NEW MODELS OF ANIMAL MOVEMENT
By
ANDREW M. HEIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA 2013 ⃝c 2013 Andrew M. Hein
2 To my parents, brothers, and sister
3 ACKNOWLEDGMENTS I want to begin by thanking my committee chair, Jamie Gillooly, for his guidance, encouragement, and his infectious enthusiasm for ideas. I will continue to strive to emulate his willingness to consider any scientific question without being intimidated by paradigm. I also want to thank my committee co-chair, Scott McKinley, for his constant willingness to collaborate and for his commitment to rigorous logic in science.
The afternoons spent at his chalk board have been among my most educational and enjoyable experiences as a graduate student.
The work presented in this dissertation benefitted greatly from discussions with my committee members Doug Levey, Bob Holt, and Jose Principe, and also with Mary Christman and Ben Bolker. Individual chapters were greatly improved by comments from S. P. Vogel, T. Bohrmann, A. P. Allen, and J. H. Brown, J. Casas, M. Vergassola,
I. Couzin, A. Brockmeier, E. Kriminger, and many others. I am very grateful for funding from a University of Florida Alumni Fellowship, a National Science Foundation Graduate Research Fellowship under Grant No. DGE-0802270, and the National Science
Foundation under Grant 0801544 in the Quantitative Spatial Ecology, Evolution and
Environment Program at the University of Florida.
I could not have completed this work without the encouragement and support of my family and friends. I especially want to thank my brother, Luke. I also owe special thanks to Gabriela Blohm, who spent many long hours discussing ideas with me and exhibited a saintly patience when I had a new idea or discovery that I could not help but share with someone. Finally, I want to thank my parents: my father, for encouraging my philosophical tendencies, and my mother for always reminding me of the right to pursue my curiosity.
4 TABLE OF CONTENTS page
ACKNOWLEDGMENTS ...... 4
LIST OF TABLES ...... 8
LIST OF FIGURES ...... 9 ABSTRACT ...... 10
CHAPTER
1 INTRODUCTION ...... 12
1.1 New Models of Animal Movement: Constraints of Physics, Constraints of Information ...... 13 1.2 Biomechanics, Energetics, and Animal Migration ...... 14 1.3 Sensory Information and Models of Animal Movement ...... 15 1.4 Linking Movement Behavior and Encounter Rates of Interacting Species . 16
2 ENERGETIC AND BIOMECHANICAL CONSTRAINTS ON ANIMAL MIGRATION DISTANCE ...... 17
2.1 Model Development ...... 18 2.1.1 Parameterizing Model for Walking, Swimming, and Flying Migrants 19 2.1.2 Model Predictions ...... 22 2.2 Materials and Methods ...... 22 2.3 Results ...... 24 2.4 Discussion ...... 26
3 SENSING AND DECISION-MAKING IN RANDOM SEARCH ...... 33
3.1 Model Development ...... 34 3.1.1 Searching Without Olfactory Data ...... 36 3.1.2 Incorporating Olfactory Data to Make Search Decisions ...... 37 3.1.3 Interpreting Scent Signals ...... 38 3.2 Materials and Methods ...... 39 3.2.1 Scent Propagation ...... 39 3.2.2 Simulation Details ...... 40 3.3 Results ...... 40 3.3.1 Visual-Olfactory Predators Find Targets Faster and More Reliably Than Visual Predators ...... 40 3.3.2 Visual-Olfactory Predators Learn From No-Signal Events ...... 42 3.3.3 Visual-Olfactory Predators Concentrate Search Effort Near Targets 43 3.4 Discussion ...... 43
5 4 SENSORY INFORMATION AND ENCOUNTER RATES OF INTERACTING SPECIES ...... 50
4.1 Materials and Methods ...... 51 4.1.1 Encounter Rate and Search Behavior: Some Definitions ...... 51 4.1.2 Framework for Modeling Movement Decisions ...... 52 4.1.2.1 Sensory signals and search behavior ...... 52 4.1.2.2 Perfect sensing and response ...... 53 4.1.2.3 Purely random search ...... 54 4.1.2.4 Imperfect sensing and response ...... 55 4.1.3 Encounter Rate Simulations ...... 56 4.1.4 Estimation of Scaling Regimes and Exponents ...... 57 4.2 Results ...... 58 4.2.1 Encounter Rates of Purely Random Predators are Near-linear in Prey Density ...... 58 4.2.2 Encounter Rates of Signal-modulated Predators Change Nonlinearly with Prey Density ...... 59 4.2.3 Sensory Response Allows Predators to Encounter Nearby Targets more Frequently ...... 60 4.3 Discussion ...... 60
5 CONCLUSIONS ...... 67
APPENDIX
A MIGRATION MODEL DERIVATION, SENSITIVITY, AND STATISTICAL ANALYSES ...... 71
A.1 General distance equation ...... 71 A.1.1 Walking ...... 71 A.1.2 Swimming ...... 72 A.1.3 Flying ...... 72 A.2 Parameter estimation and model sensitivity ...... 74 A.2.1 Estimation of p0 ...... 74 A.2.2 Sensitivity analysis ...... 74
B DERIVATION OF DISTRIBUTIONS, A NOTE ON THE USE OF BAYES’ RULE, AND SUPPLEMENTARY SIMULATION RESULTS ...... 83
B.1 True Distance Distribution (TDD) and a Comment on the Use of Bayes’ Rule ...... 83 B.2 Robustness of Results to Search Conditions ...... 84 B.2.1 Target Density ...... 84 B.2.2 Signal Emission Rate ...... 84 B.2.3 Variation in Predator Scanning Times ...... 85 B.3 The Role of No-signal Events ...... 85
6 C MODEL OF SCENT PROPAGATION AND DEPENDENCE OF REGIME TRANSITIONS ON SIGNAL PROPAGATION LENGTH ...... 90
C.1 Scent Propagation ...... 90 C.2 Dependence of Regime Break on Signal Propagation Length ...... 91 C.3 Encounter Rate of a Predator with Perfect Sensing and Response, and Non-Zero Encounter Radius ...... 91 C.4 Encounter Probabilities in the Sparse Regime ...... 92
REFERENCES ...... 95
BIOGRAPHICAL SKETCH ...... 112
7 LIST OF TABLES Table page
A-1 Empirical values of the normalization constant ...... 75
A-2 Sensitivity of distance equations to variation in input parameters...... 76
A-3 Body mass and migration distance data ...... 76
8 LIST OF FIGURES Figure page
2-1 Schematic of migration process ...... 30
2-2 Migration distances ...... 31
2-3 Number of body lengths traveled ...... 31 2-4 Observed and predicted migration distances ...... 32
3-1 Schematic of predator search behavior ...... 46
3-2 Mean predator search times and variability about mean search time ...... 47
3-3 Typical search paths of simulated predators ...... 48 3-4 Information gain as a function of the ratio of visual to olfactory radius ...... 49
3-5 Area-restricted-search behavior of visual and visual-olfactory predators .... 49
4-1 Perfect sensing and response ...... 63
4-2 Scan points during search ...... 64
4-3 Encounter rates of purely random and signal-modulated predators ...... 64 4-4 Encounters rate of signal-modulated predators ...... 65
4-5 Empirical encounter probability as a function of target density ...... 66
B-1 Searchs time at low density ...... 87
B-2 Search times with reduced emission rate ...... 87 B-3 Search times and scanning phase length ...... 88
B-4 Likelihood funcions ...... 89
B-5 Search time with conditional response to olfactory signals ...... 89
C-1 Breakpoint between linear and sublinear regime ...... 94
9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NEW MODELS OF ANIMAL MOVEMENT
By
Andrew M. Hein
August 2013
Chair: James F. Gillooly Cochair: Scott A. McKinley Major: Zoology
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
10 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.
11 CHAPTER 1 INTRODUCTION
The phenomenon of movement in general, and animal movement in particular, has
fascinated biologists for centuries (e.g. [1, 2]). Traditionally, animal movement has been studied either through detailed empirical work on particular species, or through highly abstracted mathematical models. Only recently, advances in fields such as sensory biology and biomechanics are beginning to facilitate the integration of organismal biology and mathematical theory of animal movement behavior.
Despite a rich history of investigation by theoreticians, many of the general mathematical models used to describe animal movement at the macro-scale rely on assumptions that are somewhat restrictive. For instance, some of the earliest models of animal movement were adopted from particle collision models in chemistry and used to predict encounter rates between predatory animals and their prey. These classical encounter rate models, developed by the pioneering theoretical biologist, Alfred Lotka and others, assume that predators and prey move randomly and independently of one another [3]. Lotka himself noted the inconsistency between this conception of
animal movement behavior, and the movement of animals in nature [3]. Of course,
generality often comes at the price of strong assumptions and the willingness of early ecologists to pay that price led to an enormous amount of development in the fields
of spatial ecology and coupled population dynamics (e.g., [4]). Still, ecologists like
Lotka and the visionary theoretician, John Skellam, imagined future work on animal
movement that would relax some of their own simplifying assumptions to allow for more
realistic depictions of organismal physiology and behavior [3, 5]. Accomplishing this goal requires an understanding of the elements of physiology and decision-making behavior
that are most relevant to animal movement. Since the development of early movement
models, researchers working in the areas of biomechanics and sensory biology have
made huge strides toward understanding the energetics of locomotion and the physics,
12 transmission, and processing of sensory signals. Developments in biomechanics theory, for example, have made it possible to write equations for the energetic costs of
locomotion as functions of speed and body size (e.g., [6, 7]). Empirical and theoretical
studies of sensory biology have gone a long way toward revealing how animals use
information to make movement decisions (e.g., [8–11]). These advances provide first-principles from which to derive new models of animal movement. In the chapters
that follow, I describe my attempt to contribute such models, and to use them to learn
about both the origins and the implications of animal movement behavior.
1.1 New Models of Animal Movement: Constraints of Physics, Constraints of Information
The way an animal moves around its environment must be determined, at least in part, by both the physical context of that movement and the background of information the animal has at its disposal. To better understand physical and informational constraints on animal movement, my collaborators and I have performed three theoretical studies to characterize these constraints in some generality. In Chapter
2, I describe our investigation of the biomechanical and energetic constraints related to one of the most well-studied movements that animals make: migration. We show how body mass–a fundamental characteristic of all animals–influences the maximum distances that migrating animals travel, through its effect on the physics of locomotion.
The data and models that we develop demonstrate that the dominant effect of body mass on migration distance emerges despite the differences among migratory species. One of the most interesting results of this study is the prediction that walking migrants of all sizes travel, on average, the same number of body lengths during migration (about
1.5 ×105 body lengths), as do swimming species of all sizes (1.7 × 106 body lengths).
Interestingly, this relationship does not hold for flying migrants, and the biomechanics of flight provide an explanation for this difference. A second problem is understanding how
to model animal search behavior in the presence of sensory signals [12]. Researchers
13 studying search and foraging movements have traditionally modeled movement using random walks. There has been much debate about what the most appropriate random walk models are. The assumption that underlies much of this work is that animals cannot get much useful information about the locations of their targets when target density is low. Thus, an animal must adopt some sort of statistical movement behavior that does not depend on the use of sensory cues [13]. In Chapter 3, we re-evaluate this assumption using a simulation model. In particular, we study the case of a searching predator that can measure only noisy olfactory cues from prey. We show that, so long as the range at which the predator gets noisy sensory data from prey is longer than the range at which it can capture prey, the predator can benefit tremendously from incorporating even minimal sensory data into movement behavior. We further show that a capacity for sensing and decision-making gives rise to commonly observed behaviors such as area-restricted search in regions that contain prey [14, 15]. A third and final question relates to how features of an animal’s environment influence its movement behavior. In Chapter 4, we study how movement behaviors of searching animals change as the density and spatial configuration of their targets change. Using simple mathematical models of sensing and decision-making along with simulations, we study the relationship between searcher-target encounter rate, and target density. The resulting relationships differ from classical mass-action models of species interactions, but are consistent with recent empirical data on prey encounter rates of predatory birds and fish. This study reveals the strong links between sensory data, movement behavior, and encounter rates of interacting species. Below I elaborate on the motivations for, and
findings of these investigations before describing them in full detail in Chapters 2-4.
1.2 Biomechanics, Energetics, and Animal Migration
Animal migration is one of the great wonders of nature, but the factors that determine how far migrants travel remain poorly understood. To address this issue, we develop a new quantitative model of animal migration and use it to describe the
14 maximum migration distance of walking, swimming and flying migrants. The model combines biomechanics and metabolic scaling to show how maximum migration distance is constrained by body size for each mode of travel. The model also indicates that the number of body lengths travelled by walking and swimming migrants should be approximately invariant of body size. Data from over 200 species of migratory birds, mammals, fish, and invertebrates support the central conclusion of the model that body size drives variation in maximum migration distance among species through its effects on metabolism and the cost of locomotion.
1.3 Sensory Information and Models of Animal Movement
Many organisms locate resources in environments in which sensory signals are rare, noisy, and lack directional information. Recent studies of search in such environments model search behavior using random walks (e.g., Levy walks) that match empirical movement distributions. We extend this modeling approach to include searcher responses to noisy sensory data. We explore the consequences of incorporating such sensory measurements into search behavior using simulations of a visual-olfactory predator in search of prey. Our results show that including even a simple response to noisy sensory data can dominate other features of random search, resulting in lower mean search times and decreased risk of long intervals between target encounters. In particular, we show that a lack of signal is not a lack of information.
Searchers that receive no signal can quickly abandon target-poor regions. On the other hand, receiving a strong signal leads a searcher to concentrate search effort near targets. These responses cause simulated searchers to concentrate search efforts near targets. This area-restricted search [15] behavior is a dominant feature of search movements of real predators such as oceanic birds [14, 16], which appear to use sensory signals to focus search efforts in productive areas and to avoid areas that lack prey. The model thus reveals that qualitatively realistic movement behavior can emerge even from very simple sensing and decision-making.
15 1.4 Linking Movement Behavior and Encounter Rates of Interacting Species
Most mobile animals search for resources, mates, and prey with the aid of sensory cues. The searching animal measures sensory data and presumably adjusts its search behavior based on those data. Yet, classical models of species encounter rates assume that searchers move independently of their targets. The assumption of independent movement leads to the familiar encounter rate kinetics used in modeling species interactions. Here, we use the example of predator-prey interactions to study how encounter rates change when predators use sensory information to find prey. We show that, even when predators pursue prey using only noisy, directionless odor signals, the resulting encounter rate equations differ qualitatively from those derived by classic theory of species interactions. Critically, predator sensory response lowers the sensitivity of encounter rate to prey density when prey density is low. This finding holds over a wide range of assumptions about predatory sensory capabilities, prey capture behavior, and the degree to which prey are clustered in the environment. Our results demonstrate how the exchange of information among interacting organisms can fundamentally alter the rates of physical interactions in biological systems.
16 CHAPTER 2 ENERGETIC AND BIOMECHANICAL CONSTRAINTS ON ANIMAL MIGRATION DISTANCE Each year, diverse species from around the planet set out on migrations ranging
from a few to thousands of kilometers in length [17–19]. Biologists have long hypothesized
that this variation in migration distance among species might be governed by differences
in basic species characteristics such as morphology and body size [1]. Although much
progress has been made in understanding how these characteristics are related to the mechanics of locomotion and to the migratory capabilities of individual species (e.g.
[20, 21]), success in understanding variation in migration distance among species has
been limited. This is because current models often require detailed information on the
morphology and behavior of migrants (e.g., [20, 22] ). This requirement has precluded a
quantitative analysis to determine the extent to which shared functional characteristics such as body size could be responsible for observed variation in migration distances
among species. As a result, the need for general theory and cross-species analyses of
migration has been strongly emphasized in recent years [23, 24].
Here, we present a model to describe constraints on animal migration distance. The model expands on past approaches [7, 25, 26] by incorporating (1) the body
mass-dependence of the cost of locomotion, (2) dynamic changes in the body masses
of migrants as they utilize stored fuel and (3) scaling of morphological characteristics
and maintenance metabolism among migrants of different body masses. In contrast
to past approaches, the model assumes that the number of re-fuelling stops made by migrants is unknown and may vary substantially among species. This facilitates
This chapter appeared as an article in the journal, Ecology Letters: Hein, A. M., C. Hou, and J. F. Gillooly. 2012. Energetic and biomechanical constraints on animal migration distance. Ecol. Lett. 15:104–110. Its reproduction here is authorized under the journal’s copyright policy.
17 prediction of statistical patterns of migration distance among species, even when the details of migratory behavior of individual species are unknown.
2.1 Model Development
We treat migration as a process in which a migrant travels a distance of YT (km) by breaking the journey into a series of N legs of length Yi , where i ∈ {1, 2, ..., N}
, Fig. 2-1A). Describing variation in migration distance among species thus requires
describing the processes that determine Yi , while accounting for among-species
variation in N. To accomplish this, we begin by making four simplifying assumptions (see Appendix A for detailed derivation and alternative assumptions). We assume (i)
that the total rate of energy use by a migrating animal, Ptot (W), is the sum of the rate
of energy use for general maintenance, Pmtn, and that required for locomotion, Ploc
(i.e. Ptot = Pmtn + Ploc = −dG/dt, where G = Joules of stored fuel energy), (ii) that migrants using a particular mode of locomotion are geometrically similar, such that linear
morphological characteristics (e.g. lengths of appendages) are proportional to M1/3 and
surface areas are proportional to M2/3 (where M is body mass (kg),[27] (iii) that migrant
metabolism provides the power required for locomotion, and (iv) that the number of
refueling stops made by individuals of each species is independent of body mass. During any given leg of a migration, the rate of change in migration distance per
unit change in body mass can be expressed as dYi /dM = (dYi /dt)(dtc/dG) =
−1 −vc/(Pmtn + Ploc ), where v is travel speed (m s ) and c is the energy density of
stored fuel (Joules kg−1). The distance traveled on a particular leg can be obtained by
integrating this expression from initial mass at the beginning of the leg, M0 (kg), to final
mass after all fuel energy has been used, M0(1 − f ), where f is the ratio of initial fuel M mass to 0, ∫ M0(1−f ) −v(M, )c Y = β dM. i P (M) + P (M, ) (2–1) M0 mtn loc β
Here, v, Pmtn, and Ploc have been rewritten to show their dependence on body mass
and on a small set of morphological traits, β (lengths and surface areas, e.g. wingspan,
18 body cross-sectional area), which determine the energetic cost of locomotion. This formulation allows for changes in speed and rate of energy use as the migrant loses
stored fuel mass.
Equation (2–1) can be used to predict how Yi varies among species by specifying
appropriate functions for v(M, β), Pmtn(M), and Ploc (M, β). We assume that Pmtn scales
3/4 with body mass as Pmtn = p0M , both within and among individuals, where p0 is a
normalization constant that varies by taxon [28, 29]. Biomechanics theory provides a
means of expressing Ploc and v as functions of M and β for migrants using a particular mode of locomotion (see below).
Generalizing to multi-leg migrations. Total distance traveled over the course of ∑N migration is given by the sum, i=1 Yi , where N is the number of migratory legs traveled by a given species (Fig. 2-1). N is unknown for the majority of migratory species.
To account for variation in N among species, we treat N as a random quantity with expected value, N . We treat Yi as fixed for a given species because we are interested in maximum migration distance. Following the law of iterated expectation, the expected distance traveled over N migratory legs is [ ] ∑N YT = E Yi = NYi , (2–2) i=1
where the operator, E, denotes the expected value [30]. Equation (2–2) shows that YT is
proportional to Yi , which is given by Equation (2–1).
2.1.1 Parameterizing Model for Walking, Swimming, and Flying Migrants
The model developed above is general and applies to migrants using any mode of
locomotion. Here, we parameterize the model for the three dominant modes of migratory
locomotion (walking, swimming, flight) by using standard models of locomotion to
describe the Ploc and v terms in Equation (2–1) (biomechanical models described in
19 detail in Appendix A). For walking migrants, Ploc can be described by
gM Pwalk = γ v, (2–3) Lc
−1 −1 where Lc is stride length (m), v is walking speed (m s ), γ is a cost coefficient (J N ),
and g is the acceleration due to gravity (m s−2,[31]) The only morphological variable in
Equation (2–3) is Lc , which is proportional to leg length [32]. We assume that walking migrants travel at speeds, v [33] and that they maintain these speeds over the course of
migration.
The power required for swimming can be described by the resistive model,
A v 2.8 P = b , swim δ 0.2 (2–4) Lb
2 where δ is a dimensionless cost coefficient, Ab is body cross-sectional area (m ),
−1 Lb is body length (m), and v is swimming speed (m s ,[6]). The set of relevant
morphological variables, β, is Ab and Lb. We assume that migrants swim at speeds that minimize the ratio, Ptot /v.
Power required for flight near minimum power speed can be described by the equation
2 −2 −1 3 P y = (1 + κ)[θM Lw v + ϕAbvf ], (2–5)
where κ is a dimensionless profile power coefficient, θ and ϕ are cost coefficients
2 (Appendix A), Ab is body cross sectional area (m ), Lw is wingspan (m), and κ is
2 proportional to Aw /Lw , where Aw is wing area [7]. The set of relevant morphological
variables, β, is therefore Ab, Lw , and Aw . We assume flying migrants travel at speeds
that minimize P y /vf [7].
Substituting Equations (2–3)-(2–5), corresponding migration speeds, and the
mass-dependence of maintenance metabolism into Equation (2–1) allows Yi to be
expressed as a function of initial mass M0, p0, and β for each mode of locomotion. In
each of the biomechanical models described above, the power required for locomotion
20 depends, in part, on a set of morphological lengths and areas, β, that do not change
as the migrant uses stored fuel to power migration. The dependence of Yi on β can
be eliminated by expressing morphological variables in terms of M0 based on the
assumption of geometric similarity (i.e. lengths, surface areas).
Substituting functions for Yi (Appendix A) into Equation (2–2) yields expressions for the expected maximum migration distances of walking
0.34 YT = y0M0 , (2–6)
swimming
−0.64 0.3 YT = y0p0 M0 , (2–7)
and flying [ ] p + k M0.42 Y = y log 0 1 0 T 0 0.42 (2–8) p0 + k2M0 migrants. Here y0 is a proportionality constant that varies by mode of locomotion, and k1 and k2 are empirical constants. Differences in the functional forms of Equations
(2–6) through (2–8) are caused by differences in the way Ploc depends on mass in
walking, swimming, and flying migrants. In the case of Equation (2–8), the predicted
relationship does not follow a simple power function in M0. This is because the cost of
flight increases more rapidly with increasing body mass than does the cost of walking or
swimming. The variable, p0, does not appear in the final form of the equation for walking
migrants because here we only consider the distance traveled by walking mammals,
for which p0 is roughly constant [34]. The exponents of the mass terms in Equations (2–6) through (2–8) describe how maximum migration distance changes as a function of M0 and reflect the mass-dependence of maintenance and locomotory metabolism.
The constant, y0, describes effects of mass-independent factors, such as the number of migratory legs, that affect the absolute distances traveled by migrants but do not affect the scaling of migration distance with body mass. The metabolic normalization constant, p0, and the morphological constants k1 and k2 can be estimated from empirical
21 measurements (see Materials and Methods). The framework described here uses body mass (Fig. 2-1B box a), morphology (Fig. 2-1B box b) and mode of locomotion (Fig.
2-1B box c) to determine migratory speed, and the metabolic costs of locomotory and
maintenance metabolism (Fig. 2-1B box d). Equation (2–1) ensures that changes in
speed and metabolism as the migrant uses stored fuel (Fig. 2-1B box e) are explicitly
incorporated into the prediction of Yi (Fig. 2-1B box f).
2.1.2 Model Predictions
Equations (2–6) through (2–8) make several quantitative predictions that can be
tested against data. First, each equation predicts that, after normalizing for p0, a single
curve can be used to describe expected maximum migration distance (in km) as a
function of M0 for species using each mode of locomotion. Second, each equation predicts how the number of body lengths traveled–a measure of relative distance [35]–varies with body mass. Migration distance and body length scale similarly with
1/3 mass in walking and swimming animals (i.e. YT roughly proportional to M0 , body
1/3 length ∝ M0 ) such that the number of body lengths traveled during migration, Ybl , is
1/3 1/3 0 described by Ybl = YT /(body length) ∝ M0 /M0 ∝ M0 . Thus, after normalizing for
differences in p0, the number of body lengths traveled by walking and swimming animals
should be approximately invariant with respect to M0. In flying animals, however, dividing
1/3 Equation (2–8) by M0 indicates that Ybl should decrease with increasing mass for all but the smallest flying migrants.
2.2 Materials and Methods
To evaluate the model, published measurements of maximum migration distances of
terrestrial mammals, fish, marine mammals, and flying insects and birds were collected.
Data from studies that met five criteria were included in the analysis: (1) reported movements could be considered to-and-fro migration or one-way migration [36], (2)
individuals were directly tracked by mark-recapture, telemetry or other means, groups
of individuals were tracked by repeated observation over the course of migration, or a
22 reliable estimate of distance traveled could otherwise be established, (3) maximum travel distances, maps, tracks or other information that allowed direct calculation of minimum estimates of the distances traveled by individual animals were reported, (4) there did not exist strong but indirect evidence from other studies (e.g. sightings of unmarked individuals, stable isotope data) suggesting that the maximum reported migration distance was substantially shorter than true maximum migration distance, and (5) in the case of flying species, studies reported migration distances of species that rely, at least partially, on flapping flight. The fifth criterion was imposed because the biomechanical model of flight used to derive our predictions applies most directly to flapping flight.
Migration distance and body mass data were included from a large dataset [37] for which all of the selection criteria could not be verified for all species. Including these data did not qualitatively affect our conclusions (see Results).
We estimated the constants k1 and k2 in Equation (2–8) using empirical studies of the morphology of flying insects and birds; however, the general form of Equation (2–8) and the resulting predictions are not strongly affected by variation in the empirical values used to estimate k1 and k2 (Appendix A). Empirical estimates of p0 were used in
Equations (2–7) throught (2–8) (Appendix A). Body mass data were used to estimate body lengths based on allometric equations (swimming mammals: [38]; others: [27]).
Body lengths were used to convert migration distance (km) into units of body lengths. To evaluate our first prediction, we fitted Equations (2–6) through (2–8) to migration distance data from walking (n = 33), swimming (n = 32), and flying migrants (n = 141), respectively. Equations (2–6) and (2–7) were fitted to log10-transformed distance and body mass data using ordinary least squares. Equation (2–8) was fitted to log10-transformed distance and body mass data using non-linear least squares (Gauss-Newton algorithm). Equations (2–6) through (2–8) have the general form:
d YT = y0h(M0 , p0), where h is a known function, y0 is a constant, and d is a scaling exponent. For each equation, two models were fitted: a model in which y0 was fitted
23 as a free parameter but d was set to the predicted value (i.e. d = 0.34, 0.3, 0.42, for
walking, swimming, and flying migrants, respectively), and a model in which both y0
and d were fitted. Model r 2 values reported below are based on the former method.
The latter method was used to generate 95% profile confidence intervals for the d
parameter. Prior to fitting, body mass values of swimming and flying animals were
0.3 −0.64 normalized to account for differences in p0 according to the equations Mnorm = M0 p0
0.42 −1 and Mnorm = M0 p0 , respectively. To test our second prediction–that the number of body lengths traveled was invariant of mass in walking and swimming migrants, but
decreased with mass in flying migrants–we fitted log10-transformed migration distance
(in body lengths) as a function of log10-transformed body mass (kg) using a quadratic
2 regression of the form, log10(Ybl ) = γ0 + γ1 log10(M0) + γ2 log10(M0) , where γi are regression coefficients [39]. Species were separated based on mode of locomotion
and by taxonomic groups differing in p0 (i.e. walking mammals, fish, marine mammals,
flying insects, and passerine and non-passerine birds were fitted separately). Statistical analyses were implemented using the nlme package [40] in R [41].
2.3 Results
Model predictions were evaluated using extensive data on maximum migration distances of animals from around the world (n = 206 species, Appendix A). Consistent
with our first prediction, maximum migration distance (km) varies systematically with
body mass for walking, swimming, and flying migrants (Fig. 2-2; r2 = 0.57, 0.65,
0.19, for walking, swimming, and flying species, respectively). The solid lines show predicted migration distance based on Equations (2–6) throught (2–8). There is a
tight correspondence between predicted relationships (solid lines) and fitted models
that treat both y0 and scaling exponents as free parameters (dashed lines and 95% confidence bands). In the case of walking and swimming animals, the data support model predictions of linear relationships in log-log space, with observed scaling exponents close to those predicted by Equations (2–6) and (2–7) (walking: predicted
24 = 0.34, observed = 0.36 95%CI [0.25,0.48]; swimming: predicted = 0.3, observed = 0.34 [0.28,0.41]). In the case of flying animals, data support the prediction that the
relationship is non-linear in log-log space reflecting the rapidly rising cost of flight
with increasing mass (Fig. 2-2C). Again, the observed mass exponent is close to
that predicted by Equation (2–8) (predicted = 0.42, observed = 0.43 [0.36,0.49]). Consistent with our second prediction, the number of body lengths traveled by swimming
and walking animals is independent of body mass (Fig. 2-3). On average, walking
mammals travel 1.5 × 105 body lengths (Fig. 2-3A). The slope and curvature terms
in the quadratic regression model does not differ from zero in walking mammals (n
= 33, p > 0.22) indicating that the number of body lengths traveled is uncorrelated with body mass in this group. Swimming animals travel an average of 1.7 × 106 body
lengths in a one-way migratory journey. The mean distance traveled by fish (triangles
in Fig. 3B) exceeds that traveled by swimming mammals (squares in Fig. 2-3B) by a
factor of 4 (fish: 2.1 × 106 body lengths; marine mammals: 5.3 × 105 body lengths, see Discussion), but the number of body lengths traveled is independent of mass in each
of these groups (slope and curvature does not differ from zero, fish: n = 20, p > 0.38;
swimming mammals: n = 12, p > 0.43). In flying migrants, the number of body lengths
migrated declines clearly with increasing body mass (Fig. 2-3C). In non-passerine birds
(n = 80), coefficients of linear and quadratic terms were both negative, and significantly
−5 different from zero (γ1 = -0.59, γ2 = -0.19, p < 2.2 × 10 ). In passerine birds (n =
45) and flying insects (n = 16) the γ1 term was negative and distinguishable from zero
−5 (passerines: γ1 = -0.63, p = 5.4 × 10 ; insects: γ1 = -0.16, p = 0.034). Results for flying migrants confirm our prediction that larger flying migrants generally travel fewer body lengths over the course of migration. The number of body lengths traveled decreases with increasing mass such that the smallest insects and birds travel around 1.4 × 108 body lengths whereas the largest birds travel around 5.2 × 106 body lengths. In other
25 words, the number of body lengths covered by moths, dragonflies, and hummingbirds is roughly 25-times that traveled by the largest ducks and geese.
A sensitivity analysis indicates that the agreement between model predictions and data are robust to deviations from geometric similarity and changes in the values of morphological and biomechanical parameters used to derive Equations (2–6)–(2–8) (Appendix A). In particular, the value of the exponent in metabolic scaling relationships has been a topic of much debate, with different authors reporting different exponents depending on the particular dataset and taxon studied and the method of analysis
(e.g. [34, 42]). However, sensitivity analysis shows that the shape of our predicted relationships, and the agreement between predictions and data are largely insensitive to changes in the value of the metabolic scaling exponent assumed (Appendix A).
Including data from [37] did not significantly change the estimate of the mass exponent
(0.36 95% CI [0.26,0.43] without data from [37], 0.43 [0.36,0.48] with data from [37]).
Including data from [37] decreased the model r2 from 0.37 to 0.19. 2.4 Discussion
When observed migration distances are plotted against predictions of Equations
(2–6) through (2–8), points from all three groups cluster around a 1:1 line (Fig. 2-4). The data shown in Figure 2-4 suggest that variation in maximum migration distances among species as distinct as Blue Whales (Balaenoptera musculus), Wildebeest
(Connochaetes taurinus), and Bar-tailed Godwits (Limosa lapponica) appears to be driven, in part, by the basic differences in metabolism, morphology, and biomechanics described by our model. The variation explained by the model reflects the influence of constraints on energetics and biomechanics imposed by body mass. There is a large body of work describing how morphology [6, 27], biomechanics [6, 21], and basic energetic properties such as maintenance metabolism [43, 44] are linked to body mass. Our model extends results of these studies by specifying how these quantities influence maximum migration distance of diverse species, thereby linking body mass
26 to migration distance. Our results show that constraints imposed by body mass are detectable in migration distance data, despite variation in migration distance among
species with similar body masses (i.e. variation about predicted relationships shown in
Figs. 2-2–2-4).
Migration distance data highlight the important role of basic differences in energetics in driving differences in migration distance among taxa. For example, the
number of body lengths traveled during migration is independent of body mass within
both swimming mammals and fish; however, fish travel an average of 4 times the
number of body lengths traveled by swimming mammals. Equation (2–7) shows that the
distances traveled by these groups depend on the metabolic normalization constant, p0, which describes mass-independent differences in the maintenance metabolic rates of
fish and marine mammals. In these groups, p0 differs by a factor of roughly 9.1 (p0 ≈
−3/4 −3/4 3.9 W kg in marine mammals, p0 ≈ 0.43 W kg in fish, see Appendix A), whereas
body length exhibits a similar relationship with mass in both groups (l ≈ 0.44M1/3) suggesting that the number of body lengths migrated by fish is greater by a factor of
(9.1)0.64 = 4.1, which is very close to the observed factor of 4. Thus, the difference
in the mean number of body lengths traveled by these groups may be driven by basic
differences in the cost of maintenance metabolism. Data also reveal patterns that do
not appear to be caused by the energetic and biomechanical factors considered here. For example, swimming is significantly less costly than flight in terms of the energy
required to travel a given distance [45], yet virtually all flying organisms travel distances
that are as great or greater than those traveled by most swimming species (Fig 2-4).
Whether this pattern is driven by differences in migratory behavior or other ecological or
evolutionary factors remains unknown and will likely be a fruitful area of future research. It is worth noting that other hypotheses may provide alternative explanations for
some of the qualitative patterns observed in migration distance data. For example, the
model predicts that migration distance (km) of larger flying species does not depend
27 strongly on mass. An increase in mass from 10−6 kg to 10−3 kg, increases expected migration distance by a factor of more than 8, whereas an increase in mass from 10−2
kg to 10 kg increases expected migration distance by a factor of less than 2. This
occurs because the energetic cost of flight increases rapidly with increasing mass
to the degree that the increasing fuel mass that can be carried by larger migrants provides a diminishing increase in migration distance. An alternative explanation for
this observation is that many subtropical and temperate habitats in the northern and
southern hemispheres are separated by 5 × 103 km –1 × 104 km and that many flying
migrants may not be under selection to migrate greater distances. In general, the
relationship between the distances traveled by migrants and the global distribution of suitable migratory habitats is poorly known but may ultimately influence the distances
traveled by many species.
While model predictions are supported by data, there is substantial unexplained
variation in Figures 2-2–2-4. Investigating why particular species deviate from predictions may be an effective way to identify ecological and evolutionary factors
that drive differences in migration distance but are not currently included in our model.
Our model ignores variation in fuel and morphology of species with similar masses and
does not consider the possibility that some migrants may seek to minimize the time
spent migrating. Two additional factors, in particular, are likely to contribute to observed residual variation. First, differences in the number migratory legs among otherwise
similar species will lead to variation in migration distance among species as indicated
by Equation (2–2). Second, species that interact strongly with abiotic currents during migration are likely to deviate from model predictions. The lack of information regarding the type and number of refueling stops made by migratory species, and the lack of information about the manner in which many flying and swimming migrants interact with abiotic currents represents an important gap in current knowledge. In the case of some well-studied species such as the arctic tern (Sterna paradisaea), it is clear that these
28 variables are important in facilitating extremely long-distance migrations. Individuals of this species stop at multiple highly productive foraging sites to refuel during migration
[18]. This species is also known to track global wind systems thereby taking advantage
of favorable air currents. In the case of species that migrate against abiotic currents,
migration distances might be expected to be shorter than our model predicts. Indeed, many of the swimming migrants that fall below the predicted line in Figure 2-2, are
anadromous fish such as shad (Alosa sapidissima), alewife (Alosa pseudoharengus), and river lamprey (Lampetra fluviatilis) that swim against water currents during upriver migrations. Increased understanding of the interactions between migrants and abiotic currents and the number of migratory stopovers will allow for extensions of the model that could further improve our understanding of the reasons for inter-specific differences in migration distance. In its current form, the model presented here provides a general expectation on maximum migration distance, which can be seen as a metric against which the distances traveled by particular species can be compared. The body sizes of migratory animals vary by over 11 orders of magnitude. The model presented here makes specific quantitative predictions about how this variation in size drives patterns of migration distance among species. It attributes differences in the distances traveled by migrants to systematic differences in metabolism and morphological traits that are tightly coupled to body size, and to differences in the underlying mechanics of walking, swimming, and flight. In doing so, it provides an analytically tractable framework for studying the influence of energetics and biomechanics on migration distance that is consistent with data on species ranging from the smallest migratory insects to the largest whales.
29 N A Yt = Yi i =1
Y1 Y2 Y3 YN ...
B a b Body Mass ( M0 ) Morphology ( )
d e P and P Mass loss as fuel mtn loc is used
c Mode of f Y locomotion i
Figure 2-1. (A) Total migration distance is the sum of the distances traveled on each of N migratory legs. (B) Migration distance on a single migratory leg. Body mass (a), morphology (b) and mode of locomotion (c) govern the rate at which a migrant uses stored fuel energy (d). This rate changes as migrant loses fuel mass (e), and determines the maximum distance covered during a single leg (f, Equation (2–1)). The relationship between a and b is governed by the mass-dependence of morphology. Total rate of energy use (d) is determined by the mass-dependence of maintenance metabolism and by the biomechanics of locomotion (Equations (2–3)-(2–3)).
30 iue2-3. Figure 2-2. Figure
Body lengths traveled Migration Distance (km) 10 10 10 10 nyadaeteeoentcretdfor corrected not therefore are and only in differences aesBadC oyms snraie codn oteequations the to according normalized is is mass mass M body body C, A, and panel B In panels parameters. with free (C) as regression fitted nonlinear intervals exponent or confidence B) 95% (A, and linear curves from fit best represent bands confidence ie eoema ubro oylntstaee yseisuigeach using species by traveled locomotion. lengths of body mode of number (diamonds). mean birds insects denote non-passerine flying Lines and C) (squares), and birds (squares), passerine mammals, mammals walking (triangles), and A) (triangles) by fish migration swimming during B) traveled lengths body of Number Equations of fits on based curves predicted flying are C) ( lines and Solid mammals A) insects. marine for and and mass birds fish body swimming normalized B) of mammals, function walking a as distance migration Maximum 10 10 10 10 10 10 2–6 0 1 2 3 4 5 3 5 7 9 norm 01 010 10 10 10 ● 2024 2 0 −2 )–( ● 10 ● ●● ● ● ● ● ● ● ● ● = ● −1 2–8 M 10 0 0.3 ● ● 0 odt with data to ) ● ● ● ● ● ● 10 p ● p ● ● − ● ● ● ● ● ● ● 1 ● ● ● ● ● ●●● 0 ● ● ● ● ● ● 0.64 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● mn rus aao akn nml r rmmammals from are animals walking on Data groups. among 10 ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● and ● ● A ● 10 A ● ● 3 M y 01 10 10 10 norm 0 10 te safe aaee.Dse ie and lines Dashed parameter. free a as fitted 202 0 −2 0 Normalised body mass body Normalised = Body mass Body (kg) 31 10 M 0.5 0 0.42 p 10 0 − 10 1 p 1 B 0 epciey ocretfor correct to respectively, , 4 . 10 10 1.5 B 6 10 10 −7 y −2.5 0 10 n h asscaling mass the and 10 −5 −2 10 10 −1.5 −3 10 M 10 −1 0 −1 10 k) In (kg). 10 −0.5 C C 5 10 4 10
3 ● 10 ●●● ●●● ● ●● ● ● ●●●
2 ● ●●●●● 10 ● ●● ● ● ●● Observeddistance (km)
1 ● ● 10 ●●● 0 10
10 0 10 1 10 2 10 3 10 4 10 5
Predicted distance (km)
Figure 2-4. Observed and predicted migration distances for the walking, swimming, and flying animals shown in Figure 2-2. Data from walking mammals (green circles), swimming fish (blue triangles) and marine mammals (blue squares), and flying insects (red triangles), passerine birds (red squares), and non-passerine birds (red diamonds) are shown. Black points and illustrations show the well-studied migrants Connochaetes taurinus (Wildebeest), Balaenoptera musculus (Blue Whale), and Limosa lapponica (Bar-tailed Godwit). Solid line indicates 1:1 line.
32 CHAPTER 3 SENSING AND DECISION-MAKING IN RANDOM SEARCH
Organisms routinely locate targets in complex environments. They can do this
by following gradients in the strength of sensory signals, provided such gradients are available and reliably lead toward targets [46]. But this is not always the case. In many
natural settings sensory signals are infrequent, noisy, and contain little directional
information [11]. For example, moths, sharks, and sea birds search environments that contain scent cues emitted by prey or mates, but these cues are often extremely sparse and subject to large fluctuations [9, 10, 47]. Under such sparse-signal conditions, it is not clear what behaviors allow organisms to efficiently and reliably locate resources.
Researchers have developed much of the theory of sparse-signal search by
studying mathematical models of searching organisms [12, 13, 48–51]. The dominant
paradigm for developing such models emerged from the random foraging hypothesis–the idea that searchers can encounter targets efficiently by adopting statistical movement
strategies that can be described as random walks ([12, 48], see [9, 11, 52] for alternative
approaches). This hypothesis, which has been applied to searching organisms ranging
from bees [12] to sea turtles [53], is often invoked when it is not possible or practical for
searchers to remember explicit spatial locations [48] and the typical distances between targets exceeds the searcher’s sensory range [54]. This framework has been used to
compare the performance of searchers moving according to different kinds of random
walk behavior. In particular, many studies have tried to determine whether searchers
moving according to Levy´ walks outperform searchers that move according to other types of random walk strategies (e.g. [13, 49–51]).
This chapter appeared as an article in the journal, Proceedings of the National Academy of Science: Hein, A. M. and S. A. McKinley. 2012. Sensing and decision-making in random search. Proc. Natl. Acad. Sci. USA. 109:12070–12074. Its reproduction here is authorized under the journal’s copyright policy.
33 If models are to yield insight into the behavior of searching organisms in nature, they must be simple enough to be studied, but should also capture the dominant features of search behavior. Implicit in the random foraging approach is the assumption that changes in a searchers’ movement behavior in response to sensory data are second-order effects, and that search behavior and performance are dominated by the features of the intrinsic (random) search strategy that the searcher employs. Here we explore an alternative hypothesis: that sensory processes can have a dominant effect on search performance, even when sensory signals are rare, noisy, and lack directional information.
Below we develop a general mathematical framework for modeling search decision-making. As in past models, the framework allows a searching organism to make movement decisions based on an intrinsic movement strategy (e.g. Levy´ walk), but allows such decisions to be modified based on noisy sensory data. It thus provides an explicit way to model changes in behavior in response to sensory measurements. We explore the effect of incorporating sensory data into search decisions using individual-based simulations of searching predators. We compare search times of simulated predators that make search decisions using random strategies alone (Levy´ walk and a novel diffusive strategy), to predators that modify their search behavior based on olfactory measurements. 3.1 Model Development
To study search decision-making, we consider an idealized model of a predator in search of prey. We wish to compare the behavior and performance of predators that search using a single intrinsic random strategy to predators that adaptively change their search behavior using the incomplete information gained from sensory measurements.
To evoke a strong intuition we consider two types of predator: a visual predator that makes movement decisions based on an intrinsic strategy and locates prey through a short-range, high acuity sense (vision), and a visual-olfactory predator that changes its
34 search behavior based on noisy olfactory data and detects prey at short range using vision. Predators wander through a large (periodic) two-dimensional habitat in which
the mean distance between prey is large. We assume prey emit a scent that can be
detected by nearby predators. Similar to previous approaches (e.g. [55] ), we assume
that search is divided into two phases: a local scanning phase and a movement phase (Figure 3-1A, [56]).
During the scanning phase, the predator locates any prey within its vision distance rv (Fig. 3-1A, solid inner circle) with probability one. This reflects the high local acuity
of vision. Visual-olfactory predators also scan for olfactory signals. The duration of
the scanning phase is denoted τv and τo for visual and visual-olfactory predators respectively. τv includes the time needed to visually search a region of radius rv and reorient before taking another step. τo includes the time taken to collect and process olfactory signals, visually search a region of radius rv , and reorient before taking another step. We define the olfactory radius ro (Fig. 3-1A, dashed outer circle) as the distance
where the predator registers an average of one scent signal per scanning period τo (see
below). We assume that each prey item emits scent at rate λ. During the movement
phase, the predator travels in a random uniform direction, a distance of l, at speed v.
Visual predators draw the step length l from a prescribed step length distribution θ(l),
examples of which are described in the next subsection. Visual-olfactory predators draw from a modified step length distribution defined below by Equation (3–1).
During the movement phase, we assume that the predator cannot locate prey or detect scent signals. Additionally, we assume that the predator only responds to the most recent scent signal information and does not store information about the locations it has visited. We study this limiting case where sensory signals are rare, lack directional information, and are not remembered by the predator because this is the scenario in which random search strategies are often invoked. We thus evaluate the scenario in which noisy sensory data are least likely to yield improvement over purely
35 random search. However, we point out that more sophisticated strategies are possible if predators remember past signal encounters or previously visited locations [11, 46, 57].
3.1.1 Searching Without Olfactory Data
To model predator movements, we begin with a model of decision-making in the
absence of any interaction with olfactory data. Researchers typically model the decision
process of random searchers by selecting two actions from prescribed probability
distributions: a step length l, and a turn angle ϕ. The details of these distributions
determine asymptotic properties of the search and strategies are often categorized by this asymptotic behavior: diffusive behavior, in which long-term mean-squared
displacement (MSD) scales linearly with time, and superdiffusive behavior in which MSD
increases superlinearly with time. An important feature of these strategies is that, unless
the searcher encounters a target, the distributions that define how searcher moves (i.e. the distributions of l and ϕ) are fixed. They are not altered in response to sensory
measurements.
We model the movements of visual predators using two types of strategies: a Levy´
strategy and a novel diffusive strategy. For both, we take the distribution of turn angles
between successive steps to be iid ϕ ∼ unif(0, 2π) [12]. The Levy´ strategy draws step
α−1 −α lengths from a Pareto distribution, θL(l) = (α − 1)lm l , with tail with parameter α and minimum step length lm (Fig. 3-1B solid curve, superdiffusive for 1 < α < 3 [12]). For
the second strategy, we introduce a new step-length distribution which we call the true
distance distribution (TDD) θT (l): a greedy strategy wherein the predator selects step lengths from the probability distribution of the distance to the nearest prey item (Fig.
3-1B dashed curve, see Supplementary Information (SI) Text for further discussion).
When prey are distributed according to a Poisson spatial process with intensity η in
−ηπl2 two dimensions, the TDD is given by the Rayleigh distribution θT (l) = 2ηπle . This strategy is quite distinct from the Levy´ strategy (compare curves in Fig. 3-1B) and later serves to illustrate the strong homogenizing effect of olfactory data on search behavior.
36 3.1.2 Incorporating Olfactory Data to Make Search Decisions
The key distinction between visual and olfactory senses in our model is that the
visual sense yields perfect information about the location of prey whereas the olfactory sense does not. Thus, including olfactory measurements allows us to model a predator’s
ability to gather and respond to partial information about target positions gleaned from
sensory measurements. Below we develop a model for incorporating olfactory signals
into search decision-making, but note that this framework could be modified to model
responses to other types of sensory cues. We hypothesize that predators utilize olfactory data through two steps. First,
a predator uses a signal observation to estimate the likely distance to the nearest
prey. Second, the predator modifies its intrinsic tendency to move in a particular way
(represented by θ(l)) based on this information. In keeping with recent models of
olfactory search, simulated predators collect olfactory data for τo units of time and
encounter H ∈ {0, 1, 2, ...} detectable units of scent [11, 57]. In order to act optimally,
a predator must make movement decisions based on two distinct uncertainties. First,
the predator’s distance to the nearest target is uncertain and is characterized by the
probability distribution ν. Second, for a particular ν, the optimal step length distribution θ is also uncertain. Identifying optimal predator behavior requires calculating a Bayesian
posterior for the distance distribution ν|H , and then determining the associated optimal step length distribution θ|H . This remains an unsolved and perhaps intractable problem.
Instead, we approximate this process. We wish to capture two elements of search decision-making: an intrinsic tendency to move in a particular way θ(l), and a likelihood function P(H = h|l) that translates an observed scent signal h into information about the distance to the nearest prey. A natural model for signal response that incorporates these features is a Bayesian update
37 of the step length distribution θ itself:
P(H = h|l) θ(l) θ(l|H = h) = ∫ ∞ . (3–1) P(H = h|l) (l) dl 0 θ
We refer to this as “signal-modulation” of the step length distribution θ(l). This approximation to the optimal strategy yields significant improvement in search performance (see Appendix B for further elaboration). 3.1.3 Interpreting Scent Signals
We assume the predator can estimate or intuit the probability of registering h units
of scent in τo units of time, as a function of its distance to the nearest prey. This amounts to being able to estimate the likelihood function P(H = h|l), which depends on the
process of scent propagation.
In the complex environments where many species search, turbulent fluctuations
in fluid velocity cause large local fluctuations in scent concentration [58]. When a prevailing wind or water current is present, predators can gain additional information
about the location of a scent source by measuring the velocity of the current [11, 47].
We consider the more difficult scenario in which there is no prevailing current. Under
these conditions, we model scent arrival as packets that appear at the prey position
x0 according to a Poisson arrival process and then move as a Brownian motion. From the predator’s perspective, this is equivalent to encountering a random number of units
of scent, H ∼ Pois(τo R(|x − xo|)), at its location x during a scanning phase of length
τo , where R is the rate of scent arrival defined by Equation (C–2) (see Materials and
Methods). Denoting l = |x − x0|, under these assumptions, the likelihood of h encounters
is
h −τo R(l) P(H = h|l) = [τo R(l)] e /h! (3–2)
Equation (3–2) depends on values of several physical parameters (e.g. the rate
at which detectable patches of scent decay) that may be difficult for a predator to infer
38 from measurements of its physical environment. We therefore take a qualitative view in prescribing the parameters of scent propagation. The most important qualitative feature
is the length scale ro , which corresponds to the distance at which a predator will register on average one unit of scent per scanning period τo . Heuristically, this is the distance at
which the predator is likely to detect a faint, yet non-trivial scent. A second qualitative
restriction is the expected number of encounters per unit τo at a distance of one body
length from the prey λa. Given these two measurements, the likelihood function can be
estimated.
The quantities ro and λa are much more readily measurable by a searching
organism than are the explicit parameters in Equation (C–2). It thus seems likely that these quantities may constitute part of an organism’s “olfactory search image” [59],
and may serve as the direct measurements useful for reinforcement learning.
3.2 Materials and Methods
3.2.1 Scent Propagation
To see how R(l) depends on the distance between predator and prey, let u(x) represent the mean concentration of scent at predator position x emitted by a prey
item located at position x0. An expression for the steady-state diffusion process without
advection is given by 0 = Du(x) − µu(x) + λδ(x0), where D represents the combined
molecular and turbulent diffusivity (m2s−1), µ represents the rate of dissolution of scent patches (s−1), and λ represents the rate of scent emission at the prey (s−1). In two dimensions, the mean rate of scent patch encounters by a predator of linear size a √ x R(l) = 2πD u(l) = µ located at is given by − ln(aψ) where ψ D [11]. This implies
λK0(ψl) R(l) = 2 , (3–3) −πψ ln(ψa)
where K0 represents a modified Bessel function of the second kind.
39 3.2.2 Simulation Details
The SI Text shows the robustness of results to changes in model parameters. For
each of the four search strategies (visual Levy,´ visual TDD, visual-olfactory Levy,´ and visual-olfactory TDD), we performed simulations in which predators explored a periodic
environment with 100 prey. Prey were positioned according to a Poisson point process
with the mean distance between prey chosen to achieve the desired density. In each
scanning phase, h was generated as a deviate from a Poisson distribution with mean
given by the product of τo and Equation (C–2) summed over all prey. In each simulation, the searcher was positioned at a random location and allowed to move through the
environment until it came within a distance of rv of a prey item during its scanning phase. For each strategy, we performed 1000 simulations and recorded the time until
first prey encounter in each simulation. Predators were assumed to travel at a constant speed of one body length per unit time. Environments were constructed so that prey density had a mean of 1 prey per 106 squared body lengths, a realistic low density for prey, but qualitative results hold for lower prey densities (see Appendix B). In the case of
the Levy´ strategies, we repeated simulations across a range of α values from α = 1.2
to α = 3. Note that the optimal value of α for the Levy´ predator was α = 3 for which the long-term behavior is expected to be Gaussian [12]. In all figures, Levy´ strategies with
the optimal value of α are shown unless otherwise noted.
3.3 Results
3.3.1 Visual-Olfactory Predators Find Targets Faster and More Reliably Than Visual Predators
Figure 3-2A shows mean search times of simulated visual and visual-olfactory predators (search time = time until first target encounter). Visual predators that use the Levy´ strategy (Fig. 3-2A, solid line, see also Materials and Methods) have lower mean search times than predators that use the TDD strategy (Fig. 3-3A, dashed line).
However, when conditions are such that the olfactory radius ro is greater than the vision
40 radius rv , visual-olfactory predators find prey faster than their visual counterparts (Fig. 3-2A; circles represent results from visual-olfactory Levy´ with optimal α, where optimal α
was in the range 2.6-3.0 for all ro /rv ; diamonds represent visual-olfactory TDD strategy).
Mean search time of visual-olfactory predators continues to decrease as the distance
over which prey scents can be detected increases. Visual-olfactory predators have lower mean search times than visual predators
primarily because they rarely search for long periods of time without finding prey.
Figure 3-2B shows that the tails of the search time distributions for the visual-olfactory
predators (Fig. 3-2B, circles) decay roughly exponentially at a rate that is much faster
than the decay rate of the visual predators (Fig. 3-2B, squares). At least two factors contribute to the difference in performance between the two
predator types. First, visual-olfactory predators learn from “no-signal” events. They
respond to these events by leaving regions that do not contain targets. Second, as has
been observed in many species in nature [14, 47], visual-olfactory predators perform area-restricted search [15] and concentrate search effort in regions that contain prey.
Below we discuss how both of these behaviors emerge naturally through responses to
sensory signals.
To characterize changes in predator behavior in response to sensory data in the
following sections, we use a metric of information gain: the Kullback-Leibler divergence (KL, [60]). The magnitude of the change in behavior of a visual-olfactory predator
when it receives a signal of strength h relative to its intrinsic behavior θ(l), is given by ∫ KL = θ(l|h) log(θ(l|h)/θ(l)) dl. A literal interpretation of the quantity KL is the following: suppose an observer must decide, based on empirical data, whether a searcher is using olfactory data or not. The KL gives a mean rate of gain of information obtained by observing a visual-olfactory searcher moving in response to a signal of magnitude h. In
regimes where the signal contains little useful information (for example when ro /rv ≈
1 and h =0), the behavior is not modified greatly from θ(l). The resulting KL value is
41 small. However, when information is substantial (say when h = 5, for small ro /rv ) the KL is larger.
3.3.2 Visual-Olfactory Predators Learn From No-Signal Events
Figure 3-3 shows typical search paths of the four strategies through a target field in the regime where ro > rv . When searching such an environment, a predator will frequently be too far from prey to receive scent signals. For example, the inset panels in Figure 3-3C and 3-3D show that the number of signals received in scanning phases
is typically zero, with signals of greater than zero only occurring when the predator is close to prey. Intuitively, it may seem that a predator gains little information from these
no-signal events. Yet, by not receiving a scent signal, the predator gains a vital piece of
information: prey are not likely to be nearby.
Figure 3-4A shows step length distributions of visual-olfactory predators after receiving no signal. Both strategies exhibit a low probability of making small steps. The
Levy´ strategy in particular, is strongly affected; Figure 3-1B shows that this strategy
has a high probability of taking small steps between re-orientations. Yet, when the
visual-olfactory Levy´ predator receives no signal, it is unlikely to make a small step (Fig.
3-4A, Figure B-4). Figure 3-4B shows that when h = 0, KL increases as the olfaction
radius becomes larger. In fact, as ro /rv becomes large, both strategies change more in response to no-signal events than when h = 5 (Fig. 3-4B, circles (h = 0) cross above
squares (h = 5) for both strategies). For ro /rv sufficiently large, the change in behavior
in response to no-signal events allow visual-olfactory predators to avoid performing area restricted search (ARS) when they are far from prey (Figure 3-5). The visual Levy´
predator, on the other hand, spends 24% of its steps in ARS but only 2.4% in ARS near
targets. Avoiding these wasted steps strongly affects search time. Even by responding
only to no-signal events and ignoring cases in which h > 0, a visual-olfactory Levy´
predator can find prey much more rapidly than a visual Levy´ predator (Fig. B-5).
42 The observation that no-signal events contain valuable information is qualitatively similar to an observation from optimal foraging theory regarding a forager searching a discrete patch for hidden resources. In that scenario, the more time the forager spends in the patch without encountering resources, the more certain it becomes that the patch does not contain resources [61]. Our model extends this idea to searchers moving through continuous spatial environments using two sensory modalities and reveals that the change in a searchers behavior in response to no-signal events depends critically on the length scales of these sensory modalities.
3.3.3 Visual-Olfactory Predators Concentrate Search Effort Near Targets
From Figure 3-3A and 3-3B, it is clear that visual predators behave similarly in regions that are near and far from prey. Visual-olfactory predators, on the other hand, make more short exploratory steps in the vicinity of prey (Fig. 3-3C,3-3D). The strong change in strategy that occurs when a visual-olfactory predator receives a nonzero scent signal is reflected in the large value of KL for all values of ro /rv (Fig. 3-3B). Both visual-olfactory strategies increase their probability of making a short step when they encounter a nonzero scent signal (Fig 3-3A). Because of this, visual-olfactory predators perform ARS near targets and are more likely to encounter nearby prey than are visual predators (Figure 3-5).
3.4 Discussion
The framework presented here allows one to include responses to partial information gained from noisy sensory measurements when modeling random search. Our results reveal that analysis of the length scales of sensory modalities, in this case ro and rv , is crucial to determining whether such a sensory response will dominate search performance. The distinction between different types of intrinsic strategies (e.g. Levy´ vs TDD [49, 50]) is important when it is genuinely not possible to learn about resources from a distance (ro /rv ≤ 1). However, when ro /rv >1, searchers that dynamically modify their behavior in response to sensory data experience a qualitative improvement
43 in search performance. This holds over a wide range of the parameters of the scent model and other features of predator behavior (Figures B-2). This finding suggests a
connection between sensing, decision-making, and search performance, even under
sparse-signal conditions.
Moreover, behaviors such as area-restricted search near prey [14] emerge naturally from responses to sensory information. Visual-olfactory predators preform this behavior
in our simulations by turning more frequently when they receive scent cues. Historically,
ARS has been explained as a consequence of a predator concentrating search effort
in areas where it has previously found prey. This is beneficial if prey are clustered in
space [15]. Yet, we show that this behavior can also emerge when prey are not spatially clustered, if predators change their movement behavior in response to noisy sensory
data. Recent evidence suggests that some species may initiate area-restricted search
in this way. For example, wandering albatrosses appear to alter turning patterns after
encountering prey scent, effectively concentrating their search effort in local regions [47]. Greater frigatebirds forage primarily in highly productive mesoscale eddies [16]. They
appear to track these eddies, at least in part, using scent cues.
In our simulations Levy´ predators intersperse periods of local search with
large-scale relocation movements. Movements of many species including foraging
marine fish and reptiles [53], and ants in search of colony-mates [62] exhibit this qualitative pattern [48, 53, 62]. This is often cited as a feature of Levy´ walks that makes
them effective strategies for encountering targets. Yet, our results show that Levy´
predators spend much of their time searching locally in regions that do not contain
prey (Fig. 3-5). On the other hand, visual-olfactory predators appropriately match their
behavior to their proximity to targets, leading to shorter search times. In light of our results, a natural hypothesis is that searching organisms utilize different movement
behaviors depending on their perceived distance to targets. It has been shown that
strategies that mix movements with different length scales can outperform strategies that
44 draw movements from a single distribution, but that such mixed movement behavior can be difficult to distinguish from a Levy´ strategy [51]. Indeed, recent analyses have begun to find evidence of mixed behaviors in movement data (e.g. [63]). Our framework provides a means of studying how such mixed behaviors can emerge through interactions with sensory information.
45 A B ro 3 rv 10 -3 3 ) l 10 θ( -9 ) l 2 θ( l1 10 1 10 2 10 4 l
l2 0 1 0500 1000 1500 body lengths ( l)
Figure 3-1. Schematic of predator search. A) During the scanning phase of the search, a prey encounter occurs if the predator is within a radius of rv (solid inner circle) of a prey item. The predator also detects scent signals emitted by prey within a radius of ro (dashed outer circle) at an average rate of ≥ 1 per τo units of time. The predator then turns a random uniform angle between 0 and 2π. During the movement phase, the predator moves a distance of l units determined by its step length distribution. B) Step length distributions corresponding to visual Levy´ (solid curve, α = 3, lm = 1 body length) and TDD (η = 1/(1000)2 body lengths, dashed curve) strategies. Inset shows distributions on log-log scale.
46 4 A 5 ×10 5 4
● ●●
3 ×10 3 ● ● ●
4 ● ● ● meansearch time ● ● 10 0 2 4 6 8 10 olfaction radius/vision radius
●
1000 B ●
● ●
●
10 100 ● frequency1 + ● ● ● ● ● ●● ● ● ● ● ● ● 1 0 1×10 5 2×10 5 3×10 5 search time
Figure 3-2. Predator search times. A) Mean search time as a function of the ratio of the olfactory radius (ro ) to vision radius (rv ). Solid orange line (visual Levy),´ dashed blue line (visual TDD), orange circles (visual-olfactory Levy),´ and blue diamonds (visual-olfactory TDD) each represent mean search time of 1000 replicate simulations. Confidence bands represent ±2 SEM. The following parameters values were used: a = 1, rv = lm = 50a, τv = 1 s, τo = 30 s, mean inter-target distance was 1000a, and λa = 100 units of scent per τo (see text for description of parameters, also SI Text). B) Empirical distribution of search times of visual Levy´ (orange solid line, squares), visual TDD (blue dashed line, squares), visual-olfactory Levy´ (orange solid line, circles), and visual-olfactory TDD (blue dashed line, circles) strategies. In the case of the visual-olfactory strategies, frequencies are shown for ro /rv = 4. Note the large number of searches resulting in long search times for visual predators.
47 A B
C D ) ) 20 h 20 h 15 15 5 10 scent signals ( signals scent scent signals ( signals scent 0 0 5 10
Figure 3-3. Typical search paths through a scent field with log10(1 + mean number of scent encounters per unit τo ) indicated by grayscale (darker grey denotes more encounters). In white regions, mean number of encounters is effectively zero. Paths for A) visual Levy,´ B) visual TDD, C) visual-olfactory Levy,´ and D) visual-olfactory TDD are shown. Color scale of path changes from blue to red with increasing time. Inset panels in C and D show the number of hits received during each scanning period with colors corresponding to colors in search paths. ro /rv = 4 in all panels; all other parameters as in Fig. 2A.
48 AB 15
10 ● ● ● ● ● ● ● ● ● h = ● ● ) 5 ● ● ● ●
h ● ● ●
-1 ● ● ● ●
= ●
10 ●
10 ● H | ● l l -3
( ● 5 θ 10 h = 0 -5 ● 0 Informationgain (KL) 10 0 1000 2000 0 5 10 15 body lengths ( l) olfaction radius/ vision radius
Figure 3-4. Effect of olfactory data on step lengths. A) Step length distributions after signal modulation when h = 0 and when h = 5. B) Information gain as measured by the Kullback-Leibler divergence between the visual strategy and the corresponding visual-olfactory strategy when h = 0 (squares) and h = 5 (circles) as a function of ro /rv . Dashed curves represent visual-olfactory TDD strategy. Solid curves represent visual-olfactory Levy´ strategy. Note the increasing information gain when h = 0. In both panels α, lm, and ν as in Fig. 1B, all other parameters as in Fig. 2A.
V Lévy V-O Lévy V TDD V-O TDD
0 10 20 30 0 20 40 60 80 % steps spent in % proximity area-restricted search events resulting in prey capture
Figure 3-5. Effect of olfactory data on area-restricted search (ARS). Left bars show % steps spent performing ARS. Shaded regions show the % of ARS searches that occur within 4rv of prey. For the visual Levy´ predator (top bar), contrast the large % of steps spent in ARS, with the small fraction of these steps spent near prey (top bar, shaded region). ARS defined as any period in which predator makes ≥ 5 consecutive steps within a region of radius 4rv . Right bars show % of proximity events in which predator locates prey. Bars for visual-olfactory (V-O) predators show that they successfully locate nearby prey more frequently than do visual (V) predators. Proximity events defined to by any period of ≥ 1 consecutive steps within 2rv of a target.
49 CHAPTER 4 SENSORY INFORMATION AND ENCOUNTER RATES OF INTERACTING SPECIES
Classical models of species interactions assume that encounters are governed by a process akin to mass-action; individuals move along random linear trajectories and encounter one another when they come within a critical distance [3, 64]. Under these assumptions, an individual searcher encounters targets at a rate proportional to the density of targets ρ [3, 65]. Recent work has extended the study of encounter rates to consider searchers that follow movement paths that are not linear trajectories, encounter targets probabilistically, destroy targets after encounters, and search intermittently [66–68]. Under a variety of circumstances, these models too predict that a searcher will encounter its targets at a rate proportional to target density (for a list of conditions, see [67]). A vital assumption both of older and newer models is that the searching organism moves independently of the locations of targets. In the context of predator-prey interactions, this implies for instance, that predators do not alter their movement behavior in response to sensory cues emitted by their prey.
Of course, the assumption that searchers move independently of targets is made for mathematical convenience. The question is whether models that rely on this assumption capture the salient features of encounter rate kinetics in nature. Empirical studies have shown that shutting down particular sensory modalities such as chemosensing or flow sensing can dramatically decrease search performance (e.g., [10]), and that sensory cues appear to influence both small-scale [69] and large-scale [16, 47] search behavior. While such studies more rigorously confirm the intuition that the use of sensory data should improve search performance, little is known about how sensing can influence the qualitative relationship between encounter rate and target density. Here, we argue that sensory response can have a dominant effect on the rate of encounters between searchers and their targets, not only by increasing encounter rate, but also by qualitatively changing the dependence of encounter rate on target density.
50 Below we adopt the language and intuition associated with predators searching for prey. We assume that a predator samples the environment for sensory cues passively
emitted by prey, and adjusts its movement behavior according to explicit mathematical
models presented below. This approach builds on a recently developed framework for
modeling search decision-making [70] to model the flow of sensory information from prey to predators. We consider three scenarios: (1) perfect sensing and response: the
predator can ascertain the precise locations of prey from the sensory data it receives
and responds optimally, (2) imperfect sensing and response: the predator detects
noisy scent signals emitted by prey and alters its movement behavior in response,
and (3) purely random search: the predator does not use sensory information to guide its movement decisions. Models (1) and (2) represent upper and lower bounds,
respectively, on the acquisition and use of information about prey positions. Our central
finding is that there is a systematic shift away from a linear encounter rate function at
both of these bounds, suggesting that the collection and use of any form of sensory data may fundamentally alter encounter rate kinetics. We discuss the role of information in
governing predator-prey encounter rates, but note that our general methodology could
be applied to rates of encounters in other types of ecological interactions (e.g., between
mates, competitors, mutualists).
4.1 Materials and Methods
4.1.1 Encounter Rate and Search Behavior: Some Definitions
Studies of biological search strategies typically describe how the type of movement
behavior used by a searching organism affects the time needed to encounter its first
target τ, or the rate of target encounters . For consistency with past work, we define
as the prey encounter rate of a single predator (e.g., [# prey] per [predator hour], [67]). We assume that predator density is low enough that does not depend on the
density of predators, and instead, depends only on the density of prey ρ. We define two
encounter rate functions: the mean first encounter rate (ρ), and the mean encounter
51 rate after k encounters k (ρ). The latter is often referred to as the encounter rate associated with destructive search [67, 68], emphasizing that the activity of the searcher alters the target landscape. In past studies, the non-destructive search rate is often defined in terms of random variable τ which represents the time required to find the
first target. The empirical first encounter rate is then defined to be (ρ) = 1/τ where τ indicates an average over many trials.
4.1.2 Framework for Modeling Movement Decisions
We consider an idealized model of a searching predator in a two-dimensional environment. We assume that the predator moves at a constant speed v that is much greater than the speed of its prey. In this case, it is sensible to model prey as if they are not moving, at least for the duration of the predator’s search. In the following sections, we further assume that prey density is low, and that handling time is therefore negligible relative to search time. As in past approaches, the predator divides its search into two phases: a scanning phase and a movement phase [55, 70]. This intermittency reflects the observed tradeoff between locomotion and perceptual acuity (e.g., [71]), and the intermittent nature of sampling through major sensory modalities [72]. During the scanning phase, the predator collects sensory data h, and encounters any prey within a radius re with probability one. During the movement phase, the predator moves a distance ℓ at an angle θ. The process the predator uses to determine ℓ and θ constitutes its search strategy.
4.1.2.1 Sensory signals and search behavior
To relate a predator’s search behavior to the information it acquires from sensory signals, we adapt a recently developed framework for modeling search decision-making
[70]. The framework has two essential features. First, the predator’s movement behavior in the absence of any sensory data is modeled by an intrinsic movement distribution
γ(ℓ, θ). Second, the predator uses a decoding function to extract information from the sensory data it collects and modify its intrinsic movement behavior.
52 During the movement phase of the search, predator movements are modeled by drawing from the distribution
P{H = h | ℓ, θ}γ(ℓ, θ) γ(ℓ, θ|H = h) = ∫ ∫ , (4–1) 2π ∞ P{H = h | , } ( , )d d 0 0 ℓ θ γ ℓ θ ℓ θ
where h is the sensory data collected in the previous scanning phase, and P{H =
h | ℓ, θ} is the likelihood of observing H = h, given that the target is a distance of ℓ and angle θ from the predator’s current position. Rather than associating a deterministic
action with a particular value of the signal h, we model movement decisions as actions
drawn from a probability distribution to capture the inherent variability in decision-making
[73]. The intrinsic movement distribution can be interpreted as an evolved behavior
that the predator uses in the absence of useful sensory information [13]. The decoding function, on the other hand, represents an evolved mechanism for interpreting and
moving based on sensory input, H [70]. While P{H = h | ℓ, θ} is formally a likelihood
function, we refer to it as a decoding function to emphasize that it represents a means
of interpreting and using signal data. As we show below, the three strategies we wish to
consider can be framed by specifying appropriate decoding functions. 4.1.2.2 Perfect sensing and response
Suppose the predator detects sensory observations h and, regardless of the value of h, is able to perceive the precise locations of prey. Then the decoding function in
Equation (4–1) is a point mass at the location of the nearest prey (note that a “traveling
salesman” solution to this problem could outperform such a greedy searcher, but
is computationally intractable when the number of prey is not small). In this case,
movements are taken from the distribution γ(ℓ, θ | H = h) = δ(ℓnp, θnp), where δ denotes the delta function and ℓnp and θnp are the distance and angle between the predator’s current position and the location of the nearest prey. In each movement phase, the predator moves along a linear trajectory from its current position to the position of the nearest prey (Fig. 4-1). In this case, the form of the intrinsic movement distribution is
53 unimportant, so long as it satisfies certain technical mathematical requirements such as
being continuous and having non-zero mass at (ℓnp, θnp).
When the predator moves directly from one prey to the next, it will encounter prey at a mean rate that is inversely proportional to the mean distance between prey, which we denote d. Assuming prey are distributed according to a Poisson spatial process, √ ≈ v/d, or equivalently (ρ) ≈ 2v ρ. Formally, this calculation requires that prey are
replenished and redistributed after each encounter and that there is no net decrease
in prey density. It also assumes that the encounter radius is zero. To relax the latter
assumption, note that a predator must move an average distance of 0.5ρ−1/2(1 − √ erf(re πρ)) so that its nearest prey is within its encounter radius re (see Appendix C). It √ −1/2 −1 follows that the encounter rate is (ρ) = v[0.5ρ (1 − erf(re πρ))] (Fig. 4-1, inset
panel, blue curve and points). When density is such that the mean distance between
targets is similar to re, encounter rate changes linearly with prey density (see Appendix √ C). However, as density approaches zero, this function approaches (ρ) = 2v ρ (Fig. 4-1, inset panel, orange curve). So unlike in the case encounter rate models that
assume predators move independently of prey, a predator with perfect sensing and
response will encounter prey at a rate that is proportional to the square root of prey
density when density is low.
4.1.2.3 Purely random search
We note that it is possible to formulate a search behavior that does not rely on
sensory data using the Bayesian framework of Equation (4–1) by assuming that the decoding function P{H = h | ℓ, θ} = 1 for all ℓ and θ. Each time a predator moves,
it draws a step length and turn angle from the distribution defined by Equation (4–1),
which is just the intrinsic movement distribution γ(ℓ, θ) when the decoding function is
uniform. In this interpretation of the purely random search scheme predator movements
may be independent of sensory signals in the environment for any of three reasons: (1) the predator cannot detect and/or neurally encode the signal, (2) the predator can detect
54 the signal but cannot extract information from the encoded signal, or (3) the predator has the sensory and neural machinery for encoding and decoding signals, but does not
use the information it to make movement decisions. While the latter possibility seems
unlikely and would be hard to verify experimentally, the former two lead to testable
hypotheses about the mechanism behind directed and undirected predator movements. 4.1.2.4 Imperfect sensing and response
For the signal-modulated predator [70], we focus on the case of a predator that
receives noisy scent signals that lack directional information, a scenario encountered by species like sharks, lobsters, and crabs that use scent signals to find prey in turbulent
environments [10, 74]. We assume that in a given time interval t0, the predator will encounter a number of detectable scent patches drawn from a Poisson distribution. The mean parameter depends on the distance to targets in the vicinity. We assume that all targets have the same intensity of signal emission and the rate of arrivals at a distance
ℓ is given by a function R(ℓ). As in past approaches, we assume R(ℓ) is given by the steady state solution to the diffusion equation describing the diffusion and dissipation of scent without advection (see Appendix C,[11, 70]). In this case, the decoding function is
given by the likelihood e−R(to ℓ)R(t )h P{H = h | , } = o ℓ , ℓ θ h! (4–2)
where h represents the number of detectable scent arrivals in some fixed amount of time to.
This model of olfactory search behavior has two salient features. The first is that,
because there is no directional information inherent in the signal, the predator always
draws turn angles from the same distribution (Uniform on [0, 2π]), regardless of the
signal it receives. Second, the predator has no memory of past movements or signal encounters. Such information could help the predator compute its position relative to its
prey [11] but we eliminate this possibility. The purpose of this simplified model is to study
55 the effect of minimal signal information and a minimal amount of signal processing on predator-prey encounter rates.
4.1.3 Encounter Rate Simulations
We compare the behavior of a predator that moves according to a purely random
strategy to a predator with imperfect sensing and response. In both cases, we assume
that the intrinsic movement behavior is described by a symmetric two-dimensional
Pareto distribution. Because of the symmetry we can separately draw the turn angle
θ ∼ Unif(0, 2π) and the move length ℓ ∼ γ(ℓ), where γ(ℓ) is the density of a Pareto random variable,
α−1 α γ(ℓ) = (α − 1)ℓm ℓ , (4–3)
ℓm is a minimum move length, and α is a parameter that determines whether the walk is superdiffusive (α ∈ (1, 3)). We use a Pareto distribution with a power law tail to model
intrinsic movement behavior because it has been argued that such a distribution may
have evolved as a statistical movement strategy for locating resources when sensory
data are not useful [12]. In each simulation, we placed a single predator in a prey environment and populated the environment with a Poisson number of prey with a mean of 600. The size of the environment was then scaled to achieve the desired prey density. In the first set of simulations, prey positions were generated using a Poisson point process. We then recorded the time required for the predator to encounter the first prey and used this to compute encounter rate (ρ). This is consistent with a scenario in which predators search for and capture a single prey item, and then cease to forage for a period of time, during which prey redistribute themselves in the environment. When predators encounter and destroy multiple prey in succession, they can create local zones of prey depletion. To determine whether the scaling of encounter rate is sensitive to such a local depletion effect, we allowed predators to encounter and destroy 32 prey items.
We then computed k (ρ) = k/τk , where τk was the mean time required to encounter
56 k = 32 ≈ 5.3% of the prey present on the environment. Finally, to determine whether the scaling of the encounter rate depends on the distribution of targets, we generated prey
distributions according to a highly clustered point process that we will call a preferential
attachment model. Briefly, N prey were generated by drawing from a Poisson distribution
mean 600. The size of the environment was then scaled to achieve the desired prey density. A fraction of the N prey were chosen to act as seed points and placed uniformly
at random on the space. The remaining prey were each assigned as daughters to one ∑ of the seed points iteratively with probability ni / i ni . Positions of daughters were
assigned uniformly within a circle of radius ri around the seed point, where ri was chosen so that all clusters had the same local prey density. We repeated simulations to compute (ρ) and k (ρ) for k = 32 in the highly clustered environments generated by this model.
We simulated predator exploring environments with prey densities ranging from
0.5-100 prey per 106 squared predator body lengths. This range was based on realistic low prey densities encountered by predator species in nature [75–77]. All simulations
were performed using Matlab.
4.1.4 Estimation of Scaling Regimes and Exponents
As in previous investigations (e.g., [66]), we expected that (ρ) would be a linear function of ρ for the purely random predator. On the other hand, as shown above, the predator with perfect sensing and response has an encounter rate function with several scaling regimes in the range of densities that interest us: one in which encounter rate is √ proportional to ρ, and one in which enconter rate is proportional to ρ. To accommodate
these functional forms, we assumed that locally, encounter rate can be described by a
power function of the form (ρ) = ηρβ. This allows for both linear and sublinear scaling.
To determine whether simulated predators had multiple scaling regimes we fitted (i)
a single power function, (ii) a segmented function with two distinct scaling regimes, and (iii) a segmented function with three distinct scaling regimes. Prior to fitting, we
57 log transformed density and encounter rate data from search simulations. We used a recently developed statistical method for simultaneously estimating both the break points between distinct scaling regimes and the scaling exponents in each regime [78]. We compared the fits of these three models by comparing AIC values. Statistical analyses were conducted using the Segmented package [79] in R [41]. 4.2 Results
There is a dramatic difference between movement patterns of predators that use sensory data and those that do not. As is evident from Figure 4-2, signal-modulated predators concentrate scanning effort near prey (Fig. 4-2A), whereas purely random predators scan roughly uniformly over the environment (Fig. 4-2B). Signal-modulated predators have this advantage because they move short distances between scans when they receive strong sensory signals and move long distances when they measure weak signals [70]. This behavior improves search efficiency, but perhaps more importantly, it leads to a qualitatively different relationship between the encounter rate of signal-modulated predators and their prey (Fig. 4-3A). As expected from past work on random search [66, 67], purely random predators encounter prey at a rate that scales nearly linearly with ρ across all prey densities. The encounter rate of signal-modulated predators, on the other hand, is strongly nonlinear in ρ (compare Fig. 4-3A yellow points to blue triangles). In particular, at low but realistic prey densities (Fig. 4-3A blue curve), the encounter rate of signal-modulated predators changes sublinearly with changing prey density. This anomalous scaling makes the search efficiency of signal-modulated predators more robust with respect to changes in prey density.
4.2.1 Encounter Rates of Purely Random Predators are Near-linear in Prey Density
Predators that used a purely random search strategy encountered prey at a rate that was nearly proportional to prey density (Figure 4-3A, yellow circles; (R) =
0.036ρ1.12; 95%CI forβ = [1.09, 1.15]). This near-linear scaling held when prey were
58 clustered and also when predators encountered and destroyed multiple prey per search (β ∈ [1.05, 1.11]). The encounter rate function did not show evidence of multiple scaling regimes (AIC of model with single regime − model with multiple regimes ≤ − 3.61).
4.2.2 Encounter Rates of Signal-modulated Predators Change Nonlinearly with Prey Density
Across all densities studied, predators that use sensory data to make movement
decisions encounter prey at a higher rate than predators that do not use sensory cues (Fig. 4-3). As prey density increased, the encounter rate of signal-modulated
predators increases non-linearly and clearly displays multiple scaling regimes (Fig. 4-3, blue triangles; AIC single regime - AIC three regimes = 682). At the lowest densities, encounter rates increased linearly or superlinearly with prey density. For the particular parameter values explored here, there is a transition to a second scaling regime at
ρ ≈ 1.7; however, the exact transition depends on the length scale of scent detection
(Fig C-1). In the second, intermediate regime, which covers low but realistic prey
densities, signal-modulated predators encounter prey at a rate proportional to ρβ, where
0 < β < 1. The value of the scaling exponent β = 0.56, is close the square-root scaling exhibited at low densities by the searcher with perfect sensing response. For higher
densities, data indicated a third regime, in which encounter rate increased superlinearly
with prey density (β =1.3); however, this upper regime is of less interest because it
corresponds to environments where prey are relatively dense and search behavior
becomes less important. The qualitative form of the encounter rate function of signal-modulated predators in
a uniform prey environment was preserved when prey were highly clustered, and when
predators encountered and destroyed multiple prey items in a single search. Figure
4-4 shows that the mean encounter rate after k encounters k (ρ) exhibited near-linear regimes at relatively high and low densities, and sublinear regimes at intermediate densities (β ∈ [0.44,0.54] in intermediate regime).
59 4.2.3 Sensory Response Allows Predators to Encounter Nearby Targets more Frequently
In addition to concentrating scanning effort near prey, signal-modulated predators also encounter nearby prey more frequently than purely random predators. To see this, we compute the empirical probability of encountering a nearby prey as the fraction of times a predator moves within a distance of ro of one or more prey and then encounters one or more of the prey before moving a distance of ≥ 2ro from them (P{τhit < τexit }, Figure 4-5A, upper diagram). Figure 4-5A shows that the empirical encounter probability of signal modulated predators (Fig. 4-5A blue points) is higher for all prey densities, and approaches 1 for prey densities above 10, indicating that signal-modulated predators do not miss nearby targets when density is high. Purely random predators frequently miss prey even as prey density approaches 100 (as ρ approaches 100, the typical distance between adjacent prey approaches the encounter radius re = 50 body lengths and predators frequently encounter prey without moving at all). At low density, encounter probabilities of both types of predator approach constant values. For the signal-modulated predator, this value is 0.17, similar to the value of 0.23 predicted for a Brownian searcher with constant diffusivity (see Appendix C). Figure 4-5B shows that this minimum encounter frequency is roughly three times higher for signal modulated predators than for purely random predators.
4.3 Discussion
Our results demonstrate that the use of information about the position of targets fundamentally alters the relationship between encounter rates and target density. This is true even when sensory cues contain a minimal amount of information about target locations, and searchers do not remember past signals. This finding is robust to a range of assumptions about target distribution, capture behavior, and the length over which searchers detect scent signals.
60 Reaching any general understanding of the effect of sensory data on species encounter rates is challenging. Searching organisms collect a wide variety of sensory
data and there is a a general lack of knowledge about how they use these data to make
decisions [73]. Here, we have taken the approach of studying two limiting cases of the collection and use of sensory data and one intermediate case. In the limit of perfect sensing and response, predators encounter prey at a rate proportional to the square root of prey density at low prey density. At the opposite extreme, a predator that does not use sensory information encounters prey at a rate that is nearly proportional to prey density, as expected from past treatments of encounter rate that assume that predators move independently of prey [64, 66, 67]. The intermediate case turns out to be telling: when we perturb information-free search behavior by introducing only a limited capacity for
sensing and decision-making based on a noisy, directionless signal, the encounter rate
function immediately departs from the linearity expected when predators move without
using information. Clearly, most species in nature use search behaviors that lie somewhere between
a perfect sensor with perfect response and the memoryless random walker studied in
our simulations. However, both of these extremes use sensory data to guide movement
decisions and both depart from mass-action kinetics in biologically interesting ways.
Not only do predators that use sensory information encounter prey more often, but the sublinear scaling of encounter rate with prey density reduces the sensitivity of
predators to changes in prey density. This increased robustness provides an ecological
mechanism through which sensory response may allow predators to cope with
fluctuations in prey density. Recent empirical studies lend some support to the idea
that sensing may lead to sublinear encounter rate functions in nature. These studies report that encounter rates of predatory fish and birds appear to change sublinearly with
prey density [80, 81]. We suggest that predator sensory response is a likely cause of
this pattern.
61 Our results show that introducing a response to even relatively information-poor, noisy sensory signals qualitatively alters the relationship between predator-prey encounter rate and prey density. Behaviors such as area-restricted search emerge naturally from our model of search behavior, even in the absence of signal gradients, complex signal processing, and memory of past signal and target encounters [70]. The framework we introduce here can be used to understand the connection between information and the encounter rates that are so critical to many core concepts in ecology and biological search.
62 A B 80 ●
re
● ● ●
● ●
● ●● ●● ●● Encounter rate (prey / hr) rate Encounter ●● ● 0 20 40 60 0 10 20 30 Prey density Figure 4-1. A) Predator with perfect sensing and response, searching in a two-dimensional environment. After collecting sensory data, the predator moves along a linear trajectory toward the nearest prey and encounters the prey when it comes within a distance of re. B) Mean encounter rate from −1 simulations (re = 50 body lengths, v = 1 bl s , points show mean of 100 replicates at each density). Prey distribution is randomly generated from a Poisson point process in each simulation. Blue curve shows theoretical√ mean encounter rate (see text), which approaches (ρ) = 2v ρ for low prey density (red curve). For ρ > 25, the typical distance between nearest prey is less than 2re and predators begin to encounter prey frequently without having to search. In Figures 4-1 through 4-5 , density is expressed as prey per 106 squared predator body lengths.
63 A B
Figure 4-2. Prey (red points) and locations where predator scans for prey (blue points) for A) signal-modulated and B) purely-random predators. Scan points are semitransparent so darker color indicates locations where predator has scanned more frequently. Data represent searches in which a predator made 1000 consecutive movements without destroying prey.
A B
● ● Purely random Signal−modulated ● ● ● ● ● ●
3.6 5.4 ● ● ● ● ● ● ●
●
●
Encounter rate (prey / hr) rate Encounter ●
● Ratio (signal-mod/purely random) ● ● ● ● ●●●●● 0 1.8 0 2 4 6 8 10
0.5 10 20 30 0.51 5 30 (710) (160) (110) (90) Prey density Prey density (distance between nearest prey) Figure 4-3. A) Purely random (yellow circles) and signal-modulated predators (blue triangles, k = 1) searching in uniform (Poisson) prey environment. Each point represents mean encounter rate from 1000 replicate simulations. In simulations shown, the following parameters were used: re = ℓm = 50 body lengths, v = 1 body length per second, ro = 500 body lengths, α = 2. Scent emission rate at prey location was set to 100 (see Appendix C). B) Ratio of encounter rates shown in A (rate of signal modulated predator divided by rate of purely random predator).
64 uniform prey, k = 1 clustered prey, k = 1 uniform prey, k = 32 15
) clustered prey, k = 32 5 1 / hr , prey ( Encounter rate 0.05 0.25
0.5 5 25 75
Prey density Figure 4-4. Mean encounter rate of signal-modulated predators in uniform (Poisson) and clustered (preferential attachment) prey environments. Predators encounter and destroy k prey items per search. Each point represents mean of 1000 replicate simulations. Parameters as in Fig. 4-3. Encounter rate is lower in clustered environment with k = 1 because clusters are far from one another and it can take predators a long time to locate a cluster. When k = 32, encounter rate is higher because the predator can encounter nearby targets after it locates the cluster.
65 A 1.0 0.6 0.8 0.4 0.2 0 0.1 1 10 100
prey density
B 3 4 Ratio (signal-mod/purely random) 1 2
0.1 1 10 100 prey density Figure 4-5. A) Empirical encounter probability as a function of target density. Parameters as in Fig 4-3. Upper diagram shows predator that encounters prey before exiting region of radius 2ro . Lower diagram shows predator that exits before encountering prey. B) Ratio of encounter probability of signal-modulated predator to encounter probability of purely random predator.
66 CHAPTER 5 CONCLUSIONS
Biologists have long strived to understand why animals move in the ways they do
[1, 2]. In the preceding chapters, I have described new approaches for studying animal movement behavior that incorporate either the physics of locomotion (Chapter 2), or
the process of information acquisition and use (Chapter 3 and 4). The purpose of these
studies is not to simply add complexity to previous mathematical models. Rather, it is to
explore whether constraints imposed by the physical nature of animal locomotion and
by the availability of sensory information can play a dominant role in determining how animals move. In particular, the studies presented above reveal: (1) that animal body
size appears to constrain the maximum distance traveled during migratory movements
through its effect on metabolism and the cost of locomotion, (2) that the use of even
minimal amounts of sensory information in movement decision-making can lead searching animals to concentrate their search effort near targets, and (3) that the
use of sensory information to guide movement behavior can increase the robustness of
search performance to changes in the environment.
In Chapter 2, we developed a general mathematical framework to model the distances that animals travel during migration. The model, and the extensive dataset we collected to test it, revealed some patterns and predictions that were new to the
field of animal movement. Although theoretical studies had previously discussed the possibility that migration distance might be systematically correlated with body size, no general empirical relationship between migration distance and the body masses of species had been established. Thus, the striking correlation between migration distance and the body masses of walking, swimming, and flying animals is a new contribution in itself. Two predictions were particularly interesting and well-supported by migration distance data. First, because of the relationship between body mass, the energetic cost of transport, and fuel storage, our model predicted that the number of body lengths
67 traveled by walking migrants and the number of body lengths traveled by swimming migrants should each be independent of body mass. Second, unlike the energy required
for walking and running, the energy required for flight increases extremely rapidly with
increasing body mass. Because of this, the increase in maximum migration distance
with increasing body mass becomes smaller and smaller for flying large migrants. This prediction too was supported by migration distance data.
In Chapter 3 we explored how animal movement behavior might be affected by the collection and use of sensory data from the environment. Two findings were particularly interesting. First, we found that the well-documented behavior refered to as area-restricted-search, in which animals concentrate their search effort near targets, can be induced by assuming an extremely limited sensory response on the part of a searching animal. A second finding is that, contrary to intuition, searching animals can gain a lot of information by sampling for sensory data and receiving no signal. Thus, no signal does not mean no information. Responding to no-signal data by moving long distances seems to provide searchers a way of avoiding wasted search effort in regions that lack targets.
In Chapter 4, we considered how searching animals could affect their encounter rates by using sensory information from targets to make movement decisions. In particular, we explored whether the relationship between encounter rate and target density was qualitatively different when predators searched with and without using sensory information from targets. The results of this study demonstrated that sensory response and the flow of information to a searching animal can drastically alter its rate of encounters with targets. Interestingly, we found that using sensory information to locate targets changes the way a searching animal moves, its encounter rate, and the sensitivity of its encounter rate to changes in target density.
The questions I have attempted to answer in the preceding chapters are important ones. However, the new questions that came to light through the studies described
68 above are just as important. With respect to physical constraints on migration distance, the model we proposed assumes that the number of refueling stops a migrant makes is, on average, independent of its body mass. At present, the paucity of data makes it difficult to evaluate this assumption. However, if this assumption is correct and the average number of migratory stops does not depend on body mass, one might wonder why this should be so. Such a pattern would constitute an interesting life history invariant, and would surely beg for a mechanistic explanation. Our model falls short of providing such an explanation. A second question raised by the migration model and data, is the question of why so many species appear to travel distances that are similar to their theoretical maxima. Indeed, just because larger species can migrate farther, on average, than small species doesn’t mean they must do so. Yet, it is clear from our data that species that are large do tend to migrate farther than those that are small.
This raises some intriguing questions about the evolutionary drivers of migration. Might there be selection for species to migrate as far as they can? Are there other evolutionary processes that could explain this pattern? The studies of sensing and decision-making also raise interesting questions while leaving others unanswered. For instance, in modeling decision-making in response to sensory data, we assumed that a searching animal has evolved a means of interpreting the scent signals it measures to tell it something about where its target is located. But what if environmental conditions are so variable that it is not practical to have such a reflex-like response to a particular value of a signal? Instead, learning may become necessary. Indeed, one hypothesis for the origin of complex neural mechanisms for learning and decision-making is that the ability of organisms to move provides them with the capability of affecting their interactions with a variable external environment by moving to a new place. This immediately creates a link between the ability to perceive the environment by collecting sensory input and the need to learn to make the correct movement decision using that input [82]. Thus it will
69 be interesting to determine how our results are affected when the capacity of animals to learn from past experience is incorporated.
Animal movement behavior is inherently complex. Yet, it can provide an intricate and powerful model system for exploring some of the most profound unsolved problems in biology including understanding the evolution and behavior in the face of physical constraints, the emergence and and maintenance of learning, and the processes that underlie organismal decision-making. Solutions to these problems and many others await future investigation.
70 APPENDIX A MIGRATION MODEL DERIVATION, SENSITIVITY, AND STATISTICAL ANALYSES
A.1 General distance equation
Here we provide a detailed derivation of the migration distance equations for
walking, swimming, and flying migrants presented in the main text (Equations(2–6)-(2–8)).
For each, we begin by expressing maximum migration distance on a single migratory
leg, Yi , as a function of total power, Ptot , speed, v, and energy density, c: ∫ M0(1−f ) −v c i = dM Y M0 Ptot
where Ptot = Pmtn + Ploc , M0 is initial mass at the beginning of the migratory leg, and f is the ratio of fuel mass to M0 at the beginning of the leg. To solve for Yi , we specify
0.75 functions describing Pmtn, Ploc , and v. For Pmtn, we assume Pmtn = p0M as described in the main text. Derivations of walking, swimming, and flying equations are given below.
Constants in biomechanical Equations (2–3)–(2–4) in the main text have been expanded
to more explicitly show their physical basis.
A.1.1 Walking
To estimate the power required for walking, we use Equation (2–3) described in the
main text. Empirical evidence strongly supports the predictions of this model [31, 83].
Combining this model with Equation (A.1) and integrating from initial to final mass gives ( ) −1 −1 0.25 p0vwalk +γgLc M0 Yi,walk = yw Lc ln −1 −1 0.25 0.25 p0vwalk +γgLc M0 (1−f )
where yw is a constant. Based on our assumption of geometric similarity, Lc ∝
0.33 M0 , because stride length is typically proportional to leg length [32]. We assume that
0.1 vwalk ∝ M0 among species but that it is fixed for an individual migrant [33]. Substituting
these terms for Lc and vwalk gives an expression for the mass-dependence of Yi , ( ) 0.02 0.33 p0+c1M0 Yi,walk = yw M0 ln 0.02 0.25 p0+c2M0 (1−f )
where c1 and c2 are constants. The logarithmic component of Equation (A.1.1)
contributes little to the shape of the function in the biologically relevant range of M0, and
71 0.02 0.02 can be accurately approximated as, ln[(p0 + c1M0 )/(p0 + c3M0 )] ≈ ln[(p0 + c1)/(p0 +
0.01 c3)]M . Thus, Equation (A.1.1) can be rewritten as a power function in M0, ( ) 0.34 p0+c1 i,walk ∝ yw M ln p0 Y 0 p0+c3 For walking mammals, is roughly constant and so 0.34 Yi,walk ∝ M .
A.1.2 Swimming
To estimate Ploc for swimming migrants, we use a standard resistive model of
swimming locomotion (Equation (2–4) in the main text, [84]). The cost of locomotion is
proportional to drag times speed, so locomotory power can be expressed as = α D v Pswim η t where η is dimensionless conversion efficiency from stored fuel energy to muscle
power output, and α is a dimensionless correction constant [84, 85]. We assume that
boundary layer flow around the swimming migrants considered here is approximately turbulent [86]. Given this assumption, drag on a swimming migrant of total length,
1.8 CAbv Lb, is given by Dt = α 0.2 , where C is constant determined by water density and Lb dynamic viscosity and Ab is a characteristic area (here taken to be body cross-sectional area, see [6, 84] for detailed discussion of this model). We take v to be the speed that
minimizes Ptot /v [84], and assume that as a swimming migrant burns fuel, changes in
body cross-sectional area, Ab, are small enough to be ignored. Substituting expressions
for Pmtn, Pswim, and vswim into Equation (A.1) gives, ( ) 0.2 0.36 Lb −0.64 0.52 0.28 i,swim ∝ p M [1 − (1 − f ) ] Y Ab 0 0 To recover the interspecific scaling equation from Equation (A.1.2), we note that
0.33 0.67 l ∝ M0 , Ab ∝ M0 , and therefore
−0.64 0.30 Yi,swim = ys p0 M0
where ys is a constant.
A.1.3 Flying
Locomotory power of an animal in steady horizontal flight can be expressed as the
sum of three components: the power required to remain aloft (induced power, Pind ), the
72 power required to overcome drag on the body (parasite power, Ppar ), and the power
required to overcome drag on the wings (profile power, Ppro )P y = Pind + Ppar + Ppro ,
2 2ω(Mg) −1 ρaAbCd 3 where Pind = 2 v , Ppar = v , Ppro = κ(Pind + Ppar ), ω is a dimensionless ηπLw ρa η2 induced power factor, g is the acceleration due to gravity, η is dimensionless conversion
efficiency from stored fuel energy to muscle power output, ρa is the density of air, Lw
is wingspan, Cd is a dimensionless drag coefficient, and Ab is body cross-sectional
area [7]. This formulation expresses Ppro as a dimensionless profile power factor (κ)
times the sum of the induced and parasite power [7]. We follow [7] in assuming that
2 κ ∝ Aw /Lw = 1/wing aspect ratio, where Aw = wing plan area [7]. This model is discussed in detail in [7]. v is taken to be the speed that minimizes the ratio of induced and parasite power to speed. At this speed, locomotory power is described by the ( ) 3 6 6 0.25 −1 ω g AbCd M 1.5 equation P y = (1 + κ)1.05 η 2 6 = k0M , where k0 is constant for ρaW an individual migrant. Before substituting P y and v into Equation (A.1), we make the
additional assumption that, as a flying migrant burns fuel, changes in body frontal area,
Ab, are small enough to be ignored [22]. Under this assumption, maximum migration ( ) 0.75 p0+k0M0 distance during a single leg is given by Yi, y = yf ln 0.75 0.75 , where yf is a p0+k0(1−f ) M0 constant. To recover the body mass scaling of maximum migration distance, we assume values for the constants and morphological variables that determine k0. Specifically, we
0.33 0.67 assume Lw = 1.1M0 [87], Aw = 0.16M0 [87], η = 0.23 [25], ω = 1.2 [7], ρa = 0.98
0.67 [88], Ab = 0.0081M0 [89], g = 9.8, and Cd = 0.2 [6], and κ = 1.1 [7]. Data on maximum fuel fractions of flying migrants prior to departure are available [90–99], and indicate
a mean value of f = 0.59 among species, assuming a mixture of 90% lipid and 10% ( ) 0.42 p0+k1M0 protein is used as fuel [45]. Substituting these values gives Yi, y = yf ln 0.42 , p0+k2M0
where k1 = 60 and k2 = 31.
73 A.2 Parameter estimation and model sensitivity
A.2.1 Estimation of p0
The metabolic normalization constant, p0 varies among broad taxonomic groups
[27]. We used published estimates of p0 for walking mammals, swimming fish, flying insects, non-passerine birds, and passerine birds (Table A-1). For swimming mammals,
we assume that p0 is equal to that observed in terrestrial mammals. For fish, the
◦ estimate of p0 given in Table A-1 is based on body temperatures of 20 C. We did not
have data on fish body temperatures during migration so we did attempt to correct for
deviations from this temperature. Flying insects exhibit core body temperatures between
33◦C and 45◦C, even during short flights [25, 100]. We assume that flying insects
◦ operate at body temperatures of 40 C during migration flights. We therefore corrected p0 given by [101] from 25◦C to 40◦C following the UTD correction described in [102].
A.2.2 Sensitivity analysis
The derivation of equations for walking, swimming, and flying animals described
above requires assuming values and body mass dependencies of a number of
morphological and biomechanical parameters. An analysis of the sensitivity of migration
distance equations to the particular parameter values assumed in the derivation is given in Table A-2. In particular, the sensitivity analysis focused on two important properties
of distance equations: the predicted body mass scaling exponent, d, and the r 2 statistic
computed after fitting the equation to data. From Table A-2, it as apparent that changes
in the scaling of morphological variables and maintenance metabolism, and changes in
the value of p0 have only minor effects on the predicted mass dependence of maximum migration distance and the model r 2.
To evaluate sensitivity, each parameter tested was individually increased or decreased by 10% relative to the value used in the original derivation of distance equations. In the case of some parameters, larger changes in parameter values were explored based on values reported in the literature. r 2 statistics were computed by fitting
74 equations to maximum migration distance data assuming homoscedastic errors as described in the Statistical analysis section above. In the case of the flying equation, assuming departures from geometric similarity in body frontal area (Ab),wingspan (Lw ), or wing plan area (Aw ) result in changes in the functional form of Equation (2–8) with
respect to M0. However, these changes in functional form cause only minor changes in the shape of the predicted function, and consequently result in only minor changes
in the agreement between the model and data as indicated by r 2 values. Because
of changes in functional form, the scaling exponent, d, is no longer the only variable
affecting the mass-scaling of YT , and it is therefore omitted from Table A-2. Parameters
that only affect the y0 term in Equations (2–6)–(2–8) (main text) were omitted from the
sensitivity analysis. Additionally, increasing or decreasing the value of f , Cd , Ab, W , Aw parameters by 10% did not change the predicted mass dependence of the equation for
flying animals, and did not result in detectable changes in r 2 values relative to the values used in the original derivation of the flight equation described above (i.e. r 2 = 0.19 for all parameter combinations).
Table A-1. Empirical values of the normalization constant p0. Taxon p0 value reference fish (20◦C) 0.43 [103] marine mammals 3.9 Assumed terrestrial mammals 3.9 [104] birds 3.6 [105] (non-passerines) birds (passerines) 6.3 [105] flying insects (40◦C) 1.9 [101]
75 Table A-2. Sensitivity of distance equations to variation in input parameters. The Parameter value column shows minimum and maximum value of the corresponding parameter used to determine sensitivity. The r2 column indicates the r 2 value computed after increasing or decreasing the corresponding parameter and fitting the new equation to data . The d column indicates the value of the body mass scaling exponent after increasing or decreasing the corresponding parameter. * d approximated as described above. Taxon Parameter Parameter value r2 d min/max min/max min/max Walking 0.3 0.36 Lc Lc ∝ M0 /M0 0.57/0.57 0.35/0.33* 0.08 0.23 vwalk vwalk ∝ M0 /M0 0.57/0.57 0.33/0.39* 0.67 0.83 Pmtn Pmtn ∝ M0 /M0 0.57/0.56 0.38/0.3* Swimming 0.30 0.36 Lb Lb ∝ M0 /M0 0.65/0.65 0.30/0.31 0.6 0.74 As As ∝ M0 /M0 0.66/0.61 0.32/0.27 0.67 0.83 Pmtn Pmtn ∝ M0 /M0 0.66/0.56 0.35/0.25 p0 0.39/0.47 (fish) 0.66/0.61 - 3/6 (marine mammals) Flying 0.67 0.83 Pmtn Pmtn ∝ M0 /M0 0.15/0.15 0.5/0.34 0.6 0.74 Ab Ab ∝ M0 /M0 0.16/0.2 - 0.3 0.36 Lw Lw ∝ M0 /M0 0.2/0.1 - 0.6 0.74 Aw Aw ∝ M0 /M0 0.15/0.21 -
p0 1.7/2.1 (insects) 0.19/0.18 - 3.5/4.2 (non-passerines) 5.7/6.9 (passerines)
Table A-3. Body mass and migration distance data. Mass is mean body mass. Distance is maximum reported migration distance. * mass assumed based on similarity in size to Anax junius.
Species Movement Mass (kg) Mass ref. Distance (Km) Distance ref. Mode
Acrocephalus scirpaceus Flying 0.011 [37] 6000 [37] Agelaius phoenicus Flying 0.052 [37] 2500 [37]
76 Agrotis infusa Flying 0.00033 [106] 800 [107] Agrotis ipsilon Flying 2.00E-04 [108] 1800 [109] Anas acuta Flying 0.94 [110] 5500 [111] Anas crecca Flying 0.35 [37] 5000 [37] Anas discors Flying 0.4 [37] 11000 [37] Anas querquedula Flying 0.33 [37] 9000 [37] Anax junius Flying 0.0012 [99] 2800 [112] Anser caerulescens atlantica Flying 3.5 [37] 5000 [37] Anser caerulescens Flying 2.5 [37] 5000 [37] Anser erythropus Flying 1.9 [113] 4000 [113] Anser indicus Flying 2.2 [110] 1200 [114] Anthus spinoletta Flying 0.024 [37] 1500 [37] Aphis fabae Flying 8.80E-07 [97] 1300 [115] Apus apus Flying 0.042 [37] 12000 [37] Archilochus colubris Flying 0.0033 [37] 6000 [37] Arenaria interpres Flying 0.14 [116] 5700 [116] Aythya ferina Flying 0.9 [37] 7500 [37] Aythya fuligula Flying 0.66 [37] 4500 [37] Branta bernicla Flying 1.4 [37] 6500 [37] Branta canadensis Flying 3.5 [117] 3500 [118] Branta hutchinsii Flying 2 [119] 3500 [118] Branta leucopus Flying 1.8 [37] 3200 [37] Bucephala clangula Flying 0.92 [37] 3000 [37] Calcarius lapponicus Flying 0.035 [37] 6500 [37] Calidris canutus Flying 0.15 [120] 16000 [120] Calidris mauri Flying 0.047 [121] 3200 [121] Calidris tenuirostris Flying 0.24 [94] 5400 [122] Caprimulgus vociferus Flying 0.053 [37] 6000 [37] Ceyx picta Flying 0.015 [37] 2000 [37] Chaetura pelagica Flying 0.024 [37] 10000 [37] Charadrius falklandicus Flying 0.05 [37] 3600 [37] Charadrius vociferus Flying 0.095 [37] 10000 [37] Chlidonias niger Flying 0.07 [37] 10000 [37] Chordeiles minor Flying 0.062 [37] 11000 [37] Chrysococcyx lucidus Flying 0.036 [37] 5500 [37] Ciconia nigra Flying 6 [37] 6500 [37] Clangula hyemalis Flying 0.87 [37] 5000 [37]
77 Coracias garrulus Flying 0.15 [37] 10000 [37] Crex crex Flying 0.17 [37] 10000 [37] Cuculus canorus Flying 0.11 [37] 12000 [37] Cygnus columbianus Flying 6.8 [110] 5900 [123] Cygnus cygnus Flying 9.4 [110] 2000 [124] Danaus plexippus Flying 0.00057 [98] 3600 [125] Dendroica kirklandii Flying 0.014 [37] 1900 [37] Dendroica striata Flying 0.015 [110] 12000 [35] Dolichonyx oryzivorus Flying 0.042 [37] 11000 [37] Falco naumanni Flying 0.7 [37] 8600 [37] Falco peregrinus Flying 0.7 [110] 8600 [126] Falco sparverius Flying 0.12 [37] 6000 [37] Ficedula hypoleuca Flying 0.016 [37] 7000 [37] Fringilla coelebs Flying 0.026 [37] 5000 [37] Fulica atra Flying 0.84 [37] 4000 [37] Gallinago gallinago Flying 0.082 [37] 3500 [37] Grus grus Flying 9.8 [37] 4800 [37] Grus americana Flying 6.9 [37] 4000 [37] Grus canadensis Flying 4.4 [37] 4000 [37] Halcyon sancta Flying 0.043 [37] 3900 [37] Helicoverpa zea Flying 0.00021 [127] 1600 [128] Hemianax ephippiger Flying 0.001 * 3000 [129] Hirundo rustica Flying 0.019 [37] 12000 [37] Hirundo spilodera Flying 0.021 [37] 2500 [37] Hylocichla mustelina Flying 0.051 [130] 4600 [130] Junco hyemalis Flying 0.022 [37] 4000 [37] Lanius collurio Flying 0.01 [37] 11000 [37] Larus fuscus Flying 0.8 [37] 6500 [37] Larus ridibundus Flying 0.28 [37] 4000 [37] Lathamus discolor Flying 0.062 [37] 2500 [37] Limosa lapponica Flying 0.37 [93] 12000 [131] Luscinia luscinia Flying 0.024 [110] 8500 [132] Luscinia svecica Flying 0.02 [37] 6000 [37] Mergus albellus Flying 0.68 [37] 4500 [37] Merops apiaster Flying 0.052 [37] 10000 [37] Merops nubicus Flying 0.06 [37] 12000 [37] Merops ornatus Flying 0.026 [37] 4800 [37]
78 Molothrus ater Flying 0.044 [37] 2000 [37] Motacilla flava Flying 0.022 [37] 8000 [37] Musca vetustissima Flying 1.40E-05 [133] 600 [134] Muscicapa striata Flying 0.022 [37] 13000 [37] Muscisaxicola macloviana Flying 0.022 [37] 5000 [37] Myzomela sanguinolenta Flying 0.024 [37] 2500 [37] Nomophila noctuella Flying 2.10E-05 [135] 2400 [136] Notiochelidon cyanoleuca Flying 0.012 [37] 8000 [37] Numenius borealis Flying 0.26 [37] 14000 [37] Numenius tennuirostris Flying 0.45 [37] 6000 [37] Nysius vinitor Flying 3.90E-06 [137] 300 [138] Oceanites oceanicus Flying 0.04 [37] 15000 [37] Oenanthe oenanthe Flying 0.033 [37] 1400 [37] Olor buccinator Flying 9.8 [37] 2500 [37] Pantala flavescens Flying 8.80E-05 [139] 3500 [140] Patagona gigas Flying 0.018 [37] 800 [37] Phalaenoptilus nuttalii Flying 0.052 [37] 4000 [37] Philemon citreofularis Flying 0.15 [37] 2400 [37] Philemon corniculatus Flying 0.18 [37] 1600 [37] Philomachus pugnax Flying 0.065 [37] 15000 [37] Phoebis sennae Flying 0.00016 [37] 1500 [37] Phoenicurus phoenicurus Flying 0.02 [37] 6000 [37] Phylloscopus trochilus Flying 0.0087 [110] 15000 [141] Piranga olivacea Flying 0.029 [37] 7000 [37] Plectrophenax nivialis Flying 0.048 [37] 6000 [37] Pluvialis fulva Flying 0.12 [37] 13000 [37] Pogonocichla stellata Flying 0.021 [37] 200 [37] Progne subis Flying 0.049 [130] 7600 [130] Pseudaletia unipuncta Flying 0.00019 [142] 1600 [143] Puffinus puffinus Flying 0.46 [37] 12000 [37] Puffinus tenuirostris Flying 0.56 [110] 12000 [35] Pyrocephalus rubinus Flying 0.014 [37] 4000 [37] Riparia riparia Flying 0.012 [37] 10000 [37] Sarkidiornis melantos Flying 2 [37] 3900 [37] Schistocerca gregaria Flying 0.002 [144] 5000 [145] Selasphorus rufus Flying 0.0037 [146] 3900 [147] Selasphorus sasin Flying 0.0032 [110] 810 [146]
79 Sphyrapicus varius Flying 0.05 [37] 3500 [37] Spodoptera exigua Flying 5.40E-05 [148] 3700 [149] Stellula calliope Flying 0.0028 [37] 5000 [37] Sterna dougallii Flying 0.11 [37] 6000 [37] Sterna fuscata Flying 0.18 [37] 10000 [37] Sterna maxima Flying 0.45 [37] 8000 [37] Sterna paradisaea Flying 0.013 [18] 38000 [18] Streptopelia turtur Flying 0.15 [37] 6000 [37] Sturnus vulgaris Flying 0.082 [37] 1000 [37] Sylvia borin Flying 0.018 [110] 7000 [132] Sylvia communis Flying 0.018 [37] 9000 [37] Tachycineta bicolor Flying 0.02 [37] 5500 [37] Tadorna ferruginea Flying 1.2 [110] 3800 [114] Terpsiphone viridis Flying 0.015 [37] 1800 [37] Thalasseus bergii Flying 0.3 [37] 1600 [37] Tringa glareola Flying 0.068 [37] 5000 [37] Tringa stagnatalis Flying 0.078 [37] 6500 [37] Tringa totanus Flying 0.12 [37] 6500 [37] Turdus ilaris Flying 0.098 [37] 5000 [37] Turdus iliacus Flying 0.055 [37] 6500 [37] Turdus migratorius Flying 0.077 [37] 6400 [37] Tyrannus forficatus Flying 0.043 [37] 4000 [37] Upupa epops Flying 0.07 [37] 5000 [37] Urania fulgens Flying 0.00042 [150] 1900 [151] Vanellus vanellus Flying 0.24 [37] 4500 [37] Vireo olivaceous Flying 0.019 [37] 10000 [37] Zonotrichia albicollis Flying 0.026 [37] 4500 [37] Zosterops lateralis Flying 0.018 [37] 2000 [37] Alosa aestivalis Swimming 0.29 [152] 140 [153] Alosa pseudoharengus Swimming 0.28 [152] 140 [153] Alosa sapidissima Swimming 1 [154] 370 [154] Balaena mysticetus Swimming 69000 [155] 5800 [156] Balaenoptera musculus Swimming 99000 [155] 8700 [157] Carcharodon carcharias Swimming 550 [158, 159] 11000 [158] Cetorhinus maximus Swimming 3900 [160] 9500 [161] Clupea harengus Swimming 0.16 [162] 1500 [163] Cololabis saira Swimming 0.18 [164] 500 [165]
80 Delphinapterus leucas Swimming 1400 [155] 2200 [166] Eschrichtius robustus Swimming 30000 [155] 7300 [167] Eubalaena glacialis Swimming 28000 [155] 5700 [168] Hippoglossus stenolepis Swimming 300 [169] 1200 [170] Isurus oxyrinchus Swimming 58 [159, 171] 2400 [171] Lamna ditropis Swimming 98 [159, 172] 3000 [172] Lampetra fluviatilis Swimming 0.06 [173] 100 [174] Megaptera novaeangliae Swimming 30000 [155] 8500 [175] Mirounga leonina Swimming 320 [176] 3000 [176] Odobenus rosmarus Swimming 1000 [177] 1800 [178] Oncorhynchus keta Swimming 3.9 [174] 2500 [179] Oncorhynchus nerka Swimming 2.5 [180] 1100 [180] Oncorhynchus tshawytscha Swimming 15 [174] 1100 [174] Petromyzon marinus Swimming 0.88 [181] 280 [181] Physeter macropcephalus Swimming 45000 [182] 5000 [183] Pleuronectes platessa Swimming 1 [184] 280 [184] Prionace glauca Swimming 6.7 [159, 185] 3200 [185] Rhincodon typus Swimming 34000 [186] 13000 [187] Scomber scombrus Swimming 0.7 [188] 2200 [189] Thunnus orientalis Swimming 200 [190] 7600 [191] Thunnus thunnus Swimming 240 [192] 12000 [192] Tursiops truncatus Swimming 140 [193] 1100 [194] Xiphias gladius Swimming 22 [195, 196] 2500 [195] Acinonyx jubatus Walking 42 [197] 40 [198] Alces alces Walking 480 [117] 200 [199] Antidorcas marsupialis Walking 32 [200] 360 [201] Antilocapra americana Walking 55 [202] 260 [203] Camelus bactrianus Walking 690 [155] 200 [204] Canis lupus Walking 37 [205] 500 [206] Capra sibirica Walking 130 [207] 100 [207] Capreolus capreolus Walking 27 [208] 84 [209] Capreolus pygargus Walking 40 [210] 500 [210] Cervus canadensis Walking 270 [117] 190 [211] Cervus nippon Walking 53 [155] 100 [212] Connochaetes taurinus Walking 140 [213] 400 [213] Crocuta crocuta Walking 59 [214] 80 [215] Dicrostonyx groenlandicus Walking 0.054 [155] 5.4 [216]
81 Equus zebra Walking 240 [217] 100 [218] Kobus kob Walking 79 [155] 350 [219] Lemmus lemmus Walking 0.1 [220] 32 [221] Lemmus sibiricus Walking 0.052 [155] 5.4 [216] Lepus californicus Walking 3 [205] 35 [222] Lepus timidus Walking 2.4 [223] 10 [223] Loxodonta africana Walking 3900 [155] 240 [224] Microtus fortis Walking 0.068 [155] 5 [225] Odocoileus hemionus Walking 65 [117] 160 [203] Odocoileus virginianus Walking 76 [226] 52 [227] Ovibos moschatus Walking 480 [228] 320 [229] Ovis canadensis Walking 71 [230] 40 [231] Peromiscus leucopus Walking 0.021 [155] 15 [232] Procapra guttorosa Walking 28 [233] 280 [234] Puma concolor Walking 50 [235] 50 [236] Rangifer tarandus Walking 76 [237] 1200 [238] Saiga tatarica Walking 39 [239] 500 [239] Ursus americanus Walking 100 [205] 140 [229] Vulpes fulva Walking 5.4 [205] 65 [229]
82 APPENDIX B DERIVATION OF DISTRIBUTIONS, A NOTE ON THE USE OF BAYES’ RULE, AND SUPPLEMENTARY SIMULATION RESULTS B.1 True Distance Distribution (TDD) and a Comment on the Use of Bayes’ Rule
The TDD is calculated assuming that the searcher is located at the origin and prey
are distributed according to a Poisson spatial process with intensity η. In the absence of any further information, we can compute the density of the distance to the nearest
target L. To do this, for a subset A ⊂ R2, denote the number of targets in A by N(A).
By definition P(N(A) = k) = e−η|A|(η|A|)k /k! where |A| is the area of A. It follows that
−ηπl2 P(L > l) = 1 − P(N(Bl ) = 0) = 1 − e , where Bl denotes the ball of radius l centered at the origin. The TDD density therefore satisfies
d (l) = − P(L l) = 2 le−ηπl2 . θT dt > ηπ (B–1)
This is the Rayleigh distribution and is notable because it increases until its mode at √ 1/ 2ηπ, after which the density decays rapidly like a Gaussian.
In Equation (B–1) in the main text we introduced a modification of an intrinsic step length distribution that improves search performance by incorporating signal data. The modification has the form of a Bayesian posterior distribution, but strictly speaking, this is not an implementation of Bayes’ Rule. A more probabilistically rigorous approach to incorporating signal data would be the following. After conducting an olfactory scan, an ideal predator would use the TDD as a prior to compute a Bayesian posterior distribution
ν|H for the distance to the target. Let the prior ν(l) = θT (l), then the posterior distribution for the distance to the target is
P(H = h|l) ν(l) ν(l|H = h) = ∫ ∞ (B–2) P(H = h|l) (l) dl 0 ν
where the likelihood function, P(H = h|l)), is computed as described in the main text. Identifying the optimal strategy hinges on whether it is possible to characterize an optimal step length distribution for a given ν(l|H = h). One might try to pose this as
83 a variational problem. Let P denote the space of all probability densities on R+ then a signal-modulation strategy can be defined in terms of a functional : P → P. So, using this notation, the two functionals studied in this work are TDD which is simply the
identity functional and Levy which satisfies Levy (ν) ≡ θL for all ν. For an appropriate set of strategies , one seeks an optimal strategy, ∗ = argmin∈{E[τ]} where τ is the random hitting time of the target by a searcher using strategy
B.2 Robustness of Results to Search Conditions
B.2.1 Target Density
In the simulations presented in the main text, we assume target density is one
prey per 106 square predator body lengths. This density is a realistic low prey density based on field estimates of prey densities for a variety of predators (e.g. [75, 240],[214]
and references therein). However, to determine whether our results hold at even lower
prey densities, we repeated simulations after decreasing prey density by an order of
magnitude (i.e. one prey per 107 square predator body lengths). Results from these low
density simulations are shown in Figure B-1. As in Figure 3-2A in the main text, mean search times of the visual-olfactory Levy´ and visual-olfactory TDD strategies decrease
rapidly as the ratio of the olfactory radius to the vision radius (ro /rv ) increases above
one. Moreover, these two strategies exhibit similar performance for large ro /rv as in the
results shown in the main text for higher prey density. B.2.2 Signal Emission Rate
Simulations presented in the main text were conducted assuming the mean number
of scent encounters per τo units of time was equal to 100 at a distance of one predator
body length from a prey item (i.e. λa = 100). We repeated simulations after reducing
λa to 10 encounters per τo units of time. Results are consistent with those presented in the main text (Fig. B-2). Mean search times of visual-olfactory Levy´ and TDD predators
decrease with increasing ro /rv . Search times of these two strategies also become more
similar for large ro /rv .
84 B.2.3 Variation in Predator Scanning Times
Between successive steps visual predators pause for τv units of time before taking another movement step, whereas visual-olfactory predators pause for τo units of time. Typical pause durations between successive movements of a wide variety of animals in the field range from ≈ 1 s to ≈ 60 s [56]. Here we explore the robustness of the qualitative patterns shown in the main text to changes in the duration of the scanning phase for both visual-olfactory and visual predators. Scanning times may affect the relative performance of search strategies because some strategies (e.g. visual Levy)´
pause more frequently than others. Moreover, differences between τv and τo determine
the relative amounts of time spent scanning by visual and visual-olfactory predators.
Figure B-3 shows mean search time as a function of ro /rv for a range of values of τo and
τv . In all panels, mean search time decreases with increasing ro /rv and mean search times of visual-olfactory Levy´ and TDD are substantially shorter than mean search times of the visual strategies for at least some range of ro /rv . It is worth noting that the relative
performance of the Levy´ strategies versus the TDD strategies does depend on the absolute value of τv and τo . This is because Levy´ strategies tend to go into the scanning
phase more often and search times of these strategies therefore depend more strongly on scanning times. Note that visual strategies sometimes outperform visual-olfactory
strategies for small ro /rv . This tends to occur when τo > τv because visual-olfactory
predators spend more time scanning, even though they spend a similar amount of time
moving. B.3 The Role of No-signal Events
In the main text, we discuss the potential importance of no-signal events, in which
the searching predator samples for olfactory signals and receives zero signal (i.e. h = 0). Figure 3-4 in the main text shows how the behavior of visual-olfactory predators
can be altered when h = 0, depending on the length scale at which olfactory signals
are transmitted. This effect can be understood in more detail by looking at the effect of
85 no-signal events on the likelihood function P(H = 0|l), which is shown in Figure B-4. When h = 0, the likelihood that the source is nearby is very low. When this likelihood is
multiplied by a visual strategy θ(l), the result is a visual-olfactory strategy that has a low probability of making a short step. Figure B-4 shows that as ro /rv increases, this region
of low probability becomes larger, effectively increasing the minimum step size that a signal modulated strategy will take after a no-signal event. The effect of zero signal
on the Levy´ strategy is particularly strong because the probability of making relatively
short steps is large, but the likelihood that a source is nearby given that h = 0 is low.
Because of this property, this strategy is much more strongly influenced by receiving
no signal than the TDD strategy. Another common model used in simulations of animal movement, the exponential step length distribution, also has this property.
To further explore the effect of no-signal events, we performed the following
modification to the simulations described in the main text. We began with a predator
that samples for olfactory signals during the scanning phase as the visual-olfactory predators do. If the predator received a signal of h > 0, the next step length was drawn
from a Pareto distribution as described for the visual Levy´ strategy in the main text.
However, when h = 0, the predator drew a step length from the distribution resulting from applying Equation [1] in the main text, with h = 0. In other words, the predator
behaved as a visual Levy´ predator when h > 0 but as a visual-olfactory Levy´ predator when h = 0. This is a convenient way to determine whether using no-signal events to exclude local regions of space is sufficient to improve search performance, or whether it is also necessary to use events where h > 0. Results of this simulation show that altering behavior in response to no-signal events alone is sufficient to improve search performance at low target density (Figure B-5). For example, when ro /rv ≈ 20 predators that respond with visual-olfactory behavior when h = 0 have mean search times that are
33% shorter than mean search time of visual Levy´ predators.
86 6
2 ×10 2 ● ● 6 ●
●
1 ×10 1 ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● meansearch time ● ● ● ● 0 0 5 10 15 r /r o v
Figure B-1. Mean search times for visual Levy´ (orange line), visual TDD (blue line), visual-olfactory Levy´ (orange circles) and visual-olfactory TDD (blue circles) predators. 200 replicate simulations were performed for each combination of strategy × ro /rv . The following parameters values were used: a = 1, rv = lm = 50a, mean inter-target distance was 3162a, τv = 1 s, τo = 30 s, and λa = 100 units of scent per τo . 4 ●
5 ×10 5 ●● 4 ● ● ● ● ● ● ● ● 3 ×10 3 ● ● ● 4 ● ● ● ● ● ● ● ● meansearch time 1 ×10 1 0 5 10 r /r o v
Figure B-2. Mean search time with reduced rate of scent emission. Symbols as in Fig. B-1. The following parameters values were used: a = 1, rv = lm = 50a, mean inter-target distance was 1000a, τv = 1 s, τo = 30 s, and λa = 10 units of scent per τo . Each point represents mean of 200 replicate simulations.
87 iueB-3. Figure
ersnsma f10 elct simulations. replicate 1000 of mean represents was distance used: were values parameters the of of values combinations Fig. on in results as of Symbols dependence parameters. of lack showing matrix Plot mean search time
8 ×10 3 3 ×10 4 1 ×10 5 8 ×10 3 3 ×10 4 1 ×10 5 8 ×10 3 3 ×10 4 1 ×10 5 10 8 6 4 2 0 10 8 6 4 2 0 00246810 8 6 4 2 0 10 8 6 4 2 0 τ v =1 1000 τ v a and and ,
τ 3 3 ×10 4 5 3 3 ×10 4 5 o 8 ×10 1 ×10 8 ×10 1 ×10 00246810 8 6 4 2 0 10 8 6 4 2 0 λ aaeesrnigfo o30 h following The 300. to 1 from ranging parameters a τ v =6 τ =60 100 = a 88 r B-1 o 1 = /r v nt fsetper scent of units aesrpeetdifferent represent Panels . , r v =
3 4 5 l 8 ×10 3 ×10 1 ×10 m v 50 = =300 a eninter-target mean , τ o ahpoint Each . τ τ τ o o o =1 =300 =60 τ o and τ v 1.0 0.8 0.6 0.4
Likelihood 0.2 0.0 0 200 400 600 l (body lengths)
Figure B-4. Likelihood functions (P(H = 0|l)) resulting from receiving h = 0 scent signals in a particular scanning period. When the ratio of olfactory to vision radius is small (solid black curve: ro /rv = 0.25; dashed green curve: ro /rv = 1), encountering zero units of scent reduces the likelihood only very near the predator. As ro /rv increases, the likelihood becomes small for many body lengths from the predator (dotted dark blue curve: ro /rv = 5; dot-dashed light blue curve: ro /rv = 10). Parameters as in figure B-3 with τo = 30 and τv = 1. 5
3.5×10 ● ● ● ● ● 5 ● 2×10 4 meansearch time 5×10 0 5 10 15 r /r o v
Figure B-5. Mean search time of a visual-olfactory Levy´ searcher that alters visual behavior only when h = 0. Parameters as in Fig. B-1. Solid line indicates visual Levy´ predator. Dashed line indicates visual-olfactory Levy´ predator from Fig. B-1. Each point represents mean of 200 replicate simulations.
89 APPENDIX C MODEL OF SCENT PROPAGATION AND DEPENDENCE OF REGIME TRANSITIONS ON SIGNAL PROPAGATION LENGTH C.1 Scent Propagation
We model scent propagation in turbulence as packets that appear at the prey position x0 according to a Poisson arrival process and move as a Brownian motion. From the predator’s perspective, this is equivalent to encountering a random number of units of scent, H ∼ Pois(to R(|x−xo|)), at its location x during a scanning phase of length to , where R is the rate of scent arrival. Denoting ℓ = |x − x0|, under these assumptions,
h −to R(ℓ) the likelihood of h encounters is P(H = h|ℓ) = [to R(ℓ)] e /h!. To derive R(ℓ), let u(x) represent the mean concentration of scent at predator position x emitted by a prey item located at position x0. The steady-state diffusion process without advection is described by
0 = Du(x) − µu(x) + λδ(x0) (C–1) where D represents the combined molecular and turbulent diffusivity (m2s−1), µ represents the rate of dissolution of scent patches (s−1), and λ represents the rate of scent emission at the prey (s−1). In two dimensions, the rate of scent patch encounters √ a x R(l) = 2πD u( ) = µ by a predator of linear size located at is given by − ln(aψ) ℓ where ψ D . This implies
λK0(ψℓ) R(ℓ) = 2 (C–2) −πψ ln(ψa)
where K0 represents a modified Bessel function of the second kind. Two terms are sufficient to characterize the scent environment: the typical propagation length ro , which corresponds to the distance at which a predator will register on average one unit of scent per scanning period, and the expected number of encounters per unit to at a distance of one body length from the prey.
90 C.2 Dependence of Regime Break on Signal Propagation Length
To determine whether the density at which linear regimes transitioned to non-linear
regimes depended on the length scale of predator scent detection, we repeated
simulations to compute (ρ) over a range of values of the olfaction radius ro . Figure
C-1 shows that the prey density at which the linear regime transitions to a sublinear
regime decreases as ro increases. Thus, when prey scent propagates over a longer distance, the sublinear scaling of encounter rate persists to lower prey density.
C.3 Encounter Rate of a Predator with Perfect Sensing and Response, and Non-Zero Encounter Radius
Suppose that a predator is located at the origin of a two-dimensional environment containing prey distributed according to a Poisson spatial process with intensity ρ.
The distance between the predator and the nearest target is given by the Rayleigh
distribution, which has density
2 p(ℓ) = 2ρπℓe−ρπℓ . (C–3)
We wish to compute the expected time it takes for a predator with perfect sensing and response and sped v, to reach the encounter radius re of this nearest target. That is to say, if R ∼ Rayleigh(ρ), we aim to calculate max(R − re, 0).
∫ ∞ max(R − re, 0) = (ℓ − re)p(ℓ)dℓ re 1 √ = √ (1 − erf(re πρ)) , 2 ρ
∫ x 2 = √2 e−z dz where erf(x) π 0 . The expected hitting time (and therefore the encounter rate) is the product of speed and the inverse of this quantity, and expanding in the small and large ρ reveals three distinct scaling regimes: for small ρ, the encounter rate is proportional to the square root
of prey density; for order one values of prey density, the scaling is linear; and for very
large ρ the scaling is exponential.
91 To observe the square root scaling, simply note that erf(x) → 0 as x → 0. It follows √ that (ρ) ∼ 2 ρ in this regime. For larger ρ, the error function expands as follows √ 2 ρ r 2πρ (ρ) = √ ∼ 2πre ρ e e . 1 − erf(re πρ)
Because re and ρ are small in the parameter regime of interest, there is a range of
ρ, roughly from 10 to 100, for which encounter rate scales roughly linearly with ρ (i.e.
2 ere πρ ≈ 1). This is seen in Figure 4-1. As ρ becomes large, the scaling is exponential; however, for the cases of interest here (i.e. relatively low prey density), the exponential regime is not relevant.
C.4 Encounter Probabilities in the Sparse Regime
When prey density is very sparse, each prey target exists essentially in isolation.
This is why the empirically observed probability of encounter with nearby targets
stabilizes for low prey density (see Fig 4-5.) When completely isolated, the encounter
probability is simply the probability of hitting a circle of radius re before exiting a
concentric circle of radius 2ro starting from an intermediate circle of radius ro . This
is an exactly solvable problem for certain classical random processes, but we do not yet
have the analytical tools to solve such a problem for our imperfectly sensing predators.
We can however, get a rough sense of how the encounter probability scales with the
fundamental ratio ro /re by looking at the form of the solution for a standard Brownian
motion diffusing in the above geometry. For this purpose, we consider the Bessel
process R(t) that corresponds to the radial distance of a two-dimensional Brownian
motion with diffusivity D from the origin, which satisfies following Itoˆ form stochastic
differential equation [241]
D √ dR(t) = dt + 2DdW (t), R(0) = r. R
92 The probability that this process hits the level re before 2ro is given by the solution to the ODE D Dp′′(r) + p′(r) = 0 r
with p(re) = 1 and p(2ro ) = 0. The solution is readily shown to be
ln(2r ) − ln(2r) p(r) = o ln(2ro ) − ln(re)
which, plugging in the initial condition r(0) = ro yields
ln 2 p(ro ) = . (C–4) ln 2 + ln( ro ) re
The approximation is successful because in the presence of signal, the likelihood function in the Bayesian update, Equation (4–1), truncates the power law tail of the
default Pareto distribution to be exponential instead. Random walks with exponential
jump tails are diffusive in character, meaning that Brownian motion can give a somewhat
authentic scaling in ro and re. Furthermore, note that the hitting probability for Brownian motion is insensitive to its diffusivity, meaning we do not have to attempt to tune the Brownian motion to match the imperfectly sensing searcher. On the other hand, the effective diffusivity of the imperfectly sensing searcher is certainly state dependent because larger signal magnitudes lead to shorter jump lengths. A further defect of the
Brownian approximation is that it will always overestimate the encounter probability because the imperfect searcher will occasionally experience zero signal hits when
somewhat distant from the target. This means imperfectly sensing searchers will
occasionally sample from the jump distribution with heavy tail and increase its chance of
escape before reaching the target.
93 1.7 1.5 1.3 Densityregime at transition 1.1
500 600 700 800 900 1000
olfaction radius (body lengths)
Figure C-1. Breakpoint between low density linear regime and sublinear regime as a function of the predator olfaction radius ro .
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111 BIOGRAPHICAL SKETCH Andrew Hein grew up in Auburn, Alabama, near the Tallapoosa river. He became interested in understanding living things at a young age, under the tutelage of a nameless creek behind his parents’ house. After a brief and unsuccessful career as a taxidermist, he entered grammar school, where he was an average student. Eventually, he attended and graduated from Auburn University in Zoology. After working as a biologist in Panama, he moved to Gainesville, Florida to pursue a Ph.D. He received his
Ph.D. from the University of Florida in the summer of 2013.
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