1):(nyr 1),3]/(N[(npyr+1):(nyr1),2]+ N[(npyr+1):(nyr 1),3]),mu5=mu5[(npyr+1):(nyr 1)], mu10=mu10[(npyr+1):(nyr 1)],mu15=mu15[(npyr+1):(nyr 1)])) rm(list(pop,len,mat)) } S ocial Control of Transition for W ell mixed P opulations FunSMT< function(f,inits) { # extract starting parameters R< in its$R npyr< inits$npyr nyr< inits$nyr+npyr+1 Fyr< inits$Fyr alw< inits$alw blw< inits$blw delta< inits$delta cmale< inits$cmale h2< inits$h2 b1start< inits$b1start betsd1< inits$betsd1 b2start< inits$b2start betsd2< inits$betsd2 b3start< inits$b3start betsd3< inits$betsd3 lim< inits$lim Mlorlen< inits$Mlorlen mk< inits$mk mug< inits$mug mugsd< mug*inits$mugcv muh< inits$muh muhsd< muh*inits$muhcv juvM< inits$juvM[3] # stockrecruit steepness< 0.8 # stockrecruit steepness bha< 4*steepness/(1steepness) # Maximum recruit survival bhb< bha/R # scalar for maximum number of recruits # create storage parameters pop< matrix(NA,nrow=nyr,ncol=R) len< matrix(NA,nrow=nyr,ncol=R) mat< matrix(NA,nrow=nyr,ncol=R) male< matrix(NA, nrow=nyr,ncol=R) k< list() PAGE 149 149 g< list() linf< list() M< list() k[[1]]< array(NA,c(nyr,R)) # female vbk for each individual in each year k[[2]]< array(NA,c(nyr,R)) # male vbk for each individual in each year g[[1]]< array(NA,c(nyr,R)) # female r eproductive investment for each # individual in each year g[[2]]< array(NA,c(nyr,R)) # male reproductive investment for each # individual in each year linf[[1]]< array(NA,c(nyr,R)) # female linf for each individual in each year linf[[2]]< array(NA,c(nyr,R)) # male linf for each individual in each year h< array(NA,c(nyr,R)) # juvenile growth rate for each individual in # each year M[[1]]< array(NA,c(nyr,R)) # female natural mortality for each individual in # each year M[[2]]< array(NA,c(nyr,R)) # male natural mortality for each individual in # each year beta1< array(NA,c(nyr,R)) # intercept for the maturation norm for each # individual in each year beta2< array(NA,c(nyr,R)) # slope of the maturation norm for each # individual in each year beta3< array(NA,c(nyr,R)) # length required to transition for each # individual in each year mub1< vector(length=nyr) # mean intercept (beta1) for each year mub2< vector(length=nyr) # mean slope (beta2) for each year mub3< vector(length=nyr) # mean timing of transition (beta3) for each # year newmales< rep(0,length=nyr) # number of new in each year matlen< rep(NA,length=nyr) # mean length of maturing individuals sdmatlen< rep(NA,length=nyr) # sd of length of maturing individuals Ltran< array(NA,c(nyr,R)) # lengthat transition matrix mulmale< rep(0,length=nyr) # mean length of all males muNmale< rep(0,length=nyr) # mean length of new males sdNma le< rep(0,length=nyr) # standard deviation of new male lengths N< array(0,c(nyr,3)) # total numbers, number s of females, and # number s of male per year eggs< vector(length=nyr) # eggs produced in each year feggs< vector(length=nyr) # egg s fertilized in each year Rp1< NULL # recruits for the next year (plus 1) sdb1< NULL # estimated standard deviation for beta1 in year y sdb2< NULL # same for beta2 sdb3< NULL # same for beta3 # start simulation yr=1 PAGE 150 150 pop[yr,]< 1 # set recruits in year as alive g [[1]][yr,]< rnorm(R,mug,mugsd) # assign reproductive investment female g[[2]][yr,]< g[[1]][yr,]*1.2 # reproductive investment male k[[1]][yr,]< log(1+g[[1]][yr,]/3) # k for females k[[2]][yr,]< log(1+g[[2]][yr,]/3) # k males h[yr,]< rnorm(R,muh,muhsd) # juvenile growth rate linf[[1]][yr,]< 3*h[yr,]/g[[1]][yr,] # linf females linf[[2]][yr,]< 3*h[yr,]/g[[2]][yr,] # linf males M[[1]][yr,]< mk*k[[1]][yr,] # natural mortality M[[2]][yr,]< mk*k[[2]][yr,] # natural mortality len[yr,]< h[yr,] # length at age 1 mat[yr,]< 0 # set initial maturities as immature male[yr,]< 0 # set initial males as not males # individual relationship for timing of maturation mub1[yr]< b1start # mean intercept for maturation norm mub2[yr]< b2start # mean slope for maturation norm beta1[yr,]< rnorm(R,b1start,betsd1) # random intercept for each individual beta2[yr,]< r norm(R,b2start,betsd2) # random slope for each individual # add for next year mub1[yr+1]< b1start # mean intercept for maturation norm mub2[yr+1]< b2start # mean slope for maturation norm beta1[yr+1,]< rnorm(R,b1start,betsd1) # random intercept for each individual beta2[yr+1,]< r norm(R,b2start,betsd2) # random slope f or each individual Rp1[yr]< R # recruits for next year for(yr in 2:(nyr 1)) { # set up recruits g [[1]][yr,]< rnorm(R,mug,mugsd) # reproductive investment female g[[2]][yr,]< g[[1]][yr,]*1.2 # reproductive investment male k[[1]][yr,]< lo g(1+g[[1]][yr,]/3) # k females k[[ 2]][yr,]< log(1+g[[2]][yr,]/3) # k males h[yr,]< rnorm(R,muh,muhsd) # juvenile growth rate linf[[1]][yr,]< 3*h[yr,]/g[[1]][yr,] # linf females linf[[2]][yr,]< 3*h[yr,]/g[[2]][yr,] # linf males PAGE 151 151 M[[1]][y r, ]< mk*k[[1]][yr,] # female natural mortality M[[2]][yr,]< mk*k[[2]][yr,] # male natural mortality # deal with all other fish alive take< which(pop==1) # take fish that are alive # survive takejf< which(male[take]==0) # juveniles and female s takem< which(male[take]==1) # males Mlor< vector(length=length(take)) # lorenzen mortality Mlor[takejf]< log((1+pmax(1,Mlorlen/len[take[takejf]])* (exp(k[[1]][take[takejf]]) 1))^ (M[[1]][take[takejf]]/k[[1]][take[takejf]])) # lorenzen for females if(length(takem)>0) { Mlor[takem]< log((1+pmax(1,Mlorlen/len[take[takem]])* (exp(k[[2]][take[takem]]) 1))^ (M[[2]][take[takem]]/k[[2]][take[takem]])) # lorenzen for males } if(yr<=Fyr) # no fishing at beginning, starts after Fyr { surv< rbinom(length(take),1,exp( MlorjuvM*(1 mat[take]))) # survive }else{ surv< rbinom(length(take),1,exp( MlorjuvM*(1 mat[take]) F[yr]*(len[take]>=lim))) # survival from fishing } pop[take]< surv # update survivors take2< which(pop==1) # take the survivors # mature take3< which(pop==1&mat==0) ages< yr (take31)%%nyr # ages of individuals that are alive takemat< which(len[take3]>=(beta1[take3] beta2[take3]*ages)) # individuals that are larger than the inherited length at maturation mat[take3[takemat]]< 1 # set to mature matlen[yr]< mean(len[take3[takemat]]) # mean length of newly matured individs sdmatlen[yr]< sd(len[take3[takemat]]) # stdev of newly matured individuals # transition if(yr>10) # only run transition if there are at least 10 females { PAGE 152 152 # given relative size of others and male size distribution, # randomly choose if they transition tran< take2[which(mat[take2]==1&male[take2]==0)] # individuals that can possibly transition all females males < take2[which(male[take2]==1)] # individuals that are already males fec< (g[[1]][tran]*alw*l en[tran]^blw) # fecundity if re m ained female fecm< cmale*(g[[2]][tran]*alw*len[tran]^blw) # fecundity if male fecmtot< sum(cmale*(g[[2]][males]*alw*len[males]^blw)) # total current male fecundity ord< order(len[tran],decreasing=TRUE) # order by length newmalelen< NULL jj< 1 # counters i< 1 xx< 0 while(xx<=100) # run loop as long as xx is less than 100 { fm< pmin(fecm[ord[i]], # max sperm production sum(fec[ ord[i]])* # female fecundity alw*len[tran[ord[i]]]^delta/ # contribution of male i (alw*len[tran[ord[i]]]^delta+ # total male fec if individual transitions sum(alw*len[males]^delta))) # total fec of all males ff< fec[ord[i]] # fec of individual if stays female if(fm[length(fm)]>ff[length(fm)]) # transition if fm is greater than ff { male[tran[ord[i]]]< 1 # set individual to male males[length(males)+1]< tran[ord[i]] # add new male to males vector newmales[yr]=newmales[yr]+1 # count of new males in year yr newmalelen[jj]< len[tran[ord[i]]] # keep track of new male lengths fec[which(ord==i)]=0 # remove female fecundity of individual that transitioned jj=jj+1 # moves counter forward if(i==length(tran)) # stop loop if gone through all individuals { xx=1000000 # set large to abort loop when no more females lef t }else{ i=i+1 # counter 2 moves forward } }else{ xx< xx+1 # exit loop if fm is less than ff (they do not transition) } } mulmale[yr]< mean(len[which(male==1&pop==1)]) muNmale[yr]< mean(newmalelen) sdNmale[yr]< sd(newmalelen) if(yr>50)ltranall< c(ltranall,newmalelen) PAGE 153 153 } # grow take2jf< which(male[take2]==0) # females and juveniles take2m< which(male[take2]==1) # males len[take2[take2jf]]< pmin(linf[[1]][take2[take2jf]], # max length len[take2[t ake2jf]]+ # prior length ifelse(mat[take2[take2jf]]==0,h[take2[take2jf]], # if juvenile then linear (linf[[1]][take2[take2jf]] len[take2[take2jf]])* # if female than vb (1 exp( k[[1]][take2[take2jf]])))) # last part of vb if(length( take2m)>0) { len[take2[take2m]]< pmin(linf[[1]][take2[take2m]], # max length len[take2[take2m]]+ # prior length ifelse(len[take2[take2m]]>=linf[[2]][take2[take2m]], 0, # if larger than male linf, zero growth (linf[[2]] [take2[take2m]] len[take2[take2m]])* # otherwise they grow (1 exp( k[[2]][take2[take2m]])))) } take4< which(pop==1) ages< yr (take41)%%nyr # ages of individuals that are alive mu5[yr]< mean(len[take4[which(ages==5)]]) # mean length at age 5 mu10[yr]< mean(len[take4[which(ages==10)]]) # mean length at age 10 mu15[yr]< mean(len[take4[which(ages==15)]]) # mean length at age 15 # reproduce if(yr<=50) { Rp1[yr]< R # individual relationship for timing of maturation # beta values for next year's age 1 beta1[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b1start,betsd1) beta2[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b2start,betsd2) mub1[yr+1]< mean(beta1[yr+1,1:Rp1[yr]]) mub2[yr+1]< mean(beta2[yr+1,1:Rp1[yr]]) sdb1[yr+1]< sd(beta1[yr+1,1:Rp1[yr]]) sdb2[yr+1]< sd(beta2[yr+1,1:Rp1[yr]]) }else{ takefem< which(pop==1&mat==1&male==0) takemale< which(pop==1&male==1) PAGE 154 154 if(length(takefem)==0|length(takemale)==0) { Rp1[yr]< 0 }else{ betwtf < (g[[1]][takefem]*alw*len[takefem]^blw) # fec based weight eggs[yr]< sum(betwtf) bmt1< cmale*(g[[2]][takemale]*alw*len[takemale]^blw) # total number of 'eggs' male can fertilize bmt2< sum(betwtf)*alw*len[takemale]^delta/sum(alw*len[takemale]^delta) # number 'eggs' male has access to betwtm< pmin(bmt1,bmt2) # contribution of that male feggs[yr]< min(eggs[yr],sum(betwtm)) Rp1[yr]< ifelse(yr< =50,1,round(bha*feggs[yr]/(1+bhb*feggs[yr]))) # recruits for next year bet1f< sum(beta1[takefem]*betwtf/sum(betwtf)) bet1m< sum(beta1[takemale]*betwtm/sum(betwtm)) bet2f< sum(beta2[takefem]*betwtf/sum(betwtf)) bet2m< sum(beta2[takemale]*betwtm/sum(betwtm)) b1< mean(c(bet1f,bet1m)) # response, new beta1 b2< mean(c(bet2f,bet2m)) # response, new beta2 parent1< sample(takefem,round(Rp1[yr]),replace=TRUE, prob=betwtf/sum(betwtf)) parent2< ifelse(length(takemale)==rep(1,round(Rp1[yr])), rep(takemale,round(Rp1[yr])), sample(takemale,round(Rp1[yr]), replace=TRUE,prob=betwtm/sum(betwtm))) pb1< rowMeans(cbind(beta1[parent1],beta1[parent2])) pb1std< apply(cbind(beta1[parent1],beta1[parent2]),1,FUN=sd) mb1< h2*(pb1b1)+b1 pb2< rowMeans(cbind(beta2[parent1],beta2[parent2])) pb2std< apply(cbind(beta2[parent1],beta2[parent2]),1,FUN=sd) mb2< h2*(pb2b2)+b2 # beta values for next year's age 1 beta1[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb1,betsd1)#pb1std) PAGE 155 155 beta2[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb2,betsd2)#pb2std) mub1[yr+1]< mean(beta1[yr+1,1:Rp1[yr]]) mub2[yr+1]< mean(beta2[yr+1,1:Rp1[yr]]) sdb1[yr+1]< sd(beta1[yr+1,1:Rp1[yr]]) sdb2[yr+1]< sd(beta2[y r+1,1:Rp1[yr]]) } } # add age 1 fish to the population pop[yr,]< ifelse(yr<=rep(50,R),1,c(rep(1,Rp1[yr 1]),rep(0,R Rp1[yr 1]))) # set recruits in year as alive # do this at the end so you don't have to run over age 1's in simulation le n[yr,]< h[yr,] # length at age 1 mat[yr,]< 0 # set initial maturities as immature male[yr,]< 0 # ... with males N[yr,1]< sum(pop,na.rm=TRUE) N[yr,2]< sum(pop[which(mat==1&male==0)]) N[yr,3]< sum(pop[which(male==1)]) if(sum(pop,na.rm=TRUE)==0) { mub1[(yr+1):nyr]< NA mub2[(yr+1):nyr]< NA newmales[(yr+1):nyr]< NA matlen[(yr+1):nyr]< NA mulmale[(yr+1):nyr]< NA muNmale[(yr+1):nyr]< NA N[(yr+1):nyr,]< 0 eggs[(yr+1):nyr]< 0 feggs[(yr+1):nyr]< 0 mu5[(yr+1):nyr]< NA mu10[(yr+1):nyr]< NA mu15[(yr+1):nyr]< NA break } } return(list(betas=cbind(mub1[(npyr+1):(nyr 1)],mub2[(npyr+1):(nyr 1)]), newmales=newmales[(npyr+1):(nyr 1)], matlen=matlen[(npyr+1):(nyr 1)],mulmale=mulmale[(npyr+ 1):(nyr1)], muNmale=muNmale[(npyr+1):(nyr 1)], N=N[(npyr+1):(nyr1),],eggs=eggs[(npyr+1):(nyr 1)], feggs=feggs[(npyr+1):(nyr 1)], SR=N[(npyr+1):(nyr 1),3]/(N[(npyr+1):(nyr1),2]+ N[(npyr+1):(nyr 1),3]),mu5=mu5[(npyr+1):(nyr 1)], PAGE 156 156 mu10=mu10[(npy r+1):(nyr1)],mu15=mu15[(npyr+1):(nyr 1)])) rm(list(pop,len,mat)) } S ocial Control of Transition for P opulations with a Lag FunMTS< function(f,inits) # mature, transition, survive { # extract starting parameters R< inits$R npyr< inits$npyr nyr< inits$nyr+npyr+1 Fyr< inits$Fyr alw< inits$alw blw< inits$blw delta< inits$delta cmale< inits$cmale h2< inits$h2 b1start< inits$b1start betsd1< inits$betsd1 b2start< inits$b2start betsd2< inits$betsd2 b3start< inits$b3start betsd3< inits$bet sd3 lim< inits$lim Mlorlen< inits$Mlorlen mk< inits$mk mug< inits$mug mugsd< mug*inits$mugcv muh< inits$muh muhsd< muh*inits$muhcv juvM< inits$juvM[3] # stockrecruit steepness< 0.8 # stockrecruit steepness bha< 4*steepness/(1steepness) # Maximum recruit survival bhb< bha/R # scalar for maximum number of recruits # create storage parameters pop< matrix(NA,nrow=nyr,ncol=R) len< matrix(NA,nrow=nyr,ncol=R) mat< matrix(NA,nrow=nyr,ncol=R) male< matrix(NA,nrow=nyr,ncol=R) k< list() g< list() linf< list() M< list() PAGE 157 157 k[[1]]< array(NA,c(nyr,R)) # female vbk for each individual in each year k[[2]]< array(NA,c(nyr,R)) # male vbk for each individual in each year g[[1]]< array(NA,c(nyr,R)) # female reproductive investment for eac h # individual in each year g[[2]]< array(NA,c(nyr,R)) # male reproductive investment for each # individual in each year linf[[1]]< array(NA,c(nyr,R)) # female linf for each individual in each year linf[[2]]< array(NA,c(nyr,R)) # male linf for each individual in each year h< array(NA,c(nyr,R)) # juvenile growth rate for each individual in # each year M[[1]]< array(NA,c(nyr,R)) # female natural mortality for each individual in # each year M[[2]]< array(NA,c(nyr,R)) # male natural mortality for each individual in # each year beta1< array(NA,c(nyr,R)) # intercept for the maturation norm for each # individual in each year beta2< array(NA,c(nyr,R)) # slope of the maturation norm for each # individual in each year beta3< array(NA,c(nyr,R)) # length required to transition for each # individual in each year mub1< vector(length=nyr) # mean intercept (beta1) for each year mub2< vector(length=nyr) # mean slope (beta2) for each year mub3< vector(length=nyr) # mean timing of transition (beta3) for each # year newmales< rep(0,length=nyr) # number of new in each year matlen< rep(NA,length=nyr) # mean length of maturing individuals sdmatlen< re p(NA,length=nyr) # sd of length of maturing individuals Ltran< array(NA,c(nyr,R)) # lengthat transition matrix mulmale< rep(0,length=nyr) # mean length of all males muNmale< rep(0,length=nyr) # mean length of new males sdNmale< rep(0,length=nyr) # st andard deviation of new male lengths N< array(0,c(nyr,3)) # total numbers, number s of females, and # number s of male per year eggs< vector(length=nyr) # eggs produced in each year feggs< vector(length=nyr) # eggs fertilized in each year Rp1< NULL # recruits for the next year (plus 1) sdb1< NULL # estimated standard deviation for beta1 in year y sdb2< NULL # same for beta2 sdb3< NULL # same for beta3 # start simulation yr=1 pop[yr,]< 1 # set recruits in year as alive g [[1]][yr,]< rnorm(R,mug,mugsd) # assign reproductive investment female PAGE 158 158 g[[2]][yr,]< g[[1]][yr,]*1.2 # reproductive investment male k[[1]][yr,]< log(1+g[[1]][yr,]/3) # k for females k[[2]][yr,]< log(1+g[[2]][yr,]/3) # k males h[yr,]< rnorm(R,muh,muhsd) # juvenile growth rate linf[[1]][yr,]< 3*h[yr,]/g[[1]][yr,] # linf females linf[[2]][yr,]< 3*h[yr,]/g[[2]][yr,] # linf males M[[1]][yr,]< mk*k[[1]][yr,] # natural mortality M[[2]][yr,]< mk*k[[2]][yr,] # natural mortality len[yr,]< h[yr,] # length at age 1 mat[yr,]< 0 # set initial maturities as immature male[yr,]< 0 # set initial males as not males # individual relationship for timing of maturation mub1[yr]< b1start # mean intercept for maturation norm m ub2[yr]< b2start # mean slope for maturation norm beta1[yr,]< rnorm(R,b1start,betsd1) # random intercept for each individual beta2[yr,]< r norm(R,b2start,betsd2) # random slope for each individual # add for next year mub1[yr+1]< b1start # mean int ercept for maturation norm mub2[yr+1]< b2start # mean slope for maturation norm beta1[yr+1,]< rnorm(R,b1start,betsd1) # random intercept for each individual beta2[yr+1,]< r norm(R,b2start,betsd2) # random slope for each individual Rp1[yr]< R # recruits for next year for(yr in 2:(nyr 1)) { # set up recruits g [[1]][yr,]< rnorm(R,mug,mugsd) # reproductive investment female g[[2]][yr,]< g[[1]][yr,]*1.2 # reproductive investment male k[[1]][yr,]< log(1+g[[1]][yr,]/3) # k females k[[ 2]][yr,]< log(1+g[[2]][yr,]/3) # k males h[yr,]< rnorm(R,muh,muhsd) # juvenile growth rate linf[[1]][yr,]< 3*h[yr,]/g[[1]][yr,] # linf females linf[[2]][yr,]< 3*h[yr,]/g[[2]][yr,] # linf males M[[1]][y r,]< mk*k[[1]][yr,] # female natural mort ality M[[2]][yr,]< mk*k[[2]][yr,] # male natural mortality PAGE 159 159 # mature take3< which(pop==1&mat==0) ages< yr (take31)%%nyr # ages of individuals that are alive takemat< which(len[take3]>=(beta1[take3] beta2[take3]*ages)) # individuals th at are larger than the inherited length at maturation mat[take3[takemat]]< 1 # set to mature matlen[yr]< mean(len[take3[takemat]]) # get mean length of newly matured individuals sdmatlen[yr]< sd(len[take3[takemat]]) # get stdev of newly matured individuals # transition if(yr>10)#sum(mat[1:(yr1),]*pop[1:(yr 1),])>50) # only run if at least 10 females { # given relative size of others and male size distribution, # need to randomly choose if they transition tran< take2[wh ich(mat[take2]==1&male[take2]==0)] # individuals that can possibly transition all females males < take2[which(male[take2]==1)] # individuals that are already males fec< (g [[1]][tran]*alw*len[tran]^blw) # total fecundity if remains female fecm< cmale*(g[[2]][tran]*alw*len[tran]^blw) # fecundity if male fecmtot< sum(cmale*(g[[2]][males]*alw*len[males]^blw)) # total current male fecundity ord< order(len[tran],decreasing=TRUE) newmalelen< NULL jj< 1 i< 1 xx< 0 whi le(xx<=100) { fm< pmin(fecm[ord[i]], # max sperm production sum(fec[ ord[i]])* # female fecundity alw*len[tran[ord[i]]]^delta/ # relative contribution of indiv id ual i (alw*len[tran[ord[i]]]^delta+ # total male fec if individual transitions sum(alw*len[males]^delta))) # total fec of all males ff< fec[ord[i]] # fec of individual if stays female if(fm[length(fm)]>ff[length(fm)]) # if fm is greater than ff, then they become male { male[tran[ord[i]]]< 1 # set individual to male males[length(males)+1]< tran[ord[i]] # add new male to males vector newmales[yr]=newmales[yr]+1 # kee ps count of new males in year newmalelen[jj]< len[tran[ord[i]]] # keeps track of new male l engths fec[which(ord==i)]=0 # removes female fecundity if transitioned jj=jj+1 # moves counter forward if(i==length(tran)) # catch to stop loop if gone through all individs PAGE 160 160 { xx=1000000 # set large to abort loop when no more females left }else{ i=i+1 # counter 2 moves forward } }else{ xx< xx+1 # exit loop if fm is less than ff (they do not transition) } } mulmale[yr]< mean(len[which(male==1&pop==1)]) muNmale[yr]< mean(newmalelen) } # survive take< which(pop==1) # take fish that are alive # survive takejf< which(male[take]==0) # juveniles and females takem< which(male[take]==1) # males Mlor< vector(length=length(take)) # lorenzen mortality Mlor[takejf]< log((1+pmax(1,Mlorlen/len[take[takejf]])* (exp(k[[1]][take[takejf]]) 1))^ (M[[1]][take[takejf]]/k[[1]][take[takejf]])) # lorenzen for females if(length(takem)>0) { Mlor[takem]< log((1+pmax(1,Mlorlen/len[take[takem]])* (exp(k[[2]][take[takem]]) 1))^ (M[[2]][take[takem]]/k[[2]][take[takem]])) # lorenzen for males } if(yr<=Fyr) # no fishing at beginning, starts after 50 yrs { surv< rbinom(length(take),1,exp( MlorjuvM*(1 mat[take]))) # survived }else{ surv< r binom(length(take),1,exp( MlorjuvM*(1 mat[take]) F[yr]*(len[take]>=lim))) # those that survived from natural and fishing } pop[take]< surv # update survivors take2< which(pop==1) # take the survivors # grow PAGE 161 161 take2jf< which(male[take2]==0) # females and juveniles take2m< which(male[take2]==1) # males len[take2[take2jf]]< pmin(linf[[1]][take2[take2jf]], # max length len[take2[take2jf]]+ # prior length ifelse(mat[take2[take2jf]]==0,h[take2[take2jf]], # if juvenile then linear (linf[[1]][take2[take2jf]] len[take2[take2jf]])* # if female than vb (1 exp( k[[1]][take2[take2jf]])))) # last part of vb if(length(take2m)>0) { len[take2[take2m]]< pmin(linf[[1]][take2[take2m]], # max length l en[take2[take2m]]+ # prior length ifelse(len[take2[take2m]]>=linf[[2]][take2[take2m]], 0, # if larger than male linf, zero growth (linf[[2]][take2[take2m]] len[take2[take2m]])* # otherwise they grow (1 exp ( k[[2]][take2[ta ke2m]])))) } take4< which(pop==1) ages< yr (take41)%%nyr # ages of individuals that are alive mu5[yr]< mean(len[take4[which(ages==5)]]) # mean length at age 5 mu10[yr]< mean(len[take4[which(ages==10)]]) # mean length at age 10 mu15[y r]< mean(len[take4[which(ages==15)]]) # mean length at age 15 # reproduce if(yr<=50) { Rp1[yr]< R # individual relationship for timing of maturation # beta values for next year's age 1 beta1[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b1start,betsd1) beta2[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],b2start,betsd2) mub1[yr+1]< mean(beta1[yr+1,1:Rp1[yr]]) mub2[yr+1]< mean(beta2[yr+1,1:Rp1[yr]]) sdb1[yr+1]< sd(beta1[yr+1,1:Rp1[yr]]) sdb2[yr+1]< sd(beta2[yr+1,1:Rp1[yr] ]) }else{ takefem< which(pop==1&mat==1&male==0) takemale< which(pop==1&male==1) if(length(takefem)==0|length(takemale)==0) { PAGE 162 162 Rp1[yr]< 0 }else{ betwtf< (g[[1]][takefem]*alw*len[takefem]^blw) # fec based weight eggs[yr] < sum(betwtf) bmt1< cmale*(g[[2]][takemale]*alw*len[takemale]^blw) # total number of 'eggs' male can fertilize bmt2< sum(betwtf)*alw*len[takemale]^delta/sum(alw*len[takemale]^delta) # number 'eggs' male has access to betwtm< pmin(bmt1,bmt2) # contribution of that male feggs[yr]< min(eggs[yr],sum(betwtm)) Rp1[yr]< ifelse(yr<=50,1,round(bha*feggs[yr]/(1+bhb*feggs[yr]))) # recruits for next year # mean value of beta1 in the female population bet1 f< sum(beta1[takefem]*betwtf/sum(betwtf)) bet1m< sum(beta1[takemale]*betwtm/sum(betwtm)) # ... for males bet2f< sum(beta2[takefem]*betwtf/sum(betwtf)) # ... for beta1 bet2m< sum(beta2[takemale]*betwtm/sum(betwtm)) b1< mean(c(bet1f,bet1m )) # mean of beta1 for population b2< mean(c(bet2f,bet2m)) # ... for beta2 # assigns the parents to recruits parent1< sample(takefem,round(Rp1[yr]),replace=TRUE, prob=betwtf/sum(betwtf)) parent2< ifelse(length(takemale)==rep(1,roun d(Rp1[yr])), rep(takemale,round(Rp1[yr])), sample(takemale,round(Rp1[yr]),replace=TRUE,prob=betwtm/sum(betwtm))) # takes the mean and standard deviation of the parental values for beta1 pb1< rowMeans(cbind(beta1[parent1],beta1[parent2])) pb1std< apply(cbind(beta1[parent1],beta1[parent2]),1,FUN=sd) # calculates new beta1 for offspring using heritability mb1< h2*(pb1b1)+b1 # takes the mean and standard deviation of the parental values for beta2 pb2< rowMeans(cbind(beta2[parent1],beta2[parent2])) pb2std< apply(cbind(beta2[parent1],beta2[parent2]),1,FUN=sd) # calculates new beta2 for offspring using heritability mb2< h2*(pb2b2)+b2 PAGE 163 163 # beta values for next year's age 1 beta1[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb1,betsd1)#pb1std) beta2[yr+1,1:Rp1[yr]]< rnorm(Rp1[yr],mb2,betsd2)#pb2std) mub1[yr+1]< mean(beta1[yr+1,1:Rp1[yr]]) mub2[yr+1]< mean(beta2[yr+1,1:Rp1[yr]]) sdb1[yr+1]< sd(beta1[yr+1,1:Rp1[yr]]) sdb 2[yr+1]< sd(beta2[yr+1,1:Rp1[yr]]) } } # add age 1 fish to the population pop[yr,]< ifelse(yr<=rep(50,R),1,c(rep(1,Rp1[yr 1]),rep(0,R Rp1[yr 1]))) # set recruits in year as alive # pop[yr,is.na(pop[yr,])]< 0 # do this at the end so you don't have to run over age 1's in simulation len[yr,]< h[yr,] # length at age 1 mat[yr,]< 0 # set initial maturities as immature male[yr,]< 0 # ... with males N[yr,1]< sum(pop,na.rm=TRUE) N[yr,2]< sum(pop[which(mat==1&male==0)]) N[yr,3]< sum(pop[which(male==1)]) if(sum(pop,na.rm=TRUE)==0) { mub1[(yr+1):nyr]< NA mub2[(yr+1):nyr]< NA newmales[(yr+1):nyr]< NA matlen[(yr+1):nyr]< NA mulmale[(yr+1):nyr]< NA muNmale[(yr+1):nyr]< NA N[(yr+1):nyr,]< 0 eggs[(yr+1):ny r]< 0 feggs[(yr+1):nyr]< 0 mu5[(yr+1):nyr]< NA mu10[(yr+1):nyr]< NA mu15[(yr+1):nyr]< NA break } } return(list(betas=cbind(mub1[(npyr+1):(nyr 1)], mub2[(npyr+1):(nyr1)]), newmales=newmales[(npyr+1):(nyr 1)], matlen=matlen[(npyr+ 1):(nyr1)],mulmale=mulmale[(npyr+1):(nyr 1)], muNmale=muNmale[(npyr+1):(nyr 1)], PAGE 164 164 N=N[(npyr+1):(nyr1),],eggs=eggs[(npyr+1):(nyr 1)], feggs=feggs[(npyr+1):(nyr 1)], SR=N[(npyr+1):(nyr 1),3]/(N[(npyr+1):(nyr1),2]+ N[(npyr+1):(nyr 1),3]),mu5=mu5[(npyr+1):(nyr 1)], mu10=mu10[(npyr+1):(nyr 1)],mu15=mu15[(npyr+1):(nyr 1)])) rm(list(pop,len,mat)) } PAGE 165 165 LIST OF REFERENCES Adams, S., and Williams, A.J. 2001. 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North Am. J. Fish. Manag. 35(6): 1121 1131. Wi ngfield, C.A. 1940. The effect of certain environmental factors on the growth of Brown Trout (Salmo trutta L.). J. Exp. Biol. 17(4): 435 448. Ye, Y., and Dennis, D. 2009. How reliable are the abundance indices derived from commercial catch effort standardi zation? Can. J. Fish. Aquat. Sci. 66(7): 1169 1178. doi:10.1139/F09 070. Zimmer, C., and Emlen, D.J. 2013. Evolution: Making Sense of Life. Roberts Publishers Greenwood Village, CO . PAGE 182 182 BIOGRAPHICAL SKETCH Bryan Matthias attended the University of Wisconsin Stevens Point and graduated in 2010 with a Bachelor of Science degree. He double majored in fisheries and biology with a minor in natural science He graduated in 2012 with Master of Science degree in fis heries and aquatic sciences from the University of Florida. Bryan earned a Doctor of Philosophy in fisheries and aquatic s cience from the University of Florida in 2016. 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