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New Models of Animal Movement

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Material Information

Title:
New Models of Animal Movement
Physical Description:
1 online resource (2 p.)
Language:
english
Creator:
Hein, Andrew M
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Zoology, Biology
Committee Chair:
Gillooly, James F
Committee Co-Chair:
Mckinley, Scott
Committee Members:
Levey, Douglas J
Holt, Robert D
Principe, Jose C

Subjects

Subjects / Keywords:
migration -- movement -- search
Biology -- Dissertations, Academic -- UF
Genre:
Zoology thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Movement is an iconic feature of life; microorganisms swim up chemical gradients, motile predators search their environments for prey, and migratory animals make journeys that can take them across the planet. Advances in biomechanics and sensory biology have created opportunities to develop new mathematical models of animal movement that incorporate organismal biomechanics and sensory physiology. Such models are useful for understanding the ecological and evolutionary drivers of animal movement behavior, and also for predicting basic ecological rates and scales–for example, the rate of interactions among moving predators and their prey, or the spatial scale of movements made by seasonal migrants. This dissertation is an attempt to develop such general models, and to use them to learn about both the origins and the implications of animal movement behavior. In Chapter 2, I began by investigating the physical constraints related to one of the most well studied movements that animals make: migration. I used a mathematical model to show how body mass influences the maximum distances that migrants travel through its effect on locomotion. I confirmed model predictions using a new global-scale dataset of animal migration distances. In Chapter 3, I sought to better understand how to model animal search behavior in the presence of noisy sensory signals, and how sensory information might affect the movement behavior of a searching animal. I developed a new mathematical framework for modeling the use of sensory data in movement decision-making. Results showed that even a minimal capacity for sensing can give rise to movement behaviors that are commonly observed in nature, such as concentrated search effort near prey. Finally, in Chapter 4, I studied how movement behavior of searching animals changes as the density of their targets change. This work revealed that the ability of animals to gather and respond to sensory information can enable them to encounter prey at rates that differ fundamentally from those predicted by encounter rate models that ignore the use of sensory data.
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.
Statement of Responsibility:
by Andrew M Hein.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Gillooly, James F.
Local:
Co-adviser: Mckinley, Scott.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045239:00001

MISSING IMAGE

Material Information

Title:
New Models of Animal Movement
Physical Description:
1 online resource (2 p.)
Language:
english
Creator:
Hein, Andrew M
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Zoology, Biology
Committee Chair:
Gillooly, James F
Committee Co-Chair:
Mckinley, Scott
Committee Members:
Levey, Douglas J
Holt, Robert D
Principe, Jose C

Subjects

Subjects / Keywords:
migration -- movement -- search
Biology -- Dissertations, Academic -- UF
Genre:
Zoology thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Movement is an iconic feature of life; microorganisms swim up chemical gradients, motile predators search their environments for prey, and migratory animals make journeys that can take them across the planet. Advances in biomechanics and sensory biology have created opportunities to develop new mathematical models of animal movement that incorporate organismal biomechanics and sensory physiology. Such models are useful for understanding the ecological and evolutionary drivers of animal movement behavior, and also for predicting basic ecological rates and scales–for example, the rate of interactions among moving predators and their prey, or the spatial scale of movements made by seasonal migrants. This dissertation is an attempt to develop such general models, and to use them to learn about both the origins and the implications of animal movement behavior. In Chapter 2, I began by investigating the physical constraints related to one of the most well studied movements that animals make: migration. I used a mathematical model to show how body mass influences the maximum distances that migrants travel through its effect on locomotion. I confirmed model predictions using a new global-scale dataset of animal migration distances. In Chapter 3, I sought to better understand how to model animal search behavior in the presence of noisy sensory signals, and how sensory information might affect the movement behavior of a searching animal. I developed a new mathematical framework for modeling the use of sensory data in movement decision-making. Results showed that even a minimal capacity for sensing can give rise to movement behaviors that are commonly observed in nature, such as concentrated search effort near prey. Finally, in Chapter 4, I studied how movement behavior of searching animals changes as the density of their targets change. This work revealed that the ability of animals to gather and respond to sensory information can enable them to encounter prey at rates that differ fundamentally from those predicted by encounter rate models that ignore the use of sensory data.
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.
Statement of Responsibility:
by Andrew M Hein.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Gillooly, James F.
Local:
Co-adviser: Mckinley, Scott.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045239:00001


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E c o l o g y L e t t e r s : C o p y r i g h t A s s i g n m e n t F o r m A u t h o r s n a m e : . . A u t h o r s a d d r e s s : T i t l e o f a r t i c l e ( A r t i c l e ) : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M a n u s c r i p t n o ( i f k n o w n ) : . N a m e s o f a l l a u t h o r s i n t h e o r d e r i n w h i c h t h e y a p p e a r i n t h e A r t i c l e : . . . . . T o e n a b l e B l a c k w e l l P u b l i s h i n g L t d ( B l a c k w e l l P u b l i s h i n g ) a n d t h e C e n t r e N a t i o n a l d e l a R e c h e r c h e S c i e n t i f i q u e t o p u b l i s h y o u r A r t i c l e i n E c o l o g y L e t t e r s ( t h e J o u r n a l ) t h e o w n e r s h i p o f c o p y r i g h t m u s t b e e s t a b l i s h e d T h e A r t i c l e i s d e e m e d t o i n c l u d e a l l m a t e r i a l s u b m i t t e d f o r p u b l i c a t i o n w i t h t h e e x c e p t i o n o f L e t t e r s a n d i n c l u d e s t h e t e x t f i g u r e s t a b l e s a u t h o r c o n t a c t d e t a i l s a n d a l l s u p p l e m e n t a r y m a t e r i a l a c c o m p a n y i n g t h e A r t i c l e P l e a s e r e a d t h i s f o r m c a r e f u l l y s i g n a t t h e b o t t o m ( i f y o u r e m p l o y e r o w n s c o p y r i g h t i n y o u r w o r k a r r a n g e f o r y o u r e m p l o y e r t o s i g n w h e r e m a r k e d ) a n d r e t u r n t h e O R I G I N A L t o t h e a d d r e s s b e l o w a s q u i c k l y a s p o s s i b l e ( U S F e d e r a l G o v e r n m e n t a u t h o r s p l e a s e n o t e : y o u r A r t i c l e i s i n t h e p u b l i c d o m a i n ) Y o u r A r t i c l e w i l l n o t b e p u b l i s h e d u n l e s s a C o p y r i g h t A s s i g n m e n t F o r m h a s b e e n s i g n e d a n d r e c e i v e d b y B l a c k w e l l P u b l i s h i n g P l e a s e n o t e : Y o u r e t a i n t h e f o l l o w i n g r i g h t s t o r e u s e t h e A r t i c l e a s l o n g a s y o u d o n o t s e l l o r r e p r o d u c e t h e A r t i c l e o r a n y p a r t o f i t f o r c o m m e r c i a l p u r p o s e s ( i e f o r m o n e t a r y g a i n o n y o u r o w n a c c o u n t o r o n t h a t o f a t h i r d p a r t y o r f o r i n d i r e c t f i n a n c i a l g a i n b y a c o m m e r c i a l e n t i t y ) T h e s e r i g h t s a p p l y w i t h o u t n e e d i n g t o s e e k p e r m i s s i o n f r o m B l a c k w e l l P u b l i s h i n g o r t h e C e n t r e N a t i o n a l d e l a R e c h e r c h e S c i e n t i f i q u e P r i o r t o a c c e p t a n c e : W e a s k t h a t a s p a r t o f t h e p u b l i s h i n g p r o c e s s y o u a c k n o w l e d g e t h a t t h e A r t i c l e h a s b e e n s u b m i t t e d t o t h e J o u r n a l Y o u w i l l n o t p r e j u d i c e a c c e p t a n c e i f y o u u s e t h e u n p u b l i s h e d A r t i c l e i n f o r m a n d c o n t e n t a s s u b m i t t e d f o r p u b l i c a t i o n i n t h e J o u r n a l i n t h e f o l l o w i n g w a y s : o s h a r i n g p r i n t o r e l e c t r o n i c c o p i e s o f t h e A r t i c l e w i t h c o l l e a g u e s ; o p o s t i n g a n e l e c t r o n i c v e r s i o n o f t h e A r t i c l e o n y o u r o w n p e r s o n a l w e b s i t e o n y o u r e m p l o y e r s w e b s i t e / r e p o s i t o r y a n d o n f r e e p u b l i c s e r v e r s i n y o u r s u b j e c t a r e a A f t e r a c c e p t a n c e : P r o v i d e d t h a t y o u g i v e a p p r o p r i a t e a c k n o w l e d g e m e n t t o t h e J o u r n a l C e n t r e N a t i o n a l d e l a R e c h e r c h e S c i e n t i f i q u e a n d B l a c k w e l l P u b l i s h i n g a n d f u l l b i b l i o g r a p h i c r e f e r e n c e f o r t h e A r t i c l e w h e n i t i s p u b l i s h e d y o u m a y u s e t h e a c c e p t e d v e r s i o n o f t h e A r t i c l e a s o r i g i n a l l y s u b m i t t e d f o r p u b l i c a t i o n i n t h e J o u r n a l a n d u p d a t e d t o i n c l u d e a n y a m e n d m e n t s m a d e a f t e r p e e r r e v i e w i n t h e f o l l o w i n g w a y s : o y o u m a y s h a r e p r i n t o r e l e c t r o n i c c o p i e s o f t h e A r t i c l e w i t h c o l l e a g u e s ; o y o u m a y u s e a l l o r p a r t o f t h e A r t i c l e a n d a b s t r a c t w i t h o u t r e v i s i o n o r m o d i f i c a t i o n i n p e r s o n a l c o m p i l a t i o n s o r o t h e r p u b l i c a t i o n s o f y o u r o w n w o r k ; o y o u m a y u s e t h e A r t i c l e w i t h i n y o u r e m p l o y e r s i n s t i t u t i o n o r c o m p a n y f o r e d u c a t i o n a l o r r e s e a r c h p u r p o s e s i n c l u d i n g u s e i n c o u r s e p a c k s ; o 1 2 m o n t h s a f t e r p u b l i c a t i o n y o u m a y p o s t a n e l e c t r o n i c v e r s i o n o f t h e A r t i c l e o n y o u r o w n p e r s o n a l w e b s i t e o n y o u r e m p l o y e r s w e b s i t e / r e p o s i t o r y a n d o n f r e e p u b l i c s e r v e r s i n y o u r s u b j e c t a r e a E l e c t r o n i c v e r s i o n s o f t h e a c c e p t e d A r t i c l e m u s t i n c l u d e a l i n k t o t h e p u b l i s h e d v e r s i o n o f t h e A r t i c l e t o g e t h e r w i t h t h e f o l l o w i n g t e x t : T h e d e f i n i t i v e v e r s i o n i s a v a i l a b l e a t w w w b l a c k w e l l s y n e r g y c o m P l e a s e n o t e t h a t y o u a r e n o t p e r m i t t e d t o p o s t t h e B l a c k w e l l P u b l i s h i n g P D F v e r s i o n o f t h e A r t i c l e o n l i n e A l l r e q u e s t s b y t h i r d p a r t i e s t o r e u s e t h e A r t i c l e i n w h o l e o r i n p a r t w i l l b e h a n d l e d b y B l a c k w e l l P u b l i s h i n g A n y p e r m i s s i o n f e e s w i l l b e r e t a i n e d b y t h e J o u r n a l A l l r e q u e s t s t o a d a p t s u b s t a n t i a l p a r t s o f t h e A r t i c l e i n a n o t h e r p u b l i c a t i o n ( i n c l u d i n g p u b l i c a t i o n b y B l a c k w e l l P u b l i s h i n g ) w i l l b e s u b j e c t t o y o u r a p p r o v a l ( w h i c h i s d e e m e d t o b e g i v e n i f w e h a v e n o t h e a r d f r o m y o u w i t h i n 4 w e e k s o f y o u r a p p r o v a l b e i n g s o u g h t b y u s w r i t i n g t o y o u a t y o u r l a s t n o t i f i e d a d d r e s s ) P l e a s e a d d r e s s a n y q u e r i e s t o j o u r n a l s r i g h t s @ o x o n b l a c k w e l l p u b l i s h i n g c o m I n s i g n i n g t h i s A g r e e m e n t : 1 Y o u h e r e b y w a r r a n t t h a t t h i s A r t i c l e i s a n o r i g i n a l w o r k h a s n o t b e e n p u b l i s h e d b e f o r e a n d i s n o t b e i n g c o n s i d e r e d f o r p u b l i c a t i o n e l s e w h e r e i n i t s f i n a l f o r m e i t h e r i n p r i n t e d o r e l e c t r o n i c f o r m ; 2 Y o u h e r e b y w a r r a n t t h a t y o u h a v e o b t a i n e d p e r m i s s i o n f r o m t h e c o p y r i g h t h o l d e r t o r e p r o d u c e i n t h e A r t i c l e ( i n a l l m e d i a i n c l u d i n g p r i n t a n d e l e c t r o n i c f o r m ) m a t e r i a l n o t o w n e d b y y o u a n d t h a t y o u h a v e a c k n o w l e d g e d t h e s o u r c e ; 3 Y o u h e r e b y w a r r a n t t h a t t h i s A r t i c l e c o n t a i n s n o v i o l a t i o n o f a n y e x i s t i n g c o p y r i g h t o r o t h e r t h i r d p a r t y r i g h t o r a n y m a t e r i a l o f a n o b s c e n e i n d e c e n t l i b e l l o u s o r o t h e r w i s e u n l a w f u l n a t u r e a n d t h a t t o t h e b e s t o f y o u r k n o w l e d g e t h i s A r t i c l e d o e s n o t i n f r i n g e t h e r i g h t s o f o t h e r s ; 4 Y o u h e r e b y w a r r a n t t h a t i n t h e c a s e o f a m u l t i a u t h o r e d A r t i c l e y o u h a v e o b t a i n e d i n w r i t i n g a u t h o r i z a t i o n t o e n t e r i n t o t h i s A g r e e m e n t o n t h e i r b e h a l f a n d t h a t a l l c o a u t h o r s h a v e r e a d a n d a g r e e d t h e t e r m s o f t h i s A g r e e m e n t ; 5 Y o u w a r r a n t t h a t a n y f o r m u l a o r d o s a g e g i v e n i s a c c u r a t e a n d w i l l n o t i f p r o p e r l y f o l l o w e d i n j u r e a n y p e r s o n ; 6 Y o u w i l l i n d e m n i f y a n d k e e p i n d e m n i f i e d t h e E d i t o r s C e n t r e N a t i o n a l d e l a R e c h e r c h e S c i e n t i f i q u e a n d B l a c k w e l l P u b l i s h i n g a g a i n s t a l l c l a i m s a n d e x p e n s e s ( i n c l u d i n g l e g a l c o s t s a n d e x p e n s e s ) a r i s i n g f r o m a n y b r e a c h o f t h i s w a r r a n t y a n d t h e o t h e r w a r r a n t i e s o n y o u r b e h a l f i n t h i s A g r e e m e n t B y s i g n i n g t h i s A g r e e m e n t y o u a g r e e t h a t B l a c k w e l l P u b l i s h i n g m a y a r r a n g e f o r t h e A r t i c l e t o b e :

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P u b l i s h e d i n t h e a b o v e J o u r n a l a n d s o l d o r d i s t r i b u t e d o n i t s o w n o r w i t h o t h e r r e l a t e d m a t e r i a l ; P u b l i s h e d i n m u l t i c o n t r i b u t o r b o o k f o r m o r o t h e r e d i t e d c o m p i l a t i o n s b y B l a c k w e l l P u b l i s h i n g ; R e p r o d u c e d a n d / o r d i s t r i b u t e d ( i n c l u d i n g t h e a b s t r a c t ) t h r o u g h o u t t h e w o r l d i n p r i n t e d e l e c t r o n i c o r a n y o t h e r m e d i u m w h e t h e r n o w k n o w n o r h e r e a f t e r d e v i s e d i n a l l l a n g u a g e s a n d t o a u t h o r i z e t h i r d p a r t i e s ( i n c l u d i n g R e p r o d u c t i o n R i g h t s O r g a n i z a t i o n s ) t o d o t h e s a m e ; Y o u a g r e e t o B l a c k w e l l P u b l i s h i n g u s i n g a n y i m a g e s f r o m t h e A r t i c l e o n t h e c o v e r o f t h e J o u r n a l a n d i n a n y m a r k e t i n g m a t e r i a l Y o u a u t h o r i z e B l a c k w e l l P u b l i s h i n g t o a c t o n y o u r b e h a l f t o d e f e n d t h e c o p y r i g h t i n t h e A r t i c l e i f a n y o n e s h o u l d i n f r i n g e i t a l t h o u g h t h e r e i s n o o b l i g a t i o n o n B l a c k w e l l P u b l i s h i n g t o a c t i n t h i s w a y B l a c k w e l l P u b l i s h i n g u n d e r t a k e s t h a t e v e r y c o p y o f t h e A r t i c l e p u b l i s h e d b y B l a c k w e l l P u b l i s h i n g w i l l i n c l u d e t h e f u l l b i b l i o g r a p h i c r e f e r e n c e f o r y o u r A r t i c l e t o g e t h e r w i t h t h e c o p y r i g h t s t a t e m e n t B O X A : t o b e c o m p l e t e d i f c o p y r i g h t b e l o n g s t o y o u Y o u h e r e b y a s s i g n t o B l a c k w e l l P u b l i s h i n g a n d t h e C e n t r e N a t i o n a l d e l a R e c h e r c h e S c i e n t i f i q u e c o p y r i g h t i n t h e A r t i c l e i n c l u d i n g t h e a b s t r a c t f o r t h e f u l l p e r i o d o f c o p y r i g h t a n d a l l r e n e w a l s e x t e n s i o n s r e v i s i o n s a n d r e v i v a l s t h r o u g h o u t t h e w o r l d i n a n y f o r m a n d i n a l l l a n g u a g e s B l a c k w e l l P u b l i s h i n g m a y a s s i g n t h e r i g h t s g r a n t e d i n t h i s C o p y r i g h t A s s i g n m e n t F o r m B O X B : t o b e c o m p l e t e d i f c o p y r i g h t b e l o n g s t o y o u r e m p l o y e r ( e g H M S O C S I R O ) T h e c o p y r i g h t h o l d e r g r a n t s B l a c k w e l l P u b l i s h i n g a n d t h e C e n t r e N a t i o n a l d e l a R e c h e r c h e S c i e n t i f i q u e a n e x c l u s i v e l i c e n c e t o p u b l i s h t h e A r t i c l e i n c l u d i n g t h e a b s t r a c t i n p r i n t e d a n d e l e c t r o n i c f o r m i n a l l l a n g u a g e s a n d t o a d m i n i s t e r s u b s i d i a r y r i g h t s a g r e e m e n t s w i t h t h i r d p a r t i e s f o r t h e f u l l p e r i o d o f c o p y r i g h t a n d a l l r e n e w a l s e x t e n s i o n s r e v i s i o n s a n d r e v i v a l s P r i n t N a m e o f C o p y r i g h t h o l d e r : T h i s w i l l b e p r i n t e d o n t h e c o p y r i g h t l i n e o n e a c h p a g e o f t h e A r t i c l e I t i s y o u r r e s p o n s i b i l i t y t o p r o v i d e t h e c o r r e c t i n f o r m a t i o n o f t h e c o p y r i g h t h o l d e r B O X C : t o b e c o m p l e t e d i f y o u a r e a U S F e d e r a l G o v e r n m e n t e m p l o y e e Y o u c e r t i f y t h a t t h e A r t i c l e i s i n t h e p u b l i c d o m a i n N o l i c e n c e t o p u b l i s h i s t h e r e f o r e n e c e s s a r y S i g n a t u r e ( o n b e h a l f o f a l l c o a u t h o r s ( i f a n y ) ) P r i n t n a m e : . D a t e : . I f y o u r e m p l o y e r c l a i m s c o p y r i g h t i n y o u r w o r k t h i s f o r m m u s t a l s o b e s i g n e d b e l o w b y a p e r s o n a u t h o r i z e d t o s i g n f o r a n d o n b e h a l f o f y o u r e m p l o y e r a s c o n f i r m a t i o n t h a t y o u r e m p l o y e r a c c e p t s t h e t e r m s o f t h i s l i c e n c e S i g n a t u r e ( o n b e h a l f o f t h e e m p l o y e r o f t h e a u t h o r ( s ) ) P r i n t n a m e : . . P r i n t n a m e o f e m p l o y e r : D a t e : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T h e r i g h t s c o n v e y e d i n t h i s a s s i g n m e n t w i l l o n l y a p p l y u p o n a c c e p t a n c e o f y o u r A r t i c l e f o r p u b l i c a t i o n D a t a P r o t e c t i o n : T h e P u b l i s h e r m a y s t o r e y o u r n a m e a n d c o n t a c t d e t a i l s i n e l e c t r o n i c f o r m a t i n o r d e r t o c o r r e s p o n d w i t h y o u a b o u t t h e p u b l i c a t i o n o f y o u r A r t i c l e i n t h e J o u r n a l W e w o u l d l i k e t o c o n t a c t y o u f r o m t i m e t o t i m e w i t h i n f o r m a t i o n a b o u t n e w B l a c k w e l l p u b l i c a t i o n s a n d s e r v i c e s i n y o u r s u b j e c t a r e a ( F o r E u r o p e a n c o n t r i b u t o r s t h i s m a y i n v o l v e t r a n s f e r o f y o u r p e r s o n a l d a t a o u t s i d e t h e E u r o p e a n E c o n o m i c A r e a ) P l e a s e c h e c k t h e f o l l o w i n g b o x e s i f y o u a r e h a p p y t o b e c o n t a c t e d i n t h i s w a y : ( c o n v e n t i o n a l m a i l i n g ) ( v i a e m a i l ) P l e a s e r e t u r n t h e s i g n e d f o r m t o : ( a f a x t o + 6 5 6 5 1 1 8 2 8 8 i s a c c e p t a b l e b u t t h e o r i g i n a l m u s t f o l l o w w i t h i n 7 d a y s ) P r o d u c t i o n E d i t o r E c o l o g y L e t t e r s B l a c k w e l l P u b l i s h i n g S e r v i c e s S i n g a p o r e P t e L t 6 0 0 N o r t h B r i d g e R o a d # 0 5 0 1 P a r k v i e w S q u a r e S i n g a p o r e 1 8 8 7 7 8



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NEWMODELSOFANIMALMOVEMENTByANDREWM.HEINADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013AndrewM.Hein 2

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Tomyparents,brothers,andsister 3

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ACKNOWLEDGMENTS Iwanttobeginbythankingmycommitteechair,JamieGillooly,forhisguidance,encouragement,andhisinfectiousenthusiasmforideas.Iwillcontinuetostrivetoemulatehiswillingnesstoconsideranyscienticquestionwithoutbeingintimidatedbyparadigm.Ialsowanttothankmycommitteeco-chair,ScottMcKinley,forhisconstantwillingnesstocollaborateandforhiscommitmenttorigorouslogicinscience.Theafternoonsspentathischalkboardhavebeenamongmymosteducationalandenjoyableexperiencesasagraduatestudent.TheworkpresentedinthisdissertationbenettedgreatlyfromdiscussionswithmycommitteemembersDougLevey,BobHolt,andJosePrincipe,andalsowithMaryChristmanandBenBolker.IndividualchaptersweregreatlyimprovedbycommentsfromS.P.Vogel,T.Bohrmann,A.P.Allen,andJ.H.Brown,J.Casas,M.Vergassola,I.Couzin,A.Brockmeier,E.Kriminger,andmanyothers.IamverygratefulforfundingfromaUniversityofFloridaAlumniFellowship,aNationalScienceFoundationGraduateResearchFellowshipunderGrantNo.DGE-0802270,andtheNationalScienceFoundationunderGrant0801544intheQuantitativeSpatialEcology,EvolutionandEnvironmentProgramattheUniversityofFlorida.Icouldnothavecompletedthisworkwithouttheencouragementandsupportofmyfamilyandfriends.Iespeciallywanttothankmybrother,Luke.IalsoowespecialthankstoGabrielaBlohm,whospentmanylonghoursdiscussingideaswithmeandexhibitedasaintlypatiencewhenIhadanewideaordiscoverythatIcouldnothelpbutsharewithsomeone.Finally,Iwanttothankmyparents:myfather,forencouragingmyphilosophicaltendencies,andmymotherforalwaysremindingmeoftherighttopursuemycuriosity. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1NewModelsofAnimalMovement:ConstraintsofPhysics,ConstraintsofInformation .................................. 13 1.2Biomechanics,Energetics,andAnimalMigration .............. 14 1.3SensoryInformationandModelsofAnimalMovement ........... 15 1.4LinkingMovementBehaviorandEncounterRatesofInteractingSpecies 16 2ENERGETICANDBIOMECHANICALCONSTRAINTSONANIMALMIGRATIONDISTANCE ...................................... 17 2.1ModelDevelopment .............................. 18 2.1.1ParameterizingModelforWalking,Swimming,andFlyingMigrants 19 2.1.2ModelPredictions ............................ 22 2.2MaterialsandMethods ............................. 22 2.3Results ..................................... 24 2.4Discussion ................................... 26 3SENSINGANDDECISION-MAKINGINRANDOMSEARCH .......... 33 3.1ModelDevelopment .............................. 34 3.1.1SearchingWithoutOlfactoryData ................... 36 3.1.2IncorporatingOlfactoryDatatoMakeSearchDecisions ...... 37 3.1.3InterpretingScentSignals ....................... 38 3.2MaterialsandMethods ............................. 39 3.2.1ScentPropagation ........................... 39 3.2.2SimulationDetails ........................... 40 3.3Results ..................................... 40 3.3.1Visual-OlfactoryPredatorsFindTargetsFasterandMoreReliablyThanVisualPredators ......................... 40 3.3.2Visual-OlfactoryPredatorsLearnFromNo-SignalEvents ...... 42 3.3.3Visual-OlfactoryPredatorsConcentrateSearchEffortNearTargets 43 3.4Discussion ................................... 43 5

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4SENSORYINFORMATIONANDENCOUNTERRATESOFINTERACTINGSPECIES ....................................... 50 4.1MaterialsandMethods ............................. 51 4.1.1EncounterRateandSearchBehavior:SomeDenitions ...... 51 4.1.2FrameworkforModelingMovementDecisions ............ 52 4.1.2.1Sensorysignalsandsearchbehavior ........... 52 4.1.2.2Perfectsensingandresponse ............... 53 4.1.2.3Purelyrandomsearch .................... 54 4.1.2.4Imperfectsensingandresponse .............. 55 4.1.3EncounterRateSimulations ...................... 56 4.1.4EstimationofScalingRegimesandExponents ........... 57 4.2Results ..................................... 58 4.2.1EncounterRatesofPurelyRandomPredatorsareNear-linearinPreyDensity .............................. 58 4.2.2EncounterRatesofSignal-modulatedPredatorsChangeNonlinearlywithPreyDensity ............................ 59 4.2.3SensoryResponseAllowsPredatorstoEncounterNearbyTargetsmoreFrequently ............................ 60 4.3Discussion ................................... 60 5CONCLUSIONS ................................... 67 APPENDIX AMIGRATIONMODELDERIVATION,SENSITIVITY,ANDSTATISTICALANALYSES ...................................... 71 A.1Generaldistanceequation ........................... 71 A.1.1Walking ................................. 71 A.1.2Swimming ................................ 72 A.1.3Flying .................................. 72 A.2Parameterestimationandmodelsensitivity ................. 74 A.2.1Estimationofp0 ............................. 74 A.2.2Sensitivityanalysis ........................... 74 BDERIVATIONOFDISTRIBUTIONS,ANOTEONTHEUSEOFBAYES'RULE,ANDSUPPLEMENTARYSIMULATIONRESULTS ................ 83 B.1TrueDistanceDistribution(TDD)andaCommentontheUseofBayes'Rule ....................................... 83 B.2RobustnessofResultstoSearchConditions ................ 84 B.2.1TargetDensity .............................. 84 B.2.2SignalEmissionRate .......................... 84 B.2.3VariationinPredatorScanningTimes ................. 85 B.3TheRoleofNo-signalEvents ......................... 85 6

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CMODELOFSCENTPROPAGATIONANDDEPENDENCEOFREGIMETRANSITIONSONSIGNALPROPAGATIONLENGTH .............. 90 C.1ScentPropagation ............................... 90 C.2DependenceofRegimeBreakonSignalPropagationLength ....... 91 C.3EncounterRateofaPredatorwithPerfectSensingandResponse,andNon-ZeroEncounterRadius .......................... 91 C.4EncounterProbabilitiesintheSparseRegime ................ 92 REFERENCES ....................................... 95 BIOGRAPHICALSKETCH ................................ 112 7

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LISTOFTABLES Table page A-1Empiricalvaluesofthenormalizationconstant .................. 75 A-2Sensitivityofdistanceequationstovariationininputparameters. ........ 76 A-3Bodymassandmigrationdistancedata ...................... 76 8

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LISTOFFIGURES Figure page 2-1Schematicofmigrationprocess ........................... 30 2-2Migrationdistances ................................. 31 2-3Numberofbodylengthstraveled .......................... 31 2-4Observedandpredictedmigrationdistances ................... 32 3-1Schematicofpredatorsearchbehavior ...................... 46 3-2Meanpredatorsearchtimesandvariabilityaboutmeansearchtime ...... 47 3-3Typicalsearchpathsofsimulatedpredators .................... 48 3-4Informationgainasafunctionoftheratioofvisualtoolfactoryradius ...... 49 3-5Area-restricted-searchbehaviorofvisualandvisual-olfactorypredators .... 49 4-1Perfectsensingandresponse ............................ 63 4-2Scanpointsduringsearch .............................. 64 4-3Encounterratesofpurelyrandomandsignal-modulatedpredators ....... 64 4-4Encountersrateofsignal-modulatedpredators .................. 65 4-5Empiricalencounterprobabilityasafunctionoftargetdensity .......... 66 B-1Searchstimeatlowdensity ............................. 87 B-2Searchtimeswithreducedemissionrate ..................... 87 B-3Searchtimesandscanningphaselength ..................... 88 B-4Likelihoodfuncions .................................. 89 B-5Searchtimewithconditionalresponsetoolfactorysignals ............ 89 C-1Breakpointbetweenlinearandsublinearregime ................. 94 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNEWMODELSOFANIMALMOVEMENTByAndrewM.HeinAugust2013Chair:JamesF.GilloolyCochair:ScottA.McKinleyMajor:ZoologyMovementisaniconicfeatureoflife;microorganismsswimupchemicalgradients,motilepredatorssearchtheirenvironmentsforprey,andmigratoryanimalsmakejourneysthatcantakethemacrosstheplanet.Advancesinbiomechanicsandsensorybiologyhavecreatedopportunitiestodevelopnewmathematicalmodelsofanimalmovementthatincorporateorganismalbiomechanicsandsensoryphysiology.Suchmodelsareusefulforunderstandingtheecologicalandevolutionarydriversofanimalmovementbehavior,andalsoforpredictingbasicecologicalratesandscalesforexample,therateofinteractionsamongmovingpredatorsandtheirprey,orthespatialscaleofmovementsmadebyseasonalmigrants.ThisDissertationisanattempttodevelopsuchgeneralmodels,andtousethemtolearnaboutboththeoriginsandtheimplicationsofanimalmovementbehavior.InChapter2,Ibeganbyinvestigatingthephysicalconstraintsrelatedtooneofthemostwellstudiedmovementsthatanimalsmake:migration.Iusedamathematicalmodeltoshowhowbodymassinuencesthemaximumdistancesthatmigrantstravelthroughitseffectonlocomotion.Iconrmedmodelpredictionsusinganewglobal-scaledatasetofanimalmigrationdistances.InChapter3,Isoughttobetterunderstandhowtomodelanimalsearchbehaviorinthepresenceofnoisysensorysignals,andhowsensoryinformationmightaffectthemovementbehaviorofasearchinganimal.Idevelopedanewmathematicalframeworkformodelingtheuseofsensorydatainmovementdecision-making.Resultsshowed 10

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thatevenaminimalcapacityforsensingcangiverisetomovementbehaviorsthatarecommonlyobservedinnature,suchasconcentratedsearcheffortnearprey.Finally,inChapter4,Istudiedhowmovementbehaviorofsearchinganimalschangesasthedensityoftheirtargetschange.Thisworkrevealedthattheabilityofanimalstogatherandrespondtosensoryinformationcanenablethemtoencounterpreyatratesthatdifferfundamentallyfromthosepredictedbyencounterratemodelsthatignoretheuseofsensorydata. 11

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CHAPTER1INTRODUCTIONThephenomenonofmovementingeneral,andanimalmovementinparticular,hasfascinatedbiologistsforcenturies(e.g.[ 1 2 ]).Traditionally,animalmovementhasbeenstudiedeitherthroughdetailedempiricalworkonparticularspecies,orthroughhighlyabstractedmathematicalmodels.Onlyrecently,advancesineldssuchassensorybiologyandbiomechanicsarebeginningtofacilitatetheintegrationoforganismalbiologyandmathematicaltheoryofanimalmovementbehavior.Despitearichhistoryofinvestigationbytheoreticians,manyofthegeneralmathematicalmodelsusedtodescribeanimalmovementatthemacro-scalerelyonassumptionsthataresomewhatrestrictive.Forinstance,someoftheearliestmodelsofanimalmovementwereadoptedfromparticlecollisionmodelsinchemistryandusedtopredictencounterratesbetweenpredatoryanimalsandtheirprey.Theseclassicalencounterratemodels,developedbythepioneeringtheoreticalbiologist,AlfredLotkaandothers,assumethatpredatorsandpreymoverandomlyandindependentlyofoneanother[ 3 ].Lotkahimselfnotedtheinconsistencybetweenthisconceptionofanimalmovementbehavior,andthemovementofanimalsinnature[ 3 ].Ofcourse,generalityoftencomesatthepriceofstrongassumptionsandthewillingnessofearlyecologiststopaythatpriceledtoanenormousamountofdevelopmentintheeldsofspatialecologyandcoupledpopulationdynamics(e.g.,[ 4 ]).Still,ecologistslikeLotkaandthevisionarytheoretician,JohnSkellam,imaginedfutureworkonanimalmovementthatwouldrelaxsomeoftheirownsimplifyingassumptionstoallowformorerealisticdepictionsoforganismalphysiologyandbehavior[ 3 5 ].Accomplishingthisgoalrequiresanunderstandingoftheelementsofphysiologyanddecision-makingbehaviorthataremostrelevanttoanimalmovement.Sincethedevelopmentofearlymovementmodels,researchersworkingintheareasofbiomechanicsandsensorybiologyhavemadehugestridestowardunderstandingtheenergeticsoflocomotionandthephysics, 12

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transmission,andprocessingofsensorysignals.Developmentsinbiomechanicstheory,forexample,havemadeitpossibletowriteequationsfortheenergeticcostsoflocomotionasfunctionsofspeedandbodysize(e.g.,[ 6 7 ]).Empiricalandtheoreticalstudiesofsensorybiologyhavegonealongwaytowardrevealinghowanimalsuseinformationtomakemovementdecisions(e.g.,[ 8 11 ]).Theseadvancesproviderst-principlesfromwhichtoderivenewmodelsofanimalmovement.Inthechaptersthatfollow,Idescribemyattempttocontributesuchmodels,andtousethemtolearnaboutboththeoriginsandtheimplicationsofanimalmovementbehavior. 1.1NewModelsofAnimalMovement:ConstraintsofPhysics,ConstraintsofInformationThewayananimalmovesarounditsenvironmentmustbedetermined,atleastinpart,byboththephysicalcontextofthatmovementandthebackgroundofinformationtheanimalhasatitsdisposal.Tobetterunderstandphysicalandinformationalconstraintsonanimalmovement,mycollaboratorsandIhaveperformedthreetheoreticalstudiestocharacterizetheseconstraintsinsomegenerality.InChapter2,Idescribeourinvestigationofthebiomechanicalandenergeticconstraintsrelatedtooneofthemostwell-studiedmovementsthatanimalsmake:migration.Weshowhowbodymassafundamentalcharacteristicofallanimalsinuencesthemaximumdistancesthatmigratinganimalstravel,throughitseffectonthephysicsoflocomotion.Thedataandmodelsthatwedevelopdemonstratethatthedominanteffectofbodymassonmigrationdistanceemergesdespitethedifferencesamongmigratoryspecies.Oneofthemostinterestingresultsofthisstudyisthepredictionthatwalkingmigrantsofallsizestravel,onaverage,thesamenumberofbodylengthsduringmigration(about1.5105bodylengths),asdoswimmingspeciesofallsizes(1.7106bodylengths).Interestingly,thisrelationshipdoesnotholdforyingmigrants,andthebiomechanicsofightprovideanexplanationforthisdifference.Asecondproblemisunderstandinghowtomodelanimalsearchbehaviorinthepresenceofsensorysignals[ 12 ].Researchers 13

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studyingsearchandforagingmovementshavetraditionallymodeledmovementusingrandomwalks.Therehasbeenmuchdebateaboutwhatthemostappropriaterandomwalkmodelsare.Theassumptionthatunderliesmuchofthisworkisthatanimalscannotgetmuchusefulinformationaboutthelocationsoftheirtargetswhentargetdensityislow.Thus,ananimalmustadoptsomesortofstatisticalmovementbehaviorthatdoesnotdependontheuseofsensorycues[ 13 ].InChapter3,were-evaluatethisassumptionusingasimulationmodel.Inparticular,westudythecaseofasearchingpredatorthatcanmeasureonlynoisyolfactorycuesfromprey.Weshowthat,solongastherangeatwhichthepredatorgetsnoisysensorydatafrompreyislongerthantherangeatwhichitcancaptureprey,thepredatorcanbenettremendouslyfromincorporatingevenminimalsensorydataintomovementbehavior.Wefurthershowthatacapacityforsensinganddecision-makinggivesrisetocommonlyobservedbehaviorssuchasarea-restrictedsearchinregionsthatcontainprey[ 14 15 ].Athirdandnalquestionrelatestohowfeaturesofananimal'senvironmentinuenceitsmovementbehavior.InChapter4,westudyhowmovementbehaviorsofsearchinganimalschangeasthedensityandspatialcongurationoftheirtargetschange.Usingsimplemathematicalmodelsofsensinganddecision-makingalongwithsimulations,westudytherelationshipbetweensearcher-targetencounterrate,andtargetdensity.Theresultingrelationshipsdifferfromclassicalmass-actionmodelsofspeciesinteractions,butareconsistentwithrecentempiricaldataonpreyencounterratesofpredatorybirdsandsh.Thisstudyrevealsthestronglinksbetweensensorydata,movementbehavior,andencounterratesofinteractingspecies.BelowIelaborateonthemotivationsfor,andndingsoftheseinvestigationsbeforedescribingtheminfulldetailinChapters2-4. 1.2Biomechanics,Energetics,andAnimalMigrationAnimalmigrationisoneofthegreatwondersofnature,butthefactorsthatdeterminehowfarmigrantstravelremainpoorlyunderstood.Toaddressthisissue,wedevelopanewquantitativemodelofanimalmigrationanduseittodescribethe 14

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maximummigrationdistanceofwalking,swimmingandyingmigrants.Themodelcombinesbiomechanicsandmetabolicscalingtoshowhowmaximummigrationdistanceisconstrainedbybodysizeforeachmodeoftravel.Themodelalsoindicatesthatthenumberofbodylengthstravelledbywalkingandswimmingmigrantsshouldbeapproximatelyinvariantofbodysize.Datafromover200speciesofmigratorybirds,mammals,sh,andinvertebratessupportthecentralconclusionofthemodelthatbodysizedrivesvariationinmaximummigrationdistanceamongspeciesthroughitseffectsonmetabolismandthecostoflocomotion. 1.3SensoryInformationandModelsofAnimalMovementManyorganismslocateresourcesinenvironmentsinwhichsensorysignalsarerare,noisy,andlackdirectionalinformation.Recentstudiesofsearchinsuchenvironmentsmodelsearchbehaviorusingrandomwalks(e.g.,Levywalks)thatmatchempiricalmovementdistributions.Weextendthismodelingapproachtoincludesearcherresponsestonoisysensorydata.Weexploretheconsequencesofincorporatingsuchsensorymeasurementsintosearchbehaviorusingsimulationsofavisual-olfactorypredatorinsearchofprey.Ourresultsshowthatincludingevenasimpleresponsetonoisysensorydatacandominateotherfeaturesofrandomsearch,resultinginlowermeansearchtimesanddecreasedriskoflongintervalsbetweentargetencounters.Inparticular,weshowthatalackofsignalisnotalackofinformation.Searchersthatreceivenosignalcanquicklyabandontarget-poorregions.Ontheotherhand,receivingastrongsignalleadsasearchertoconcentratesearcheffortneartargets.Theseresponsescausesimulatedsearcherstoconcentratesearcheffortsneartargets.Thisarea-restrictedsearch[ 15 ]behaviorisadominantfeatureofsearchmovementsofrealpredatorssuchasoceanicbirds[ 14 16 ],whichappeartousesensorysignalstofocussearcheffortsinproductiveareasandtoavoidareasthatlackprey.Themodelthusrevealsthatqualitativelyrealisticmovementbehaviorcanemergeevenfromverysimplesensinganddecision-making. 15

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1.4LinkingMovementBehaviorandEncounterRatesofInteractingSpeciesMostmobileanimalssearchforresources,mates,andpreywiththeaidofsensorycues.Thesearchinganimalmeasuressensorydataandpresumablyadjustsitssearchbehaviorbasedonthosedata.Yet,classicalmodelsofspeciesencounterratesassumethatsearchersmoveindependentlyoftheirtargets.Theassumptionofindependentmovementleadstothefamiliarencounterratekineticsusedinmodelingspeciesinteractions.Here,weusetheexampleofpredator-preyinteractionstostudyhowencounterrateschangewhenpredatorsusesensoryinformationtondprey.Weshowthat,evenwhenpredatorspursuepreyusingonlynoisy,directionlessodorsignals,theresultingencounterrateequationsdifferqualitativelyfromthosederivedbyclassictheoryofspeciesinteractions.Critically,predatorsensoryresponselowersthesensitivityofencounterratetopreydensitywhenpreydensityislow.Thisndingholdsoverawiderangeofassumptionsaboutpredatorysensorycapabilities,preycapturebehavior,andthedegreetowhichpreyareclusteredintheenvironment.Ourresultsdemonstratehowtheexchangeofinformationamonginteractingorganismscanfundamentallyaltertheratesofphysicalinteractionsinbiologicalsystems. 16

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CHAPTER2ENERGETICANDBIOMECHANICALCONSTRAINTSONANIMALMIGRATIONDISTANCEEachyear,diversespeciesfromaroundtheplanetsetoutonmigrationsrangingfromafewtothousandsofkilometersinlength[ 17 19 ].Biologistshavelonghypothesizedthatthisvariationinmigrationdistanceamongspeciesmightbegovernedbydifferencesinbasicspeciescharacteristicssuchasmorphologyandbodysize[ 1 ].Althoughmuchprogresshasbeenmadeinunderstandinghowthesecharacteristicsarerelatedtothemechanicsoflocomotionandtothemigratorycapabilitiesofindividualspecies(e.g.[ 20 21 ]),successinunderstandingvariationinmigrationdistanceamongspecieshasbeenlimited.Thisisbecausecurrentmodelsoftenrequiredetailedinformationonthemorphologyandbehaviorofmigrants(e.g.,[ 20 22 ]).Thisrequirementhasprecludedaquantitativeanalysistodeterminetheextenttowhichsharedfunctionalcharacteristicssuchasbodysizecouldberesponsibleforobservedvariationinmigrationdistancesamongspecies.Asaresult,theneedforgeneraltheoryandcross-speciesanalysesofmigrationhasbeenstronglyemphasizedinrecentyears[ 23 24 ].Here,wepresentamodeltodescribeconstraintsonanimalmigrationdistance.Themodelexpandsonpastapproaches[ 7 25 26 ]byincorporating(1)thebodymass-dependenceofthecostoflocomotion,(2)dynamicchangesinthebodymassesofmigrantsastheyutilizestoredfueland(3)scalingofmorphologicalcharacteristicsandmaintenancemetabolismamongmigrantsofdifferentbodymasses.Incontrasttopastapproaches,themodelassumesthatthenumberofre-fuellingstopsmadebymigrantsisunknownandmayvarysubstantiallyamongspecies.Thisfacilitates Thischapterappearedasanarticleinthejournal,EcologyLetters:Hein,A.M.,C.Hou,andJ.F.Gillooly.2012.Energeticandbiomechanicalconstraintsonanimalmigrationdistance.Ecol.Lett.15:104.Itsreproductionhereisauthorizedunderthejournal'scopyrightpolicy. 17

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predictionofstatisticalpatternsofmigrationdistanceamongspecies,evenwhenthedetailsofmigratorybehaviorofindividualspeciesareunknown. 2.1ModelDevelopmentWetreatmigrationasaprocessinwhichamigranttravelsadistanceofYT(km)bybreakingthejourneyintoaseriesofNlegsoflengthYi,wherei2f1,2,...,Ng,Fig. 2-1 A).DescribingvariationinmigrationdistanceamongspeciesthusrequiresdescribingtheprocessesthatdetermineYi,whileaccountingforamong-speciesvariationinN.Toaccomplishthis,webeginbymakingfoursimplifyingassumptions(seeAppendix A fordetailedderivationandalternativeassumptions).Weassume(i)thatthetotalrateofenergyusebyamigratinganimal,Ptot(W),isthesumoftherateofenergyuseforgeneralmaintenance,Pmtn,andthatrequiredforlocomotion,Ploc(i.e.Ptot=Pmtn+Ploc=)]TJ /F3 11.955 Tf 9.3 0 Td[(dG=dt,whereG=Joulesofstoredfuelenergy),(ii)thatmigrantsusingaparticularmodeoflocomotionaregeometricallysimilar,suchthatlinearmorphologicalcharacteristics(e.g.lengthsofappendages)areproportionaltoM1=3andsurfaceareasareproportionaltoM2=3(whereMisbodymass(kg),[ 27 ](iii)thatmigrantmetabolismprovidesthepowerrequiredforlocomotion,and(iv)thatthenumberofrefuelingstopsmadebyindividualsofeachspeciesisindependentofbodymass.Duringanygivenlegofamigration,therateofchangeinmigrationdistanceperunitchangeinbodymasscanbeexpressedasdYi=dM=(dYi=dt)(dtc=dG)=)]TJ /F3 11.955 Tf 9.3 0 Td[(vc=(Pmtn+Ploc),wherevistravelspeed(ms)]TJ /F4 7.97 Tf 6.59 0 Td[(1)andcistheenergydensityofstoredfuel(Jouleskg)]TJ /F4 7.97 Tf 6.59 0 Td[(1).Thedistancetraveledonaparticularlegcanbeobtainedbyintegratingthisexpressionfrominitialmassatthebeginningoftheleg,M0(kg),tonalmassafterallfuelenergyhasbeenused,M0(1)]TJ /F3 11.955 Tf 12.48 0 Td[(f),wherefistheratioofinitialfuelmasstoM0, Yi=ZM0(1)]TJ /F7 7.97 Tf 6.59 0 Td[(f)M0)]TJ /F3 11.955 Tf 9.3 0 Td[(v(M,)c Pmtn(M)+Ploc(M,)dM.(2)Here,v,Pmtn,andPlochavebeenrewrittentoshowtheirdependenceonbodymassandonasmallsetofmorphologicaltraits,(lengthsandsurfaceareas,e.g.wingspan, 18

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bodycross-sectionalarea),whichdeterminetheenergeticcostoflocomotion.Thisformulationallowsforchangesinspeedandrateofenergyuseasthemigrantlosesstoredfuelmass.Equation( 2 )canbeusedtopredicthowYivariesamongspeciesbyspecifyingappropriatefunctionsforv(M,),Pmtn(M),andPloc(M,).WeassumethatPmtnscaleswithbodymassasPmtn=p0M3=4,bothwithinandamongindividuals,wherep0isanormalizationconstantthatvariesbytaxon[ 28 29 ].BiomechanicstheoryprovidesameansofexpressingPlocandvasfunctionsofMandformigrantsusingaparticularmodeoflocomotion(seebelow).Generalizingtomulti-legmigrations.Totaldistancetraveledoverthecourseofmigrationisgivenbythesum,PNi=1Yi,whereNisthenumberofmigratorylegstraveledbyagivenspecies(Fig. 2-1 ).Nisunknownforthemajorityofmigratoryspecies.ToaccountforvariationinNamongspecies,wetreatNasarandomquantitywithexpectedvalue,N.WetreatYiasxedforagivenspeciesbecauseweareinterestedinmaximummigrationdistance.Followingthelawofiteratedexpectation,theexpecteddistancetraveledoverNmigratorylegsis YT=E"NXi=1Yi#=NYi,(2)wheretheoperator,E,denotestheexpectedvalue[ 30 ].Equation( 2 )showsthatYTisproportionaltoYi,whichisgivenbyEquation( 2 ). 2.1.1ParameterizingModelforWalking,Swimming,andFlyingMigrantsThemodeldevelopedaboveisgeneralandappliestomigrantsusinganymodeoflocomotion.Here,weparameterizethemodelforthethreedominantmodesofmigratorylocomotion(walking,swimming,ight)byusingstandardmodelsoflocomotiontodescribethePlocandvtermsinEquation( 2 )(biomechanicalmodelsdescribedin 19

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detailinAppendix A ).Forwalkingmigrants,Ploccanbedescribedby Pwalk=gM Lcv,(2)whereLcisstridelength(m),viswalkingspeed(ms)]TJ /F4 7.97 Tf 6.59 0 Td[(1),isacostcoefcient(JN)]TJ /F4 7.97 Tf 6.58 0 Td[(1),andgistheaccelerationduetogravity(ms)]TJ /F4 7.97 Tf 6.59 0 Td[(2,[ 31 ])TheonlymorphologicalvariableinEquation( 2 )isLc,whichisproportionaltoleglength[ 32 ].Weassumethatwalkingmigrantstravelatspeeds,v[ 33 ]andthattheymaintainthesespeedsoverthecourseofmigration.Thepowerrequiredforswimmingcanbedescribedbytheresistivemodel, Pswim=Abv2.8 L0.2b,(2)whereisadimensionlesscostcoefcient,Abisbodycross-sectionalarea(m2),Lbisbodylength(m),andvisswimmingspeed(ms)]TJ /F4 7.97 Tf 6.59 0 Td[(1,[ 6 ]).Thesetofrelevantmorphologicalvariables,,isAbandLb.Weassumethatmigrantsswimatspeedsthatminimizetheratio,Ptot=v.Powerrequiredforightnearminimumpowerspeedcanbedescribedbytheequation Py=(1+)[M2L)]TJ /F4 7.97 Tf 6.58 0 Td[(2wv)]TJ /F4 7.97 Tf 6.58 0 Td[(1+Abv3f],(2)whereisadimensionlessprolepowercoefcient,andarecostcoefcients(Appendix A ),Abisbodycrosssectionalarea(m2),Lwiswingspan(m),andisproportionaltoAw=L2w,whereAwiswingarea[ 7 ].Thesetofrelevantmorphologicalvariables,,isthereforeAb,Lw,andAw.WeassumeyingmigrantstravelatspeedsthatminimizePy=vf[ 7 ].SubstitutingEquations( 2 )-( 2 ),correspondingmigrationspeeds,andthemass-dependenceofmaintenancemetabolismintoEquation( 2 )allowsYitobeexpressedasafunctionofinitialmassM0,p0,andforeachmodeoflocomotion.Ineachofthebiomechanicalmodelsdescribedabove,thepowerrequiredforlocomotion 20

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depends,inpart,onasetofmorphologicallengthsandareas,,thatdonotchangeasthemigrantusesstoredfueltopowermigration.ThedependenceofYioncanbeeliminatedbyexpressingmorphologicalvariablesintermsofM0basedontheassumptionofgeometricsimilarity(i.e.lengths,surfaceareas).SubstitutingfunctionsforYi(Appendix A )intoEquation( 2 )yieldsexpressionsfortheexpectedmaximummigrationdistancesofwalking YT=y0M0.340,(2)swimming YT=y0p)]TJ /F4 7.97 Tf 6.59 0 Td[(0.640M0.30,(2)andying YT=y0logp0+k1M0.420 p0+k2M0.420(2)migrants.Herey0isaproportionalityconstantthatvariesbymodeoflocomotion,andk1andk2areempiricalconstants.DifferencesinthefunctionalformsofEquations( 2 )through( 2 )arecausedbydifferencesinthewayPlocdependsonmassinwalking,swimming,andyingmigrants.InthecaseofEquation( 2 ),thepredictedrelationshipdoesnotfollowasimplepowerfunctioninM0.Thisisbecausethecostofightincreasesmorerapidlywithincreasingbodymassthandoesthecostofwalkingorswimming.Thevariable,p0,doesnotappearinthenalformoftheequationforwalkingmigrantsbecausehereweonlyconsiderthedistancetraveledbywalkingmammals,forwhichp0isroughlyconstant[ 34 ].TheexponentsofthemasstermsinEquations( 2 )through( 2 )describehowmaximummigrationdistancechangesasafunctionofM0andreectthemass-dependenceofmaintenanceandlocomotorymetabolism.Theconstant,y0,describeseffectsofmass-independentfactors,suchasthenumberofmigratorylegs,thataffecttheabsolutedistancestraveledbymigrantsbutdonotaffectthescalingofmigrationdistancewithbodymass.Themetabolicnormalizationconstant,p0,andthemorphologicalconstantsk1andk2canbeestimatedfromempirical 21

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measurements(seeMaterialsandMethods).Theframeworkdescribedhereusesbodymass(Fig. 2-1 Bboxa),morphology(Fig. 2-1 Bboxb)andmodeoflocomotion(Fig. 2-1 Bboxc)todeterminemigratoryspeed,andthemetaboliccostsoflocomotoryandmaintenancemetabolism(Fig. 2-1 Bboxd).Equation( 2 )ensuresthatchangesinspeedandmetabolismasthemigrantusesstoredfuel(Fig. 2-1 Bboxe)areexplicitlyincorporatedintothepredictionofYi(Fig. 2-1 Bboxf). 2.1.2ModelPredictionsEquations( 2 )through( 2 )makeseveralquantitativepredictionsthatcanbetestedagainstdata.First,eachequationpredictsthat,afternormalizingforp0,asinglecurvecanbeusedtodescribeexpectedmaximummigrationdistance(inkm)asafunctionofM0forspeciesusingeachmodeoflocomotion.Second,eachequationpredictshowthenumberofbodylengthstraveledameasureofrelativedistance[ 35 ]varieswithbodymass.Migrationdistanceandbodylengthscalesimilarlywithmassinwalkingandswimminganimals(i.e.YTroughlyproportionaltoM1=30,bodylength/M1=30)suchthatthenumberofbodylengthstraveledduringmigration,Ybl,isdescribedbyYbl=YT=(bodylength)/M1=30=M1=30/M00.Thus,afternormalizingfordifferencesinp0,thenumberofbodylengthstraveledbywalkingandswimminganimalsshouldbeapproximatelyinvariantwithrespecttoM0.Inyinganimals,however,dividingEquation( 2 )byM1=30indicatesthatYblshoulddecreasewithincreasingmassforallbutthesmallestyingmigrants. 2.2MaterialsandMethodsToevaluatethemodel,publishedmeasurementsofmaximummigrationdistancesofterrestrialmammals,sh,marinemammals,andyinginsectsandbirdswerecollected.Datafromstudiesthatmetvecriteriawereincludedintheanalysis:(1)reportedmovementscouldbeconsideredto-and-fromigrationorone-waymigration[ 36 ],(2)individualsweredirectlytrackedbymark-recapture,telemetryorothermeans,groupsofindividualsweretrackedbyrepeatedobservationoverthecourseofmigration,ora 22

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reliableestimateofdistancetraveledcouldotherwisebeestablished,(3)maximumtraveldistances,maps,tracksorotherinformationthatalloweddirectcalculationofminimumestimatesofthedistancestraveledbyindividualanimalswerereported,(4)theredidnotexiststrongbutindirectevidencefromotherstudies(e.g.sightingsofunmarkedindividuals,stableisotopedata)suggestingthatthemaximumreportedmigrationdistancewassubstantiallyshorterthantruemaximummigrationdistance,and(5)inthecaseofyingspecies,studiesreportedmigrationdistancesofspeciesthatrely,atleastpartially,onappingight.Thefthcriterionwasimposedbecausethebiomechanicalmodelofightusedtoderiveourpredictionsappliesmostdirectlytoappingight.Migrationdistanceandbodymassdatawereincludedfromalargedataset[ 37 ]forwhichalloftheselectioncriteriacouldnotbeveriedforallspecies.Includingthesedatadidnotqualitativelyaffectourconclusions(seeResults).Weestimatedtheconstantsk1andk2inEquation( 2 )usingempiricalstudiesofthemorphologyofyinginsectsandbirds;however,thegeneralformofEquation( 2 )andtheresultingpredictionsarenotstronglyaffectedbyvariationintheempiricalvaluesusedtoestimatek1andk2(Appendix A ).Empiricalestimatesofp0wereusedinEquations( 2 )throught( 2 )(Appendix A ).Bodymassdatawereusedtoestimatebodylengthsbasedonallometricequations(swimmingmammals:[ 38 ];others:[ 27 ]).Bodylengthswereusedtoconvertmigrationdistance(km)intounitsofbodylengths.Toevaluateourrstprediction,wettedEquations( 2 )through( 2 )tomigrationdistancedatafromwalking(n=33),swimming(n=32),andyingmigrants(n=141),respectively.Equations( 2 )and( 2 )werettedtolog10-transformeddistanceandbodymassdatausingordinaryleastsquares.Equation( 2 )wasttedtolog10-transformeddistanceandbodymassdatausingnon-linearleastsquares(Gauss-Newtonalgorithm).Equations( 2 )through( 2 )havethegeneralform:YT=y0h(Md0,p0),wherehisaknownfunction,y0isaconstant,anddisascalingexponent.Foreachequation,twomodelsweretted:amodelinwhichy0wastted 23

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asafreeparameterbutdwassettothepredictedvalue(i.e.d=0.34,0.3,0.42,forwalking,swimming,andyingmigrants,respectively),andamodelinwhichbothy0anddweretted.Modelr2valuesreportedbelowarebasedontheformermethod.Thelattermethodwasusedtogenerate95%prolecondenceintervalsforthedparameter.Priortotting,bodymassvaluesofswimmingandyinganimalswerenormalizedtoaccountfordifferencesinp0accordingtotheequationsMnorm=M0.30p)]TJ /F4 7.97 Tf 6.59 0 Td[(0.640andMnorm=M0.420p)]TJ /F4 7.97 Tf 6.59 0 Td[(10,respectively.Totestoursecondpredictionthatthenumberofbodylengthstraveledwasinvariantofmassinwalkingandswimmingmigrants,butdecreasedwithmassinyingmigrantswettedlog10-transformedmigrationdistance(inbodylengths)asafunctionoflog10-transformedbodymass(kg)usingaquadraticregressionoftheform,log10(Ybl)=0+1log10(M0)+2log10(M0)2,whereiareregressioncoefcients[ 39 ].Specieswereseparatedbasedonmodeoflocomotionandbytaxonomicgroupsdifferinginp0(i.e.walkingmammals,sh,marinemammals,yinginsects,andpasserineandnon-passerinebirdswerettedseparately).Statisticalanalyseswereimplementedusingthenlmepackage[ 40 ]inR[ 41 ]. 2.3ResultsModelpredictionswereevaluatedusingextensivedataonmaximummigrationdistancesofanimalsfromaroundtheworld(n=206species,Appendix A ).Consistentwithourrstprediction,maximummigrationdistance(km)variessystematicallywithbodymassforwalking,swimming,andyingmigrants(Fig. 2-2 ;r2=0.57,0.65,0.19,forwalking,swimming,andyingspecies,respectively).ThesolidlinesshowpredictedmigrationdistancebasedonEquations( 2 )throught( 2 ).Thereisatightcorrespondencebetweenpredictedrelationships(solidlines)andttedmodelsthattreatbothy0andscalingexponentsasfreeparameters(dashedlinesand95%condencebands).Inthecaseofwalkingandswimminganimals,thedatasupportmodelpredictionsoflinearrelationshipsinlog-logspace,withobservedscalingexponentsclosetothosepredictedbyEquations( 2 )and( 2 )(walking:predicted 24

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=0.34,observed=0.3695%CI[0.25,0.48];swimming:predicted=0.3,observed=0.34[0.28,0.41]).Inthecaseofyinganimals,datasupportthepredictionthattherelationshipisnon-linearinlog-logspacereectingtherapidlyrisingcostofightwithincreasingmass(Fig. 2-2 C).Again,theobservedmassexponentisclosetothatpredictedbyEquation( 2 )(predicted=0.42,observed=0.43[0.36,0.49]).Consistentwithoursecondprediction,thenumberofbodylengthstraveledbyswimmingandwalkinganimalsisindependentofbodymass(Fig. 2-3 ).Onaverage,walkingmammalstravel1.5105bodylengths(Fig. 2-3 A).Theslopeandcurvaturetermsinthequadraticregressionmodeldoesnotdifferfromzeroinwalkingmammals(n=33,p>0.22)indicatingthatthenumberofbodylengthstraveledisuncorrelatedwithbodymassinthisgroup.Swimminganimalstravelanaverageof1.7106bodylengthsinaone-waymigratoryjourney.Themeandistancetraveledbysh(trianglesinFig.3B)exceedsthattraveledbyswimmingmammals(squaresinFig. 2-3 B)byafactorof4(sh:2.1106bodylengths;marinemammals:5.3105bodylengths,seeDiscussion),butthenumberofbodylengthstraveledisindependentofmassineachofthesegroups(slopeandcurvaturedoesnotdifferfromzero,sh:n=20,p>0.38;swimmingmammals:n=12,p>0.43).Inyingmigrants,thenumberofbodylengthsmigrateddeclinesclearlywithincreasingbodymass(Fig. 2-3 C).Innon-passerinebirds(n=80),coefcientsoflinearandquadratictermswerebothnegative,andsignicantlydifferentfromzero(1=-0.59,2=-0.19,p<2.210)]TJ /F4 7.97 Tf 6.58 0 Td[(5).Inpasserinebirds(n=45)andyinginsects(n=16)the1termwasnegativeanddistinguishablefromzero(passerines:1=-0.63,p=5.410)]TJ /F4 7.97 Tf 6.58 0 Td[(5;insects:1=-0.16,p=0.034).Resultsforyingmigrantsconrmourpredictionthatlargeryingmigrantsgenerallytravelfewerbodylengthsoverthecourseofmigration.Thenumberofbodylengthstraveleddecreaseswithincreasingmasssuchthatthesmallestinsectsandbirdstravelaround1.4108bodylengthswhereasthelargestbirdstravelaround5.2106bodylengths.Inother 25

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words,thenumberofbodylengthscoveredbymoths,dragonies,andhummingbirdsisroughly25-timesthattraveledbythelargestducksandgeese.AsensitivityanalysisindicatesthattheagreementbetweenmodelpredictionsanddataarerobusttodeviationsfromgeometricsimilarityandchangesinthevaluesofmorphologicalandbiomechanicalparametersusedtoderiveEquations( 2 )( 2 )(Appendix A ).Inparticular,thevalueoftheexponentinmetabolicscalingrelationshipshasbeenatopicofmuchdebate,withdifferentauthorsreportingdifferentexponentsdependingontheparticulardatasetandtaxonstudiedandthemethodofanalysis(e.g.[ 34 42 ]).However,sensitivityanalysisshowsthattheshapeofourpredictedrelationships,andtheagreementbetweenpredictionsanddataarelargelyinsensitivetochangesinthevalueofthemetabolicscalingexponentassumed(Appendix A ).Includingdatafrom[ 37 ]didnotsignicantlychangetheestimateofthemassexponent(0.3695%CI[0.26,0.43]withoutdatafrom[ 37 ],0.43[0.36,0.48]withdatafrom[ 37 ]).Includingdatafrom[ 37 ]decreasedthemodelr2from0.37to0.19. 2.4DiscussionWhenobservedmigrationdistancesareplottedagainstpredictionsofEquations( 2 )through( 2 ),pointsfromallthreegroupsclusterarounda1:1line(Fig. 2-4 ).ThedatashowninFigure 2-4 suggestthatvariationinmaximummigrationdistancesamongspeciesasdistinctasBlueWhales(Balaenopteramusculus),Wildebeest(Connochaetestaurinus),andBar-tailedGodwits(Limosalapponica)appearstobedriven,inpart,bythebasicdifferencesinmetabolism,morphology,andbiomechanicsdescribedbyourmodel.Thevariationexplainedbythemodelreectstheinuenceofconstraintsonenergeticsandbiomechanicsimposedbybodymass.Thereisalargebodyofworkdescribinghowmorphology[ 6 27 ],biomechanics[ 6 21 ],andbasicenergeticpropertiessuchasmaintenancemetabolism[ 43 44 ]arelinkedtobodymass.Ourmodelextendsresultsofthesestudiesbyspecifyinghowthesequantitiesinuencemaximummigrationdistanceofdiversespecies,therebylinkingbodymass 26

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tomigrationdistance.Ourresultsshowthatconstraintsimposedbybodymassaredetectableinmigrationdistancedata,despitevariationinmigrationdistanceamongspecieswithsimilarbodymasses(i.e.variationaboutpredictedrelationshipsshowninFigs. 2-2 2-4 ).Migrationdistancedatahighlighttheimportantroleofbasicdifferencesinenergeticsindrivingdifferencesinmigrationdistanceamongtaxa.Forexample,thenumberofbodylengthstraveledduringmigrationisindependentofbodymasswithinbothswimmingmammalsandsh;however,shtravelanaverageof4timesthenumberofbodylengthstraveledbyswimmingmammals.Equation( 2 )showsthatthedistancestraveledbythesegroupsdependonthemetabolicnormalizationconstant,p0,whichdescribesmass-independentdifferencesinthemaintenancemetabolicratesofshandmarinemammals.Inthesegroups,p0differsbyafactorofroughly9.1(p03.9Wkg)]TJ /F4 7.97 Tf 6.59 0 Td[(3=4inmarinemammals,p00.43Wkg)]TJ /F4 7.97 Tf 6.59 0 Td[(3=4insh,seeAppendix A ),whereasbodylengthexhibitsasimilarrelationshipwithmassinbothgroups(l0.44M1=3)suggestingthatthenumberofbodylengthsmigratedbyshisgreaterbyafactorof(9.1)0.64=4.1,whichisveryclosetotheobservedfactorof4.Thus,thedifferenceinthemeannumberofbodylengthstraveledbythesegroupsmaybedrivenbybasicdifferencesinthecostofmaintenancemetabolism.Dataalsorevealpatternsthatdonotappeartobecausedbytheenergeticandbiomechanicalfactorsconsideredhere.Forexample,swimmingissignicantlylesscostlythanightintermsoftheenergyrequiredtotravelagivendistance[ 45 ],yetvirtuallyallyingorganismstraveldistancesthatareasgreatorgreaterthanthosetraveledbymostswimmingspecies(Fig 2-4 ).Whetherthispatternisdrivenbydifferencesinmigratorybehaviororotherecologicalorevolutionaryfactorsremainsunknownandwilllikelybeafruitfulareaoffutureresearch.Itisworthnotingthatotherhypothesesmayprovidealternativeexplanationsforsomeofthequalitativepatternsobservedinmigrationdistancedata.Forexample,themodelpredictsthatmigrationdistance(km)oflargeryingspeciesdoesnotdepend 27

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stronglyonmass.Anincreaseinmassfrom10)]TJ /F4 7.97 Tf 6.59 0 Td[(6kgto10)]TJ /F4 7.97 Tf 6.59 0 Td[(3kg,increasesexpectedmigrationdistancebyafactorofmorethan8,whereasanincreaseinmassfrom10)]TJ /F4 7.97 Tf 6.59 0 Td[(2kgto10kgincreasesexpectedmigrationdistancebyafactoroflessthan2.Thisoccursbecausetheenergeticcostofightincreasesrapidlywithincreasingmasstothedegreethattheincreasingfuelmassthatcanbecarriedbylargermigrantsprovidesadiminishingincreaseinmigrationdistance.Analternativeexplanationforthisobservationisthatmanysubtropicalandtemperatehabitatsinthenorthernandsouthernhemispheresareseparatedby5103km104kmandthatmanyyingmigrantsmaynotbeunderselectiontomigrategreaterdistances.Ingeneral,therelationshipbetweenthedistancestraveledbymigrantsandtheglobaldistributionofsuitablemigratoryhabitatsispoorlyknownbutmayultimatelyinuencethedistancestraveledbymanyspecies.Whilemodelpredictionsaresupportedbydata,thereissubstantialunexplainedvariationinFigures 2-2 2-4 .Investigatingwhyparticularspeciesdeviatefrompredictionsmaybeaneffectivewaytoidentifyecologicalandevolutionaryfactorsthatdrivedifferencesinmigrationdistancebutarenotcurrentlyincludedinourmodel.Ourmodelignoresvariationinfuelandmorphologyofspecieswithsimilarmassesanddoesnotconsiderthepossibilitythatsomemigrantsmayseektominimizethetimespentmigrating.Twoadditionalfactors,inparticular,arelikelytocontributetoobservedresidualvariation.First,differencesinthenumbermigratorylegsamongotherwisesimilarspecieswillleadtovariationinmigrationdistanceamongspeciesasindicatedbyEquation( 2 ).Second,speciesthatinteractstronglywithabioticcurrentsduringmigrationarelikelytodeviatefrommodelpredictions.Thelackofinformationregardingthetypeandnumberofrefuelingstopsmadebymigratoryspecies,andthelackofinformationaboutthemannerinwhichmanyyingandswimmingmigrantsinteractwithabioticcurrentsrepresentsanimportantgapincurrentknowledge.Inthecaseofsomewell-studiedspeciessuchasthearctictern(Sternaparadisaea),itisclearthatthese 28

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variablesareimportantinfacilitatingextremelylong-distancemigrations.Individualsofthisspeciesstopatmultiplehighlyproductiveforagingsitestorefuelduringmigration[ 18 ].Thisspeciesisalsoknowntotrackglobalwindsystemstherebytakingadvantageoffavorableaircurrents.Inthecaseofspeciesthatmigrateagainstabioticcurrents,migrationdistancesmightbeexpectedtobeshorterthanourmodelpredicts.Indeed,manyoftheswimmingmigrantsthatfallbelowthepredictedlineinFigure 2-2 ,areanadromousshsuchasshad(Alosasapidissima),alewife(Alosapseudoharengus),andriverlamprey(Lampetrauviatilis)thatswimagainstwatercurrentsduringuprivermigrations.Increasedunderstandingoftheinteractionsbetweenmigrantsandabioticcurrentsandthenumberofmigratorystopoverswillallowforextensionsofthemodelthatcouldfurtherimproveourunderstandingofthereasonsforinter-specicdifferencesinmigrationdistance.Initscurrentform,themodelpresentedhereprovidesageneralexpectationonmaximummigrationdistance,whichcanbeseenasametricagainstwhichthedistancestraveledbyparticularspeciescanbecompared.Thebodysizesofmigratoryanimalsvarybyover11ordersofmagnitude.Themodelpresentedheremakesspecicquantitativepredictionsabouthowthisvariationinsizedrivespatternsofmigrationdistanceamongspecies.Itattributesdifferencesinthedistancestraveledbymigrantstosystematicdifferencesinmetabolismandmorphologicaltraitsthataretightlycoupledtobodysize,andtodifferencesintheunderlyingmechanicsofwalking,swimming,andight.Indoingso,itprovidesananalyticallytractableframeworkforstudyingtheinuenceofenergeticsandbiomechanicsonmigrationdistancethatisconsistentwithdataonspeciesrangingfromthesmallestmigratoryinsectstothelargestwhales. 29

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Figure2-1. (A)TotalmigrationdistanceisthesumofthedistancestraveledoneachofNmigratorylegs.(B)Migrationdistanceonasinglemigratoryleg.Bodymass(a),morphology(b)andmodeoflocomotion(c)governtherateatwhichamigrantusesstoredfuelenergy(d).Thisratechangesasmigrantlosesfuelmass(e),anddeterminesthemaximumdistancecoveredduringasingleleg(f,Equation( 2 )).Therelationshipbetweenaandbisgovernedbythemass-dependenceofmorphology.Totalrateofenergyuse(d)isdeterminedbythemass-dependenceofmaintenancemetabolismandbythebiomechanicsoflocomotion(Equations( 2 )-( 2 )). 30

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Figure2-2. MaximummigrationdistanceasafunctionofnormalizedbodymassforA)walkingmammals,B)swimmingshandmarinemammalsandC)yingbirdsandinsects.SolidlinesarepredictedcurvesbasedontsofEquations( 2 )( 2 )todatawithy0ttedasafreeparameter.Dashedlinesandcondencebandsrepresentbesttcurvesand95%condenceintervalsfromlinear(A,B)ornonlinearregression(C)withy0andthemassscalingexponentttedasfreeparameters.InpanelA,bodymassisM0(kg).InpanelsBandC,bodymassisnormalizedaccordingtotheequationsMnorm=M0.30p)]TJ /F4 7.97 Tf 6.58 0 Td[(0.64andMnorm=M0.420p)]TJ /F4 7.97 Tf 6.58 0 Td[(10,respectively,tocorrectfordifferencesinp0amonggroups.Dataonwalkinganimalsarefrommammalsonlyandarethereforenotcorrectedforp0. Figure2-3. NumberofbodylengthstraveledduringmigrationbyA)walkingmammals,B)swimmingsh(triangles)andmammals(squares),andC)yinginsects(triangles),passerinebirds(squares),andnon-passerinebirds(diamonds).Linesdenotemeannumberofbodylengthstraveledbyspeciesusingeachmodeoflocomotion. 31

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Figure2-4. Observedandpredictedmigrationdistancesforthewalking,swimming,andyinganimalsshowninFigure 2-2 .Datafromwalkingmammals(greencircles),swimmingsh(bluetriangles)andmarinemammals(bluesquares),andyinginsects(redtriangles),passerinebirds(redsquares),andnon-passerinebirds(reddiamonds)areshown.Blackpointsandillustrationsshowthewell-studiedmigrantsConnochaetestaurinus(Wildebeest),Balaenopteramusculus(BlueWhale),andLimosalapponica(Bar-tailedGodwit).Solidlineindicates1:1line. 32

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CHAPTER3SENSINGANDDECISION-MAKINGINRANDOMSEARCHOrganismsroutinelylocatetargetsincomplexenvironments.Theycandothisbyfollowinggradientsinthestrengthofsensorysignals,providedsuchgradientsareavailableandreliablyleadtowardtargets[ 46 ].Butthisisnotalwaysthecase.Inmanynaturalsettingssensorysignalsareinfrequent,noisy,andcontainlittledirectionalinformation[ 11 ].Forexample,moths,sharks,andseabirdssearchenvironmentsthatcontainscentcuesemittedbypreyormates,butthesecuesareoftenextremelysparseandsubjecttolargeuctuations[ 9 10 47 ].Undersuchsparse-signalconditions,itisnotclearwhatbehaviorsalloworganismstoefcientlyandreliablylocateresources.Researchershavedevelopedmuchofthetheoryofsparse-signalsearchbystudyingmathematicalmodelsofsearchingorganisms[ 12 13 48 51 ].Thedominantparadigmfordevelopingsuchmodelsemergedfromtherandomforaginghypothesistheideathatsearcherscanencountertargetsefcientlybyadoptingstatisticalmovementstrategiesthatcanbedescribedasrandomwalks([ 12 48 ],see[ 9 11 52 ]foralternativeapproaches).Thishypothesis,whichhasbeenappliedtosearchingorganismsrangingfrombees[ 12 ]toseaturtles[ 53 ],isofteninvokedwhenitisnotpossibleorpracticalforsearcherstorememberexplicitspatiallocations[ 48 ]andthetypicaldistancesbetweentargetsexceedsthesearcher'ssensoryrange[ 54 ].Thisframeworkhasbeenusedtocomparetheperformanceofsearchersmovingaccordingtodifferentkindsofrandomwalkbehavior.Inparticular,manystudieshavetriedtodeterminewhethersearchersmovingaccordingtoLevywalksoutperformsearchersthatmoveaccordingtoothertypesofrandomwalkstrategies(e.g.[ 13 49 51 ]). Thischapterappearedasanarticleinthejournal,ProceedingsoftheNationalAcademyofScience:Hein,A.M.andS.A.McKinley.2012.Sensinganddecision-makinginrandomsearch.Proc.Natl.Acad.Sci.USA.109:12070.Itsreproductionhereisauthorizedunderthejournal'scopyrightpolicy. 33

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Ifmodelsaretoyieldinsightintothebehaviorofsearchingorganismsinnature,theymustbesimpleenoughtobestudied,butshouldalsocapturethedominantfeaturesofsearchbehavior.Implicitintherandomforagingapproachistheassumptionthatchangesinasearchers'movementbehaviorinresponsetosensorydataaresecond-ordereffects,andthatsearchbehaviorandperformancearedominatedbythefeaturesoftheintrinsic(random)searchstrategythatthesearcheremploys.Hereweexploreanalternativehypothesis:thatsensoryprocessescanhaveadominanteffectonsearchperformance,evenwhensensorysignalsarerare,noisy,andlackdirectionalinformation.Belowwedevelopageneralmathematicalframeworkformodelingsearchdecision-making.Asinpastmodels,theframeworkallowsasearchingorganismtomakemovementdecisionsbasedonanintrinsicmovementstrategy(e.g.Levywalk),butallowssuchdecisionstobemodiedbasedonnoisysensorydata.Itthusprovidesanexplicitwaytomodelchangesinbehaviorinresponsetosensorymeasurements.Weexploretheeffectofincorporatingsensorydataintosearchdecisionsusingindividual-basedsimulationsofsearchingpredators.Wecomparesearchtimesofsimulatedpredatorsthatmakesearchdecisionsusingrandomstrategiesalone(Levywalkandanoveldiffusivestrategy),topredatorsthatmodifytheirsearchbehaviorbasedonolfactorymeasurements. 3.1ModelDevelopmentTostudysearchdecision-making,weconsideranidealizedmodelofapredatorinsearchofprey.Wewishtocomparethebehaviorandperformanceofpredatorsthatsearchusingasingleintrinsicrandomstrategytopredatorsthatadaptivelychangetheirsearchbehaviorusingtheincompleteinformationgainedfromsensorymeasurements.Toevokeastrongintuitionweconsidertwotypesofpredator:avisualpredatorthatmakesmovementdecisionsbasedonanintrinsicstrategyandlocatespreythroughashort-range,highacuitysense(vision),andavisual-olfactorypredatorthatchangesits 34

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searchbehaviorbasedonnoisyolfactorydataanddetectspreyatshortrangeusingvision.Predatorswanderthroughalarge(periodic)two-dimensionalhabitatinwhichthemeandistancebetweenpreyislarge.Weassumepreyemitascentthatcanbedetectedbynearbypredators.Similartopreviousapproaches(e.g.[ 55 ]),weassumethatsearchisdividedintotwophases:alocalscanningphaseandamovementphase(Figure 3-1 A,[ 56 ]).Duringthescanningphase,thepredatorlocatesanypreywithinitsvisiondistancerv(Fig. 3-1 A,solidinnercircle)withprobabilityone.Thisreectsthehighlocalacuityofvision.Visual-olfactorypredatorsalsoscanforolfactorysignals.Thedurationofthescanningphaseisdenotedvandoforvisualandvisual-olfactorypredatorsrespectively.vincludesthetimeneededtovisuallysearcharegionofradiusrvandreorientbeforetakinganotherstep.oincludesthetimetakentocollectandprocessolfactorysignals,visuallysearcharegionofradiusrv,andreorientbeforetakinganotherstep.Wedenetheolfactoryradiusro(Fig. 3-1 A,dashedoutercircle)asthedistancewherethepredatorregistersanaverageofonescentsignalperscanningperiodo(seebelow).Weassumethateachpreyitememitsscentatrate.Duringthemovementphase,thepredatortravelsinarandomuniformdirection,adistanceofl,atspeedv.Visualpredatorsdrawthesteplengthlfromaprescribedsteplengthdistribution(l),examplesofwhicharedescribedinthenextsubsection.Visual-olfactorypredatorsdrawfromamodiedsteplengthdistributiondenedbelowbyEquation( 3 ).Duringthemovementphase,weassumethatthepredatorcannotlocatepreyordetectscentsignals.Additionally,weassumethatthepredatoronlyrespondstothemostrecentscentsignalinformationanddoesnotstoreinformationaboutthelocationsithasvisited.Westudythislimitingcasewheresensorysignalsarerare,lackdirectionalinformation,andarenotrememberedbythepredatorbecausethisisthescenarioinwhichrandomsearchstrategiesareofteninvoked.Wethusevaluatethescenarioinwhichnoisysensorydataareleastlikelytoyieldimprovementoverpurely 35

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randomsearch.However,wepointoutthatmoresophisticatedstrategiesarepossibleifpredatorsrememberpastsignalencountersorpreviouslyvisitedlocations[ 11 46 57 ]. 3.1.1SearchingWithoutOlfactoryDataTomodelpredatormovements,webeginwithamodelofdecision-makingintheabsenceofanyinteractionwitholfactorydata.Researcherstypicallymodelthedecisionprocessofrandomsearchersbyselectingtwoactionsfromprescribedprobabilitydistributions:asteplengthl,andaturnangle.Thedetailsofthesedistributionsdetermineasymptoticpropertiesofthesearchandstrategiesareoftencategorizedbythisasymptoticbehavior:diffusivebehavior,inwhichlong-termmean-squareddisplacement(MSD)scaleslinearlywithtime,andsuperdiffusivebehaviorinwhichMSDincreasessuperlinearlywithtime.Animportantfeatureofthesestrategiesisthat,unlessthesearcherencountersatarget,thedistributionsthatdenehowsearchermoves(i.e.thedistributionsofland)arexed.Theyarenotalteredinresponsetosensorymeasurements.Wemodelthemovementsofvisualpredatorsusingtwotypesofstrategies:aLevystrategyandanoveldiffusivestrategy.Forboth,wetakethedistributionofturnanglesbetweensuccessivestepstobeiidunif(0,2)[ 12 ].TheLevystrategydrawssteplengthsfromaParetodistribution,L(l)=()]TJ /F8 11.955 Tf 12.14 0 Td[(1)l)]TJ /F4 7.97 Tf 6.59 0 Td[(1ml)]TJ /F10 7.97 Tf 6.59 0 Td[(,withtailwithparameterandminimumsteplengthlm(Fig. 3-1 Bsolidcurve,superdiffusivefor1<<3[ 12 ]).Forthesecondstrategy,weintroduceanewstep-lengthdistributionwhichwecallthetruedistancedistribution(TDD)T(l):agreedystrategywhereinthepredatorselectssteplengthsfromtheprobabilitydistributionofthedistancetothenearestpreyitem(Fig. 3-1 Bdashedcurve,seeSupplementaryInformation(SI)Textforfurtherdiscussion).WhenpreyaredistributedaccordingtoaPoissonspatialprocesswithintensityintwodimensions,theTDDisgivenbytheRayleighdistributionT(l)=2le)]TJ /F10 7.97 Tf 6.59 0 Td[(l2.ThisstrategyisquitedistinctfromtheLevystrategy(comparecurvesinFig. 3-1 B)andlaterservestoillustratethestronghomogenizingeffectofolfactorydataonsearchbehavior. 36

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3.1.2IncorporatingOlfactoryDatatoMakeSearchDecisionsThekeydistinctionbetweenvisualandolfactorysensesinourmodelisthatthevisualsenseyieldsperfectinformationaboutthelocationofpreywhereastheolfactorysensedoesnot.Thus,includingolfactorymeasurementsallowsustomodelapredator'sabilitytogatherandrespondtopartialinformationabouttargetpositionsgleanedfromsensorymeasurements.Belowwedevelopamodelforincorporatingolfactorysignalsintosearchdecision-making,butnotethatthisframeworkcouldbemodiedtomodelresponsestoothertypesofsensorycues.Wehypothesizethatpredatorsutilizeolfactorydatathroughtwosteps.First,apredatorusesasignalobservationtoestimatethelikelydistancetothenearestprey.Second,thepredatormodiesitsintrinsictendencytomoveinaparticularway(representedby(l))basedonthisinformation.Inkeepingwithrecentmodelsofolfactorysearch,simulatedpredatorscollectolfactorydataforounitsoftimeandencounterH2f0,1,2,...gdetectableunitsofscent[ 11 57 ].Inordertoactoptimally,apredatormustmakemovementdecisionsbasedontwodistinctuncertainties.First,thepredator'sdistancetothenearesttargetisuncertainandischaracterizedbytheprobabilitydistribution.Second,foraparticular,theoptimalsteplengthdistributionisalsouncertain.IdentifyingoptimalpredatorbehaviorrequirescalculatingaBayesianposteriorforthedistancedistributionjH,andthendeterminingtheassociatedoptimalsteplengthdistributionjH.Thisremainsanunsolvedandperhapsintractableproblem.Instead,weapproximatethisprocess.Wewishtocapturetwoelementsofsearchdecision-making:anintrinsictendencytomoveinaparticularway(l),andalikelihoodfunctionP(H=hjl)thattranslatesanobservedscentsignalhintoinformationaboutthedistancetothenearestprey.AnaturalmodelforsignalresponsethatincorporatesthesefeaturesisaBayesianupdate 37

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ofthesteplengthdistributionitself: (ljH=h)=P(H=hjl)(l) R10P(H=hjl)(l)dl.(3)Werefertothisassignal-modulationofthesteplengthdistribution(l).Thisapproximationtotheoptimalstrategyyieldssignicantimprovementinsearchperformance(seeAppendix B forfurtherelaboration). 3.1.3InterpretingScentSignalsWeassumethepredatorcanestimateorintuittheprobabilityofregisteringhunitsofscentinounitsoftime,asafunctionofitsdistancetothenearestprey.ThisamountstobeingabletoestimatethelikelihoodfunctionP(H=hjl),whichdependsontheprocessofscentpropagation.Inthecomplexenvironmentswheremanyspeciessearch,turbulentuctuationsinuidvelocitycauselargelocaluctuationsinscentconcentration[ 58 ].Whenaprevailingwindorwatercurrentispresent,predatorscangainadditionalinformationaboutthelocationofascentsourcebymeasuringthevelocityofthecurrent[ 11 47 ].Weconsiderthemoredifcultscenarioinwhichthereisnoprevailingcurrent.Undertheseconditions,wemodelscentarrivalaspacketsthatappearatthepreypositionx0accordingtoaPoissonarrivalprocessandthenmoveasaBrownianmotion.Fromthepredator'sperspective,thisisequivalenttoencounteringarandomnumberofunitsofscent,HPois(oR(jx)]TJ /F15 11.955 Tf 12.57 0 Td[(xoj)),atitslocationxduringascanningphaseoflengtho,whereRistherateofscentarrivaldenedbyEquation( C )(seeMaterialsandMethods).Denotingl=jx)]TJ /F15 11.955 Tf 11.89 0 Td[(x0j,undertheseassumptions,thelikelihoodofhencountersis P(H=hjl)=[oR(l)]he)]TJ /F10 7.97 Tf 6.59 0 Td[(oR(l)=h!(3)Equation( 3 )dependsonvaluesofseveralphysicalparameters(e.g.therateatwhichdetectablepatchesofscentdecay)thatmaybedifcultforapredatortoinfer 38

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frommeasurementsofitsphysicalenvironment.Wethereforetakeaqualitativeviewinprescribingtheparametersofscentpropagation.Themostimportantqualitativefeatureisthelengthscalero,whichcorrespondstothedistanceatwhichapredatorwillregisteronaverageoneunitofscentperscanningperiodo.Heuristically,thisisthedistanceatwhichthepredatorislikelytodetectafaint,yetnon-trivialscent.Asecondqualitativerestrictionistheexpectednumberofencountersperunitoatadistanceofonebodylengthfromthepreya.Giventhesetwomeasurements,thelikelihoodfunctioncanbeestimated.ThequantitiesroandaaremuchmorereadilymeasurablebyasearchingorganismthanaretheexplicitparametersinEquation( C ).Itthusseemslikelythatthesequantitiesmayconstitutepartofanorganism'solfactorysearchimage[ 59 ],andmayserveasthedirectmeasurementsusefulforreinforcementlearning. 3.2MaterialsandMethods 3.2.1ScentPropagationToseehowR(l)dependsonthedistancebetweenpredatorandprey,letu(x)representthemeanconcentrationofscentatpredatorpositionxemittedbyapreyitemlocatedatpositionx0.Anexpressionforthesteady-statediffusionprocesswithoutadvectionisgivenby0=Du(x))]TJ /F9 11.955 Tf 12.24 0 Td[(u(x)+(x0),whereDrepresentsthecombinedmolecularandturbulentdiffusivity(m2s)]TJ /F4 7.97 Tf 6.58 0 Td[(1),representstherateofdissolutionofscentpatches(s)]TJ /F4 7.97 Tf 6.59 0 Td[(1),andrepresentstherateofscentemissionattheprey(s)]TJ /F4 7.97 Tf 6.59 0 Td[(1).Intwodimensions,themeanrateofscentpatchencountersbyapredatoroflinearsizealocatedatxisgivenbyR(l)=2D )]TJ /F4 7.97 Tf 7.99 0 Td[(ln(a )u(l)where =p D[ 11 ].Thisimplies R(l)=2K0( l) )]TJ /F9 11.955 Tf 9.29 0 Td[( ln( a),(3)whereK0representsamodiedBesselfunctionofthesecondkind. 39

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3.2.2SimulationDetailsTheSITextshowstherobustnessofresultstochangesinmodelparameters.Foreachofthefoursearchstrategies(visualLevy,visualTDD,visual-olfactoryLevy,andvisual-olfactoryTDD),weperformedsimulationsinwhichpredatorsexploredaperiodicenvironmentwith100prey.PreywerepositionedaccordingtoaPoissonpointprocesswiththemeandistancebetweenpreychosentoachievethedesireddensity.Ineachscanningphase,hwasgeneratedasadeviatefromaPoissondistributionwithmeangivenbytheproductofoandEquation( C )summedoverallprey.Ineachsimulation,thesearcherwaspositionedatarandomlocationandallowedtomovethroughtheenvironmentuntilitcamewithinadistanceofrvofapreyitemduringitsscanningphase.Foreachstrategy,weperformed1000simulationsandrecordedthetimeuntilrstpreyencounterineachsimulation.Predatorswereassumedtotravelataconstantspeedofonebodylengthperunittime.Environmentswereconstructedsothatpreydensityhadameanof1preyper106squaredbodylengths,arealisticlowdensityforprey,butqualitativeresultsholdforlowerpreydensities(seeAppendix B ).InthecaseoftheLevystrategies,werepeatedsimulationsacrossarangeofvaluesfrom=1.2to=3.NotethattheoptimalvalueoffortheLevypredatorwas=3forwhichthelong-termbehaviorisexpectedtobeGaussian[ 12 ].Inallgures,Levystrategieswiththeoptimalvalueofareshownunlessotherwisenoted. 3.3Results 3.3.1Visual-OlfactoryPredatorsFindTargetsFasterandMoreReliablyThanVisualPredatorsFigure 3-2 Ashowsmeansearchtimesofsimulatedvisualandvisual-olfactorypredators(searchtime=timeuntilrsttargetencounter).VisualpredatorsthatusetheLevystrategy(Fig. 3-2 A,solidline,seealsoMaterialsandMethods)havelowermeansearchtimesthanpredatorsthatusetheTDDstrategy(Fig. 3-3 A,dashedline).However,whenconditionsaresuchthattheolfactoryradiusroisgreaterthanthevision 40

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radiusrv,visual-olfactorypredatorsndpreyfasterthantheirvisualcounterparts(Fig. 3-2 A;circlesrepresentresultsfromvisual-olfactoryLevywithoptimal,whereoptimalwasintherange2.6-3.0forallro=rv;diamondsrepresentvisual-olfactoryTDDstrategy).Meansearchtimeofvisual-olfactorypredatorscontinuestodecreaseasthedistanceoverwhichpreyscentscanbedetectedincreases.Visual-olfactorypredatorshavelowermeansearchtimesthanvisualpredatorsprimarilybecausetheyrarelysearchforlongperiodsoftimewithoutndingprey.Figure 3-2 Bshowsthatthetailsofthesearchtimedistributionsforthevisual-olfactorypredators(Fig. 3-2 B,circles)decayroughlyexponentiallyataratethatismuchfasterthanthedecayrateofthevisualpredators(Fig. 3-2 B,squares).Atleasttwofactorscontributetothedifferenceinperformancebetweenthetwopredatortypes.First,visual-olfactorypredatorslearnfromno-signalevents.Theyrespondtotheseeventsbyleavingregionsthatdonotcontaintargets.Second,ashasbeenobservedinmanyspeciesinnature[ 14 47 ],visual-olfactorypredatorsperformarea-restrictedsearch[ 15 ]andconcentratesearcheffortinregionsthatcontainprey.Belowwediscusshowbothofthesebehaviorsemergenaturallythroughresponsestosensorysignals.Tocharacterizechangesinpredatorbehaviorinresponsetosensorydatainthefollowingsections,weuseametricofinformationgain:theKullback-Leiblerdivergence(KL,[ 60 ]).Themagnitudeofthechangeinbehaviorofavisual-olfactorypredatorwhenitreceivesasignalofstrengthhrelativetoitsintrinsicbehavior(l),isgivenbyKL=R(ljh)log((ljh)=(l))dl.AliteralinterpretationofthequantityKListhefollowing:supposeanobservermustdecide,basedonempiricaldata,whetherasearcherisusingolfactorydataornot.TheKLgivesameanrateofgainofinformationobtainedbyobservingavisual-olfactorysearchermovinginresponsetoasignalofmagnitudeh.Inregimeswherethesignalcontainslittleusefulinformation(forexamplewhenro=rv1andh=0),thebehaviorisnotmodiedgreatlyfrom(l).TheresultingKLvalueis 41

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small.However,wheninformationissubstantial(saywhenh=5,forsmallro=rv)theKLislarger. 3.3.2Visual-OlfactoryPredatorsLearnFromNo-SignalEventsFigure 3-3 showstypicalsearchpathsofthefourstrategiesthroughatargeteldintheregimewherero>rv.Whensearchingsuchanenvironment,apredatorwillfrequentlybetoofarfrompreytoreceivescentsignals.Forexample,theinsetpanelsinFigure 3-3 Cand 3-3 Dshowthatthenumberofsignalsreceivedinscanningphasesistypicallyzero,withsignalsofgreaterthanzeroonlyoccurringwhenthepredatorisclosetoprey.Intuitively,itmayseemthatapredatorgainslittleinformationfromtheseno-signalevents.Yet,bynotreceivingascentsignal,thepredatorgainsavitalpieceofinformation:preyarenotlikelytobenearby.Figure 3-4 Ashowssteplengthdistributionsofvisual-olfactorypredatorsafterreceivingnosignal.Bothstrategiesexhibitalowprobabilityofmakingsmallsteps.TheLevystrategyinparticular,isstronglyaffected;Figure 3-1 Bshowsthatthisstrategyhasahighprobabilityoftakingsmallstepsbetweenre-orientations.Yet,whenthevisual-olfactoryLevypredatorreceivesnosignal,itisunlikelytomakeasmallstep(Fig. 3-4 A,Figure B-4 ).Figure 3-4 Bshowsthatwhenh=0,KLincreasesastheolfactionradiusbecomeslarger.Infact,asro=rvbecomeslarge,bothstrategieschangemoreinresponsetono-signaleventsthanwhenh=5(Fig. 3-4 B,circles(h=0)crossabovesquares(h=5)forbothstrategies).Forro=rvsufcientlylarge,thechangeinbehaviorinresponsetono-signaleventsallowvisual-olfactorypredatorstoavoidperformingarearestrictedsearch(ARS)whentheyarefarfromprey(Figure 3-5 ).ThevisualLevypredator,ontheotherhand,spends24%ofitsstepsinARSbutonly2.4%inARSneartargets.Avoidingthesewastedstepsstronglyaffectssearchtime.Evenbyrespondingonlytono-signaleventsandignoringcasesinwhichh>0,avisual-olfactoryLevypredatorcanndpreymuchmorerapidlythanavisualLevypredator(Fig. B-5 ). 42

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Theobservationthatno-signaleventscontainvaluableinformationisqualitativelysimilartoanobservationfromoptimalforagingtheoryregardingaforagersearchingadiscretepatchforhiddenresources.Inthatscenario,themoretimetheforagerspendsinthepatchwithoutencounteringresources,themorecertainitbecomesthatthepatchdoesnotcontainresources[ 61 ].Ourmodelextendsthisideatosearchersmovingthroughcontinuousspatialenvironmentsusingtwosensorymodalitiesandrevealsthatthechangeinasearchersbehaviorinresponsetono-signaleventsdependscriticallyonthelengthscalesofthesesensorymodalities. 3.3.3Visual-OlfactoryPredatorsConcentrateSearchEffortNearTargetsFromFigure 3-3 Aand 3-3 B,itisclearthatvisualpredatorsbehavesimilarlyinregionsthatarenearandfarfromprey.Visual-olfactorypredators,ontheotherhand,makemoreshortexploratorystepsinthevicinityofprey(Fig. 3-3 C, 3-3 D).Thestrongchangeinstrategythatoccurswhenavisual-olfactorypredatorreceivesanonzeroscentsignalisreectedinthelargevalueofKLforallvaluesofro=rv(Fig. 3-3 B).Bothvisual-olfactorystrategiesincreasetheirprobabilityofmakingashortstepwhentheyencounteranonzeroscentsignal(Fig 3-3 A).Becauseofthis,visual-olfactorypredatorsperformARSneartargetsandaremorelikelytoencounternearbypreythanarevisualpredators(Figure 3-5 ). 3.4DiscussionTheframeworkpresentedhereallowsonetoincluderesponsestopartialinformationgainedfromnoisysensorymeasurementswhenmodelingrandomsearch.Ourresultsrevealthatanalysisofthelengthscalesofsensorymodalities,inthiscaseroandrv,iscrucialtodeterminingwhethersuchasensoryresponsewilldominatesearchperformance.Thedistinctionbetweendifferenttypesofintrinsicstrategies(e.g.LevyvsTDD[ 49 50 ])isimportantwhenitisgenuinelynotpossibletolearnaboutresourcesfromadistance(ro=rv1).However,whenro=rv>1,searchersthatdynamicallymodifytheirbehaviorinresponsetosensorydataexperienceaqualitativeimprovement 43

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insearchperformance.Thisholdsoverawiderangeoftheparametersofthescentmodelandotherfeaturesofpredatorbehavior(Figures B-2 ).Thisndingsuggestsaconnectionbetweensensing,decision-making,andsearchperformance,evenundersparse-signalconditions.Moreover,behaviorssuchasarea-restrictedsearchnearprey[ 14 ]emergenaturallyfromresponsestosensoryinformation.Visual-olfactorypredatorspreformthisbehaviorinoursimulationsbyturningmorefrequentlywhentheyreceivescentcues.Historically,ARShasbeenexplainedasaconsequenceofapredatorconcentratingsearcheffortinareaswhereithaspreviouslyfoundprey.Thisisbenecialifpreyareclusteredinspace[ 15 ].Yet,weshowthatthisbehaviorcanalsoemergewhenpreyarenotspatiallyclustered,ifpredatorschangetheirmovementbehaviorinresponsetonoisysensorydata.Recentevidencesuggeststhatsomespeciesmayinitiatearea-restrictedsearchinthisway.Forexample,wanderingalbatrossesappeartoalterturningpatternsafterencounteringpreyscent,effectivelyconcentratingtheirsearcheffortinlocalregions[ 47 ].Greaterfrigatebirdsforageprimarilyinhighlyproductivemesoscaleeddies[ 16 ].Theyappeartotracktheseeddies,atleastinpart,usingscentcues.InoursimulationsLevypredatorsintersperseperiodsoflocalsearchwithlarge-scalerelocationmovements.Movementsofmanyspeciesincludingforagingmarineshandreptiles[ 53 ],andantsinsearchofcolony-mates[ 62 ]exhibitthisqualitativepattern[ 48 53 62 ].ThisisoftencitedasafeatureofLevywalksthatmakesthemeffectivestrategiesforencounteringtargets.Yet,ourresultsshowthatLevypredatorsspendmuchoftheirtimesearchinglocallyinregionsthatdonotcontainprey(Fig. 3-5 ).Ontheotherhand,visual-olfactorypredatorsappropriatelymatchtheirbehaviortotheirproximitytotargets,leadingtoshortersearchtimes.Inlightofourresults,anaturalhypothesisisthatsearchingorganismsutilizedifferentmovementbehaviorsdependingontheirperceiveddistancetotargets.Ithasbeenshownthatstrategiesthatmixmovementswithdifferentlengthscalescanoutperformstrategiesthat 44

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drawmovementsfromasingledistribution,butthatsuchmixedmovementbehaviorcanbedifculttodistinguishfromaLevystrategy[ 51 ].Indeed,recentanalyseshavebeguntondevidenceofmixedbehaviorsinmovementdata(e.g.[ 63 ]).Ourframeworkprovidesameansofstudyinghowsuchmixedbehaviorscanemergethroughinteractionswithsensoryinformation. 45

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Figure3-1. Schematicofpredatorsearch.A)Duringthescanningphaseofthesearch,apreyencounteroccursifthepredatoriswithinaradiusofrv(solidinnercircle)ofapreyitem.Thepredatoralsodetectsscentsignalsemittedbypreywithinaradiusofro(dashedoutercircle)atanaveragerateof1perounitsoftime.Thepredatorthenturnsarandomuniformanglebetween0and2.Duringthemovementphase,thepredatormovesadistanceoflunitsdeterminedbyitssteplengthdistribution.B)SteplengthdistributionscorrespondingtovisualLevy(solidcurve,=3,lm=1bodylength)andTDD(=1/(1000)2bodylengths,dashedcurve)strategies.Insetshowsdistributionsonlog-logscale. 46

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Figure3-2. Predatorsearchtimes.A)Meansearchtimeasafunctionoftheratiooftheolfactoryradius(ro)tovisionradius(rv).Solidorangeline(visualLevy),dashedblueline(visualTDD),orangecircles(visual-olfactoryLevy),andbluediamonds(visual-olfactoryTDD)eachrepresentmeansearchtimeof1000replicatesimulations.Condencebandsrepresent2SEM.Thefollowingparametersvalueswereused:a=1,rv=lm=50a,v=1s,o=30s,meaninter-targetdistancewas1000a,anda=100unitsofscentpero(seetextfordescriptionofparameters,alsoSIText).B)EmpiricaldistributionofsearchtimesofvisualLevy(orangesolidline,squares),visualTDD(bluedashedline,squares),visual-olfactoryLevy(orangesolidline,circles),andvisual-olfactoryTDD(bluedashedline,circles)strategies.Inthecaseofthevisual-olfactorystrategies,frequenciesareshownforro=rv=4.Notethelargenumberofsearchesresultinginlongsearchtimesforvisualpredators. 47

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Figure3-3. Typicalsearchpathsthroughascenteldwithlog10(1+meannumberofscentencountersperunito)indicatedbygrayscale(darkergreydenotesmoreencounters).Inwhiteregions,meannumberofencountersiseffectivelyzero.PathsforA)visualLevy,B)visualTDD,C)visual-olfactoryLevy,andD)visual-olfactoryTDDareshown.Colorscaleofpathchangesfrombluetoredwithincreasingtime.InsetpanelsinCandDshowthenumberofhitsreceivedduringeachscanningperiodwithcolorscorrespondingtocolorsinsearchpaths.ro=rv=4inallpanels;allotherparametersasinFig.2A. 48

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Figure3-4. Effectofolfactorydataonsteplengths.A)Steplengthdistributionsaftersignalmodulationwhenh=0andwhenh=5.B)InformationgainasmeasuredbytheKullback-Leiblerdivergencebetweenthevisualstrategyandthecorrespondingvisual-olfactorystrategywhenh=0(squares)andh=5(circles)asafunctionofro=rv.Dashedcurvesrepresentvisual-olfactoryTDDstrategy.Solidcurvesrepresentvisual-olfactoryLevystrategy.Notetheincreasinginformationgainwhenh=0.Inbothpanels,lm,andasinFig.1B,allotherparametersasinFig.2A. Figure3-5. Effectofolfactorydataonarea-restrictedsearch(ARS).Leftbarsshow%stepsspentperformingARS.Shadedregionsshowthe%ofARSsearchesthatoccurwithin4rvofprey.ForthevisualLevypredator(topbar),contrastthelarge%ofstepsspentinARS,withthesmallfractionofthesestepsspentnearprey(topbar,shadedregion).ARSdenedasanyperiodinwhichpredatormakes5consecutivestepswithinaregionofradius4rv.Rightbarsshow%ofproximityeventsinwhichpredatorlocatesprey.Barsforvisual-olfactory(V-O)predatorsshowthattheysuccessfullylocatenearbypreymorefrequentlythandovisual(V)predators.Proximityeventsdenedtobyanyperiodof1consecutivestepswithin2rvofatarget. 49

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CHAPTER4SENSORYINFORMATIONANDENCOUNTERRATESOFINTERACTINGSPECIESClassicalmodelsofspeciesinteractionsassumethatencountersaregovernedbyaprocessakintomass-action;individualsmovealongrandomlineartrajectoriesandencounteroneanotherwhentheycomewithinacriticaldistance[ 3 64 ].Undertheseassumptions,anindividualsearcherencounterstargetsatarateproportionaltothedensityoftargets[ 3 65 ].Recentworkhasextendedthestudyofencounterratestoconsidersearchersthatfollowmovementpathsthatarenotlineartrajectories,encountertargetsprobabilistically,destroytargetsafterencounters,andsearchintermittently[ 66 68 ].Underavarietyofcircumstances,thesemodelstoopredictthatasearcherwillencounteritstargetsatarateproportionaltotargetdensity(foralistofconditions,see[ 67 ]).Avitalassumptionbothofolderandnewermodelsisthatthesearchingorganismmovesindependentlyofthelocationsoftargets.Inthecontextofpredator-preyinteractions,thisimpliesforinstance,thatpredatorsdonotaltertheirmovementbehaviorinresponsetosensorycuesemittedbytheirprey.Ofcourse,theassumptionthatsearchersmoveindependentlyoftargetsismadeformathematicalconvenience.Thequestioniswhethermodelsthatrelyonthisassumptioncapturethesalientfeaturesofencounterratekineticsinnature.Empiricalstudieshaveshownthatshuttingdownparticularsensorymodalitiessuchaschemosensingorowsensingcandramaticallydecreasesearchperformance(e.g.,[ 10 ]),andthatsensorycuesappeartoinuencebothsmall-scale[ 69 ]andlarge-scale[ 16 47 ]searchbehavior.Whilesuchstudiesmorerigorouslyconrmtheintuitionthattheuseofsensorydatashouldimprovesearchperformance,littleisknownabouthowsensingcaninuencethequalitativerelationshipbetweenencounterrateandtargetdensity.Here,wearguethatsensoryresponsecanhaveadominanteffectontherateofencountersbetweensearchersandtheirtargets,notonlybyincreasingencounterrate,butalsobyqualitativelychangingthedependenceofencounterrateontargetdensity. 50

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Belowweadoptthelanguageandintuitionassociatedwithpredatorssearchingforprey.Weassumethatapredatorsamplestheenvironmentforsensorycuespassivelyemittedbyprey,andadjustsitsmovementbehavioraccordingtoexplicitmathematicalmodelspresentedbelow.Thisapproachbuildsonarecentlydevelopedframeworkformodelingsearchdecision-making[ 70 ]tomodeltheowofsensoryinformationfrompreytopredators.Weconsiderthreescenarios:(1)perfectsensingandresponse:thepredatorcanascertainthepreciselocationsofpreyfromthesensorydataitreceivesandrespondsoptimally,(2)imperfectsensingandresponse:thepredatordetectsnoisyscentsignalsemittedbypreyandaltersitsmovementbehaviorinresponse,and(3)purelyrandomsearch:thepredatordoesnotusesensoryinformationtoguideitsmovementdecisions.Models(1)and(2)representupperandlowerbounds,respectively,ontheacquisitionanduseofinformationaboutpreypositions.Ourcentralndingisthatthereisasystematicshiftawayfromalinearencounterratefunctionatbothofthesebounds,suggestingthatthecollectionanduseofanyformofsensorydatamayfundamentallyalterencounterratekinetics.Wediscusstheroleofinformationingoverningpredator-preyencounterrates,butnotethatourgeneralmethodologycouldbeappliedtoratesofencountersinothertypesofecologicalinteractions(e.g.,betweenmates,competitors,mutualists). 4.1MaterialsandMethods 4.1.1EncounterRateandSearchBehavior:SomeDenitionsStudiesofbiologicalsearchstrategiestypicallydescribehowthetypeofmovementbehaviorusedbyasearchingorganismaffectsthetimeneededtoencounteritsrsttarget,ortherateoftargetencounters)]TJ /F1 11.955 Tf 6.78 0 Td[(.Forconsistencywithpastwork,wedene)]TJ /F1 11.955 Tf 10.1 0 Td[(asthepreyencounterrateofasinglepredator(e.g.,[#prey]per[predatorhour],[ 67 ]).Weassumethatpredatordensityislowenoughthat)]TJ /F1 11.955 Tf 10.1 0 Td[(doesnotdependonthedensityofpredators,andinstead,dependsonlyonthedensityofprey.Wedenetwoencounterratefunctions:themeanrstencounterrate\(),andthemeanencounter 51

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rateafterkencounters)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(k().Thelatterisoftenreferredtoastheencounterrateassociatedwithdestructivesearch[ 67 68 ],emphasizingthattheactivityofthesearcheraltersthetargetlandscape.Inpaststudies,thenon-destructivesearchrateisoftendenedintermsofrandomvariablewhichrepresentsthetimerequiredtondthersttarget.Theempiricalrstencounterrateisthendenedtobe\()=1=whereindicatesanaverageovermanytrials. 4.1.2FrameworkforModelingMovementDecisionsWeconsideranidealizedmodelofasearchingpredatorinatwo-dimensionalenvironment.Weassumethatthepredatormovesataconstantspeedvthatismuchgreaterthanthespeedofitsprey.Inthiscase,itissensibletomodelpreyasiftheyarenotmoving,atleastforthedurationofthepredator'ssearch.Inthefollowingsections,wefurtherassumethatpreydensityislow,andthathandlingtimeisthereforenegligiblerelativetosearchtime.Asinpastapproaches,thepredatordividesitssearchintotwophases:ascanningphaseandamovementphase[ 55 70 ].Thisintermittencyreectstheobservedtradeoffbetweenlocomotionandperceptualacuity(e.g.,[ 71 ]),andtheintermittentnatureofsamplingthroughmajorsensorymodalities[ 72 ].Duringthescanningphase,thepredatorcollectssensorydatah,andencountersanypreywithinaradiusrewithprobabilityone.Duringthemovementphase,thepredatormovesadistance`atanangle.Theprocessthepredatorusestodetermine`andconstitutesitssearchstrategy. 4.1.2.1SensorysignalsandsearchbehaviorTorelateapredator'ssearchbehaviortotheinformationitacquiresfromsensorysignals,weadaptarecentlydevelopedframeworkformodelingsearchdecision-making[ 70 ].Theframeworkhastwoessentialfeatures.First,thepredator'smovementbehaviorintheabsenceofanysensorydataismodeledbyanintrinsicmovementdistribution(`,).Second,thepredatorusesadecodingfunctiontoextractinformationfromthesensorydataitcollectsandmodifyitsintrinsicmovementbehavior. 52

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Duringthemovementphaseofthesearch,predatormovementsaremodeledbydrawingfromthedistribution (`,jH=h)=PfH=hj`,g(`,) R20R10PfH=hj`,g(`,)d`d,(4)wherehisthesensorydatacollectedinthepreviousscanningphase,andPfH=hj`,gisthelikelihoodofobservingH=h,giventhatthetargetisadistanceof`andanglefromthepredator'scurrentposition.Ratherthanassociatingadeterministicactionwithaparticularvalueofthesignalh,wemodelmovementdecisionsasactionsdrawnfromaprobabilitydistributiontocapturetheinherentvariabilityindecision-making[ 73 ].Theintrinsicmovementdistributioncanbeinterpretedasanevolvedbehaviorthatthepredatorusesintheabsenceofusefulsensoryinformation[ 13 ].Thedecodingfunction,ontheotherhand,representsanevolvedmechanismforinterpretingandmovingbasedonsensoryinput,H[ 70 ].WhilePfH=hj`,gisformallyalikelihoodfunction,werefertoitasadecodingfunctiontoemphasizethatitrepresentsameansofinterpretingandusingsignaldata.Asweshowbelow,thethreestrategieswewishtoconsidercanbeframedbyspecifyingappropriatedecodingfunctions. 4.1.2.2PerfectsensingandresponseSupposethepredatordetectssensoryobservationshand,regardlessofthevalueofh,isabletoperceivethepreciselocationsofprey.ThenthedecodingfunctioninEquation( 4 )isapointmassatthelocationofthenearestprey(notethatatravelingsalesmansolutiontothisproblemcouldoutperformsuchagreedysearcher,butiscomputationallyintractablewhenthenumberofpreyisnotsmall).Inthiscase,movementsaretakenfromthedistribution(`,jH=h)=(`np,np),wheredenotesthedeltafunctionand`npandnparethedistanceandanglebetweenthepredator'scurrentpositionandthelocationofthenearestprey.Ineachmovementphase,thepredatormovesalongalineartrajectoryfromitscurrentpositiontothepositionofthenearestprey(Fig. 4-1 ).Inthiscase,theformoftheintrinsicmovementdistributionis 53

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unimportant,solongasitsatisescertaintechnicalmathematicalrequirementssuchasbeingcontinuousandhavingnon-zeromassat(`np,np).Whenthepredatormovesdirectlyfromonepreytothenext,itwillencounterpreyatameanratethatisinverselyproportionaltothemeandistancebetweenprey,whichwedenote d.AssumingpreyaredistributedaccordingtoaPoissonspatialprocess,)]TJ /F2 11.955 Tf 11.04 0 Td[(v= d,orequivalently\()2vp .Formally,thiscalculationrequiresthatpreyarereplenishedandredistributedaftereachencounterandthatthereisnonetdecreaseinpreydensity.Italsoassumesthattheencounterradiusiszero.Torelaxthelatterassumption,notethatapredatormustmoveanaveragedistanceof0.5)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2(1)]TJ /F8 11.955 Tf -417.49 -23.91 Td[(erf(rep ))sothatitsnearestpreyiswithinitsencounterradiusre(seeAppendix C ).Itfollowsthattheencounterrateis\()=v[0.5)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2(1)]TJ /F8 11.955 Tf 12.42 0 Td[(erf(rep ))])]TJ /F4 7.97 Tf 6.58 0 Td[(1(Fig. 4-1 ,insetpanel,bluecurveandpoints).Whendensityissuchthatthemeandistancebetweentargetsissimilartore,encounterratechangeslinearlywithpreydensity(seeAppendix C ).However,asdensityapproacheszero,thisfunctionapproaches\()=2vp (Fig. 4-1 ,insetpanel,orangecurve).Sounlikeinthecaseencounterratemodelsthatassumepredatorsmoveindependentlyofprey,apredatorwithperfectsensingandresponsewillencounterpreyataratethatisproportionaltothesquarerootofpreydensitywhendensityislow. 4.1.2.3PurelyrandomsearchWenotethatitispossibletoformulateasearchbehaviorthatdoesnotrelyonsensorydatausingtheBayesianframeworkofEquation( 4 )byassumingthatthedecodingfunctionPfH=hj`,g=1forall`and.Eachtimeapredatormoves,itdrawsasteplengthandturnanglefromthedistributiondenedbyEquation( 4 ),whichisjusttheintrinsicmovementdistribution(`,)whenthedecodingfunctionisuniform.Inthisinterpretationofthepurelyrandomsearchschemepredatormovementsmaybeindependentofsensorysignalsintheenvironmentforanyofthreereasons:(1)thepredatorcannotdetectand/orneurallyencodethesignal,(2)thepredatorcandetect 54

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thesignalbutcannotextractinformationfromtheencodedsignal,or(3)thepredatorhasthesensoryandneuralmachineryforencodinganddecodingsignals,butdoesnotusetheinformationittomakemovementdecisions.Whilethelatterpossibilityseemsunlikelyandwouldbehardtoverifyexperimentally,theformertwoleadtotestablehypothesesaboutthemechanismbehinddirectedandundirectedpredatormovements. 4.1.2.4ImperfectsensingandresponseForthesignal-modulatedpredator[ 70 ],wefocusonthecaseofapredatorthatreceivesnoisyscentsignalsthatlackdirectionalinformation,ascenarioencounteredbyspecieslikesharks,lobsters,andcrabsthatusescentsignalstondpreyinturbulentenvironments[ 10 74 ].Weassumethatinagiventimeintervalt0,thepredatorwillencounteranumberofdetectablescentpatchesdrawnfromaPoissondistribution.Themeanparameterdependsonthedistancetotargetsinthevicinity.Weassumethatalltargetshavethesameintensityofsignalemissionandtherateofarrivalsatadistance`isgivenbyafunctionR(`).Asinpastapproaches,weassumeR(`)isgivenbythesteadystatesolutiontothediffusionequationdescribingthediffusionanddissipationofscentwithoutadvection(seeAppendix C ,[ 11 70 ]).Inthiscase,thedecodingfunctionisgivenbythelikelihood PfH=hj`,g=e)]TJ /F7 7.97 Tf 6.59 0 Td[(R(to`)R(to`)h h!,(4)wherehrepresentsthenumberofdetectablescentarrivalsinsomexedamountoftimeto.Thismodelofolfactorysearchbehaviorhastwosalientfeatures.Therstisthat,becausethereisnodirectionalinformationinherentinthesignal,thepredatoralwaysdrawsturnanglesfromthesamedistribution(Uniformon[0,2]),regardlessofthesignalitreceives.Second,thepredatorhasnomemoryofpastmovementsorsignalencounters.Suchinformationcouldhelpthepredatorcomputeitspositionrelativetoitsprey[ 11 ]butweeliminatethispossibility.Thepurposeofthissimpliedmodelistostudy 55

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theeffectofminimalsignalinformationandaminimalamountofsignalprocessingonpredator-preyencounterrates. 4.1.3EncounterRateSimulationsWecomparethebehaviorofapredatorthatmovesaccordingtoapurelyrandomstrategytoapredatorwithimperfectsensingandresponse.Inbothcases,weassumethattheintrinsicmovementbehaviorisdescribedbyasymmetrictwo-dimensionalParetodistribution.BecauseofthesymmetrywecanseparatelydrawtheturnangleUnif(0,2)andthemovelength`(`),where(`)isthedensityofaParetorandomvariable, (`)=()]TJ /F8 11.955 Tf 11.95 0 Td[(1)`)]TJ /F4 7.97 Tf 6.59 0 Td[(1m`,(4)`misaminimummovelength,andisaparameterthatdetermineswhetherthewalkissuperdiffusive(2(1,3)).WeuseaParetodistributionwithapowerlawtailtomodelintrinsicmovementbehaviorbecauseithasbeenarguedthatsuchadistributionmayhaveevolvedasastatisticalmovementstrategyforlocatingresourceswhensensorydataarenotuseful[ 12 ].Ineachsimulation,weplacedasinglepredatorinapreyenvironmentandpopulatedtheenvironmentwithaPoissonnumberofpreywithameanof600.Thesizeoftheenvironmentwasthenscaledtoachievethedesiredpreydensity.Intherstsetofsimulations,preypositionsweregeneratedusingaPoissonpointprocess.Wethenrecordedthetimerequiredforthepredatortoencountertherstpreyandusedthistocomputeencounterrate\().Thisisconsistentwithascenarioinwhichpredatorssearchforandcaptureasinglepreyitem,andthenceasetoforageforaperiodoftime,duringwhichpreyredistributethemselvesintheenvironment.Whenpredatorsencounteranddestroymultiplepreyinsuccession,theycancreatelocalzonesofpreydepletion.Todeterminewhetherthescalingofencounterrateissensitivetosuchalocaldepletioneffect,weallowedpredatorstoencounteranddestroy32preyitems.Wethencomputed)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(k()=k= k,where kwasthemeantimerequiredtoencounter 56

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k=325.3%ofthepreypresentontheenvironment.Finally,todeterminewhetherthescalingoftheencounterratedependsonthedistributionoftargets,wegeneratedpreydistributionsaccordingtoahighlyclusteredpointprocessthatwewillcallapreferentialattachmentmodel.Briey,NpreyweregeneratedbydrawingfromaPoissondistributionmean600.Thesizeoftheenvironmentwasthenscaledtoachievethedesiredpreydensity.AfractionoftheNpreywerechosentoactasseedpointsandplaceduniformlyatrandomonthespace.Theremainingpreywereeachassignedasdaughterstooneoftheseedpointsiterativelywithprobabilityni=Pini.Positionsofdaughterswereassigneduniformlywithinacircleofradiusriaroundtheseedpoint,whereriwaschosensothatallclustershadthesamelocalpreydensity.Werepeatedsimulationstocompute\()and)]TJ /F7 7.97 Tf 6.78 -1.8 Td[(k()fork=32inthehighlyclusteredenvironmentsgeneratedbythismodel.Wesimulatedpredatorexploringenvironmentswithpreydensitiesrangingfrom0.5-100preyper106squaredpredatorbodylengths.Thisrangewasbasedonrealisticlowpreydensitiesencounteredbypredatorspeciesinnature[ 75 77 ].AllsimulationswereperformedusingMatlab. 4.1.4EstimationofScalingRegimesandExponentsAsinpreviousinvestigations(e.g.,[ 66 ]),weexpectedthat\()wouldbealinearfunctionofforthepurelyrandompredator.Ontheotherhand,asshownabove,thepredatorwithperfectsensingandresponsehasanencounterratefunctionwithseveralscalingregimesintherangeofdensitiesthatinterestus:oneinwhichencounterrateisproportionaltop ,andoneinwhichenconterrateisproportionalto.Toaccommodatethesefunctionalforms,weassumedthatlocally,encounterratecanbedescribedbyapowerfunctionoftheform\()=.Thisallowsforbothlinearandsublinearscaling.Todeterminewhethersimulatedpredatorshadmultiplescalingregimeswetted(i)asinglepowerfunction,(ii)asegmentedfunctionwithtwodistinctscalingregimes,and(iii)asegmentedfunctionwiththreedistinctscalingregimes.Priortotting,we 57

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logtransformeddensityandencounterratedatafromsearchsimulations.Weusedarecentlydevelopedstatisticalmethodforsimultaneouslyestimatingboththebreakpointsbetweendistinctscalingregimesandthescalingexponentsineachregime[ 78 ].WecomparedthetsofthesethreemodelsbycomparingAICvalues.StatisticalanalyseswereconductedusingtheSegmentedpackage[ 79 ]inR[ 41 ]. 4.2ResultsThereisadramaticdifferencebetweenmovementpatternsofpredatorsthatusesensorydataandthosethatdonot.AsisevidentfromFigure 4-2 ,signal-modulatedpredatorsconcentratescanningeffortnearprey(Fig. 4-2 A),whereaspurelyrandompredatorsscanroughlyuniformlyovertheenvironment(Fig. 4-2 B).Signal-modulatedpredatorshavethisadvantagebecausetheymoveshortdistancesbetweenscanswhentheyreceivestrongsensorysignalsandmovelongdistanceswhentheymeasureweaksignals[ 70 ].Thisbehaviorimprovessearchefciency,butperhapsmoreimportantly,itleadstoaqualitativelydifferentrelationshipbetweentheencounterrateofsignal-modulatedpredatorsandtheirprey(Fig. 4-3 A).Asexpectedfrompastworkonrandomsearch[ 66 67 ],purelyrandompredatorsencounterpreyataratethatscalesnearlylinearlywithacrossallpreydensities.Theencounterrateofsignal-modulatedpredators,ontheotherhand,isstronglynonlinearin(compareFig. 4-3 Ayellowpointstobluetriangles).Inparticular,atlowbutrealisticpreydensities(Fig. 4-3 Abluecurve),theencounterrateofsignal-modulatedpredatorschangessublinearlywithchangingpreydensity.Thisanomalousscalingmakesthesearchefciencyofsignal-modulatedpredatorsmorerobustwithrespecttochangesinpreydensity. 4.2.1EncounterRatesofPurelyRandomPredatorsareNear-linearinPreyDensityPredatorsthatusedapurelyrandomsearchstrategyencounteredpreyataratethatwasnearlyproportionaltopreydensity(Figure 4-3 A,yellowcircles;\(R)=0.0361.12;95%CIfor=[1.09,1.15]).Thisnear-linearscalingheldwhenpreywere 58

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clusteredandalsowhenpredatorsencounteredanddestroyedmultiplepreypersearch(2[1.05,1.11]).Theencounterratefunctiondidnotshowevidenceofmultiplescalingregimes(AICofmodelwithsingleregime)]TJ /F1 11.955 Tf 12.62 0 Td[(modelwithmultipleregimes)]TJ /F1 11.955 Tf 25.24 0 Td[(3.61). 4.2.2EncounterRatesofSignal-modulatedPredatorsChangeNonlinearlywithPreyDensityAcrossalldensitiesstudied,predatorsthatusesensorydatatomakemovementdecisionsencounterpreyatahigherratethanpredatorsthatdonotusesensorycues(Fig. 4-3 ).Aspreydensityincreased,theencounterrateofsignal-modulatedpredatorsincreasesnon-linearlyandclearlydisplaysmultiplescalingregimes(Fig. 4-3 ,bluetriangles;AICsingleregime-AICthreeregimes=682).Atthelowestdensities,encounterratesincreasedlinearlyorsuperlinearlywithpreydensity.Fortheparticularparametervaluesexploredhere,thereisatransitiontoasecondscalingregimeat1.7;however,theexacttransitiondependsonthelengthscaleofscentdetection(Fig C-1 ).Inthesecond,intermediateregime,whichcoverslowbutrealisticpreydensities,signal-modulatedpredatorsencounterpreyatarateproportionalto,where0<<1.Thevalueofthescalingexponent=0.56,isclosethesquare-rootscalingexhibitedatlowdensitiesbythesearcherwithperfectsensingresponse.Forhigherdensities,dataindicatedathirdregime,inwhichencounterrateincreasedsuperlinearlywithpreydensity(=1.3);however,thisupperregimeisoflessinterestbecauseitcorrespondstoenvironmentswherepreyarerelativelydenseandsearchbehaviorbecomeslessimportant.Thequalitativeformoftheencounterratefunctionofsignal-modulatedpredatorsinauniformpreyenvironmentwaspreservedwhenpreywerehighlyclustered,andwhenpredatorsencounteredanddestroyedmultiplepreyitemsinasinglesearch.Figure 4-4 showsthatthemeanencounterrateafterkencounters)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(k()exhibitednear-linearregimesatrelativelyhighandlowdensities,andsublinearregimesatintermediatedensities(2[0.44,0.54]inintermediateregime). 59

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4.2.3SensoryResponseAllowsPredatorstoEncounterNearbyTargetsmoreFrequentlyInadditiontoconcentratingscanningeffortnearprey,signal-modulatedpredatorsalsoencounternearbypreymorefrequentlythanpurelyrandompredators.Toseethis,wecomputetheempiricalprobabilityofencounteringanearbypreyasthefractionoftimesapredatormoveswithinadistanceofroofoneormorepreyandthenencountersoneormoreofthepreybeforemovingadistanceof2rofromthem(Pfhit
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Reachinganygeneralunderstandingoftheeffectofsensorydataonspeciesencounterratesischallenging.Searchingorganismscollectawidevarietyofsensorydataandthereisaagenerallackofknowledgeabouthowtheyusethesedatatomakedecisions[ 73 ].Here,wehavetakentheapproachofstudyingtwolimitingcasesofthecollectionanduseofsensorydataandoneintermediatecase.Inthelimitofperfectsensingandresponse,predatorsencounterpreyatarateproportionaltothesquarerootofpreydensityatlowpreydensity.Attheoppositeextreme,apredatorthatdoesnotusesensoryinformationencounterspreyataratethatisnearlyproportionaltopreydensity,asexpectedfrompasttreatmentsofencounterratethatassumethatpredatorsmoveindependentlyofprey[ 64 66 67 ].Theintermediatecaseturnsouttobetelling:whenweperturbinformation-freesearchbehaviorbyintroducingonlyalimitedcapacityforsensinganddecision-makingbasedonanoisy,directionlesssignal,theencounterratefunctionimmediatelydepartsfromthelinearityexpectedwhenpredatorsmovewithoutusinginformation.Clearly,mostspeciesinnatureusesearchbehaviorsthatliesomewherebetweenaperfectsensorwithperfectresponseandthememorylessrandomwalkerstudiedinoursimulations.However,bothoftheseextremesusesensorydatatoguidemovementdecisionsandbothdepartfrommass-actionkineticsinbiologicallyinterestingways.Notonlydopredatorsthatusesensoryinformationencounterpreymoreoften,butthesublinearscalingofencounterratewithpreydensityreducesthesensitivityofpredatorstochangesinpreydensity.Thisincreasedrobustnessprovidesanecologicalmechanismthroughwhichsensoryresponsemayallowpredatorstocopewithuctuationsinpreydensity.Recentempiricalstudieslendsomesupporttotheideathatsensingmayleadtosublinearencounterratefunctionsinnature.Thesestudiesreportthatencounterratesofpredatoryshandbirdsappeartochangesublinearlywithpreydensity[ 80 81 ].Wesuggestthatpredatorsensoryresponseisalikelycauseofthispattern. 61

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Ourresultsshowthatintroducingaresponsetoevenrelativelyinformation-poor,noisysensorysignalsqualitativelyalterstherelationshipbetweenpredator-preyencounterrateandpreydensity.Behaviorssuchasarea-restrictedsearchemergenaturallyfromourmodelofsearchbehavior,evenintheabsenceofsignalgradients,complexsignalprocessing,andmemoryofpastsignalandtargetencounters[ 70 ].Theframeworkweintroduceherecanbeusedtounderstandtheconnectionbetweeninformationandtheencounterratesthataresocriticaltomanycoreconceptsinecologyandbiologicalsearch. 62

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Figure4-1. A)Predatorwithperfectsensingandresponse,searchinginatwo-dimensionalenvironment.Aftercollectingsensorydata,thepredatormovesalongalineartrajectorytowardthenearestpreyandencountersthepreywhenitcomeswithinadistanceofre.B)Meanencounterratefromsimulations(re=50bodylengths,v=1bls)]TJ /F4 7.97 Tf 6.58 0 Td[(1,pointsshowmeanof100replicatesateachdensity).PreydistributionisrandomlygeneratedfromaPoissonpointprocessineachsimulation.Bluecurveshowstheoreticalmeanencounterrate(seetext),whichapproaches\()=2vp forlowpreydensity(redcurve).For>25,thetypicaldistancebetweennearestpreyislessthan2reandpredatorsbegintoencounterpreyfrequentlywithouthavingtosearch.InFigures 4-1 through 4-5 ,densityisexpressedaspreyper106squaredpredatorbodylengths. 63

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Figure4-2. Prey(redpoints)andlocationswherepredatorscansforprey(bluepoints)forA)signal-modulatedandB)purely-randompredators.Scanpointsaresemitransparentsodarkercolorindicateslocationswherepredatorhasscannedmorefrequently.Datarepresentsearchesinwhichapredatormade1000consecutivemovementswithoutdestroyingprey. Figure4-3. A)Purelyrandom(yellowcircles)andsignal-modulatedpredators(bluetriangles,k=1)searchinginuniform(Poisson)preyenvironment.Eachpointrepresentsmeanencounterratefrom1000replicatesimulations.Insimulationsshown,thefollowingparameterswereused:re=`m=50bodylengths,v=1bodylengthpersecond,ro=500bodylengths,=2.Scentemissionrateatpreylocationwassetto100(seeAppendix C ).B)RatioofencounterratesshowninA(rateofsignalmodulatedpredatordividedbyrateofpurelyrandompredator). 64

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Figure4-4. Meanencounterrateofsignal-modulatedpredatorsinuniform(Poisson)andclustered(preferentialattachment)preyenvironments.Predatorsencounteranddestroykpreyitemspersearch.Eachpointrepresentsmeanof1000replicatesimulations.ParametersasinFig. 4-3 .Encounterrateislowerinclusteredenvironmentwithk=1becauseclustersarefarfromoneanotheranditcantakepredatorsalongtimetolocateacluster.Whenk=32,encounterrateishigherbecausethepredatorcanencounternearbytargetsafteritlocatesthecluster. 65

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Figure4-5. A)Empiricalencounterprobabilityasafunctionoftargetdensity.ParametersasinFig 4-3 .Upperdiagramshowspredatorthatencounterspreybeforeexitingregionofradius2ro.Lowerdiagramshowspredatorthatexitsbeforeencounteringprey.B)Ratioofencounterprobabilityofsignal-modulatedpredatortoencounterprobabilityofpurelyrandompredator. 66

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CHAPTER5CONCLUSIONSBiologistshavelongstrivedtounderstandwhyanimalsmoveinthewaystheydo[ 1 2 ].Intheprecedingchapters,Ihavedescribednewapproachesforstudyinganimalmovementbehaviorthatincorporateeitherthephysicsoflocomotion(Chapter 2 ),ortheprocessofinformationacquisitionanduse(Chapter 3 and 4 ).Thepurposeofthesestudiesisnottosimplyaddcomplexitytopreviousmathematicalmodels.Rather,itistoexplorewhetherconstraintsimposedbythephysicalnatureofanimallocomotionandbytheavailabilityofsensoryinformationcanplayadominantroleindetermininghowanimalsmove.Inparticular,thestudiespresentedabovereveal:(1)thatanimalbodysizeappearstoconstrainthemaximumdistancetraveledduringmigratorymovementsthroughitseffectonmetabolismandthecostoflocomotion,(2)thattheuseofevenminimalamountsofsensoryinformationinmovementdecision-makingcanleadsearchinganimalstoconcentratetheirsearcheffortneartargets,and(3)thattheuseofsensoryinformationtoguidemovementbehaviorcanincreasetherobustnessofsearchperformancetochangesintheenvironment.InChapter 2 ,wedevelopedageneralmathematicalframeworktomodelthedistancesthatanimalstravelduringmigration.Themodel,andtheextensivedatasetwecollectedtotestit,revealedsomepatternsandpredictionsthatwerenewtotheeldofanimalmovement.Althoughtheoreticalstudieshadpreviouslydiscussedthepossibilitythatmigrationdistancemightbesystematicallycorrelatedwithbodysize,nogeneralempiricalrelationshipbetweenmigrationdistanceandthebodymassesofspecieshadbeenestablished.Thus,thestrikingcorrelationbetweenmigrationdistanceandthebodymassesofwalking,swimming,andyinganimalsisanewcontributioninitself.Twopredictionswereparticularlyinterestingandwell-supportedbymigrationdistancedata.First,becauseoftherelationshipbetweenbodymass,theenergeticcostoftransport,andfuelstorage,ourmodelpredictedthatthenumberofbodylengths 67

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traveledbywalkingmigrantsandthenumberofbodylengthstraveledbyswimmingmigrantsshouldeachbeindependentofbodymass.Second,unliketheenergyrequiredforwalkingandrunning,theenergyrequiredforightincreasesextremelyrapidlywithincreasingbodymass.Becauseofthis,theincreaseinmaximummigrationdistancewithincreasingbodymassbecomessmallerandsmallerforyinglargemigrants.Thispredictiontoowassupportedbymigrationdistancedata.InChapter 3 weexploredhowanimalmovementbehaviormightbeaffectedbythecollectionanduseofsensorydatafromtheenvironment.Twondingswereparticularlyinteresting.First,wefoundthatthewell-documentedbehaviorreferedtoasarea-restricted-search,inwhichanimalsconcentratetheirsearcheffortneartargets,canbeinducedbyassuminganextremelylimitedsensoryresponseonthepartofasearchinganimal.Asecondndingisthat,contrarytointuition,searchinganimalscangainalotofinformationbysamplingforsensorydataandreceivingnosignal.Thus,nosignaldoesnotmeannoinformation.Respondingtono-signaldatabymovinglongdistancesseemstoprovidesearchersawayofavoidingwastedsearcheffortinregionsthatlacktargets.InChapter 4 ,weconsideredhowsearchinganimalscouldaffecttheirencounterratesbyusingsensoryinformationfromtargetstomakemovementdecisions.Inparticular,weexploredwhethertherelationshipbetweenencounterrateandtargetdensitywasqualitativelydifferentwhenpredatorssearchedwithandwithoutusingsensoryinformationfromtargets.Theresultsofthisstudydemonstratedthatsensoryresponseandtheowofinformationtoasearchinganimalcandrasticallyalteritsrateofencounterswithtargets.Interestingly,wefoundthatusingsensoryinformationtolocatetargetschangesthewayasearchinganimalmoves,itsencounterrate,andthesensitivityofitsencounterratetochangesintargetdensity.ThequestionsIhaveattemptedtoanswerintheprecedingchaptersareimportantones.However,thenewquestionsthatcametolightthroughthestudiesdescribed 68

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abovearejustasimportant.Withrespecttophysicalconstraintsonmigrationdistance,themodelweproposedassumesthatthenumberofrefuelingstopsamigrantmakesis,onaverage,independentofitsbodymass.Atpresent,thepaucityofdatamakesitdifculttoevaluatethisassumption.However,ifthisassumptioniscorrectandtheaveragenumberofmigratorystopsdoesnotdependonbodymass,onemightwonderwhythisshouldbeso.Suchapatternwouldconstituteaninterestinglifehistoryinvariant,andwouldsurelybegforamechanisticexplanation.Ourmodelfallsshortofprovidingsuchanexplanation.Asecondquestionraisedbythemigrationmodelanddata,isthequestionofwhysomanyspeciesappeartotraveldistancesthataresimilartotheirtheoreticalmaxima.Indeed,justbecauselargerspeciescanmigratefarther,onaverage,thansmallspeciesdoesn'tmeantheymustdoso.Yet,itisclearfromourdatathatspeciesthatarelargedotendtomigratefartherthanthosethataresmall.Thisraisessomeintriguingquestionsabouttheevolutionarydriversofmigration.Mighttherebeselectionforspeciestomigrateasfarastheycan?Arethereotherevolutionaryprocessesthatcouldexplainthispattern?Thestudiesofsensinganddecision-makingalsoraiseinterestingquestionswhileleavingothersunanswered.Forinstance,inmodelingdecision-makinginresponsetosensorydata,weassumedthatasearchinganimalhasevolvedameansofinterpretingthescentsignalsitmeasurestotellitsomethingaboutwhereitstargetislocated.Butwhatifenvironmentalconditionsaresovariablethatitisnotpracticaltohavesuchareex-likeresponsetoaparticularvalueofasignal?Instead,learningmaybecomenecessary.Indeed,onehypothesisfortheoriginofcomplexneuralmechanismsforlearninganddecision-makingisthattheabilityoforganismstomoveprovidesthemwiththecapabilityofaffectingtheirinteractionswithavariableexternalenvironmentbymovingtoanewplace.Thisimmediatelycreatesalinkbetweentheabilitytoperceivetheenvironmentbycollectingsensoryinputandtheneedtolearntomakethecorrectmovementdecisionusingthatinput[ 82 ].Thusitwill 69

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beinterestingtodeterminehowourresultsareaffectedwhenthecapacityofanimalstolearnfrompastexperienceisincorporated.Animalmovementbehaviorisinherentlycomplex.Yet,itcanprovideanintricateandpowerfulmodelsystemforexploringsomeofthemostprofoundunsolvedproblemsinbiologyincludingunderstandingtheevolutionandbehaviorinthefaceofphysicalconstraints,theemergenceandandmaintenanceoflearning,andtheprocessesthatunderlieorganismaldecision-making.Solutionstotheseproblemsandmanyothersawaitfutureinvestigation. 70

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APPENDIXAMIGRATIONMODELDERIVATION,SENSITIVITY,ANDSTATISTICALANALYSES A.1GeneraldistanceequationHereweprovideadetailedderivationofthemigrationdistanceequationsforwalking,swimming,andyingmigrantspresentedinthemaintext(Equations( 2 )-( 2 )).Foreach,webeginbyexpressingmaximummigrationdistanceonasinglemigratoryleg,Yi,asafunctionoftotalpower,Ptot,speed,v,andenergydensity,c:Yi=RM0(1)]TJ /F7 7.97 Tf 6.59 0 Td[(f)M0)]TJ /F7 7.97 Tf 6.59 0 Td[(vc PtotdMwherePtot=Pmtn+Ploc,M0isinitialmassatthebeginningofthemigratoryleg,andfistheratiooffuelmasstoM0atthebeginningoftheleg.TosolveforYi,wespecifyfunctionsdescribingPmtn,Ploc,andv.ForPmtn,weassumePmtn=p0M0.75asdescribedinthemaintext.Derivationsofwalking,swimming,andyingequationsaregivenbelow.ConstantsinbiomechanicalEquations( 2 )( 2 )inthemaintexthavebeenexpandedtomoreexplicitlyshowtheirphysicalbasis. A.1.1WalkingToestimatethepowerrequiredforwalking,weuseEquation( 2 )describedinthemaintext.Empiricalevidencestronglysupportsthepredictionsofthismodel[ 31 83 ].CombiningthismodelwithEquation( A.1 )andintegratingfrominitialtonalmassgivesYi,walk=ywLclnp0v)]TJ /F13 5.978 Tf 5.75 0 Td[(1walk+gL)]TJ /F13 5.978 Tf 5.75 0 Td[(1cM0.250 p0v)]TJ /F13 5.978 Tf 5.75 0 Td[(1walk+gL)]TJ /F13 5.978 Tf 5.75 0 Td[(1cM0.250(1)]TJ /F7 7.97 Tf 6.59 0 Td[(f)0.25whereywisaconstant.Basedonourassumptionofgeometricsimilarity,Lc/M0.330,becausestridelengthistypicallyproportionaltoleglength[ 32 ].Weassumethatvwalk/M0.10amongspeciesbutthatitisxedforanindividualmigrant[ 33 ].SubstitutingthesetermsforLcandvwalkgivesanexpressionforthemass-dependenceofYi,Yi,walk=ywM0.330lnp0+c1M0.020 p0+c2M0.020(1)]TJ /F7 7.97 Tf 6.59 0 Td[(f)0.25wherec1andc2areconstants.ThelogarithmiccomponentofEquation( A.1.1 )contributeslittletotheshapeofthefunctioninthebiologicallyrelevantrangeofM0,and 71

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canbeaccuratelyapproximatedas,ln[(p0+c1M0.020)=(p0+c3M0.020)]ln[(p0+c1)=(p0+c3)]M0.01.Thus,Equation( A.1.1 )canberewrittenasapowerfunctioninM0,Yi,walk/ywM0.340lnp0+c1 p0+c3Forwalkingmammals,p0isroughlyconstantandsoYi,walk/M0.34. A.1.2SwimmingToestimatePlocforswimmingmigrants,weuseastandardresistivemodelofswimminglocomotion(Equation( 2 )inthemaintext,[ 84 ]).Thecostoflocomotionisproportionaltodragtimesspeed,solocomotorypowercanbeexpressedasPswim= Dtvwhereisdimensionlessconversionefciencyfromstoredfuelenergytomusclepoweroutput,andisadimensionlesscorrectionconstant[ 84 85 ].Weassumethatboundarylayerowaroundtheswimmingmigrantsconsideredhereisapproximatelyturbulent[ 86 ].Giventhisassumption,dragonaswimmingmigrantoftotallength,Lb,isgivenbyDt=CAbv1.8 L0.2b,whereCisconstantdeterminedbywaterdensityanddynamicviscosityandAbisacharacteristicarea(heretakentobebodycross-sectionalarea,see[ 6 84 ]fordetaileddiscussionofthismodel).WetakevtobethespeedthatminimizesPtot=v[ 84 ],andassumethatasaswimmingmigrantburnsfuel,changesinbodycross-sectionalarea,Ab,aresmallenoughtobeignored.SubstitutingexpressionsforPmtn,Pswim,andvswimintoEquation( A.1 )gives,Yi,swim/L0.2b Ab0.36p)]TJ /F4 7.97 Tf 6.59 0 Td[(0.640M0.520[1)]TJ /F8 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(f)0.28]TorecovertheinterspecicscalingequationfromEquation( A.1.2 ),wenotethatl/M0.330,Ab/M0.670,andthereforeYi,swim=ysp)]TJ /F4 7.97 Tf 6.59 0 Td[(0.640M0.300whereysisaconstant. A.1.3FlyingLocomotorypowerofananimalinsteadyhorizontalightcanbeexpressedasthesumofthreecomponents:thepowerrequiredtoremainaloft(inducedpower,Pind),the 72

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powerrequiredtoovercomedragonthebody(parasitepower,Ppar),andthepowerrequiredtoovercomedragonthewings(prolepower,Ppro)Py=Pind+Ppar+Ppro,wherePind=2!(Mg)2 L2wav)]TJ /F4 7.97 Tf 6.58 0 Td[(1,Ppar=aAbCd 2v3,Ppro=(Pind+Ppar),!isadimensionlessinducedpowerfactor,gistheaccelerationduetogravity,isdimensionlessconversionefciencyfromstoredfuelenergytomusclepoweroutput,aisthedensityofair,Lwiswingspan,Cdisadimensionlessdragcoefcient,andAbisbodycross-sectionalarea[ 7 ].ThisformulationexpressesPproasadimensionlessprolepowerfactor()timesthesumoftheinducedandparasitepower[ 7 ].Wefollow[ 7 ]inassumingthat/Aw=L2w=1=wingaspectratio,whereAw=wingplanarea[ 7 ].Thismodelisdiscussedindetailin[ 7 ].vistakentobethespeedthatminimizestheratioofinducedandparasitepowertospeed.Atthisspeed,locomotorypowerisdescribedbytheequationPy=(1+)1.05)]TJ /F4 7.97 Tf 6.59 0 Td[(1!3g6AbCdM6 2aW60.25=k0M1.5,wherek0isconstantforanindividualmigrant.BeforesubstitutingPyandvintoEquation( A.1 ),wemaketheadditionalassumptionthat,asayingmigrantburnsfuel,changesinbodyfrontalarea,Ab,aresmallenoughtobeignored[ 22 ].Underthisassumption,maximummigrationdistanceduringasinglelegisgivenbyYi,y=yflnp0+k0M0.750 p0+k0(1)]TJ /F7 7.97 Tf 6.58 0 Td[(f)0.75M0.750,whereyfisaconstant.Torecoverthebodymassscalingofmaximummigrationdistance,weassumevaluesfortheconstantsandmorphologicalvariablesthatdeterminek0.Specically,weassumeLw=1.1M0.330[ 87 ],Aw=0.16M0.670[ 87 ],=0.23[ 25 ],!=1.2[ 7 ],a=0.98[ 88 ],Ab=0.0081M0.670[ 89 ],g=9.8,andCd=0.2[ 6 ],and=1.1[ 7 ].Dataonmaximumfuelfractionsofyingmigrantspriortodepartureareavailable[ 90 99 ],andindicateameanvalueoff=0.59amongspecies,assumingamixtureof90%lipidand10%proteinisusedasfuel[ 45 ].SubstitutingthesevaluesgivesYi,y=yflnp0+k1M0.420 p0+k2M0.420,wherek1=60andk2=31. 73

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A.2Parameterestimationandmodelsensitivity A.2.1Estimationofp0Themetabolicnormalizationconstant,p0variesamongbroadtaxonomicgroups[ 27 ].Weusedpublishedestimatesofp0forwalkingmammals,swimmingsh,yinginsects,non-passerinebirds,andpasserinebirds(Table A-1 ).Forswimmingmammals,weassumethatp0isequaltothatobservedinterrestrialmammals.Forsh,theestimateofp0giveninTable A-1 isbasedonbodytemperaturesof20C.Wedidnothavedataonshbodytemperaturesduringmigrationsowedidattempttocorrectfordeviationsfromthistemperature.Flyinginsectsexhibitcorebodytemperaturesbetween33Cand45C,evenduringshortights[ 25 100 ].Weassumethatyinginsectsoperateatbodytemperaturesof40Cduringmigrationights.Wethereforecorrectedp0givenby[ 101 ]from25Cto40CfollowingtheUTDcorrectiondescribedin[ 102 ]. A.2.2SensitivityanalysisThederivationofequationsforwalking,swimming,andyinganimalsdescribedaboverequiresassumingvaluesandbodymassdependenciesofanumberofmorphologicalandbiomechanicalparameters.AnanalysisofthesensitivityofmigrationdistanceequationstotheparticularparametervaluesassumedinthederivationisgiveninTable A-2 .Inparticular,thesensitivityanalysisfocusedontwoimportantpropertiesofdistanceequations:thepredictedbodymassscalingexponent,d,andther2statisticcomputedafterttingtheequationtodata.FromTable A-2 ,itasapparentthatchangesinthescalingofmorphologicalvariablesandmaintenancemetabolism,andchangesinthevalueofp0haveonlyminoreffectsonthepredictedmassdependenceofmaximummigrationdistanceandthemodelr2.Toevaluatesensitivity,eachparametertestedwasindividuallyincreasedordecreasedby10%relativetothevalueusedintheoriginalderivationofdistanceequations.Inthecaseofsomeparameters,largerchangesinparametervalueswereexploredbasedonvaluesreportedintheliterature.r2statisticswerecomputedbytting 74

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equationstomaximummigrationdistancedataassuminghomoscedasticerrorsasdescribedintheStatisticalanalysissectionabove.Inthecaseoftheyingequation,assumingdeparturesfromgeometricsimilarityinbodyfrontalarea(Ab),wingspan(Lw),orwingplanarea(Aw)resultinchangesinthefunctionalformofEquation( 2 )withrespecttoM0.However,thesechangesinfunctionalformcauseonlyminorchangesintheshapeofthepredictedfunction,andconsequentlyresultinonlyminorchangesintheagreementbetweenthemodelanddataasindicatedbyr2values.Becauseofchangesinfunctionalform,thescalingexponent,d,isnolongertheonlyvariableaffectingthemass-scalingofYT,anditisthereforeomittedfromTable A-2 .Parametersthatonlyaffectthey0terminEquations( 2 )( 2 )(maintext)wereomittedfromthesensitivityanalysis.Additionally,increasingordecreasingthevalueoff,Cd,Ab,W,Awparametersby10%didnotchangethepredictedmassdependenceoftheequationforyinganimals,anddidnotresultindetectablechangesinr2valuesrelativetothevaluesusedintheoriginalderivationoftheightequationdescribedabove(i.e.r2=0.19forallparametercombinations). TableA-1. Empiricalvaluesofthenormalizationconstantp0. Taxonp0valuereference sh(20C)0.43[ 103 ]marinemammals3.9Assumedterrestrialmammals3.9[ 104 ]birds3.6[ 105 ](non-passerines)birds(passerines)6.3[ 105 ]yinginsects(40C)1.9[ 101 ] 75

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TableA-2. Sensitivityofdistanceequationstovariationininputparameters.TheParametervaluecolumnshowsminimumandmaximumvalueofthecorrespondingparameterusedtodeterminesensitivity.Ther2columnindicatesther2valuecomputedafterincreasingordecreasingthecorrespondingparameterandttingthenewequationtodata.Thedcolumnindicatesthevalueofthebodymassscalingexponentafterincreasingordecreasingthecorrespondingparameter.*dapproximatedasdescribedabove. TaxonParameterParametervaluer2dmin/maxmin/maxmin/max WalkingLcLc/M0.30/M0.3600.57/0.570.35/0.33*vwalkvwalk/M0.080/M0.2300.57/0.570.33/0.39*PmtnPmtn/M0.670/M0.8300.57/0.560.38/0.3*SwimmingLbLb/M0.300/M0.3600.65/0.650.30/0.31AsAs/M0.60/M0.7400.66/0.610.32/0.27PmtnPmtn/M0.670/M0.8300.66/0.560.35/0.25p00.39/0.47(sh)0.66/0.61-3/6(marinemammals)FlyingPmtnPmtn/M0.670/M0.8300.15/0.150.5/0.34AbAb/M0.60/M0.7400.16/0.2-LwLw/M0.30/M0.3600.2/0.1-AwAw/M0.60/M0.7400.15/0.21-p01.7/2.1(insects)0.19/0.18-3.5/4.2(non-passerines)5.7/6.9(passerines) TableA-3.Bodymassandmigrationdistancedata.Massismeanbodymass.Distanceismaximumreportedmigrationdistance.*massassumedbasedonsimilarityinsizetoAnaxjunius. SpeciesMovementMass(kg)Massref.Distance(Km)Distanceref.Mode AcrocephalusscirpaceusFlying0.011[ 37 ]6000[ 37 ]AgelaiusphoenicusFlying0.052[ 37 ]2500[ 37 ] 76

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AgrotisinfusaFlying0.00033[ 106 ]800[ 107 ]AgrotisipsilonFlying2.00E-04[ 108 ]1800[ 109 ]AnasacutaFlying0.94[ 110 ]5500[ 111 ]AnascreccaFlying0.35[ 37 ]5000[ 37 ]AnasdiscorsFlying0.4[ 37 ]11000[ 37 ]AnasquerquedulaFlying0.33[ 37 ]9000[ 37 ]AnaxjuniusFlying0.0012[ 99 ]2800[ 112 ]AnsercaerulescensatlanticaFlying3.5[ 37 ]5000[ 37 ]AnsercaerulescensFlying2.5[ 37 ]5000[ 37 ]AnsererythropusFlying1.9[ 113 ]4000[ 113 ]AnserindicusFlying2.2[ 110 ]1200[ 114 ]AnthusspinolettaFlying0.024[ 37 ]1500[ 37 ]AphisfabaeFlying8.80E-07[ 97 ]1300[ 115 ]ApusapusFlying0.042[ 37 ]12000[ 37 ]ArchilochuscolubrisFlying0.0033[ 37 ]6000[ 37 ]ArenariainterpresFlying0.14[ 116 ]5700[ 116 ]AythyaferinaFlying0.9[ 37 ]7500[ 37 ]AythyafuligulaFlying0.66[ 37 ]4500[ 37 ]BrantaberniclaFlying1.4[ 37 ]6500[ 37 ]BrantacanadensisFlying3.5[ 117 ]3500[ 118 ]BrantahutchinsiiFlying2[ 119 ]3500[ 118 ]BrantaleucopusFlying1.8[ 37 ]3200[ 37 ]BucephalaclangulaFlying0.92[ 37 ]3000[ 37 ]CalcariuslapponicusFlying0.035[ 37 ]6500[ 37 ]CalidriscanutusFlying0.15[ 120 ]16000[ 120 ]CalidrismauriFlying0.047[ 121 ]3200[ 121 ]CalidristenuirostrisFlying0.24[ 94 ]5400[ 122 ]CaprimulgusvociferusFlying0.053[ 37 ]6000[ 37 ]CeyxpictaFlying0.015[ 37 ]2000[ 37 ]ChaeturapelagicaFlying0.024[ 37 ]10000[ 37 ]CharadriusfalklandicusFlying0.05[ 37 ]3600[ 37 ]CharadriusvociferusFlying0.095[ 37 ]10000[ 37 ]ChlidoniasnigerFlying0.07[ 37 ]10000[ 37 ]ChordeilesminorFlying0.062[ 37 ]11000[ 37 ]ChrysococcyxlucidusFlying0.036[ 37 ]5500[ 37 ]CiconianigraFlying6[ 37 ]6500[ 37 ]ClangulahyemalisFlying0.87[ 37 ]5000[ 37 ] 77

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CoraciasgarrulusFlying0.15[ 37 ]10000[ 37 ]CrexcrexFlying0.17[ 37 ]10000[ 37 ]CuculuscanorusFlying0.11[ 37 ]12000[ 37 ]CygnuscolumbianusFlying6.8[ 110 ]5900[ 123 ]CygnuscygnusFlying9.4[ 110 ]2000[ 124 ]DanausplexippusFlying0.00057[ 98 ]3600[ 125 ]DendroicakirklandiiFlying0.014[ 37 ]1900[ 37 ]DendroicastriataFlying0.015[ 110 ]12000[ 35 ]DolichonyxoryzivorusFlying0.042[ 37 ]11000[ 37 ]FalconaumanniFlying0.7[ 37 ]8600[ 37 ]FalcoperegrinusFlying0.7[ 110 ]8600[ 126 ]FalcosparveriusFlying0.12[ 37 ]6000[ 37 ]FicedulahypoleucaFlying0.016[ 37 ]7000[ 37 ]FringillacoelebsFlying0.026[ 37 ]5000[ 37 ]FulicaatraFlying0.84[ 37 ]4000[ 37 ]GallinagogallinagoFlying0.082[ 37 ]3500[ 37 ]GrusgrusFlying9.8[ 37 ]4800[ 37 ]GrusamericanaFlying6.9[ 37 ]4000[ 37 ]GruscanadensisFlying4.4[ 37 ]4000[ 37 ]HalcyonsanctaFlying0.043[ 37 ]3900[ 37 ]HelicoverpazeaFlying0.00021[ 127 ]1600[ 128 ]HemianaxephippigerFlying0.001*3000[ 129 ]HirundorusticaFlying0.019[ 37 ]12000[ 37 ]HirundospiloderaFlying0.021[ 37 ]2500[ 37 ]HylocichlamustelinaFlying0.051[ 130 ]4600[ 130 ]JuncohyemalisFlying0.022[ 37 ]4000[ 37 ]LaniuscollurioFlying0.01[ 37 ]11000[ 37 ]LarusfuscusFlying0.8[ 37 ]6500[ 37 ]LarusridibundusFlying0.28[ 37 ]4000[ 37 ]LathamusdiscolorFlying0.062[ 37 ]2500[ 37 ]LimosalapponicaFlying0.37[ 93 ]12000[ 131 ]LuscinialusciniaFlying0.024[ 110 ]8500[ 132 ]LusciniasvecicaFlying0.02[ 37 ]6000[ 37 ]MergusalbellusFlying0.68[ 37 ]4500[ 37 ]MeropsapiasterFlying0.052[ 37 ]10000[ 37 ]MeropsnubicusFlying0.06[ 37 ]12000[ 37 ]MeropsornatusFlying0.026[ 37 ]4800[ 37 ] 78

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MolothrusaterFlying0.044[ 37 ]2000[ 37 ]MotacillaavaFlying0.022[ 37 ]8000[ 37 ]MuscavetustissimaFlying1.40E-05[ 133 ]600[ 134 ]MuscicapastriataFlying0.022[ 37 ]13000[ 37 ]MuscisaxicolamaclovianaFlying0.022[ 37 ]5000[ 37 ]MyzomelasanguinolentaFlying0.024[ 37 ]2500[ 37 ]NomophilanoctuellaFlying2.10E-05[ 135 ]2400[ 136 ]NotiochelidoncyanoleucaFlying0.012[ 37 ]8000[ 37 ]NumeniusborealisFlying0.26[ 37 ]14000[ 37 ]NumeniustennuirostrisFlying0.45[ 37 ]6000[ 37 ]NysiusvinitorFlying3.90E-06[ 137 ]300[ 138 ]OceanitesoceanicusFlying0.04[ 37 ]15000[ 37 ]OenantheoenantheFlying0.033[ 37 ]1400[ 37 ]OlorbuccinatorFlying9.8[ 37 ]2500[ 37 ]PantalaavescensFlying8.80E-05[ 139 ]3500[ 140 ]PatagonagigasFlying0.018[ 37 ]800[ 37 ]PhalaenoptilusnuttaliiFlying0.052[ 37 ]4000[ 37 ]PhilemoncitreofularisFlying0.15[ 37 ]2400[ 37 ]PhilemoncorniculatusFlying0.18[ 37 ]1600[ 37 ]PhilomachuspugnaxFlying0.065[ 37 ]15000[ 37 ]PhoebissennaeFlying0.00016[ 37 ]1500[ 37 ]PhoenicurusphoenicurusFlying0.02[ 37 ]6000[ 37 ]PhylloscopustrochilusFlying0.0087[ 110 ]15000[ 141 ]PirangaolivaceaFlying0.029[ 37 ]7000[ 37 ]PlectrophenaxnivialisFlying0.048[ 37 ]6000[ 37 ]PluvialisfulvaFlying0.12[ 37 ]13000[ 37 ]PogonocichlastellataFlying0.021[ 37 ]200[ 37 ]PrognesubisFlying0.049[ 130 ]7600[ 130 ]PseudaletiaunipunctaFlying0.00019[ 142 ]1600[ 143 ]PufnuspufnusFlying0.46[ 37 ]12000[ 37 ]PufnustenuirostrisFlying0.56[ 110 ]12000[ 35 ]PyrocephalusrubinusFlying0.014[ 37 ]4000[ 37 ]RipariaripariaFlying0.012[ 37 ]10000[ 37 ]SarkidiornismelantosFlying2[ 37 ]3900[ 37 ]SchistocercagregariaFlying0.002[ 144 ]5000[ 145 ]SelasphorusrufusFlying0.0037[ 146 ]3900[ 147 ]SelasphorussasinFlying0.0032[ 110 ]810[ 146 ] 79

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SphyrapicusvariusFlying0.05[ 37 ]3500[ 37 ]SpodopteraexiguaFlying5.40E-05[ 148 ]3700[ 149 ]StellulacalliopeFlying0.0028[ 37 ]5000[ 37 ]SternadougalliiFlying0.11[ 37 ]6000[ 37 ]SternafuscataFlying0.18[ 37 ]10000[ 37 ]SternamaximaFlying0.45[ 37 ]8000[ 37 ]SternaparadisaeaFlying0.013[ 18 ]38000[ 18 ]StreptopeliaturturFlying0.15[ 37 ]6000[ 37 ]SturnusvulgarisFlying0.082[ 37 ]1000[ 37 ]SylviaborinFlying0.018[ 110 ]7000[ 132 ]SylviacommunisFlying0.018[ 37 ]9000[ 37 ]TachycinetabicolorFlying0.02[ 37 ]5500[ 37 ]TadornaferrugineaFlying1.2[ 110 ]3800[ 114 ]TerpsiphoneviridisFlying0.015[ 37 ]1800[ 37 ]ThalasseusbergiiFlying0.3[ 37 ]1600[ 37 ]TringaglareolaFlying0.068[ 37 ]5000[ 37 ]TringastagnatalisFlying0.078[ 37 ]6500[ 37 ]TringatotanusFlying0.12[ 37 ]6500[ 37 ]TurdusilarisFlying0.098[ 37 ]5000[ 37 ]TurdusiliacusFlying0.055[ 37 ]6500[ 37 ]TurdusmigratoriusFlying0.077[ 37 ]6400[ 37 ]TyrannusforcatusFlying0.043[ 37 ]4000[ 37 ]UpupaepopsFlying0.07[ 37 ]5000[ 37 ]UraniafulgensFlying0.00042[ 150 ]1900[ 151 ]VanellusvanellusFlying0.24[ 37 ]4500[ 37 ]VireoolivaceousFlying0.019[ 37 ]10000[ 37 ]ZonotrichiaalbicollisFlying0.026[ 37 ]4500[ 37 ]ZosteropslateralisFlying0.018[ 37 ]2000[ 37 ]AlosaaestivalisSwimming0.29[ 152 ]140[ 153 ]AlosapseudoharengusSwimming0.28[ 152 ]140[ 153 ]AlosasapidissimaSwimming1[ 154 ]370[ 154 ]BalaenamysticetusSwimming69000[ 155 ]5800[ 156 ]BalaenopteramusculusSwimming99000[ 155 ]8700[ 157 ]CarcharodoncarchariasSwimming550[ 158 159 ]11000[ 158 ]CetorhinusmaximusSwimming3900[ 160 ]9500[ 161 ]ClupeaharengusSwimming0.16[ 162 ]1500[ 163 ]CololabissairaSwimming0.18[ 164 ]500[ 165 ] 80

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DelphinapterusleucasSwimming1400[ 155 ]2200[ 166 ]EschrichtiusrobustusSwimming30000[ 155 ]7300[ 167 ]EubalaenaglacialisSwimming28000[ 155 ]5700[ 168 ]HippoglossusstenolepisSwimming300[ 169 ]1200[ 170 ]IsurusoxyrinchusSwimming58[ 159 171 ]2400[ 171 ]LamnaditropisSwimming98[ 159 172 ]3000[ 172 ]LampetrauviatilisSwimming0.06[ 173 ]100[ 174 ]MegapteranovaeangliaeSwimming30000[ 155 ]8500[ 175 ]MiroungaleoninaSwimming320[ 176 ]3000[ 176 ]OdobenusrosmarusSwimming1000[ 177 ]1800[ 178 ]OncorhynchusketaSwimming3.9[ 174 ]2500[ 179 ]OncorhynchusnerkaSwimming2.5[ 180 ]1100[ 180 ]OncorhynchustshawytschaSwimming15[ 174 ]1100[ 174 ]PetromyzonmarinusSwimming0.88[ 181 ]280[ 181 ]PhysetermacropcephalusSwimming45000[ 182 ]5000[ 183 ]PleuronectesplatessaSwimming1[ 184 ]280[ 184 ]PrionaceglaucaSwimming6.7[ 159 185 ]3200[ 185 ]RhincodontypusSwimming34000[ 186 ]13000[ 187 ]ScomberscombrusSwimming0.7[ 188 ]2200[ 189 ]ThunnusorientalisSwimming200[ 190 ]7600[ 191 ]ThunnusthunnusSwimming240[ 192 ]12000[ 192 ]TursiopstruncatusSwimming140[ 193 ]1100[ 194 ]XiphiasgladiusSwimming22[ 195 196 ]2500[ 195 ]AcinonyxjubatusWalking42[ 197 ]40[ 198 ]AlcesalcesWalking480[ 117 ]200[ 199 ]AntidorcasmarsupialisWalking32[ 200 ]360[ 201 ]AntilocapraamericanaWalking55[ 202 ]260[ 203 ]CamelusbactrianusWalking690[ 155 ]200[ 204 ]CanislupusWalking37[ 205 ]500[ 206 ]CaprasibiricaWalking130[ 207 ]100[ 207 ]CapreoluscapreolusWalking27[ 208 ]84[ 209 ]CapreoluspygargusWalking40[ 210 ]500[ 210 ]CervuscanadensisWalking270[ 117 ]190[ 211 ]CervusnipponWalking53[ 155 ]100[ 212 ]ConnochaetestaurinusWalking140[ 213 ]400[ 213 ]CrocutacrocutaWalking59[ 214 ]80[ 215 ]DicrostonyxgroenlandicusWalking0.054[ 155 ]5.4[ 216 ] 81

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EquuszebraWalking240[ 217 ]100[ 218 ]KobuskobWalking79[ 155 ]350[ 219 ]LemmuslemmusWalking0.1[ 220 ]32[ 221 ]LemmussibiricusWalking0.052[ 155 ]5.4[ 216 ]LepuscalifornicusWalking3[ 205 ]35[ 222 ]LepustimidusWalking2.4[ 223 ]10[ 223 ]LoxodontaafricanaWalking3900[ 155 ]240[ 224 ]MicrotusfortisWalking0.068[ 155 ]5[ 225 ]OdocoileushemionusWalking65[ 117 ]160[ 203 ]OdocoileusvirginianusWalking76[ 226 ]52[ 227 ]OvibosmoschatusWalking480[ 228 ]320[ 229 ]OviscanadensisWalking71[ 230 ]40[ 231 ]PeromiscusleucopusWalking0.021[ 155 ]15[ 232 ]ProcapraguttorosaWalking28[ 233 ]280[ 234 ]PumaconcolorWalking50[ 235 ]50[ 236 ]RangifertarandusWalking76[ 237 ]1200[ 238 ]SaigatataricaWalking39[ 239 ]500[ 239 ]UrsusamericanusWalking100[ 205 ]140[ 229 ]VulpesfulvaWalking5.4[ 205 ]65[ 229 ] 82

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APPENDIXBDERIVATIONOFDISTRIBUTIONS,ANOTEONTHEUSEOFBAYES'RULE,ANDSUPPLEMENTARYSIMULATIONRESULTS B.1TrueDistanceDistribution(TDD)andaCommentontheUseofBayes'RuleTheTDDiscalculatedassumingthatthesearcherislocatedattheoriginandpreyaredistributedaccordingtoaPoissonspatialprocesswithintensity.Intheabsenceofanyfurtherinformation,wecancomputethedensityofthedistancetothenearesttargetL.Todothis,forasubsetAR2,denotethenumberoftargetsinAbyN(A).BydenitionP(N(A)=k)=e)]TJ /F10 7.97 Tf 6.58 0 Td[(jAj(jAj)k=k!wherejAjistheareaofA.ItfollowsthatP(L>l)=1)]TJ /F3 11.955 Tf 11.97 0 Td[(P(N(Bl)=0)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(l2,whereBldenotestheballofradiuslcenteredattheorigin.TheTDDdensitythereforesatises T(l)=)]TJ /F3 11.955 Tf 12.64 8.09 Td[(d dtP(L>l)=2le)]TJ /F10 7.97 Tf 6.59 0 Td[(l2.(B)ThisistheRayleighdistributionandisnotablebecauseitincreasesuntilitsmodeat1=p 2,afterwhichthedensitydecaysrapidlylikeaGaussian.InEquation( B )inthemaintextweintroducedamodicationofanintrinsicsteplengthdistributionthatimprovessearchperformancebyincorporatingsignaldata.ThemodicationhastheformofaBayesianposteriordistribution,butstrictlyspeaking,thisisnotanimplementationofBayes'Rule.Amoreprobabilisticallyrigorousapproachtoincorporatingsignaldatawouldbethefollowing.Afterconductinganolfactoryscan,anidealpredatorwouldusetheTDDasapriortocomputeaBayesianposteriordistributionjHforthedistancetothetarget.Lettheprior(l)=T(l),thentheposteriordistributionforthedistancetothetargetis (ljH=h)=P(H=hjl)(l) R10P(H=hjl)(l)dl(B)wherethelikelihoodfunction,P(H=hjl)),iscomputedasdescribedinthemaintext.Identifyingtheoptimalstrategyhingesonwhetheritispossibletocharacterizeanoptimalsteplengthdistributionforagiven(ljH=h).Onemighttrytoposethisas 83

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avariationalproblem.LetPdenotethespaceofallprobabilitydensitiesonR+thenasignal-modulationstrategycanbedenedintermsofafunctional:P!P.So,usingthisnotation,thetwofunctionalsstudiedinthisworkareTDDwhichissimplytheidentityfunctionalandLevywhichsatisesLevy()Lforall.Foranappropriatesetofstrategies,oneseeksanoptimalstrategy,=argmin2fE[]gwhereistherandomhittingtimeofthetargetbyasearcherusingstrategy B.2RobustnessofResultstoSearchConditions B.2.1TargetDensityInthesimulationspresentedinthemaintext,weassumetargetdensityisonepreyper106squarepredatorbodylengths.Thisdensityisarealisticlowpreydensitybasedoneldestimatesofpreydensitiesforavarietyofpredators(e.g.[ 75 240 ],[ 214 ]andreferencestherein).However,todeterminewhetherourresultsholdatevenlowerpreydensities,werepeatedsimulationsafterdecreasingpreydensitybyanorderofmagnitude(i.e.onepreyper107squarepredatorbodylengths).ResultsfromtheselowdensitysimulationsareshowninFigure B-1 .AsinFigure 3-2 Ainthemaintext,meansearchtimesofthevisual-olfactoryLevyandvisual-olfactoryTDDstrategiesdecreaserapidlyastheratiooftheolfactoryradiustothevisionradius(ro=rv)increasesaboveone.Moreover,thesetwostrategiesexhibitsimilarperformanceforlargero=rvasintheresultsshowninthemaintextforhigherpreydensity. B.2.2SignalEmissionRateSimulationspresentedinthemaintextwereconductedassumingthemeannumberofscentencountersperounitsoftimewasequalto100atadistanceofonepredatorbodylengthfromapreyitem(i.e.a=100).Werepeatedsimulationsafterreducingato10encountersperounitsoftime.Resultsareconsistentwiththosepresentedinthemaintext(Fig. B-2 ).Meansearchtimesofvisual-olfactoryLevyandTDDpredatorsdecreasewithincreasingro=rv.Searchtimesofthesetwostrategiesalsobecomemoresimilarforlargero=rv. 84

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B.2.3VariationinPredatorScanningTimesBetweensuccessivestepsvisualpredatorspauseforvunitsoftimebeforetakinganothermovementstep,whereasvisual-olfactorypredatorspauseforounitsoftime.Typicalpausedurationsbetweensuccessivemovementsofawidevarietyofanimalsintheeldrangefrom1sto60s[ 56 ].Hereweexploretherobustnessofthequalitativepatternsshowninthemaintexttochangesinthedurationofthescanningphaseforbothvisual-olfactoryandvisualpredators.Scanningtimesmayaffecttherelativeperformanceofsearchstrategiesbecausesomestrategies(e.g.visualLevy)pausemorefrequentlythanothers.Moreover,differencesbetweenvandodeterminetherelativeamountsoftimespentscanningbyvisualandvisual-olfactorypredators.Figure B-3 showsmeansearchtimeasafunctionofro=rvforarangeofvaluesofoandv.Inallpanels,meansearchtimedecreaseswithincreasingro=rvandmeansearchtimesofvisual-olfactoryLevyandTDDaresubstantiallyshorterthanmeansearchtimesofthevisualstrategiesforatleastsomerangeofro=rv.ItisworthnotingthattherelativeperformanceoftheLevystrategiesversustheTDDstrategiesdoesdependontheabsolutevalueofvando.ThisisbecauseLevystrategiestendtogointothescanningphasemoreoftenandsearchtimesofthesestrategiesthereforedependmorestronglyonscanningtimes.Notethatvisualstrategiessometimesoutperformvisual-olfactorystrategiesforsmallro=rv.Thistendstooccurwheno>vbecausevisual-olfactorypredatorsspendmoretimescanning,eventhoughtheyspendasimilaramountoftimemoving. B.3TheRoleofNo-signalEventsInthemaintext,wediscussthepotentialimportanceofno-signalevents,inwhichthesearchingpredatorsamplesforolfactorysignalsandreceiveszerosignal(i.e.h=0).Figure 3-4 inthemaintextshowshowthebehaviorofvisual-olfactorypredatorscanbealteredwhenh=0,dependingonthelengthscaleatwhicholfactorysignalsaretransmitted.Thiseffectcanbeunderstoodinmoredetailbylookingattheeffectof 85

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no-signaleventsonthelikelihoodfunctionP(H=0jl),whichisshowninFigure B-4 .Whenh=0,thelikelihoodthatthesourceisnearbyisverylow.Whenthislikelihoodismultipliedbyavisualstrategy(l),theresultisavisual-olfactorystrategythathasalowprobabilityofmakingashortstep.Figure B-4 showsthatasro=rvincreases,thisregionoflowprobabilitybecomeslarger,effectivelyincreasingtheminimumstepsizethatasignalmodulatedstrategywilltakeafterano-signalevent.TheeffectofzerosignalontheLevystrategyisparticularlystrongbecausetheprobabilityofmakingrelativelyshortstepsislarge,butthelikelihoodthatasourceisnearbygiventhath=0islow.Becauseofthisproperty,thisstrategyismuchmorestronglyinuencedbyreceivingnosignalthantheTDDstrategy.Anothercommonmodelusedinsimulationsofanimalmovement,theexponentialsteplengthdistribution,alsohasthisproperty.Tofurtherexploretheeffectofno-signalevents,weperformedthefollowingmodicationtothesimulationsdescribedinthemaintext.Webeganwithapredatorthatsamplesforolfactorysignalsduringthescanningphaseasthevisual-olfactorypredatorsdo.Ifthepredatorreceivedasignalofh>0,thenextsteplengthwasdrawnfromaParetodistributionasdescribedforthevisualLevystrategyinthemaintext.However,whenh=0,thepredatordrewasteplengthfromthedistributionresultingfromapplyingEquation[1]inthemaintext,withh=0.Inotherwords,thepredatorbehavedasavisualLevypredatorwhenh>0butasavisual-olfactoryLevypredatorwhenh=0.Thisisaconvenientwaytodeterminewhetherusingno-signaleventstoexcludelocalregionsofspaceissufcienttoimprovesearchperformance,orwhetheritisalsonecessarytouseeventswhereh>0.Resultsofthissimulationshowthatalteringbehaviorinresponsetono-signaleventsaloneissufcienttoimprovesearchperformanceatlowtargetdensity(Figure B-5 ).Forexample,whenro=rv20predatorsthatrespondwithvisual-olfactorybehaviorwhenh=0havemeansearchtimesthatare33%shorterthanmeansearchtimeofvisualLevypredators. 86

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FigureB-1. MeansearchtimesforvisualLevy(orangeline),visualTDD(blueline),visual-olfactoryLevy(orangecircles)andvisual-olfactoryTDD(bluecircles)predators.200replicatesimulationswereperformedforeachcombinationofstrategyro=rv.Thefollowingparametersvalueswereused:a=1,rv=lm=50a,meaninter-targetdistancewas3162a,v=1s,o=30s,anda=100unitsofscentpero. FigureB-2. Meansearchtimewithreducedrateofscentemission.SymbolsasinFig. B-1 .Thefollowingparametersvalueswereused:a=1,rv=lm=50a,meaninter-targetdistancewas1000a,v=1s,o=30s,anda=10unitsofscentpero.Eachpointrepresentsmeanof200replicatesimulations. 87

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FigureB-3. Plotmatrixshowinglackofdependenceofresultsonvaluesoftheoandvparameters.SymbolsasinFig. B-1 .Panelsrepresentdifferentcombinationsofvandoparametersrangingfrom1to300.Thefollowingparametersvalueswereused:a=1,rv=lm=50a,meaninter-targetdistancewas1000a,anda=100unitsofscentpero.Eachpointrepresentsmeanof1000replicatesimulations. 88

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FigureB-4. Likelihoodfunctions(P(H=0jl))resultingfromreceivingh=0scentsignalsinaparticularscanningperiod.Whentheratioofolfactorytovisionradiusissmall(solidblackcurve:ro=rv=0.25;dashedgreencurve:ro=rv=1),encounteringzerounitsofscentreducesthelikelihoodonlyverynearthepredator.Asro=rvincreases,thelikelihoodbecomessmallformanybodylengthsfromthepredator(dotteddarkbluecurve:ro=rv=5;dot-dashedlightbluecurve:ro=rv=10).Parametersasingure B-3 witho=30andv=1. FigureB-5. Meansearchtimeofavisual-olfactoryLevysearcherthataltersvisualbehavioronlywhenh=0.ParametersasinFig. B-1 .SolidlineindicatesvisualLevypredator.Dashedlineindicatesvisual-olfactoryLevypredatorfromFig. B-1 .Eachpointrepresentsmeanof200replicatesimulations. 89

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APPENDIXCMODELOFSCENTPROPAGATIONANDDEPENDENCEOFREGIMETRANSITIONSONSIGNALPROPAGATIONLENGTH C.1ScentPropagationWemodelscentpropagationinturbulenceaspacketsthatappearatthepreypositionx0accordingtoaPoissonarrivalprocessandmoveasaBrownianmotion.Fromthepredator'sperspective,thisisequivalenttoencounteringarandomnumberofunitsofscent,HPois(toR(jx)]TJ /F15 11.955 Tf 10.74 0 Td[(xoj)),atitslocationxduringascanningphaseoflengthto,whereRistherateofscentarrival.Denoting`=jx)]TJ /F15 11.955 Tf 12.16 0 Td[(x0j,undertheseassumptions,thelikelihoodofhencountersisP(H=hj`)=[toR(`)]he)]TJ /F7 7.97 Tf 6.58 0 Td[(toR(`)=h!.ToderiveR(`),letu(x)representthemeanconcentrationofscentatpredatorpositionxemittedbyapreyitemlocatedatpositionx0.Thesteady-statediffusionprocesswithoutadvectionisdescribedby 0=Du(x))]TJ /F9 11.955 Tf 11.96 0 Td[(u(x)+(x0)(C)whereDrepresentsthecombinedmolecularandturbulentdiffusivity(m2s)]TJ /F4 7.97 Tf 6.59 0 Td[(1),representstherateofdissolutionofscentpatches(s)]TJ /F4 7.97 Tf 6.59 0 Td[(1),andrepresentstherateofscentemissionattheprey(s)]TJ /F4 7.97 Tf 6.58 0 Td[(1).Intwodimensions,therateofscentpatchencountersbyapredatoroflinearsizealocatedatxisgivenbyR(l)=2D )]TJ /F4 7.97 Tf 8 0 Td[(ln(a )u(`)where =p D.Thisimplies R(`)=2K0( `) )]TJ /F9 11.955 Tf 9.29 0 Td[( ln( a)(C)whereK0representsamodiedBesselfunctionofthesecondkind.Twotermsaresufcienttocharacterizethescentenvironment:thetypicalpropagationlengthro,whichcorrespondstothedistanceatwhichapredatorwillregisteronaverageoneunitofscentperscanningperiod,andtheexpectednumberofencountersperunittoatadistanceofonebodylengthfromtheprey. 90

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C.2DependenceofRegimeBreakonSignalPropagationLengthTodeterminewhetherthedensityatwhichlinearregimestransitionedtonon-linearregimesdependedonthelengthscaleofpredatorscentdetection,werepeatedsimulationstocompute\()overarangeofvaluesoftheolfactionradiusro.Figure C-1 showsthatthepreydensityatwhichthelinearregimetransitionstoasublinearregimedecreasesasroincreases.Thus,whenpreyscentpropagatesoveralongerdistance,thesublinearscalingofencounterratepersiststolowerpreydensity. C.3EncounterRateofaPredatorwithPerfectSensingandResponse,andNon-ZeroEncounterRadiusSupposethatapredatorislocatedattheoriginofatwo-dimensionalenvironmentcontainingpreydistributedaccordingtoaPoissonspatialprocesswithintensity.ThedistancebetweenthepredatorandthenearesttargetisgivenbytheRayleighdistribution,whichhasdensity p(`)=2`e)]TJ /F10 7.97 Tf 6.58 0 Td[(`2.(C)Wewishtocomputetheexpectedtimeittakesforapredatorwithperfectsensingandresponseandspedv,toreachtheencounterradiusreofthisnearesttarget.Thatistosay,ifRRayleigh(),weaimtocalculatemax(R)]TJ /F3 11.955 Tf 11.95 0 Td[(re,0).max(R)]TJ /F3 11.955 Tf 11.95 0 Td[(re,0)=Z1re(`)]TJ /F3 11.955 Tf 11.95 0 Td[(re)p(`)d`=1 2p (1)]TJ /F1 11.955 Tf 11.96 0 Td[(erf(rep )),whereerf(x)=2 p Rx0e)]TJ /F7 7.97 Tf 6.59 0 Td[(z2dz.Theexpectedhittingtime(andthereforetheencounterrate)istheproductofspeedandtheinverseofthisquantity,andexpandinginthesmallandlargerevealsthreedistinctscalingregimes:forsmall,theencounterrateisproportionaltothesquarerootofpreydensity;fororderonevaluesofpreydensity,thescalingislinear;andforverylargethescalingisexponential. 91

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Toobservethesquarerootscaling,simplynotethaterf(x)!0asx!0.Itfollowsthat\()2p inthisregime.Forlarger,theerrorfunctionexpandsasfollows\()=2p 1)]TJ /F1 11.955 Tf 11.95 0 Td[(erf(rep )2reer2e.Becausereandaresmallintheparameterregimeofinterest,thereisarangeof,roughlyfrom10to100,forwhichencounterratescalesroughlylinearlywith(i.e.er2e1).ThisisseeninFigure 4-1 .Asbecomeslarge,thescalingisexponential;however,forthecasesofinteresthere(i.e.relativelylowpreydensity),theexponentialregimeisnotrelevant. C.4EncounterProbabilitiesintheSparseRegimeWhenpreydensityisverysparse,eachpreytargetexistsessentiallyinisolation.Thisiswhytheempiricallyobservedprobabilityofencounterwithnearbytargetsstabilizesforlowpreydensity(seeFig 4-5 .)Whencompletelyisolated,theencounterprobabilityissimplytheprobabilityofhittingacircleofradiusrebeforeexitingaconcentriccircleofradius2rostartingfromanintermediatecircleofradiusro.Thisisanexactlysolvableproblemforcertainclassicalrandomprocesses,butwedonotyethavetheanalyticaltoolstosolvesuchaproblemforourimperfectlysensingpredators.Wecanhowever,getaroughsenseofhowtheencounterprobabilityscaleswiththefundamentalratioro=rebylookingattheformofthesolutionforastandardBrownianmotiondiffusingintheabovegeometry.Forthispurpose,weconsidertheBesselprocessR(t)thatcorrespondstotheradialdistanceofatwo-dimensionalBrownianmotionwithdiffusivityDfromtheorigin,whichsatisesfollowingItoformstochasticdifferentialequation[ 241 ]dR(t)=D Rdt+p 2DdW(t),R(0)=r. 92

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Theprobabilitythatthisprocesshitsthelevelrebefore2roisgivenbythesolutiontotheODEDp00(r)+D rp0(r)=0withp(re)=1andp(2ro)=0.Thesolutionisreadilyshowntobep(r)=ln(2ro))]TJ /F8 11.955 Tf 11.95 0 Td[(ln(2r) ln(2ro))]TJ /F8 11.955 Tf 11.95 0 Td[(ln(re)which,pluggingintheinitialconditionr(0)=royields p(ro)=ln2 ln2+ln(ro re).(C)Theapproximationissuccessfulbecauseinthepresenceofsignal,thelikelihoodfunctionintheBayesianupdate,Equation( 4 ),truncatesthepowerlawtailofthedefaultParetodistributiontobeexponentialinstead.Randomwalkswithexponentialjumptailsarediffusiveincharacter,meaningthatBrownianmotioncangiveasomewhatauthenticscalinginroandre.Furthermore,notethatthehittingprobabilityforBrownianmotionisinsensitivetoitsdiffusivity,meaningwedonothavetoattempttotunetheBrownianmotiontomatchtheimperfectlysensingsearcher.Ontheotherhand,theeffectivediffusivityoftheimperfectlysensingsearcheriscertainlystatedependentbecauselargersignalmagnitudesleadtoshorterjumplengths.AfurtherdefectoftheBrownianapproximationisthatitwillalwaysoverestimatetheencounterprobabilitybecausetheimperfectsearcherwilloccasionallyexperiencezerosignalhitswhensomewhatdistantfromthetarget.Thismeansimperfectlysensingsearcherswilloccasionallysamplefromthejumpdistributionwithheavytailandincreaseitschanceofescapebeforereachingthetarget. 93

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FigureC-1. Breakpointbetweenlowdensitylinearregimeandsublinearregimeasafunctionofthepredatorolfactionradiusro. 94

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BIOGRAPHICALSKETCH AndrewHeingrewupinAuburn,Alabama,neartheTallapoosariver.Hebecameinterestedinunderstandinglivingthingsatayoungage,underthetutelageofanamelesscreekbehindhisparents'house.Afterabriefandunsuccessfulcareerasataxidermist,heenteredgrammarschool,wherehewasanaveragestudent.Eventually,heattendedandgraduatedfromAuburnUniversityinZoology.AfterworkingasabiologistinPanama,hemovedtoGainesville,FloridatopursueaPh.D.HereceivedhisPh.D.fromtheUniversityofFloridainthesummerof2013. 112