1 T HE OPTIMAL USE OF A PESTICIDE WITH INCREASING RESISTANCE: THE CASE OF CITRUS GREENING DISEASE IN FLORIDA By Yuelu Xu A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE RE QUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2014
2 Â© 2014 Yuelu Xu
3 ACKNOWLEDGMENTS I would like to take this opportunity to express my sincere gratitude to those people who have helped me in acco mplishing this thesis. First of all, I am deeply grateful to my supervisory committee chair, Dr. Kelly Grogan for her tremendous help and encouragement throughout the whole process of thesis writing. It would be impossible for me to complete the thesis wi thout her patient guidance, helpful advice and detailed instruction. Her r igorous attitude toward academic work impressed me a lot. I hope that one day I could help others as she helped me. Additionally, great thanks go to my two committee member s. I am si ncerely grateful to Dr. Gao, Zhifeng for his scientific advice, knowledge and many insightful discussions with me. I also deeply appreciate Dr. Salois, Matthew for his generous help, encouragement and many suggestions to me. Besides, I would also thank my friends for providing support and friendship that makes my two year study in University of Florida full of fun. In the end , I would like to express my deepest appreciation to my parents for their support and love. I would not have made it this far without them.
4 TABLE OF CONTENTS page ACKNOWLEDGME NTS ................................ ................................ ................................ ............... 3 LIST OF TABLES ................................ ................................ ................................ ........................... 5 LIST OF FIGURES ................................ ................................ ................................ ......................... 6 ABSTRACT ................................ ................................ ................................ ................................ ..... 7 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................... 9 Florida C itrus I ndustry ................................ ................................ ................................ .............. 9 Huanglongbing (HLB) ................................ ................................ ................................ .............. 9 Pesticide R esistance ................................ ................................ ................................ ................ 12 Research Objectives ................................ ................................ ................................ ................ 14 2 LITERATURE REVIEW ................................ ................................ ................................ ....... 15 Optimal P esticide U se u nder I ncreasing P esticide R esistance ................................ ............... 15 Optimal C ontrol of V ector borne D iseases ................................ ................................ ............ 16 3 THE MODEL ................................ ................................ ................................ ......................... 19 4 DISCUSSION ................................ ................................ ................................ ......................... 33 5 CONCLUSION ................................ ................................ ................................ ....................... 35 APPENDIX : CALCULA TION PROCEDURES OF SHADOW PRICES AND NULLCLINES 36 LIST OF REFERENCES ................................ ................................ ................................ ............... 43 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ......... 47
5 LIST OF TABLES Table page 3 1 Table of base parameter values ................................ ................................ .......................... 32
6 LIST OF FIGURES Figure page 1 1 Citrus Prod uction United States and Florida: Crop Years 1991 1992 through 2010 2011(Source: FDACS, 2012) ................................ ................................ ............................. 11 1 2 18 41.) ................................ ................................ ................................ ................................ 13 3 1 Phase diagram ................................ ................................ ................................ .......... 25 3 2 Phase diagr am ................................ ................................ ................................ .......... 26 3 3 Phase diagram ................................ ................................ ................................ ......... 28 3 4 Phase diagram ................................ ................................ ................................ ......... 29
7 Abstract of Thesis Presented to t he Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science T HE OPTIMAL USE OF A PESTICIDE WITH INCREASING RESISTANCE: THE CASE OF CITRUS GREENING DISEASE IN FLORIDA By Yu elu Xu August 2014 Chair: Kelly A. Grogan Major: Food and Resource Economics The citrus industry is one of the largest agricultural industries in the state of Florida, worth more than 1.5 billion dollars per year. In recent years, citrus production has been lower than historical averages, in large part due to Huanglongbing (HLB) or citrus greening disease. In Florida, HLB was first discovered in orange production in Miami Dade County in September 2005 and has since spread through the whole state at a rap id speed. Globally, there are at least 40 countries in which HLB affects citrus production and about 100 million trees have been infected by 2010 (National Research Council, 2010). Chemical control is the primary form of control used by Florida commercial citrus producers. With the fast spread of HLB and the increasing application of the pesticide used to treat it, pesticide resistance is already occurring (Stelinski et al., 2011). As susceptible individuals die after each application, the proportion of re sistant individuals increases. Judicious use of pesticides could delay further resistance development. Currently, the optimal amount and timing of pesticide applications, taking resistance into consideration, are not yet known for the case of citrus greeni ng disease in Florida. Previous work in this area has analyzed optimal pest control decisions in the presence of resistance when the insect damages the crop directly. This
8 study contributes to the previous literatures by building a theoretical model that d etermines the optimal strategy of chemical control, taking into account resistance for an insect that serves as a vector of the disease. The model shows that the susceptibility of pests to pesticide s might be a renewable resource in some circumstances , wh ich differs from previous work that has assumed resistance is always declining . The results of this study will help Florida citrus producers in response to the problem of pesticide resistance.
9 CHAPTER 1 INTRODUCTION Florida C itrus I ndustry The citrus indus try is one of the largest agricultural industries in the state of Florida; it is worth more than 1.5 billion dollars per year and generated at least 75,800 jobs in the 2007 08 season (Hodges and Rahmani, 2009). Oranges are the primary crop in the Florida c itrus industry, which are mainly used for making juice and concentrate. In recent years, diseases have resulted in citrus production that is below historical average yield (FDACS, 2012). Citrus producers in Florida suffered on average a 10 15% loss of frui t as a result of the fruit dropping before harvest during the 2012 13 harvest season (EPA, 2013). Among the diseases, one, called Huanglongbing (HLB) or citrus greening disease, is perhaps the worst threat worldwide, causing losses and seriously impeding g lobal citrus production. Huanglongbing (HLB) HLB was first reported in the early twentieth century in southern regions of China and reached orange production in Florida by September 2005, beginning in Miami Dade County and then spreading through the whole state at a rapid speed (Fan et al., 2009; Hodges and Spreen, 2012). The bacteria of HLB, Xylella fastidiosa , are spread by an insect, called the Asian citrus psyllid, which feeds on tree leaves. HLB spread rates are mainly affected by two factors: the age of the trees and the local vector population (Bassanezi et al., 2008; Gottwald, 2010). HLB can also be transmitted through grafting but this will not be the focus of this thesis (Halbert and Manjunath, 2004). Xylella fastidiosa infected citrus trees show nutrition deficiency because the bacteria hinder the flow of nutrition in the xylem tissues , which leads to symptoms appearing on citrus branches and leaves ( Muhammad et al., 2011). There are a series of symptoms as the disease
10 progresses. The initial symp toms of HLB include the appearance of some yellow mottles on leaves, sparsely foliated trees and twig dieback. Early detection of the disease is impeded by several factors. First, the disease has a latency period. This is the time between infection by a pa thogen and the onset of infectivity. Second, there is an incubation period, which is the time between infection by a pathogen and onset of symptom expression. The incubation and latency periods are two concurrent and related temporal processes, both begin ning at infection. The length of each is greatly affected by tree age, with the disease progressing more quickly in younger trees (Gottwald, 2010). Even after the incubation period, the initial symptoms can be challenging to detect, implying that an infect ed tree serves as a source of the disease during both the incubation period and the initial period of symptoms without the grower knowing that the tree is diseased. In the secondary stage, symptoms of HLB include small, upright leaves and yellow shoots cov ering entire leaves (Gottwald, 2010). HLB can lead to fruit drop prior to harvest, irregularly shaped fruit with a thick peel with a portion that remains green, and a bitter taste. Once the fruit s taste is affected, it is of no value for commercial use. The state of Florida accounts for 65% of the citrus production of the United States (USDA, 2013). As shown in Figure 1.1, there was a sharp drop in citrus production in Florida in the 2004 05 season. The decline can be attributed to several causes. It is m ostly attributed to a series of hurricanes that struck Florida in the summer of 2004 and the concurrent spread of citrus canker, another bacterial citrus disease, from the high winds (Irey et al., 2006). HLB was first detected in 2005, and the relatively l ow yields in more recent years are attributed primarily to HLB.
11 Figure 1 1 . Citrus Production United States and Florida: Crop Years 1991 1992 through 2010 2011(Source: FDACS, 2012) For the 2006 07 through 2010 11 seasons, the total revenue received by Florida orange producers from orange production was 16% lower than the predicted counterfactual values without HLB. HLB also has a negative impact on employment in Florida. The difference in employment between with HLB and without HLB scenarios amounts to 8,257 jobs in the past five years (Hodges and Spreen, 2012). Unfortunately, there is no cure for HLB, but there are three types of measures that growers can use to control HLB. The first type is physical control. Because citrus trees are the only long ter m reservoir of HLB bacteria, the disease can be temporarily locally eliminated by removing infected trees and then replanting trees. However, this method does not work for most producers for several reasons. First, there may be many infected and symptomati c trees that are still producing commercially viable fruit. Removing all of them would result in a large loss (Bassanezi et al., 2008). Second, there could be many diseased trees present that are not yet symptomatic because of the long incubation period s o they cannot be identified as diseased and subsequently removed. Consequently, a reservoir of the disease would still exist, lessening the
12 benefit of removing the symptomatic trees. Third, the vector of the disease can re infect fields through movement ac ross fields. The second type of control is biological control. This measure uses predators and parasites that feed on or lay eggs in psyllids to provide control. Examples include ladybeetles, lacewings and spiders, which provide some control of psyllid p opulations (UC IPM, 2013). However, there are several disadvantages of biological control even if it has less negative effects on the environment. Biological control can involve high costs, and applied predators and parasites are easily influenced by tempe rature, humidity, and environment. This can lead to low efficacy and even lower cost effectiveness. The third form of control is chemical control, which suppresses psyllid populations through soil and foliar applications of insecticides. For example, a fol iar pyrethroid insecticide can be applied to eliminate adults and immature psyllids when they are observed on trees (UC IPM, 2013). Currently, chlorpyrifos, fenpropathrin, imidacloprid and aldicarb are recommended soil and foliar insecticides, which are co mmonly used to control psyllids (Qureshi and Stansly, 2007). The most common pesticide used by Florida citrus producers is imidacloprid, which is often applied as a soil drench for control of psyllids on young, non fruit bearing trees ( Serikawa et al., 201 2). recent years, growers have begun using a foliar nutrition program to reduce the negative effect of HLB on the productivity of citrus trees. Foliar nutritional sprays do not aim to eliminate the bacterium and inoculum, but rather to maintain the prod uctivity of trees for as long as possible (Spann et al., 2011). Pesticide R esistance Pesticide resistance was discovered by Dr. Melander in 1902. He found that some Wenatchee scales were still alive after the application of 26Â° sulphur lime which is ten t imes
13 stronger than normal applications (Melander, 1914). After this finding, more and more cases of pesticide resistance were discovered. Resistance to pesticides results from genetic mutations found in some pests, which allow the pests to survive after th e application of the pesticide. These individuals then pass on their genetic traits when reproducing. Figure 1 2 . Agrochemicals, pp. 18 41.) Over the last 60 years, the number of resistant arthropod species has increased rapidly in response to the development and use of insecticides. The horizontal bars in Figure 1.2 denote the timespans over which particular insecti cide groups have been used. The dates indicate the year in which resistance was first documented for that class of pesticides. From this figure, we can see that pesticide resistance is always increasing. In the United States, pesticide resistance has resul ted in about $ 10 billion of extra pest control costs each year. Growers tend to respond to resistance by applying even larger amounts of pesticides. Despite heavier dosages of pesticides applied, agricultural production in the United States has still suffe red about a 10% loss because of pesticide resistance (Palumbi, 2001). Florida citrus, but the efficacy of chemical control may be threatened by resistance. Stelinski et al.
14 (2011 ) conducted an experiment to test susceptibility of psyllids from five different sites to imidacloprid, chlorpyriphos, thiamethoxam, malathion and fenpropathrin in 2009. In this experiment, imidacloprid had a LD 50 resistance ratio of 35, which meant the ra tio of the number of susceptible psyllids to the number of resistant psyllids is 35 in the tested field. This was the lowest level of the resistance among the tested pesticides. Psyllids showed the highest level of resistance to fenpropathrin with an LD 50 resistance ratio of 4.8. Pesticides tested in this experiment are all common types that are regularly applied by Florida citrus growers. From this experiment, we can see that pesticide resistance will potentially be a problem for citrus growers. Pesticide resistance will lead to further increased use of pesticides, which can have additional negative effects. On the farm, increased use of pesticides will increase costs. Off the farm, increased use of pesticides can result in contamination of water, lowered air quality, and other negative ecological effects. Research Objectives Judicious use of pesticides could delay further resistance development. However, there is an absence of empirical work about the optimal amount and timing of pesticide applications to control citrus greening disease in Florida taking resistance into consideration. This study will help Florida citrus producers in response to the problem of pesticide resistance. This thesis addresses two research questions: (i) In each season, how much of the pesticide should be applied? (ii) What outcomes will be caused by different initial conditions? The remainder of t his thesis is as follows . The next section discusses the relevant literature. The bioeconomic model is presented in the third section, wh ich is followed by a discussion of numerical results and sensitivity analysis of parameters based on the theoretical model. Finally, conclusions are provided.
15 CHAPTER 2 LITERATURE REVIEW Relevant literature falls into two categories: optimal pesticide use with increasing pesticide resistance and the control of vector borne diseases. Optimal P esticide U se u nder I ncreasing P esticide R esistance This group of papers have addressed the issue of pest control with pesticide resistance in a dynamic framework. Hue th and Regev (1974) modeled pest susceptibility to the pesticide as a non renewable resource implying that susceptibility can only decrease over time and it can be depleted. They set up a general theoretical model to obtain the optimal timing and amount of pesticide to apply by a standard discrete time optimal control problem. The authors found that the optimal use at any depletion of the stock of susceptibility . The optimal time path of pesticide applied varies over the season, which differs from entomological recommendations , which are usually fixed. However, this is a general theoretical model, which does not consider specific pests and crops. Regev, Shalit an d Gutierrez (1983) developed a theoretical model that considered the increasing pesticide resistance of the alfalfa weevil in alfalfa production in California. This paper modeled the conflicting role of pesticide applications on the pest population and pes ticide resistance. The authors used the proportion of pests that are immune to the pesticide to describe the pesticide resistance level. They found that growers should not increase pesticide applications to control the level of the pest population when res istance begins to develop. Instead, doing nothing could be an optimal strategy when considering costs and the external environment. The model in this study only considers the case where pests destroy the crop directly and does not consider pesticide resist ance in the vector borne disease case.
16 Munro (1997) studied the effect of human activity and genetic evolution on the economic consequences of pesticide resistance. The author built two theoretical models: one model determined the optimal dosage of the pes ticide by a myopic planner who only considered the dynamic process of the pest population. The second model considers both the pest population dynamic process and genetic evolution of pesticide resistance when obtaining the optimal plan of pest management. He concluded that, in equilibrium, considering pesticide resistance results in a lower level of the pest population. Brown, Dickinson and Kramer (2013) developed a theoretical model to obtain the optimal pesticide application amount for policymakers in re ference to the case of malaria vector control with pesticide resistance. The authors concluded that use of pesticide for the control of malaria should be kept at a low level in the long term to prevent development of resistance in the mosquito population. Although this model is in reference to a vector borne disease with consideration of pesticide resistance, it is not designed for agricultural production. Optimal C ontrol of V ector borne D iseases Several papers have addressed the issue of pest control in ca ses of vector borne diseases. Literature in this category uses theoretical models to determine the optimal strategy when managing the loss caused by vector borne diseases. Marsh, Huffaker, and Long (2000) set up a framework which investigates the optimal control of a vector borne disease with discrete time management in reference to the potato leafroll virus net necrosis in potato production. The virus, the pest vector, and the crop stock were all modeled as renewable resources. The objective function was maximized subject to crop quality standards, which were established in market contracts. The authors derived the optimal time and amount of the pesticide to apply without considering the possibility of pesticide resistance.
17 Brown, Lynch, and Zilberman (200 2) established a theoretical framework to investigate the optimal strategy to control insect transmitted plant diseases with an application to the case of examined. O ne method was to remove riparian plants, the source vegetation. The other method used Christmas trees as barriers to control insect movement. From the framework, the authors derived the optimal input use, size of barriers and amount of removal of riparian plants to control the disease. The authors determined the optimal barrier size and found that the increasing effectiveness of barriers can significantly reduce the optimal barrier size and the cost of barrier crops . When barrier effectiveness, the fraction of insects blocked by one foot of barrier, increases from 0 to 1, the profit per acre increases from $3,054 to $5,201. Unlike the model that follows, this model is a static model, and the authors did not consider chemical control. Fuller, Alston and Sanch irico (2011) set up a model to examine how vectors, the Blue Disease. They modeled the relationship between the crop production and the spatial dynamics of disease ve ctors. In this study, the authors compared the individual grower case with two two grower cases: unilateral, uncooperative control and cooperative control. They found that the cooperative strategy could mitigate the damage from Pierce s Disease more than t he other cases and would lead to less abandoned land. While previous literature considers optimal pest control decisions taking pesticide resistance into consideration for the case of an insect pest that directly damages the crop, to the best of our knowle dge, no work has yet considered pest control and pesticide resistance for the case of an insect vectored disease. This paper s main contribution is to demonstrate that pesticide
18 susceptibility is not necessarily a non renewable resource. It also demonstrat es various possible pesticide application patterns over time as a result of various starting conditions.
19 CHAPTER 3 THE MODEL This model contains four main components: the pest population and its growth, the susceptibility of the pest p opulation to the pesticide, the relationship between the pest population and disease incidence, and yield as a function of disease incidence. I present this framework in reference to Huanglongbing (HLB) in Florida citrus production, but it is generalizable to other insect vectored diseases. It is assumed that the citrus grove is a closed system, and there is no immigration or emigration of pests to or from the given space. The pesticide used in the model is assumed to be effective immediately after being ap plied. Moreover, all production factors, except the chemical used, are assumed to be at the optimal level. Let denote the total pest population at time t , denote the amount of pesticide used at time t , and denote the proportion of the total alleles which are susceptible. Following Munro (1997), the dynamic process of the total pest population can be written as: (1) . Here, is the intrinsic growth rate of the pest , and is the carrying capacity. The time path of the proportion of the total alleles that are susceptible to the pesticide can be written as: (2) = , where denotes the absolute fitness of a susceptible phenotype in the absence of the pesticide, and denotes the fitn ess of the resistant strain. Fitness means the ability of a certain genotype or phenotype, for example, resistant homozygote, susceptible homozygote, and heterozygote, to survive and reproduce with other individuals among the same species. Absolute fitness is the ratio of the number of a certain genotype or phenotype present in the population after natural
20 selection occurs to the number before selection occurs (Munro, 1997). Let A denote an allele that makes the organism susceptible , and a denotes an allele that makes the organism resistant. Then t here are three possible genotypes: AA , aa and Aa . In the model, A is dominant. Therefore, Aa has the same fitness and susceptibility as AA . Since HLB is a vector borne disease, the pest does not destroy crops dire ctly in this model, but rather serves as a disease vector. The disease influences the yield directly. The disease incidence is the most important factor to describe it. Scientists have applied logistic, exponential and Gompertz growth functions to model th e dynamic progress of HLB incidence (Bassanezi et al., 2006; Gottwald et al., 1989; Gottwald et al., 2010). Disease incidence, , is the proportion of diseased trees at time t , which is a function of the pest population. Based on Fuller (2013), the d ynamic process of can be written as (3) = . Here, is the growth rate of disease incidence. The other crucial factor is disease severity, which describes the yield effect of the disease within an individual symptomatic t ree. Here denotes the disease severity of an individual tree at time t . Following Bassanezi and Bassanezi (2008), the function can be written as (4) . Here, denotes the value of the initi al proportion of symptoms in an individual tree at time t ; is the rate of disease severity progress in infected trees. The younger a tree is, the more easily it is infected by HLB, and the faster the disease spreads through the tree. According to Bassan ezi and Bassanezi (2008), when trees are 0 2 years old, is 3.68, and is 0.2; when trees are 3 5 years old, is 1.84, and is 0.1; when trees are 6 10 years old, is 0.92, and is 0.05; and when trees are older than 10 years, is 0.69, and is 0.025.
21 On the economic side of the model, ( ) denotes the yield of crops at time t . is the tree age at time t . Therefore, the yield function can be written as (5) (6) , where is the per tree yield of one healthy tree; is the total number of trees in the grove; and m, n and q are parameters which are used to estimate the yield in trees without HLB. The dynamic optimization of the present value of profit with continuous management can be stated mathematically as (7) Subject to (8) (9) = (10) = where P is defined as output price net of other input costs; L is the unit cost of the pesticide; r is the discount rate and t is the time. is the time at which the grower will switch to the alternative technology if one is avail able. Otherwise, T represents the time at which the grower stops producing citrus. All economic parameters here are assumed to be constant. Therefore, the Hamiltonian for this problem can be written as
22 (11) , where , and are shad ow prices of the pest population, the susceptibility level and the disease incidence, respectively. The first order condition is: (12) , which can be rearranged to yield: (13) Equation (13) is the marginal condition of profit maximization. The LHS of (13) is th e marginal monetary cost and the cost imposed on future periods from the reduction in susceptibility caused by one more unit of pesticide applied and the RHS of (13) is the marginal benefit of damages avoided by applying one more unit of pesticide. In the absence of resistance, equation (13) would not have the term in brackets and the only marginal cost would be the monetary cost. Additionally, the Hamiltonian yields the following first order conditions: (14)
23 (15) (16) (17) (18) = (19) = Please see the appendix for detailed calculations pertaining to equations ( 20 ) through ( 35 ) below. From (12), (14) and (15), solving for shadow prices yields: (20) (21) From (12), (20) and (21), solving for and produces: (22) (23) According to (12), (14), (20) and (22), solving for the nullcline of and yields: (24) Solving (24) for yields: (25) ,
24 According to (12) and (24), solving for in (25): Put (26) into (25), and we can get (27) . Solving for th e nullcline of yields: (2 8 ) Solving (26) for yields: According to (12) and (26), solving for in (28) yields: Putting ( 30 ) into (2 9 ) yields: (3 1 ) , For phase diagram, occurs when (27) holds. occurs when:
25 By plotting the nullclines, equations (27) and (31) with values: , , , and , yields the phase diagram illustrated in Figure 3 1. The base biological parameter values used are from Liu and Tsai (2000), Filchak et al. (2000) and Argentine et al. (1989). Figure 3 1 . Phase diagram From Figure 3.1, we find that a possible equilibrium with a positive pest population does not exist. However, if the grower applied the pesticide level at which equals zero, he could drive the pest population to zero over an infinite time horizon. This path may not be optimal for th e grower if considering a shorter time horizon. Three other general points are of interest. If a grower starts at point A, he will get on a trajectory such that pesticide use will always increase, and the pest population will decrease. However, pesticide u se will eventually approach infinity as the pest population approaches zero. With a relatively high pesticide application rate, as found on trajectories at point A, resistance develops and prevents the elimination of the pest. Higher and higher applicatio n levels become necessary to continue reducing the pest population. If the starting point is at point B, the low level of pesticide use will keep the pest population at a high
26 level. If the starting point is at point C, the low level of pesticide use will fail to achieve the goal of pest control, and the pest population actually increases. When choosing , , and , the phase diagram is illustrated in Figure 3 2 . Figure 3 2 . Phase diagram When increasing the value of , and , the two nullclines have an intersection which i mplies an equilibrium with an infinite time horizon (Figure 3 2). Graphically, the dashed trajectories leading into the equilibrium are optimal paths if the grower considers an infinite time horizon and the solid trajectories are examples of possible traje ctories for shorter time horizons and different starting pest and pesticide application levels. The equilibrium E is a saddle point. If a grower starts from point A, pesticide use will follow the same trend as occurred with point A in figure 3 1. Pesticid e use will always increase as the pest population decreases and will have to go to infinity in order to reduce the pest population to a small level. If the starting point is at point B, the low level of pesticide use will keep the pest population at a high level. If the starting point is at point C, the low level of pesticide use cannot control the growth of the pest population. If
27 the starting point is at point D, the amount of pesticide applied will continuously increase. The pest population will initiall y increase but then begin to decrease again when the pesticide application level crosses the nullcline. According to (12), (15), (21) and (23), solving for the equation of motion for and setting it equal to zero yields: (32) S olving for the nullcline of yields: (33) S olving for the nullcline of yields: (34) = Solving (3 4 ) for yields: (35) . For the phase diagram , when (33) holds, and when (35) holds. When choo sing the same values from Figure 3 1, , , the phase diagram is illustrated in Figure 3 3 .
28 Figure 3 3 . Phase diagram Because Figure 3 3 correspond s with the phase diagram of Figure 3 1 , an equilibrium of the pest population cannot be reached ; the dashed trajectories , which would be the optimal paths for an infinite time horizon , cannot actually reach ( ). Graphically, the solid trajectories are possible trajectories if the time horizon is finite. If the starting point is at point A, the level o f pesticide use is high, resulting in a constant decrease in the susceptible proportion of pests. Thus, the proportion of the susceptible alleles continues to decrease with inc reasing use of pesticide, and the pesticide use will eventually go to infinity. If the starting point is at point B, the high starting level of the proportion of the susceptible alleles allows for relatively low and constantly declining pesticide use. Susc eptibility initially decreases but increases again as pesticide use decreases further. If the starting point is at point C, the high proportion of the susceptible alleles and low pesticide use result in declining pesticide use and increasing susceptibility . If the starting point is at point D, a low level of susceptibility and a low level of pesticide use result in increasing pesticide use, which keeps the proportion of the susceptible alleles low.
29 When choosing the same values from Figure 3 2, , , the phase diagram is illustrated in Figure 3 4 . Figure 3 4 . Phase diagram From Figure 3 4, we find that, although different values cause different shapes of phase diagram s (Figure 3 1 and Figure 3 2), Figure s 3 3 and 3 4 ha ve similar shapes and have the same explanation despite having different parameter values. We assume the citrus grower plan s to sell the land if resistance becomes too problematic . Currently, there are no alternativ e pest or disease control methods, so this assumption is relevant. For free terminal time T with a scrap value, the transversality condition to determine the optimal T can be written as : (3 6 ) . Here is the scrap value function , which is assumed to be equal to the present discounted value of the land if sold . The scrap value at time t can be written as: ,
30 where is the value of the land under its ne xt best use and is consequently the value the grower could receive by selling the land. T aking the derivative of equation (37) with respect to T , we get: This value is assumed because there are currently no alternative con trols for HLB available. Therefore, (3 6 ) can be rewritten as ( 39 ) Solving (3 7 ) for , we could get: + w here :
31 Here is the optimal pesticide use in the last period of management. Because equation ( 40 ) is too complex, the graph of equation ( 40 ) for cannot be plotted in the phase diagrams . However, all optimal trajectories should end along the curve defined by ( 40 ).
32 Table 3 1. T able of base parameter values Parameter Definition Base Value Units Min Max K Carrying capacity 2000 pests/ tree 1000 1000 0 Natural growth rate of psyllid population 0.16 pests 0.03 0.2 r Interest rate 0.050 interest 0.010 0.150 L Chemical control unit price 1000 dollars/acre 800 1200 F Absolute fitness of a susceptible phenotype in the absence of the pesticide 0.3 per pes t 0.1 0.4 F a Fitness of resistant strain 0.06 0 per pest 0.001 0.9 00
33 CHAPTER 4 DISCUSSION Figure 3 1 and Figure 3 2 demonstrate that changes in parameter values can have large effects on the possible trajectories and outcomes . The simulation of paramet ers shows that with increasing values of and , and decreasing the value of , the figure changes from Figure 3 1 where no equilibrium exists to Figure 3 2 where an equilibrium is possible . The latter occurs when the fitness of resistant pests and the growth rate both increase and the fitness of susceptible pests decrease s, relative to the values used to construct Figure 3 1 . If and increas e further and decreas es further , the equilibrium of the infinite time horizon management will mov e toward towards a higher equilibrium pest population and lower equilibrium pesticide application rate . In Munro (1997), in the interior optimum. In my model , the optimal pesticide use , under infinite time horizon management, is Figure 3 3 and Figure 3 4 are two similar phase diagram s, which show that no matter how parameters change, an equilibrium over an i nfinite time horizon always exists. However, the equilibrium in Figure 3 3 cannot be reached because of its corresponding phase diagram does not have an equilibrium with a positive pest population. The optimal use of pesticide in Figure 3 3 and Figure 3 4 will increase if increases or decreases and vice versa . Comparing phase diagrams of pesticide use and the pest population, and pesticide use and the susceptib le proportion of total alleles, we have the following findings. First, the dynamic relationship of the pest population and pesticide use, and the dynamic relationship of the proportion of the susceptible alleles and the pesticide use are similar . Second , Figure 3 3 and Figure 3 4 suggest that the proportion of the susceptible alleles to the pesticide might be a renewable resource and can increase over time , which is different from the assumption that
34 susceptibility is a nonrenewable resource and can only decline over time, as used in previous relevant literature (Regev and Hue th , 1974; Regev, Shalit and Gutierrez, 1983). Third, although the proportion of the susceptible alleles is renewable, regenerating the susceptibility may require an increase in the pest population. This may increase the spread of the disease, which will le ad to citrus losses. For the control of the pest population, results in Figure 3 1 and Figure 3 2 show that the higher the fitness of the resistant strain and the growth rate of pests are, the lower pesticide growers should apply , if the management is ov er an infinite time horizon . If the grower consider s a finite time horizon, low and high level s of the initial pest population both need high level use of the pesticide. Otherwise, the pest population cannot be under control , which will lead to a high rate of disease spread within the citrus grove . For the management of susceptibility to pesticide, results in Figure 3 3 and Figure 3 4 show that over a finite time horizon management, if the grower starts with a high level of susceptibility, he can use a low level of pesticide to preserve the susceptibility with the risk of increasing pest population or use a high level of pesticide to deplete susceptibility with a constantly decreasing pest population . If the grower starts from a low level of susceptibility , implying serious resistance to the pesticide, a very low level of pesticide use could regenerate susceptibility , but it allows for fast growth of pests. Although it is good to regenerate susceptibility, growers should make decisions based on their own need s and time horizons .
35 CHAPTER 5 CONCLUSION This paper models the optimal pesticide maximization when faced with the problem of pest control for a vector borne disease when pesticide resistance is possible. Wh i le the use of pesticide decreases the pest population directly, and impedes the speed of disease incidence s progress indirectly, it can increase the level of the pests resistance to the pesticide. Thus, a steady state may not exist, which implies that a uniform pesticide application policy is likely not optimal. Four phase diagrams in this paper show possible dynamic trajectories for different initial situations. Pesticide resistance is an important factor for pesticide input control, which is necessary to consider when managing a citrus grove in F l orida. One limitation of this study is the many assumptions about the parameters. Further research should be conducted to better parameterize the model. Specifically data on the relationship between disease inc idence and the vector population are needed. Second, data are needed to parameterize the relationship between resistance development and pesticide use for HLB in Florida specifically. Moreover, data on biological components for Florida will improve this st udy. One possible extension of the model could consider adding a spatial component because resistance is usually a regional problem. For example, the cooperation between two neighbor ing growers to control the pest population and susceptibility level would be an interesting extension to this model . Also, with the development of technology, it is possible that a more expensive, less effective pesticide or biological control will be applied. Solving for the optimal time for ol method could be another extension of this study.
36 APPENDIX CALCULATION PROCEDURES OF SHADOW PRICES AND NULLCLINES According to : ( A 1 ) , which is (12) in CHAPTER 3, we can get: (A 2 ) and (A 3 ) Tak ing the derivative of (A 2 ) with respect to time, we can get: (A 4 ) Tak ing the derivative of (A 3 ) with respect to time, we can get: (A 5 ) Referring back to (14) i n CHAPTER 3 : (A 6) . P ut ting (A 2 ) and (A 4 ) into (A 6) , we get: (A 7) Referring back to (15) i n CHAPTER 3 : (A 8) . P ut ting (A 3), (A 5 ) into (A 8) yields : (A 9) Solving (A 7) and (A 9) jointly for and , we get (A 10) , which is (20) in CHAPTER 3 , and
37 (A 11) , which is (21) in CHAPTER 3. P ut ting (A 11) into (A 4 ) , we get: (A 12) , which is (22) in CHAPTER 3. P ut ting (A 10) into (A 5 ) , we get: (A 13) , which is (23) in CHAPTER 3. To s olv e for the nullcline s of and , p ut (A 10) and (A 12) into (A 7) to get: (A 14) . Solving (A 14) for y ields : (A 15) , which is (24) in CHAPTER 3. To get phase diagram, we need to get the nullcline of Setting equal to 0 yields: (A 1 6 ) By m ultiplying both sides of (A 16) by its denominator , we get: (A 1 7 ) Assuming that , eliminate them and rearrange: (A 1 8 ) Putting expressions for , , , , and into (A 1 8 ) yields :
38 (A 19 ) Simp lify ing (A 19 ) yields : (A 2 0 ) Solving (A 21 ) for , we get two answers: (A 2 1 ) , which is (25) in CHAPTER 3 and (A 2 2 ) Because (A 2 2 ) is always less than zero, I use (A 2 1 ) . As this nullcline is for phase diagram, I need to solve for in (A 2 1 ) . Put tin g (A 16), (A 10), (A 11) in (A 1), we get: (A 2 3 )
39 Putting expressions of , , , and into (A 23), we get: (A 2 4 ) w here A = B = C = Solving (A 2 4 ) for , we get : (A 2 5 ) where Simplify ing (A 2 5 ), we get : (A 2 6 ) , which is (26) in CHAPTER 3. Putting (A 2 6 ) into (A 2 1 ), we get :
40 (A 2 7 ) , which is (27) in CHAPTER 3. T he nullcline of is: (A 2 8 ) Solving (A 28 ) for , yields: (A 29 ) As this nullcline is for phase diagr am, I need to solve for in (A 29 ). To solve for in (A 29 ) , I need to use (A 30 ) which is from the first order conditions: (A 30 ) Put ting expressions for , , , and into (A 30 ) , we get: (A 31) where Solving (A 31) for yields:
41 (A 32) , which is (3 0) in CHAPTER 3. Put ting (A 32) into (A 30), we get : (A 33) , which is (31) in CHAPTER 3. For the nullclines of and , p ut (A 3), (A 5) into (A 9) to get: (A 34) S olving for , we get: (A 35) To get phase diagram, we need to get the nullcline of by setting equal to 0 : (A 3 6 ) Assuming , eliminate them and rearrange: (A 3 7 ) Put ting expressions fo r , , , and into (A 37 ) , we get : (A 3 8 )
42 where Simplify ing (A 3 8 ), we get : (A 39 ) w here Solving (A 39 ) for , we get two answers: (A 40 ) , which is (33) in CHAPTER 3 and (A 4 1 ) Because (A 4 1 ) is always less than zero, I use (A 40 ). For the nullcline of , setting equal to zero yields: (A 4 2 ) Solving (A 4 2 ) for , we get: (A 4 3 ) , which is (35) in CHAPTER 3
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47 BIOGRAPHICAL SKETCH (06/24/1990) Yuelu Xu was born June 24, 1990, in Shanghai , China. Yuelu is a student of the Master of Science program in Food and Resource Economics Department at the University of Florida. She attended Shanghai University of Interna tional Business and Economics in China and gained in event management from 200 8 to 201 2. After successfully graduated, s he was admitted to the University of Florida, majored in food and resource economics. She received her MS from the Uni versity of Florida in the summer of 2014.