“A bold and successful attempt to illustrate the theoretical foundations of all of the subdisciplines of ecology, including basic and applied, and extending through biophysical, population, community, and ecosystem ecology. Encyclopedia of Theoretical Ecology is a compendium of clear and concise essays by the intellectual leaders across this vast breadth of knowledge.”–Harold Mooney, Stanford University

“A remarkable and indispensable reference work that also is flexible enough to provide essential readings for a wide variety of courses. A masterful collection of authoritative papers that convey the rich and fundamental nature of modern theoretical ecology.”–Simon A. Levin, Princeton University

“Theoretical ecologists exercise their imaginations to make sense of the astounding complexity of both real and possible ecosystems. Imagining a real or possible topic left out of the Encyclopedia of Theoretical Ecology has proven just as challenging. This comprehensive compendium demonstrates that theoretical ecology has become a mature science, and the volume will serve as the foundation for future creativity in this area.”–Fred Adler, University of Utah

“The editors have assembled an outstanding group of contributors who are a great match for their topics. Sometimes the author is a key, authoritative figure in a field; and at other times, the author has enough distance to convey all sides of a subject. The next time you need to introduce ecology students to a theoretical topic, you’ll be glad to have this encyclopedia on your bookshelf.”–Stephen Ellner, Cornell University

“Everything you wanted to know about theoretical ecology, and much that you didn’t know you needed to know but will now! Alan Hastings and Louis Gross have done us a great service by bringing together in very accessible form a huge amount of information about a broad, complicated, and expanding field.”–Daniel Simberloff, University of Tennessee, Knoxville

Alan Hastings is a Distinguished Professor at the University of California Davis in the Department of Environmental Science and Policy. He is the Editor-in-Chief for the Theoretical Ecology Series (Academic Press) and the Journal of Mathematical Biology. Louis Gross is a Professor of Ecology, Evolutionary Biology, and Mathematics, and Director of the Institute of Environmental Modeling at the University of Tennessee.

Encyclopedia of Theoretical Ecology

By Alan Hastings, Louis J. Gross

UNIVERSITY OF CALIFORNIA PRESS

Copyright © 2012 The Regents of the University of California
All rights reserved.
ISBN: 978-0-520-26965-1

Contents

Contents by Subject Area, xi,
Contributors, xiii,
Guide to the Encyclopedia, xxi,
Preface, xxiii,
Adaptive Behavior and Vigilance Peter A. Bednekoff, 1,
Adaptive Dynamics J. A. J. Metz, 7,
Adaptive Landscapes Max Shpak, 17,
Age Structure Tim Benton, 26,
Allee Effects Caz M. Taylor, 32,
Allometry and Growth Andrew J. Kerkhoff, 38,
Apparent Competition Robert D. Holt, 45,
Applied Ecology Cleo Bertelsmeier; Elsa Bonnaud; Stephen Gregory; Franck Courchamp, 52,
Assembly Processes James A. Drake; Paul Staelens; Daniel Wieczynski, 60,
Bayesian Statistics Kiona Ogle; Jarrett J. Barber, 64,
Behavioral Ecology B. D. Roitberg; R. G. Lalonde, 74,
Belowground Processes James Umbanhowar, 80,
Beverton–Holt Model Louis W. Botsford, 86,
Bifurcations Fabio Dercole; Sergio Rinaldi, 88,
Biogeochemistry and Nutrient Cycles Benjamin Z. Houlton, 95,
Birth–Death Models Christopher J. Dugaw, 101,
Bottom-Up Control John C. Moore; Peter C. De Ruiter, 106,
Branching Processes Linda J. S. Allen, 112,
Cannibalism Alan Hastings, 120,
Cellular Automata David E. Hiebeler, 123,
Chaos Robert F. Costantino; Robert A. Desharnais, 126,
Coevolution Brian D. Inouye, 131,
Compartment Models Donald L. DeAngelis, 136,
Computational Ecology Stuart H. Gage, 141,
Conservation Biology H. Resit Akcakaya, 145,
Continental Scale Patterns Brian A. Maurer, 152,
Cooperation, Evolution of Matthew R. Zimmerman; Richard McElreath; Peter J. Richerson, 155,
Delay Differential Equations Yang Kuang, 163,
Demography Charlotte Lee, 166,
Difference Equations Jim M. Cushing, 170,
Discounting in Bioeconomics Ram Ranjan; Jason F. Shogren, 176,
Disease Dynamics Giulio De Leo; Chelsea L. Wood, 179,
Dispersal, Animal Gabriela Yates; Mark S. Boyce, 188,
Dispersal, Evolution of Marissa L. Baskett, 192,
Dispersal, Plant Helene C. Muller-Landau, 198,
Diversity Measures Anne Chao; Lou Jost, 203,
Dynamic Programming Michael Bode; Hedley Grantham, 207,
Ecological Economics Sunny Jardine; James N. Sanchirico, 213,
Ecosystem Ecology Yiqi Luo; Ensheng Weng; Yuanhe Yang, 219,
Ecosystem Engineers Kim Cuddington, 230,
Ecosystem Services Fiorenza Micheli; Anne Guerry, 235,
Ecosystem Valuation Stephen Polasky, 241,
Ecotoxicology Valery Forbes; Peter Calow, 247,
Energy Budgets S. A. L. M. Kooijman, 249,
Environmental Heterogeneity and Plants Gordon A. Fox; Bruce E. Kendall; Susan Schwinning, 258,
Epidemiology and Epidemic Modeling Lisa Sattenspiel, 263,
Evolutionarily Stable Strategies Richard McElreath, 270,
Evolutionary Computation James W. Haefner, 272,
Facilitation Michael W. McCoy; Christine Holdredge; Brian R. Silliman; Andrew H. Altieri; Mads S. Thomsen, 276,
Fisheries Ecology Elliott Lee Hazen; Larry B. Crowder, 280,
Food Chains and Food Web Modules Kevin McCann; Gabriel Gellner, 288,
Food Webs Axel G. Rossberg, 294,
Foraging Behavior Thomas Caraco, 302,
Forest Simulators Michael C. Dietze; Andrew M. Latimer, 307,
Frequentist Statistics N. Thompson Hobbs, 316,
Functional Traits of Species and Individuals Duncan J. Irschick; Chi-Yun Kuo, 324,
Game Theory Karl Sigmund; Christian Hilbe, 330,
Gap Analysis and Presence/Absence Models Jocelyn L. Aycrigg; J. Michael Scott, 334,
Gas and Energy Fluxes Across Landscapes Dennis Baldocchi, 337,
Geographic Information Systems Michael F. Goodchild, 341,
Harvesting Theory Wayne Marcus Getz, 346,
Hydrodynamics John L. Largier, 357,
Individual-Based Ecology Steven F. Railsback; Volker Grimm, 365,
Information Criteria in Ecology Subhash R. Lele; Mark L. Taper, 371,
Integrated Whole Organism Physiology Arnold J. Bloom, 376,
Integrodifference Equations Mark Kot; Mark A. Lewis; Michael G. Neubert, 381,
Invasion Biology Mark A. Lewis; Christopher L. Jerde, 384,
Landscape Ecology Jianguo Wu, 392,
Marine Reserves and Ecosystem-Based Management Leah R. Gerber; Tara Gancos Crawford; Benjamin Halpern, 397,
Markov Chains Louis J. Gross, 404,
Mating Behavior Patricia Adair Gowaty, 408,
Matrix Models Eelke Jongejans; Hans De Kroon, 415,
Meta-Analysis Michael D. Jennions; Kerrie Mengerson, 423,
Metabolic Theory of Ecology James F. Gillooly; April Hayward; Melanie E. Moses, 426,
Metacommunities Marcel Holyoak; Jamie M. Kneitel, 434,
Metapopulations Ilkka Hanski, 438,
Microbial Communities Thomas G. Platt; Peter C. Zee; Keenan M. L. Mack; James D. Bever, 445,
Model Fitting Perry De Valpine, 450,
Movement: From Individuals to Populations Paul R. Moorcroft, 456,
Mutation, Selection, and Genetic Drift Brian Charlesworth, 463,
Networks, Ecological Anna Eklöf; Stefano Allesina, 470,
Neutral Community Ecology Stephen P. Hubbell, 478,
Niche Construction John Odling-Smee, 485,
Niche Overlap Howard V. Cornell, 489,
Nicholson–Bailey Host Parasitoid Model Cheryl J. Briggs, 489,
Nondimensionalization Roger M. Nisbet, 501,
NPZ Models Peter J. S. Franks, 505,
Ocean Circulation, Dynamics of Christopher A. Edwards, 510,
Optimal Control Theory Hien T. Tran, 519,
Ordinary Differential Equations Sebastian J. Schreiber, 523,
Pair Approximations Joshua L. Payne, 531,
Partial Differential Equations Nicholas F. Britton, 534,
Phase Plane Analysis Shandelle M. Henson, 538,
Phenotypic Plasticity Mario Pineda-Krch, 545,
Phylogenetic Reconstruction Brian C. O’Meara, 550,
Phylogeography Scott V. Edwards; Susan E. Cameron Devitt; Matthew K. Fujita, 557,
Plant Competition and Canopy Interactions E. David Ford, 565,
Population Ecology Michael B. Bonsall; Claire Dooley, 571,
Population Viability Analysis William F. Morris, 582,
Predator–Prey Models Peter A. Abrams, 587,
Quantitative Genetics Paul David Williams, 595,
Reaction–Diffusion Models Chris Cosner, 603,
Regime Shifts Reinette Biggs; Thorsten Blenckner; Carl Folke; Line Gordon; Albert Norström; Magnus Nyström; Garry Peterson, 609,
Reserve Selection and Conservation Prioritization Atte Moilanen, 617,
Resilience and Stability Michio Kondoh, 624,
Restoration Ecology Richard J. Hall, 629,
Ricker Model Eric P. Bjorkstedt, 632,
Sex, Evolution of Jan Engelstädter; Francisco Úbeda, 637,
Single-Species Population Models Karen C. Abbott; Anthony R. Ives, 641,
SIR Models Lewi Stone; Guy Katriel; Frank M. Hilker, 648,
Spatial Ecology Alan Hastings, 659,
Spatial Models, Stochastic Stephen M. Krone, 666,
Spatial Spread Alan Hastings, 670,
Species Ranges Kevin J. Gaston; Hannah S. Smith, 674,
Stability Analysis Chad E. Brassil, 680,
Stage Structure Roger M. Nisbet; Cheryl J. Briggs, 686,
Statistics in Ecology Kevin Gross, 691,
Stochasticity (Overview) Matt J. Keeling, 698,
Stochasticity, Demographic Brett A. Melbourne, 706,
Stochasticity, Environmental Jörgen Ripa, 712,
Stoichiometry, Ecological James J. Elser; Yang Kuang, 718,
Storage Effect Robin Snyder, 722,
Stress and Species Interactions Ragan M. Callaway, 726,
Succession Herman H. Shugart, 728,
Synchrony, Spatial Andrew M. Liebhold, 734,
Top-Down Control Peter C. De Ruiter; John C. Moore, 739,
Transport in Individuals Vincent P. Gutschick, 744,
Two-Species Competition Priyanga Amarasekare, 752,
Urban Ecology Mary L. Cadenasso; Steward T. A. Pickett, 765,
Glossary, 771,
Index, 803,

CHAPTER 1

A

ADAPTIVE BEHAVIOR AND VIGILANCE

PETER A. BEDNEKOFF

Eastern Michigan University, Ypsilanti

Adaptive behavior is behavior that raises an animal’s fitness in its biotic and abiotic environment. The study of adaptive behavior is mainly the study of how behavior changes with environmental changes. Fitness is the relative contribution of an organism to genes in future generations. Because understanding fitness across the entire life of an organism is a daunting task (and tracking it across several generations even more so), researchers have commonly assumed a particular relationship between behavior in the short term and fitness. This has involved examining many behaviors using many fitness proxies and modeling techniques. This entry explores the simplifying assumptions about the relationship between behavior and fitness while concentrating on a single type of behavior, watching for predators. Though the focus here is on such vigilance behavior, the approach employed may be applied to many other types of behavior.

BASELINE MODELS OF VIGILANCE BEHAVIOR

Anti-predator vigilance involves pauses in other behavior, such as feeding, in order to scan the environment for predators. Researchers commonly assume that that foraging animals lift their heads independently of one another and share information about detected attacks. Shared information about attacks is known as collective detection. The original model of anti-predator vigilance showed that in bigger groups each individual could scan less while maintaining the same probability of an undetected attack. If each individual is vigilant v proportion of the time, then an attack goes undetected when no individual is vigilant, which occurs a fraction (1 – v)n of the time, where n is the number of individuals in the group. This model did not relate this directly to fitness but determined that animals could scan less in larger groups at the same risk of an undetected attack. Subsequent studies have built upon the assumptions of independent scanning and collective detection of attacks but added dilution of risk. Here, the predator can only effectively attack one individual (or perhaps a small portion of the group), even when all individuals are unaware of the impending attack.

Other researchers have supposed that animals maximize survival while maintaining a required level of food intake. This second criterion can make sense if food increases do not impact future reproduction very much, for example, during the nongrowing season for animals that have small growth during this period. These two criteria for adaptive behavior—maximizing food intake while maintaining a set probability of surviving and minimizing risk of attack and maintaining a set level of food intake—obviously cannot both hold true simultaneously. They can, however, both be approximations of a larger truth. In general, fitness will depend on the value of food and the probability of surviving. In simple models, vigilance increases survival but decreases food intake, and thereby future reproductive output. This entry discusses models of the optimal level of vigilance that build upon the work of Parker and Hammerstein but differ from their models by allowing multiple attacks and assuming future reproduction is a linearly increasing function of foraging intake.

To develop heuristic models of vigilance in groups, assume that foragers feed for some extended time, T, before reproducing. Animals can forage at any rate between 0 and 1, and there is a direct tradeoff with vigilance—food intake is proportional to 1 – v. An animal’s total foraging intake across the period is thus proportional to (1 – v)T, and reproductive success at the end of the period is proportional to foraging intake—V = k(1 – v)T. In order to reproduce, however, animals must survive this period. If they are attacked at rate – according to a Poisson process, then the probability of survival decreases exponentially with the attack rate, period length, and the mortality per attack—S = Exp[-αTM]. The probability of dying in an attack increases with foraging rate. For a single individual the mortality rate is M = (1 – v). In a group, the mortality rate may depend on the actions of others. This is the topic of the next section.

PERFECT COLLECTIVE DETECTION

When a predator attacks a group, the focal individual dies if it is not vigilant and is not warned by other members of the group. For the effects of others, the classic assumption is perfect collective detection: the focal individual is warned of the attack if any group member is vigilant at the time. Thus, the attack succeeds only if no member of the group is vigilant at the time. If the focal individual is joined by n – 1 other individuals, each vigilant [??] of the time, this equals (1 – v)(1 – [??])n-1. If the predator must choose among these n unaware prey and does so equally, the risk is 1/n for each group member. Thus, the probability that the focal individual dies in an attack is

M = (1 – v)(1 – [??])n-1/n.

As we would expect, animals are safer when they are more vigilant and when they are in larger groups. Because the fitness of the focal individual depends on the vigilance of others, game theory is needed to find the solution. The evolutionarily stable strategy (ESS) can be found by differentiating the fitness function with respect to v, setting it to zero, and solving for v. This gives v as a function of [??]—in this case:

v* = 1 – n/αT (1 – [??])n-1.

Because an individual can be warned by others, it can rely on their vigilance to some extent. In a group where everyone else is very vigilant, the best response is to be less vigilant. In a group where others are not vigilant, the best response is to be more vigilant. In between, there is a level of vigilance that is the best response to itself. To find this evolutionarily stable strategy, set [??] = v and again find v. In this case the evolutionarily stable level of vigilance is

v* = 1 – (n/αT)1/n.

The evolutionarily stable strategy here is equivalent to the Nash equilibrium from game theory. For comparison, the optimal cooperative strategy, or Pareto equilibrium, can be found by substituting [??] = v in the fitness function, differentiating, setting this equal to zero, and then solving for v. With this first model, the optimal cooperative strategy is

vc* = 1 – (1/αT)1/n.

These expressions differ only in the numerator of the ratio in the second right-hand term. Since this second right-hand term equals the feeding rate, individuals feed at n1/n times the cooperative rate under the ESS. These results replicate findings that ESS levels of vigilance are lower than cooperative when collective detection is perfect. Analyzing n1/n as n varies shows that the difference between the ESS and cooperative solutions is greatest for n = 3 and declines with larger group sizes.

NO COLLECTIVE DETECTION

A null model indicates how anti-predator behavior would be different without collective detection. In this model, individuals that are vigilant at the start of the attack escape, and the predator targets one of the nondetectors. As the number of detectors, i, goes up, the effective group size for dilution decreases by the same number. As before, our focal individual is in danger only when it is not vigilant (1 – v) but now the effects of others must be summed across all possible numbers of detectors:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The possible number of detectors ranges from zero to n – 1—all group members other than our focal individual. Inside the summation, the factorial gives the number of ways of having i detectors, the numerator of the right-hand term gives the probability of any one such combination, and the denominator gives the dilution of risk among the n – 1 nondetectors (including the focal individual). This sum simplifies such that the overall mortality is

M = (1 – v)(1 – [??]n)/n(1 – [??]).

Once again the optimal response depends on the vigilance of others,

v* = 1 – n(1 – [??])/αT(1 – [??]n),

and the ESS occurs in the case when the optimal response is to match the vigilance of others, which yields

v* = (1 – n/αT)1/n.

(Continues…)Excerpted from Encyclopedia of Theoretical Ecology by Alan Hastings, Louis J. Gross. Copyright © 2012 The Regents of the University of California. Excerpted by permission of UNIVERSITY OF CALIFORNIA PRESS.
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