Ben-Gurion University of the Negev
From the SelectedWorks of Amos Bouskila
2010
Games Played by Predators and Prey
Amos Bouskila
Available at: http://works.bepress.com/amos_bouskila/3/
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Bouskila A. (2010) Games Played by Predators and Prey. In: Breed M.D. and
Moore J., (eds.) Encyclopedia of Animal Behavior, volume 2, pp. 6-11 Oxford:
Academic Press.
© 2010 Elsevier Ltd. All rights reserved.
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Games Played by Predators and Prey
A. Bouskila, Ben-Gurion University of the Negev, Beer Sheva, Israel
ã 2010 Elsevier Ltd. All rights reserved.
Introduction
Game theory has been very useful in the understanding
of the behavior of animals. Game theory models and the
concept of evolutionarily stable strategy (ESS) provided
sound explanations to a variety of phenomena that could
not otherwise be fully understood. A game theoretic
approach should be used to understand the behavior of
animals whenever there are reasons to believe that the
strategy or the behavior of one organism is affected by the
behavior of the other and vice versa. The mathematical
tools used to solving game theory problems generate
predictions regarding the best response of each player to
the strategy of the opponent. Initially, most game theory
models dealt with different individuals within a species.
With time, asymmetric games were analyzed, and later on,
this was expanded to include games between individuals
of different species. Predator–prey games are a special
type of asymmetric games, in which the players are
engaged in a predator–prey relationship and often belong
to different species. The players do not necessarily have to
belong to different species (e.g., as in cannibalistic relationships), but the examples in this study only refer to
predator–prey game models between different species.
As in other types of games between animals, one can
investigate the predator–prey game on two different time
scales: the game may describe situations and life-history
strategies that were selected for at the evolutionary time
scale, or it could describe behaviors and strategies
operating within the life of the individual, often termed
ecological time scale.
The Types of Predator–Prey Games
Considered
in search of alternative prey. Hugie showed that the distributions for the waiting times of the predator and the prey
have different shapes, and only rarely was the waiting time
of the predator longer than that of the prey.
Games of Spatial Distribution
Habitat selection games involve the physical location in
which the players spend their activity time. In one of the
first predator–prey games described, Iwasa analyzed the
vertical migration of zooplankton species and their predators in lakes or in the sea (great depth during daytime
and near the surface at night). Previous explanations
included effects of the physical environment or biotic
relationship between zooplankton and phytoplankton.
However, a habitat selection game based on predator
avoidance at the time of high predator efficiency better
explained many observed characteristics of the vertical
migration.
Intuitively, in habitat-selection games, the prey is
expected to concentrate its activity where its food is
abundant, while the predator seeks the habitat or microhabitat where it can capture the most prey. In fact, these
considerations are much more complicated due to the
involvement of trade-offs in the strategy of each player,
and one of the most obvious trade-offs is between food
and safety: often, the habitat that has the highest abundance of food exposes the animal to higher predation
risk. Habitat selection games are the most common
predator–prey games modeled and discussed so far. As
we shall see later, due to the nature of the stable solutions,
at times, we find the results of these games quite
unintuitive.
Games of Temporal Distribution
Games of Vigilance and Search Intensities
Predators and prey may be involved in games of temporal
nature, the most general of which are games that address
the question of when should prey and predators choose to
be active outside their shelter or roost. A different type of
game, often termed the ‘waiting game’ or ‘shell game,’ also
belongs here: a prey animal escapes into a shelter from
which it cannot know whether the predator is still waiting
outside the shelter. These models calculate simultaneously
both the optimal emergence time of the prey from the
shelter and the appropriate length of time for the predator
to wait for the prey outside the shelter, before it moves on
Even when the time of activity and its locations have been
determined, the predator and the prey may be involved in
games that determine how much effort each one should
invest in detection of the other. The prey may invest time
in vigilance, thereby sacrificing foraging time or forging
efficiency. The predator may determine how much (in
terms of energy and time) it should invest in search
activities. Search activities often involve much more
energy expenditure than resting, expose the predator to
risks of injury or risks from its own predators, and reduce
the time available for social interactions.
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Games Played by Predators and Prey
Games of Pursuit and Escape Behaviors
After the prey has been detected, both predator and prey
still need to determine how much effort to invest in
pursuit and in escape, respectively. Before predator–prey
games were investigated specifically, Stewart used a
genetic model to find the appropriate search strategy of
a predator and the corresponding escape behavior of the
prey, after it had encountered cues of the predator. Later
on, Vega-Redondo and Hasson investigated pursuit deterrence: which behaviors can a prey animal use to manipulate and reduce the pursuit motivation of the predator. In
this case, the investigators were seeking an honest signal
by the prey that would clarify to the predator that the
pursuit will not lead to a successful capture.
Games of Life-History Parameters: Growth,
Birth, and Death Rates
While the previous types of games represent the evolution of behavioral decisions of predators and prey in
situations formed while they are engaged in a game,
there are games between predators and prey in which
their life-history parameters are assumed to be determined at a larger evolutionary scale. Such games may
involve growth rate decisions or other growth decisions.
For example, Bouskila and colleagues modeled the timing
of the switch of a prey animal from one growth phase to
another, considering the ability of predators to evolve and
modify their search strategy to adjust for this change. In
addition, games have been proposed to address the coevolution of characters that affect birth and death rates of
predators and prey (such as body size) toward a stable
solution that maintains coexisting populations of the two
types of organisms.
Games of Traits: Thermal Physiology
Another set of predator–prey games at a large evolutionary scale concern the co-evolution of physiological and
morphological traits. The evolution of physiological
adaptation under a situation of a game was recently
entered into a framework of a game between competing
conspecifics, through an Ideal Free Distribution game.
This concept was expanded to a predator–prey game by
Mitchell and Angiletta, leading to conclusions regarding
the effects of the predators on the evolution of physiological specialty. For example, under severe predation pressure, prey animals are predicted to evolve toward being
generalists in thermal preference, rather than specialists.
Prey that specialize on a narrow range of temperatures
spend time in specific patches that maintain this range of
temperatures and facilitate predation. Thermal specialization among prey animals can be a stable solution only
when predation pressure is mild.
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The Type of Models Used
Various approaches have been used to model
predator–prey games. Analytical solutions are often
sought to solve the simplest games. Models are formulated
as a set of equations, including the fitness functions of
predators and prey, sometimes in matrix form. Equilibrium points are found analytically through the simultaneous calculation of the derivatives of the functions. These
models often necessitate simplifying assumptions, such as
assuming that all players within a category (predators
or prey) are identical. While this assumption has been
useful in simplifying the models, there are cases in which
more complicated elements need to be included in order
to capture the essence of the system described by the
model. In such cases, computing-intensive simulations
are employed for their solution by using a state-variable
dynamic game or an evolutionary algorithm. The former
calculates the best response of a mutant in a group and
then allows the rest of the group to copy the statedependent solution found by the mutant. This process is
cycled as many times as needed to reach a stable solution,
in which the best reply to the group’s strategy is the same
strategy. In order to adjust this game to a predator–prey
game, the state-dependent solutions are found simultaneously both for the predator and the prey. The evolutionary algorithms have a game concept embedded in their
structure, because new genomes are formed either by
genetic mutations or recombinations, and they compete
against all other genomes. A stable solution is reached
when a genome proliferates and cannot be invaded by
new genomes, and here too, the process is run simultaneously for genomes of prey and predators. Mitchell combined an evolutionary algorithm with an Individual-Based
Model, to determine the fitness consequences of the
actions taken by the different genomes. All together,
these techniques enable the incorporation of such concepts as the states of individual animals and their spatial
distribution, which have been shown in other disciplines as
very useful tools for reaching solutions of complex evolutionary problems. As it is sometimes done, Alonzo analyzed each of the games among prey individuals and
among predators separately, and then the full game
between predators and prey was analyzed and compared
to the partial games.
Empirical Work Used
Difficulties
The theory of predator–prey games was mainly developed
in recent years. Empirical studies have been conducted
only rarely in the past in a way that demonstrates that
animals indeed use a game-theory solution. Apart from
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Games Played by Predators and Prey
the fact that the theory that might have led to such
empirical studies was not very well developed, there are
objective difficulties that stem from the fact that predation
events are involved here. Predation events in general, not
necessarily in game situations, are quite rare to observe in
nature. It is thus quite complicated to design a study in a
natural system that includes predation rates or predation
events. One solution to this problem has been to transfer
such studies to seminatural conditions or even to laboratory conditions. An additional difficulty stems from the
nature of the game itself. When animals are engaged in a
game situation, simple observation cannot easily verify if
a game is going on. Observing the end result of the game
may hide the behavioral options that were not chosen or
led to unstable solutions. In order to test the existence
of the game, quite often, the game needs to be perturbed,
that is, animals need to get cues for a different situation,
and only then may their behaviors be evaluated under the
framework of game. Here, too, the ability to change the
conditions of the game are limited when done in nature
and are more likely to be done under more controlled
conditions. Thus, occasionally, the empirical work mentioned in games of predators and prey is compatible with
the concept of such games but cannot always demonstrate
the existence of the game as the best or only explanation
for the observed pattern. A few examples of field studies,
as well as studies in seminatural arenas and in the lab, are
listed as follows. Such studies are often mentioned in
theoretical work, and in some cases, have contributed to
the development of the theory of predator–prey games.
Experiments in Seminatural Field Conditions
Altwegg manipulated simultaneously the state of prey and
predators in order to analyze the effectiveness of antipredator behavior of tadpoles against their invertebrate predators while they are at different states. The study was
done outdoors, in standard tubs, and it demonstrated that
predatory rates depended both on the behavior of the
prey and the predators. A somewhat similar study was
performed by Berger-Tal and Kotler, but with vertebrates
both as prey and as predators, in a large aviary (Figure 1):
the energetic state of owls and their prey (desert gerbils)
were simultaneously manipulated in order to shed some
light on the game between these players. Unlike the
previous study, the predators in this study were not very
sensitive to the state of the prey, while the prey definitely
modified their forging behavior depending on the hunger
level of the predators.
An important consideration that predators need to
consider while they pursue prey is their own exposure
to risk of injury or risk from their own predators. One of
the empirical studies that addressed the game between
foxes and their prey involved experiments in the same
aviary mentioned earlier. Berger-Tal and colleagues
manipulated the risk of injury to foxes when they foraged
in food trays. Foxes demonstrated that they consider the
risk of injury in their decisions of time allocation, and this
has important implications for predator–prey games in
which the prey spends time in microhabitats that may
impose high risk of injury to the predator.
Laboratory Experiments
Hammond and colleagues studied habitat selection of
tadpoles and dragonflies in laboratory experiments.
Empirical Work Used
Field Observations
Quinn and Cresswell found that predators preferentially
attack prey according to their vulnerability, rather than
according to their numbers. They performed experiments
with model birds at different vulnerability positions and
recorded the attacks on these birds. This result is compatible with game theory models that predict the common
interest of nonvulnerable prey and predators, both against
vulnerable prey individuals. Bouskila manipulated presence and absence of snakes in the Mojave Desert and used
seasonal changes in activity of the snakes to study the
habitat selection of rodents and their predators. Rodents
avoided habitats in which snakes were placed, and also
habitats in which snakes are likely to choose for ambush,
even when they were not placed there. Results were
compatible with a model that described the simultaneous
habitat selection of a predator and its prey. Additionally,
the same model was also compatible with observations of
snake movements in a rich oasis embedded in a dry desert
matrix.
Figure 1 An example of a two-compartment large enclosure
suitable for observing and manipulating elements of
predator–prey games. The enclosure includes infrared sensors to
monitor transitions of foxes between the compartments as well
as electronic seed trays, to monitor individual rodent visits at
various stations. Photo: A. Bouskila.
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Games Played by Predators and Prey
They recorded choice of habitat (rich or poor in prey
resource) under three treatments: prey alone, predator
alone, and both species together. These experiments
tested and confirmed some of the predictions generated
by predator–prey game models, such as the preference of
predators for patches with abundance of prey food, which
the predators do not consume. In addition, the prediction
that prey animals are not sensitive to the density of their
own resources was confirmed too. In this study, a model
selection approach was used to choose among factors that
could potentially explain the patterns of space use that
were observed.
In some cases, experiments are performed in situations
where the predators and the prey are likely to be in a
game situation, but the experiment itself was not meant to
demonstrate or verify if the results fit any theory based on
games. For example, Dangles and colleagues describe the
optimal velocities of a spider for approaching and capturing a cricket and found that there are two speeds in which
the vulnerability of the cricket was maximized, and thus,
this approach speed was utilized by the spiders. Although
this study was performed without an underlying model to
test, it deals with simultaneous decisions of predators and
prey and may provide a basis for such a model.
Common Themes in Models and
Experiments
Cases in Which Predators Respond to Prey
Resource
In many cases, it has been found that predators should
distribute themselves according to the distribution of prey
resource, rather than according to the parameters that are
supposed to affect the predators directly. This nonintuitive
result is one of the most consistent results that emerged
from several predator–prey game models. This effect of
one player’s parameters on the second player is especially
pronounced when there is no competition or other intraspecific interactions within the population of each player.
In some of the models, when intraspecific interactions are
included, the predators are still affected by prey resource
distribution, but the prey too is affected by its food distribution. In such cases, prey distribution does not match the
distribution of resource, as we would expect according to
the Ideal Free Distribution model, rather it undermatches
the resources, that is, the proportion of animals in the rich
patches is smaller than the resource proportion. Other
considerations emerged when Hugie and Dill included
metabolic and foraging costs, and found that these caused
undermatching to the resource too.
Alonzo included the state of the players in the model
and found that another consideration emerged and caused
undermatching of prey resources by prey animals: individuals at lower states were forced to forage in the risky and
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rich food patch. Thus, the predicted resource matching
was not achieved in this game too.
Habitat preferences due to food distribution are often
traded off with safety considerations, if safety differs
among habitats. When the number of shelters or the
level of safety in a habitat is manipulated in models,
another general result emerges, namely, the prey is
strongly associated with the habitat that provided safety,
while the predators often concentrate in the habitat where
the predator has a higher success rate.
Efficient Predators Make Predator Avoidance
Ineffective
Predator prey game models have also shown that when a
predator is able to efficiently react to the change in the
prey distribution, the predator may render the prey antipredator behavior inefficient. In such cases, the prey cannot escape predation and as a result, the prey should
abandon all predator-avoidance strategies. Avoiding predation usually comes at a cost of foraging efficiency, due to
the food and safety trade-off mentioned earlier. In those
situations where the prey will gain nothing by antipredator
behaviors, the prey should attempt to collect food as much
as possible and at least grow as fast as possible.
Equally, during a thermal game where patches with
different temperatures were provided as the resource,
prey animals became indifferent to temperatures (i.e.,
evolved into generalists) when predation pressure was
high.
Another case in which antipredator behavior is predicted to be ineffective was described by Wolf and
Mangel. They analyzed a situation in which the prey
selects the antipredator behavior, while the predator
chooses the attack rate. At high attack rates, the prey
loses so much time following attacks that they are forced
to forage in the rich and dangerous habitat to avoid
starvation, basically abandoning their antipredator behavior. An interesting situation is thus predicted in such
systems: the predators should make many false attacks
(undistinguishable by the prey from true attacks) in
order to induce the prey to abandon its antipredator
behavior.
Implications and Importance of
Predator–Prey Games
Individual Behavior Implications
The recent development of models of predator–prey
games has demonstrated that the incorporation of the
game approach into the analysis of the predator–prey
relationship has important implications. For many years,
studies of predators and prey assumed, often for simplicity, that the prey is faced with a constant level of
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Games Played by Predators and Prey
predation, at least at a given time and habitat. Predators
were considered unresponsive to prey distribution or to
prey behavior. However, as in other disciplines within
animal behavior in which the game approach has been
incorporated, the development of predator–prey games
has allowed us to capture a much more precise image of
the situation by allowing predators and prey to choose
their behavior for the best strategy to both players.
Removing the assumption of rigid behavior of predators
has opened up the development of new hypotheses and a
different way to view the behavior of prey and, obviously,
predators too. The use of this approach is beginning to
change the view of predator–prey behavior in all situations in which a game is likely to take place.
Population and Community-Level Implications
Predator–prey models are usually calculated at the individual level. Nevertheless, the implications from the predicted behaviors of the individuals have been extended
at times to populations and communities. For example,
Brown and colleagues demonstrated that a predator–prey
game that predicts the activity time of each player given
that the resource for the prey is provided as a pulse,
increases the stability of population dynamics. At a larger
evolutionary scale, Brown and Vincent address coevolution of predators and prey in a community and
allow the number of predator and prey species to be
evaluated at the ESS point, as an emergent property of
the stable strategy. Depending upon conditions, the predators may either be keystone species (whose removal
may lead to prey species extinction) or may have insignificant impact on current populations of prey. Not only may
the number of species in the community be determined
by a predator–prey game, but in some cases, such a game
may lead to selection of certain characters, and in extreme
cases, even to speciation, such as toward two different
species with different thermal preference.
Future Challenges
Incorporating Realistic Assumptions
As in other types of models, the assumptions of the
predator–prey game models are introduced for simplification and to keep the models tractable. There is a wellknown trade-off between generality and specificity in
models: in general models, there might be simplifying
assumptions, but they might not be realistic for many
specific systems, or even not for any of the systems.
Now that some of the general patterns are beginning to
emerge from the current models, it seems useful to modify some of the assumptions and make them more realistic
(at the cost of generality, in some cases).
Experimental Design
In spite of the wave of game theory models of predator
and prey, there still is a great lack of empirical studies that
deal with such situations and even fewer studies that can
demonstrate that a game situation indeed exists between
predators and prey. As mentioned before, there are inherent difficulties in the design of experiments to test the
existence of a game between predators and prey. The
game often needs to be perturbed by either supplying
misleading cues or limiting the freedom of choice from
one of the players in order to check whether the reactions
of the other player are compatible with the existence of
game considerations in the interactions. For example, Sih
suggested five treatments to fully analyze a predator prey
habitat selection game: (1) prey alone, (2) predator alone,
(3) prey with restricted predator, (4) predator with
restricted prey, and (5) predators and prey free to move
between both habitats. This requirement is not impossible
to achieve, but it requires specific studies in controlled
environments, such as in the lab and in seminatural arenas
and enclosures. Such designs are beginning to be more
prevalent, and the best demonstrations will be achieved
when specific experiments will be coupled with specific
predator–prey game models, designed or adjusted to the
system in question.
See also: Empirical Studies of Predator and Prey
Behavior; Game Theory.
Multitrophic-Levels Games
Most of the current predator prey games deal with two
trophic levels. In one of the few exceptions, Rosenheim
added the consideration of a top predator and demonstrated that under such circumstances a predator–prey game
reaches very different predictions. Models have yet to be
developed in order to describe the simultaneous decisions
of several species within one of the trophic levels. The
more alternative prey species that are involved in the
game (or the more predator species), the more complicated the results may become. However, in such models,
the decisions of one player are likely to be less coupled to
those of any one player from the second trophic level.
Further Reading
Alonzo SH (2002) State-dependent habitat selection games between
predators and prey: The importance of behavioural interactions and
expected lifetime reproductive success. Evolutionary Ecology
Research 4: 759–778.
Altwegg R (2003) Hungry predators render predator-avoidance behavior
in tadpoles ineffective. Oikos 100: 311–316.
Berger-Tal O, Mukherjee S, Kotler BP, and Brown JS (2009) Look before
you leap: Is risk of injury a foraging cost? Behavioral Ecology and
Sociobiology 63: 1821–1827.
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Games Played by Predators and Prey
Bouskila A (2001) A habitat selection game of interactions
between rodents and their predators. Annales Zoologici Fennici
38: 55–70.
Bouskila A, Robinson ME, Roitberg BD, and Tenhumberg B (1998) Lifehistory decisions under predation risk: Importance of a game
perspective. Evolutionary Ecology 12: 701–715.
Brown JS, Kotler BP, and Bouskila A (2001) Ecology of fear: Foraging
games between predators and prey with pulsed resources. Annales
Zoologici Fennici 38: 71–87.
Cresswell W and Quinn J (2004) Faced with a choice, sparrowhawks
more often attack the more vulnerable prey group. Oikos
104: 71.
Hammond JI, Luttbeg B, and Sih A (2007) Predator and prey space use:
Dragonflies and tadpoles in an interactive game. Ecology 88:
1525–1535.
Hugie DM and Dill LM (1994) Fish and game: A game theoretic
approach to habitat selection by predators and prey. Journal of Fish
Biology 45: 151–169.
Iwasa Y (1982) Vertical migration of zooplankton: A game between
predator and prey. American Naturalist 120: 171–180.
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Mitchell WA (2009) Multi-behavioral strategies in a predator–prey game:
An evolutionary algorithm analysis. Oikos 118: 1073–1083.
Mitchell WA and Angilletta MJ Jr (2009) Thermal games: Frequencydependent models of thermal adaptation. Functional Ecology
23: 510–520.
Rosenheim JA (2004) Top predators constrain the habitat selection
games played by intermediate predators and their prey. Israel
Journal of Zoology 50: 129–138.
Sih A (2005) Predator–prey space use as an emergent outcome of a
behavioral response race. In: Barbosa P and Castellanos I (eds.)
Ecology of Predator–Prey Interactions, pp. 240–255. New York, NY:
Oxford University Press.
Stewart FM (1971) Evolution of dimorphism in a predator–prey model.
Theoretical Population Biology 2: 493–506.
Vega-Redondo F and Hasson O (1993) A game-theoretic model of
predator–prey signaling. Journal of Theoretical Biology 162:
309–319.
Wolf N and Mangel M (2007) Strategy, compromise, and cheating in
predator–prey games. Evolutionary Ecology Research
9: 1293–1304.
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