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Encyclopedia of Theoretical Ecology

Alan Hastings (Editor), Louis Gross (Editor)

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This major reference is an overview of the current state of theoretical ecology through a series of topical entries centered on both ecological and statistical themes. Coverage ranges across scales—from the physiological, to populations, landscapes, and ecosystems. Entries provide an introduction to broad fields such as Applied Ecology, Behavioral Ecology, Computational Ecology, Ecosystem Ecology, Epidemiology and Epidemic Modeling, Population Ecology, Spatial Ecology and Statistics in Ecology. Others provide greater specificity and depth, including discussions on the Allee effect, ordinary differential equations, and ecosystem services. Descriptions of modern statistical and modeling approaches and how they contributed to advances in theoretical ecology are also included. Succinct, uncompromising, and authoritative—a “must have” for those interested in the use of theory in the ecological sciences.
Adaptive Dynamics
Adaptive Landscapes
Adaptive Vigilance Behavior
Age Structure
Allee Effects
Allometry and Growth
Animal Dispersal see Dispersal, Animal
Apparent Competition
Applied Ecology
Assembly Processes
Bayesian Statistics
Behavioral Ecology
Belowground Processes
Beverton–Holt Model
Bifurcations
Biogeochemistry and Nutrient Cycles
Birth–Death Models
Bottom-Up Control
Branching Processes
Cannibalism
Cellular Automata
Chaos
Coevolution
Compartment Models
Competition, Apparent see Apparent Competition
Computational Ecology
Conservation Biology
Continental Scale Patterns
Cooperation, Evolution of
Delay Differential Equations
Demographic Stochasticity see Stochasticity, Demographic
Demography
Difference Equations
Differential Equations, Delay see Delay Differential Equations
Differential Equations, Ordinary see Ordinary Differential Equations
Differential Equations, Partial see Partial Differential Equations
Discounting in Bioeconomics
Disease Dynamics
Dispersal, Animal
Dispersal, Evolution of
Dispersal, Plant
Diversity Measures
Dynamic Programming
Ecological Economics
Ecological Networks see Networks, Ecological
Ecological Stoichiometry see Stoichiometry, Ecological
Economics, Ecological see Ecological Economics
Ecosystem-Based Management see Marine Reserves and Ecosystem-Based Management
Ecosystem Ecology
Ecosystem Engineers
Ecosystem Services
Ecosystem Valuation
Ecotoxicology
Energy Budgets
Environmental Heterogeneity and Plants
Environmental Stochasticity see Stochasticity, Environmental
Epidemiology and Epidemic Modeling
Evolutionarily Stable Strategies
Evolutionary Computation
Facilitation
Fisheries Ecology
Food Chains and Food Web Modules
Food Webs
Foraging Behavior
Forest Simulators
Frequentist Statistics
Functional Traits of Species and Individuals
Game Theory
Gap Analysis and Presence/Absence Models
Gas and Energy Fluxes across Landscapes
Genetic Drift see Mutation, Selection, and Genetic Drift
Geographic Information Systems
Harvesting Theory
Heterogeneity, Environmental see Environmental Heterogeneity and Plants
Hydrodynamics
Individual-Based Ecology
Information Criteria in Ecology
Integrated Whole Organism Physiology
Integrodifference Equations
Invasion Biology
Landscape Ecology
Marine Reserves and Ecosystem-Based Management
Markov Chains
Mating Behavior
Matrix Models
Meta-Analysis
Metabolic Theory of Ecology
Metacommunities
Metapopulations
Microbial Communities
Model Fitting
Movement: From Individuals to Populations
Mutation, Selection, and Genetic Drift
Networks, Ecological
Neutral Community Theory
Niche Construction
Niche Overlap
Nicholson Bailey Host Parasitoid Model
Nondimensionalization
NPZ Models
Nutrient Cycles see Biogeochemistry and Nutrient Cycles
Ocean Circulation, Dynamics of
Optimal Control Theory
Ordinary Differential Equations
Pair Approximations
Partial Differential Equations
Phase Plane Analysis
Phenotypic Plasticity
Phylogenetic Reconstruction
Phylogeography
Plant Competition and Canopy Interactions
Plant Dispersal see Dispersal, Plant
Population Ecology
Population Viability Analysis
Predator–Prey Models
Presence/Absence Models see Gap Analysis and Presence/Absence Models
Programming, Dynamic see Dynamic Programming
Quantitative Genetics
Reaction–Diffusion Models
Regime Shifts
Reserve Selection and Conservation Prioritization
Resilience and Stability
Restoration Ecology
Ricker Model
Selection see Mutation, Selection, and Genetic Drift
Sex, Evolution of
Single-Species Metapopulations see Metapopulations
Single-Species Population Models
SIR Models
Spatial Ecology
Spatial Models, Stochastic
Spatial Spread
Species Ranges
Stability Analysis
Stage Structure
Statistics in Ecology
Stochasticity
Stochasticity, Demographic
Stochasticity, Environmental
Stoichiometry, Ecological
Storage Effect
Stress and Species Interactions
Succession
Synchrony, Spatial
Top-Down Control
Toxicology, Environmental see Ecotoxicology
Transport in Individuals
Two-Species Competition
Urban Ecology
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.
“Overall, the Encyclopedia of Theoretical Ecology is an excellent resource for anybody with a wider range of interests in ecology. It actually defines the field and serves as a starting point as well as a refresher on the basic ideas and approaches in theoretical ecology across a wide range of fields. . . . Its production quality and editorial thoroughness will ensure a lasting usefulness.”—Carsten F. Dormann Basic & Applied Ecology
“This is a comprehensive, informative, well-written work that should prove highly useful to academic, special, and public libraries supporting readers interested in ecology. . . . Highly recommended.”—B. C. Thomsett-Scott, University of North Texas Choice
“The encyclopedia clearly illustrates just how important theory is to all aspects of the real world of organisms and functioning ecosystems that it is now so essential for us to protect. . . . A readily accessible, wide-ranging source of information.”—John R. Haslett, Division of Integrative Animal Biology, University of Salzburg Biological Conservation
"Remarkable and authoritative . . . includes essential readings with clear presentations."—Evelyne Bremond-Hoslet Mammalia
"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

Adaptive Vigilance Behavior

Peter A. Bednekoff

Department of Biology, Eastern Michigan University, Ypsilanti, MI 48197

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.

I. 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 non-growing 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 while 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 trade-off 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.

II. 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:

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 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:

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:

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:

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, under the ESS, individuals feed at n1/n times the cooperative rate. 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 thatthe difference between the ESS and cooperative solutions is greatest for n=3 and declines with larger group sizes.

III. 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 non-detectors. 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:

.

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 non-detectors (including the focal individual. This sum simplifies such that the overall mortality is:

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

,

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

.

This expression is negative or undefined when the expected number of attacks is less than the group size, αT n. In this situation, zero vigilance is the best solution.

In this null model, a group size effect is present, but generally weak (Fig. 1). Since a strong effect of group size on vigilance has been observed many times in nature, this result suggests that the observed group size effect on vigilance probably depends upon collective detection to some extent.

The cooperative solution for the null model is difficult to state in general but for a pair of animals. When fewer than four attacks occur on average (αT4) this expression is undefined and zero vigilance (v*=0) is the best available vigilance level. Compared to the ESS solution for a pair (Fig. 2), without collective detection the cooperative vigilance level is lower than the ESS vigilance level.

The models in this entry differ from those of Parker and Hammerstein first in allowing for multiple attacks rather than a single attack. Their equivalent model produced an ESS of zero vigilance for all group sizes greater than 1. In the models described here which include perfect collective detection, the ESS level of vigilance falls to zero when the number of expected attacks is no bigger than the group size. Thus the two models produce similar results under the correct simplifying assumptions, but quite different results are possible when multiple attacks are considered.

While earlier models suggested ESS vigilance should be zero for pairs or larger groups, observations showed non-zero levels of vigilance in groups. This mismatch led to consideration of cooperative models of vigilance. The conclusions of these early models apply only in some cases. With perfect collective detection, cooperative vigilance is generally higher than the ESS level. Without collective detection, however, cooperative vigilance is generally lower. In the model with no collective detection, the cooperative response is zero vigilance while the ESS solution is moderate levels of vigilance when between 2 and 4 attacks are expected (Fig. 2) and the ESS level of vigilance is generally higher than the cooperative level anytime two or more attacks are expected.

IV. Partial collective detection

The models in this article have thus far only examined the extremes of perfect collective detection and no collective detection. Empirical studies indicate that collective detection is real yet slow and potentially faulty. For large groups, a model of imperfect collective detection would need to keep track of the overall contagion of information. This has not been done and is beyond the scope of this article. For a pair of animals, however, a model only needs to track the flow of information from a primary detector to the other member of the pair. Here the danger for the focal animal is

The focal animal is in danger when it fails to detect an attack (1-v). If the other animal fails to detect the attack (1-), the two are equally likely to be targeted by the predator. If the other animal detects the attack (with probability ), the focal animal escapes if it gets information but is the sole target for attack if it fails to get information (1-i) from the other individual where i is the probability that the focal animal receives a warning from the other animal.

Study of vigilance generally includes the separate advantages for collective detection and risk dilution, where risk dilution occurs when the predator chooses among of a group of uninformed prey. This distinction blurs, however, when collective detection is imperfect and takes time. If different members of the group have different amounts of information during the course of the attack, dilution depends strongly on when and how predators target prey for attack. If predators target prey early in an attack, risk is spread equally among the n group members. If predators wait until later in the attack, however, they can choose among the subset of the group that has not yet detected them. Thus dilution depends on how many other individuals are similarly uninformed of the attack. The formulation given here assumes that the predator chooses a target after primary detection, but before secondary detection is complete. Thus the predator ignores any individual with their head up at the time of the attack, and chooses with equal probability among the other individuals. This gives an advantage to individuals who are vigilant at the start of the attack. Other formulations would be appropriate if predators could target individuals that were the very last to become informed. Such formulations would place even greater advantage on personal detection of attacks.

Working with the mortality expression for imperfect collective, we solve and find that the ESS level of vigilance for a member of a pair is

while the cooperative level is .

For small numbers of attacks, these expressions may be negative-so that the internal optimum is zero vigilance. With more attacks, vigilance tends to increase under either the ESS or the cooperative criterion. The effects of information flow are more complicated. With increasing information passing from detector to non-detector, the ESS level of vigilance generally decreases. Cooperative vigilance, however, increases with greater information flow at moderately low attack rates but then decreases at higher attack rates. In all cases, cooperative vigilance is higher than non-cooperative only with i1/2 (Fig. 3). Other work has shown that i1/2 is also the criterion for coordinated vigilance to be advantageous. As pointed out by Steve Lima, the widespread observation that vigilance is not coordinated is evidence against cooperation. The potential fruit of cooperation, the public goods in this case, depend critically on the information that non-detectors get from detectors of attacks. In nature this information may be incomplete or unreliable, so each individual needs to gather information for itself. Individual vigilance is both more valuable and more costly than relying on social information. Thus with multiple attacks and imperfect collective detection, the norm is for each individual to invest in vigilance.

The ESS level of vigilance decreases with greater information passing from detector to non-detector. This is seen by examining the best response to the vigilance of the other group member (Fig. 4). With little or no information, the best reaction approaches the level for a single bird (7/8). When information only has a probability one-half of reaching a non-detector, the reaction does not depend on the vigilance of the other group member. Here any increase in detection with the vigilance of the other animal is exactly offset by decrease in dilution when information fails to get through. With i1/2 the best response is considerably lower than for a lone animal, and lower when the other animal is more vigilant. Observations show that animals are often much less vigilant in pairs than alone, suggesting considerable information flow and collective detection. On the other hand, there is much less information that animals react directly to the vigilance rates of others. Perhaps the danger to one member of a group is little influenced by the vigilance rates of others, either because information flow is moderate (i≈1/2) or due to some other factor not included in this model.

V. Risk allocation

Classic models of vigilance behavior generally considered perfect collective detection and a very limited number of attacks. Some outcomes of the models differ with imperfect collective detection and multiple attacks. Multiple attacks might occur over a substantial period of time, and conditions might change during this period. Consider the situation when group size changes over time. For simplicity, assume that the focal animal spends some portion of its time, p, alone and the rest of the time, q, paired with some other animal. Other assumptions are the same as above. If individuals face various situations with different levels of risk, they may allocate vigilance and feeding behavior across the situations with v1 being the vigilance in situation 1 and v2 the vigilance in situation 2.

In contrast to previous models of risk allocation, the models presented here use a linear relationship between feeding rate and detection behavior. This assumption simplifies the models, and also leads to the solution of zero vigilance in the safer situation. The results are striking in that the safer situation in a pair switches at i=1/2. This model of risk allocation does not yield a simple ratio of feeding rates because one feeding rate is always the maximum, 1. The model, nonetheless, illustrates that vigilance depends upon the schedule of other situations an animal faces. With i1/2, the ESS level of vigilance is zero when alone and 1-(1/αTp) when in a pair. Thus allocation has reversed the modest group size effect seen above with little collective detection. With moderate to high collective detection (i1/2), the vigilance in the pair goes to zero with the vigilance of an animal feeding alone depending on how often it is in a pair.

The result is remarkable and counterintuitive in that collective detection leads to an evolutionary stable solution where neither individual is vigilant, detection does not occur, and so collective detection never happens. Thus in any attack on a pair, one individual or the other will die. Because this model assumes that the proportion of time in a pair is not affect by vigilance levels, it implicitly assumes that group members that die are replaced before the next attack. The next section explores whether this combination of assumptions leads to the surprising results.

VI. Partner loss and mutual dependence

Most models of vigilance have assumed a stable group size. Models of risk allocation assume changes in risk factors, such as group size, but that the schedule of risk factor changes is not influenced by their feeding or vigilance rates. Now consider a model where group size changes as a result of mortality. Assume all individuals start in pairs and that collective detection is perfect (i = 1). The previous results for risk allocation provide an interesting foundation in that they do not predict vigilance in pairs. Therefore vigilance in pairs is due to protecting an irreplaceable partner-which is one kind of mutual dependence in fitness.

The assumptions for what happens when a predator attacks a pair are exactly as before. If the other animal dies in an attack, however, our focal animal faces all future attacks alone. The solution involves two vigilance rates, one in a pair and one alone. Survival is summed across all routes in which the focal individual survives, each weighted by its probability. To find foraging success, note that if when the other animal dies in the ith of j attacks, the focal animal forages in a pair for on average i/(j+1) of the total time period and alone the rest of the period.

The most important result of this model is that the ESS result is to be vigilant in a pair. With simple risk allocation, the ESS solution is zero vigilance. Thus each individual invests in the survival of its irreplaceable partner because having a partner increases its own survival. Potentially this is a widespread phenomenon for animals in the non-breeding season because animals lost to predation cannot be replaced until the next breeding season. It will only have a large effect, however, when loss of another individual strongly affects the fitness of a focal individual. This could happen if animals spend the non-breeding season in small, closed groups. Given that non-breeding groups are often fluid, mutual dependence might apply more widely to mated pairs that stay together year round. If mates are difficult to replace or mating increases with experience, then each member of the pair would do well to ensure that its partner survives. Thus individuals might often be "mate guarding" but in the broad sense of protecting their partner's survival and ability to reproduce. Perhaps mutual dependence might contribute to the year round territorial behavior of tropical birds.

VII. Conclusions

In this entry the initial models of vigilance were very simple yet considered multiple attacks. This exercise was very successful because it was able to recreate results from previous models while showing that these results are not general to multiple attacks. Extending these models to partial information worked well, though general and tractable models were only found for pairs of individuals and not for larger groups. For risk allocation and mutual dependence, the rationale for the first models is limited. The models for risk allocation developed here provide a baseline for analyzing mutual dependence. Here again, the models of mutual dependence were neither as simple nor as general as the models of vigilance they build upon and other models might more readily demonstrate mutual dependence. Thus simplifying assumptions for one purpose may complicate extending the model to another purpose. To explain the logic of potential contributing factors, it can be very helpful to examine a series of very simple models, each resting on its own foundations. When examining the relative importance of and interactions between contributing factors, more complicated models may be necessary. Simple and complex models inform each other, and a range of tools can be useful on any modeling job.

Further Reading

Bednekoff, P.A. 2001. Coordination of safe, selfish sentinels based on mutual benefits. Annales Zoologici Fennici 38: 5-14.

Bednekoff, P. A. 2007. "Foraging in the face of danger." Pp. 305-329 in Foraging. eds. D.W. Stephens, J.S. Brown, R.C. Ydenberg. Chicago: University of Chicago Press.

Bednekoff, P.A. and Lima, S.L. 1998. Randomness, chaos, and confusion in the study of anti-predator vigilance. Trend Ecol. Evol. 13: 284-287.

Bednekoff, P.A. and Lima, S.L. 1998 Re-examining safety in numbers: interactions between risk dilution and collective detection depend upon predator targetting behavior. Proceedings of the Royal Society B. 265: 2021-2026.

Bednekoff, P. A., and S. L. Lima. 2004. Risk allocation and competition in foraging groups: reversed effects of competition if group size varies under risk of predation. Proceedings of the Royal Society of London Series B: Biological Sciences 271: 1491-1496.

Lima, S.L. 1990. "The influence of models on the interpretation of vigilance." Pp. 246-267 in Interpretation and explanation in the study of animal behavior: Vol. 2. Explanation, evolution and adaptation. eds. M. Bekoff and D. Jamieson. Boulder, CO: Westview Press.

Lima, S. L., and P. A. Bednekoff. 1999. Temporal variation in danger drives antipredator behavior: the predation risk allocation hypothesis. American Naturalist 153: 649-659.

Parker, G.A. Hammerstein, P. 1985. Game theory and animal behaviour. In: Evolution: Essays in Honour of John Maynard Smith (Ed. by P.J. Greenwood P.H. Harvey), pp. 73-94. Cambridge: Cambridge University Press.

Pulliam, H.R. 1973. On the advantages of flocking. J. Theor. Biol. 38: 419-422.

Treisman, M. 1975. Predation and the evolution of gregariousness. I. Models for concealment and evasion. Anim. Behav. 23: 779-800.

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