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Behavioral Ecology Vol. 13 No. 3: 427-438
© 2002 International Society for Behavioral Ecology
Temporal partitioning and aggression among foragers: modeling the effects of stochasticity and individual state
Population Biology Section, The University of Amsterdam, Kruislaan 320, 1098 SM Amsterdam, The Netherlands
Address correspondence to S.A. Richards, who is now at the National Center for Ecological Analysis and Synthesis, 735 State Street, Suite 300, Santa Barbara, CA 93101-5504. E-mail: richards{at}nceas.ucsb.edu .
Received 1 January 2001; revised 30 July 2001; accepted 19 August 2001.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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In many natural systems, individuals compete with conspecifics and heterospecifics for food and in some cases, individuals have been observed to partition their foraging times or fight over food. In this study, I investigated when it is optimal for a consumer to partition time and be aggressive. I formulated an individual-based model of foraging and used game theory to find evolutionarily stable strategies (ESSs) that maximize the probability that consumers survive each day and acquire their daily food requirements. Consumers choose when to forage and when to behave aggressively during confrontations over food. Consumers are each associated with a state variable, representing the amount of food eaten, and a dominance ranking, which describes how likely they are to forage and fight for food. The ESS is sensitive to food abundance, consumer state, and the dominance ranking. When food is abundant, temporal partitioning is often an ESS where the dominant consumer forages first; however, partitioning is unlikely to be an ESS when food abundance is low. Fights over food are typically avoided but may be part of an ESS when food abundance is low, both consumers are hungry, or the time available for foraging each day is drawing to a close. Because the ESS is sensitive to consumer state, the stochastic nature of finding food often results in considerable variation in observed foraging dynamics from one day to the next, even when consumers adopt the same state-dependent strategy each day. Results are compared with empirical observations, and I discuss implications for consumer coexistence.
Key words: aggression, coexistence, foraging, game theory, temporal partitioning.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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There have been numerous theoretical studies that investigated the conditions under which multiple species can coexist on common resources (Armstrong and McGehee, 1980
).
Coexistence may occur when there are environmental heterogeneities, and
trade-offs exist among species where each species has a competitive advantage
over some part of the heterogeneity. Spatial and temporal heterogeneities in
the resource distribution that allow species to coexist may arise from
environmental processes that act independently of the consumers (e.g.,
Chesson and Warner, 1981;
Hastings, 1980;
Levins, 1979). Heterogeneities
may also arise from the consumers themselves through their resource
consumption abilities and population dynamics (e.g.,
Armstrong and McGehee, 1980;
Brown, 1989;
Levins, 1979;
Richards et al., 2000;
Vance, 1984). Species
coexistence may also be possible in the presence of consumer behavior because
it can reinforce existing heterogeneities; however, a theoretical treatment of
the conditions necessary for coexistence is incomplete
(Chesson and Rosenzweig,
1991). One example of behavior that has the potential to promote
coexistence is temporal partitioning of foraging
(Carothers and Jaksic, 1984)
because it may reduce resource overlap or any negative interactions that may
occur during co-feeding (e.g., time wasting or even mortality during
aggressive confrontations; Case and
Gilpin, 1974). It has also been suggested that temporal
partitioning may promote the coexistence of species that rely strongly on a
common resource (Kotler et al.,
1993; Kronfeld-Schor and
Dayan, 1999; Ziv et al.,
1993).
Schoener (1974b) surveyed
modes of resource partitioning among a wide range of sympatric species and
found that time appeared to be a far less common way to partition when
compared with habitat and food type. However, temporal partitioning has been
observed in a number of communities, including those of bats
(Kunz, 1973), lizards
(Pianka, 1973), raptors
(Jaksi
, 1982), and
desert rodents (Kotler et al.,
1993; Shkolnik,
1971; Ziv et al.,
1993). Despite observations of resource temporal partitioning,
there have been relatively few theoretical investigations of the conditions
necessary for it to be an evolutionarily stable strategy that promotes species
co-existence (Brown, 1989;
Carothers and Jaksi,
1984; Kronfeld-Schor and
Dayan, 1999).
Foraging theory suggests that feeding periods should be skipped if the cost
of waiting is less than the cost of feeding (e.g.,
Brown, 1989;
Brown et al., 2001;
Schoener, 1974a). Examples of
costs incurred when waiting include loss of energy reserves and mortality from
predation or starvation. When feeding, consumers may incur mortality costs
from predators and aggressive competitors. In many situations co-feeding may
yield benefits because it may decrease predation risk (e.g., through
vigilance) or increase the effective availability of prey (e.g., through prey
flushing or herding or through gaining access to other's territories)
(Overholtzer and Motta, 2000;
Schoener, 1971). Consumers may
also derive benefits by behaving aggressively toward their competitors because
such behavior may provide opportunities for stealing food or displace
competitors, thereby providing exclusive access to resources
(Caraco, 1979;
Case and Gilpin, 1974;
Robertson et al., 1976;
Whitehouse, 1997). The costs
and benefits associated with a particular feeding behavior will typically
depend on the state of the environment (e.g., feeding passively may be
profitable when resources are abundant and competitor densities are low);
costs and benefits are also likely to depend on the state of the individual
and its competitors (Houston and McNamara,
1999). For example, feeding aggressively may be optimal for a
forager when its energy reserves are low and risk of starvation is high or
when its competitors are well fed and less likely to escalate a confrontation.
Alternatively, passive feeding, or even refraining from feeding, may be
optimal if the forager is well fed (i.e., its risk of starvation is low) or if
its competitors are hungry and likely to partake in aggressive and dangerous
confrontations.
In this study, I investigated when it is optimal for an individual to
forage, and if so, when it is optimal for it to exhibit aggression toward its
competitors. I am interested in how the state of an individual and the states
of its competitors influence the optimal foraging decision. Specifically, I
considered the situation where consumers compete for common resources that
renew each day, which occurs in a number of communities, including
nectarivores (Possingham,
1988) and desert granivores
(Kotler et al., 1993;
Mitchell et al., 1990).
Consumer behavior is explicitly modeled by allowing each consumer to choose
when it forages (which is risky) and when it seeks shelter and waits (which is
risk free). If a consumer forages at the same time as its competitors, then it
may choose to be aggressive toward them. Aggressive encounters may result in
the successful stealing of food, or death, whereas being passive may result in
the loss of food to aggressors but is always safe. Each consumer's behavior
may depend on its own state (i.e., the number of resources it has consumed
that day) and also on the state of its environment (i.e., the time of day, the
abundance of resource remaining in the environment, and the state of the other
competitors). An important component of the model is that both foraging
success and mortality are treated as stochastic processes. At any time,
consumers may be in one of a number of states, depending on how successful
they have been at foraging.
To determine the optimal foraging behavior of a consumer, I treated resource competition as a dynamic game and applied game theory (GT) and stochastic dynamic programming (SDP). Because foraging is a game, an optimal strategy is the best response to the strategies of the other consumers. Here I considered the case where there are two consumers. The reason I only considered two consumers is because it is possible to determine the optimal strategies for this case. When the number of competitors increases, the formulation of the model and its analysis becomes increasingly complex. In the Discussion, I indicate how the model's results may be influenced by the presence of many consumers. I did not distinguish whether the two consumers are of the same species; however, the model does allow the consumers to have different food encounter rates, predation risks, or dominance during confrontations. The optimal foraging strategy for both consumers was evaluated for a variety of situations, and the model shows that the profitability of co-feeding and aggression may be sensitive to many factors, including the state of both consumers, time of day, resource abundance, predation risk, and each consumer's ability to find food and fight during a confrontation. The optimal strategies assume that consumers have perfect knowledge of their environment, and to determine the advantage of having such knowledge, I also compare results with much simpler foraging strategies (e.g., always feed aggressively if hungry). I compare the model's predictions with empirical observations and discuss implications for consumer coexistence.
The model
Suppose two consumers (A and B) compete for food during some time period
each day, which I refer to as the foraging period (FP). Both consumers are
assumed not to forage outside of the FP, and during this time the resource
renews. Individuals may not forage continuously through time because they
experience high morality from predators during part of the diel cycle
(Kotler et al., 1993) or
because environmental conditions (e.g., temperature and light) govern the
periods when feeding activities are possible
(Pianka, 1973). It is also
assumed that both consumers seek a fixed caloric quota each day, which is
consistent with observations on a number of species ranging from insects to
large mammals (Williams,
1962). If a consumer's quota is not met by the end of the FP, then
it does not survive to the start of the next FP and there is no advantage when
consuming more than the quota. At any time during the FP a consumer may choose
to forage for food, which is associated with a risk of predation, or it may
choose to seek shelter, which is risk free
(Barta and Giraldeau, 2000). If
a consumer feeds in the presence of its competitor, then it may choose to
actively interfere with it by adopting aggressive behavior. For simplicity
consumers are assumed to begin foraging as soon as they leave their shelter
(i.e., there is no travel cost to and from the feeding area).
Foraging dynamics are described using a Markovian model, where the FP is
divided into T small time steps of length 
t. Let
Fi denote the number of food items consumer i
seeks each FP to meet its daily caloric requirement (Fi is
an integer and i = A or B). Here I assume that the requirement is
independent of the time spent foraging (i.e., the added metabolic cost due to
foraging is low). At the start of each FP there are F food items up
for grabs. All model parameters are presented in
Table 1.
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First, consider the case when consumer i forages alone. The rate
it locates food items obeys a type II functional response and is denoted and
defined by
 | (1) |
As mentioned in the Introduction, co-feeding may be associated with either
net costs or benefits to feeding success and risk of predation. When consumer
i is co-feeding, its feeding rate becomes
iei(f), where
i is a constant. If i < 1
then feeding rates decline, which may occur when consumers experience
interference competition (not due to aggression) or when the resource is prey
that exhibit predator avoidance behaviors at higher predator numbers.
Alternatively, the i may be greater than 1,
indicating that group feeding benefits feeding rates, and this may occur when
co-feeding increases the effective availability of prey. The rate of predation
when co-feeding is given by 
imi. If
i < 1, co-feeding decreases the risk of
predation, which may occur when predators are less likely to attack or catch
prey in a group. However, if higher numbers of active foragers are more likely
to attract predators, then it is possible that i
> 1. For simplicity, constants of proportionality are used to model the
effects of co-feeding. Alternatively, new functional responses could be
derived from time-budget models; however, such analyses are typically
cumbersome, and for the case of feeding rates their form is often only
slightly different to that of a type II functional response
(Ruxton et al., 1992).
The probability that during a time step consumer i is killed by a
predator is approximated by
 | (2) |
 | (3) |
If both consumers choose to feed during a given time step and survive
predation, then there is a probability 
they will meet (i.e., at least
one notices the foraging success of the other). Suppose a meeting occurs and
only one of the two competitors finds food during the time step. In this case
a confrontation occurs, which is resolved using the standard hawk—dove
game (Maynard Smith, 1982).
Competitors can choose to be either passive (dove) or aggressive (hawk). If
both choose dove, then the consumer who found the food item gets to consume
it. If one consumer chooses hawk and the other dove, then the hawk gets the
food item. In both of these cases neither consumer experiences a mortality
risk from the other. If both choose hawk, then a contest ensues, with the
winner taking the food and the loser facing a probability
di of being killed in the contest. Consumers are
associated with a competitive weight, wi, and the
probability i wins a contest is given by
Wi/(WA +
WB).
Stochastic dynamic programming
I use SDP to determine when consumers should choose to search for food and
when they should seek shelter during the FP. This technique identifies the
strategy that will best achieve an objective in a stochastic environment. The
strategies available to both consumers are either wait or forage and, if
foraging, whether to act as a hawk or dove during a confrontation. Their
objective is to consume Fi food items before the
end of the FP. The environment is stochastic because finding food and death
are stochastic processes. SDP has been used extensively to gain insights about
how individual behaviors can influence the success of various activities,
including migration and reproduction
(Houston and McNamara, 1999;
Mangel and Clark, 1988). SDP
has also been applied to problems involving foraging; however, most previous
studies have concentrated on the situation where an individual is feeding in
isolation. Here SDP is applied to the situation where foraging success of an
individual depends not only on its own feeding strategy but also on its
competitor's feeding strategy (see also
Barta and Giraldeau, 2000;
Houston and McNamara, 1988;
Lucas et al., 1996;
McNamara et al., 1997).
At the start of any time step the system may be in one of a number of
states. States are denoted by a triplet (xA,
xB, f), where xi
describes the state of consumer i, and f is the number of
food items remaining in the habitat. If the consumer is dead, then
xi = dead, otherwise
xi is an integer ranging from 0 to
Fi indicating the number of food items consumed
since the start of the FP. Hereafter the symbols 
and ß are
reserved for referring to system states (i.e., and ß are
triplets).
To calculate the probability that the system moves from one state to another after each time step, we need to know the foraging behaviors that both consumers adopt during the time step. Let pi denote the probability that consumer i chooses to forage and qi denote the probability that if both consumers forage at the same time and confront over food then consumer i chooses to act as a hawk. Thus the strategies adopted during a time step can be described by the vector y = (pA, pB, qA, qB), which may depend on the current state of the system and time.
Suppose at time t the system is in state . The probability
that consumer i will survive and attain its food requirements by the
end of the FP, assuming both consumers continue to forage optimally, is
denoted Ri(,t,T). Thus
Ri is the probability i achieves its
goal and from here on it is referred to as the optimal expected reward. We
wish to identify the set of optimal foraging strategies for each state and
time step, denoted y*(,t), that maximizes
Ri[(0,0,F),0,T] for both
consumers. Note that the set of strategies that maximizes
RA is not necessarily the same set that maximizes
RB. I use game theory to identify the strategies that, if
adopted, then no change in strategy will give either consumer a greater reward
(Maynard Smith, 1982). From
here on I use the term "optimal strategies" to refer to such
strategies, and in the next section I show how they are evaluated. The optimal
state-dependent expected reward for consumer i at the end of the FP
(i.e., t = T) is simply
 | (4) |
,t,T) denote
the expected reward at time T for consumer i when the system
is in state at time t and the consumers choose pure strategy
z for the next time step and thereafter choose the optimal strategy.
A pure strategy is one where a particular behavior is chosen. There are seven
pure strategies: neither consumer forages (z = 1); only consumer B
forages (z = 2); only consumer A forages (z = 3); both
forage as doves (z = 4); both forage, A as a dove and B as a hawk
(z = 5); both forage, A as a hawk and B as a dove (z = 6);
and both forage as hawks (z = 7). Let
A(ß,,z) denote the probability the system moves
from state to state ß when pure strategy z occurs. This
probability is not difficult to formulate given the probabilities
Ei and Mi, and
applying the rules of the hawk—dove game. The expected reward for
consumer i when pure strategy z occurs, is given by
 | (5) |
,t) =
[pA*(,t),
pB*(,t),
qA*(,t),
qB*(,t)]. The optimal expected
reward can then be calculated using
 | (6) |
 | (7) |
 | (8) |
 | (9) |
 | (10) |
 | (11) |
 | (12) |
 | (13) |
t, then Equations 5-13 can be used to calculate the optimal
expected reward at time t. Repeating this process we can calculate
the optimal expected rewards for every possible state of the system over the
entire FP. In the next section I use game theory to identify the set of
optimal strategies, y*(,t).
Game theory
It is obvious that if a consumer achieves its goal of finding
Fi food items, then it should choose to cease
foraging because it will not benefit from consuming more food and will face
the risk of predation. Equally obvious is that a consumer should choose to
forage if the other consumer is dead or has achieved its own goal because it
is then guaranteed of foraging alone, and foraging conditions will not improve
later in the FP. However, when both consumers are hungry the optimal decision
on when to forage is less clear. I now consider this case and use game theory
to determine the optimal foraging strategy for both consumers.
First, I identify when a consumer should adopt the hawk strategy if it
finds itself in a confrontation over food. Let
i(qA,qB,,t,T)
denote the optimal expected reward at time T for consumer i
if at time t the system is in state and consumer A and
consumer B choose to play hawk with probability qA and
qB, respectively. This expectation is given by
 | (14) |
 |
and
, that describe a Nash equilibrium
(Nash, 1951); that is, the
probabilities that describe a strategy for each player that is the best
response to the strategy of the other player. In our case a Nash equilibrium
satisfies
 | (15) |
 | (16) |
In this model the two consumers typically find themselves competing in an
asymmetric game, which generally gives a single Nash equilibrium where both
adopt a pure strategy (i.e., and are either 0 or 1)
(Maynard Smith, 1982). If the
inequalities in Equations 15 and 16 are strict, then the strategy
(
)
is an evolutionarily stable strategy (ESS). It is possible that the Nash
equilibrium is not composed of pure strategies, and the system could exhibit
cyclic behavior. In these situations it is assumed that the dynamics quickly
lead to a stable polymorphism (Bishop and
Cannings, 1978; Maynard Smith,
1982). Appendix A summarizes how the optimal strategies are
evaluated. Under certain conditions there are two Nash equilibriums (see
Appendix A)—one consumer acts as a hawk and the other as a dove. Here, I
introduce an asymmetry into the system by assuming that consumer A is always
the one who acts as a hawk. An asymmetry may be appropriate if both consumers
recognize each other from previous encounters and consumer A adopted the hawk
strategy, or if consumers use some other character to settle contests; for
example, body size (Maynard Smith,
1982). From here on I refer to consumer A as the dominant
consumer.
Now that it is known whether aggression will occur when co-feeding, we can
work out the probability a forager should choose to feed. Let
i(pA,pB,,t,T)
denote the expected reward for consumer i if during time step
t consumer A and consumer B choose to forage with probability
pA and pB, respectively. This
expectation is given by
 | (17) |
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and
are calculated, the optimal expected
reward can be calculated using
 | (18) |
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RESULTS |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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Without aggression
First, I consider the case where consumers cannot fight competitors or steal their food (i.e., playing hawk has no effect). If, in addition, co-feeding has no effect on food encounter rates or risk of mortality (i.e.,
i = i = 1), then the
optimal strategy for both consumers is to begin foraging at the start of the
FP and to continue foraging until their food requirement is met. Instead,
suppose that co-feeding has negative effects (e.g., co-feeding increases risk
of predation or reduces feeding rates). In this case it may be optimal for
consumers to partition foraging times. Temporal partitioning occurs if a
consumer, who has not met its food requirement, chooses to not forage with
certainty (i.e., 0 
xi < Fi and
pi < 1). The consumer's optimal strategy minimizes its
probability of mortality, which may come from predation during foraging or
from starvation after the FP. Thus a good strategy is one where food is
consumed quickly and the risk of not finding food before the end of the FP is
low. Because food encounter rates are higher at higher food densities, it is
always optimal for at least one consumer to start foraging at the start of the
FP. Simulations show that when food abundance is low, both consumers should
forage at time t = 0, and when food is abundant A should forage first
because of the asymmetry assumption. However, if the resource density is
intermediate and B is worse at finding food than A, then it may be optimal for
A to initially give way to B. This is because it is riskier for B to wait than
A. Thus, we may suspect that over time the system will evolve so that when
there are two Nash equilibria (either A or B forages alone), it is the worse
food seeker who forages.
To illustrate the above results, I consider the case where cofeeding
reduces food encounter rates by 50% and consumer B is the better resource
competitor. Parameter values are presented in
Table 1, except = 0,
A = B = 0.5, and aB = 0.6.
Figure 1A shows the optimal
feeding strategy for both consumers at four points in time during the FP,
assuming both foragers are alive and hungry. Initially (t = 0), B
should always give way to A; however, by time t = 20, whether B
should forage is dependent on how successful both have been (i.e., it depends
on the state of both consumers). B may have already found food by this time if
its optimal strategy between t = 0 and t = 20 is to forage.
At time t = 20, B should typically give way to A if A has found some
food, but if by chance A has been unsuccessful, then B should not wait any
longer and co-feed. The reason is that there is a good chance that A will
still be foraging for some time, which increases the chance that B will not
find its own requirements if it continues to wait. If B has already found some
food, then it may be optimal for B to give way to A again. As time goes on,
the number of states where co-feeding is optimal increases. Note that the
optimal strategies are always pure strategies—consumers choose to forage
or wait with certainty (i.e., pi = 0 or 1). By time
t = 60 both consumers should forage if hungry, regardless of the
state of their competitor. When the optimal strategy is adopted by both
consumers, the most likely outcome is that A finds its food requirement while
foraging alone, then B begins to forage and eventually finds its requirement;
it is rare that co-feeding occurs.
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Figure 1B presents a table giving the probability that both consumers survive and acquire their food requirements if they adopt the optimal strategy. In this example, I have chosen parameter values where it is likely that both consumers are successful. For the optimal strategy to be implemented, the consumers must know the state of the other. To evaluate the benefit of having such knowledge, I also calculated the probabilities of survival each FP if consumers adopt much simpler foraging strategies that require no knowledge of their competitor's state. Three strategies I considered are (1) a consumer always forages if it is hungry, (2) consumer B only forages when A has ceased foraging, and (3) consumer A only forages when B has ceased foraging. Figure 1B shows the expected survival probabilities for these strategies. As feeding is a game, the optimal strategy does not give the highest expected reward for either consumer, which would occur if the consumer's competitor always gave way. Note that both consumers are advantaged by having knowledge of the other's state (cf. expectations for the optimal and both always forage if hungry strategies). Although the differences in expected survival between strategies is always low (< 2%), these differences may give significant changes in consumer fitness if the consumers compete over many consecutive FPs before having offspring.
Suppose that consumers differ not in their ability to find food, but
instead, in their predation risk when foraging (i.e., mA
mB). In this case the optimal strategy is often for
the more risk-prone forager to give way because its benefits when foraging
alone are high. Co-feeding may increase its predation risk directly (i.e.,
i > 1), or indirectly by reducing its food
encounter rate (i.e., i < 1), thereby increasing
its exposure time to predators. The more prone forager should give way if it
is likely that its competitor will quickly achieve its own food requirement,
leaving sufficient time for the risk-prone forager to subsequently meet its
own. Thus, in this case, if the optimal strategy is described by two Nash
equilibria, either A or B forage alone, then the less risk-prone consumer is
more likely to be the one who forages. Simulations also show that co-feeding
is often less likely to be optimal early on if predation risk,
mi, is high for both competitors.
With aggression
In this section I consider cases where adopting aggressive behavior has an
effect on foraging success and mortality risk. First, I present a scenario
that has an optimal set of foraging strategies that is typical for a wide
range of parameter values. I then change key parameters and show how they
typically influence the optimal strategies.
Suppose co-feeding has no effect on food encounter rates or risk of
predation (i.e., i = i =
1). As consumer A is more likely to act as hawk, it is often in A's best
interest to co-feed because if A has not found food during a time step, it may
still receive food if B has found some. On the other hand, consumer B would
rather forage alone so that it does not risk having food stolen. However, not
feeding is costly for B because it reduces the length of time that it can find
its food requirements, and finding food gets harder as time goes on because
food becomes more scarce as it is picked off by A.
Figure 2 shows results when
consumers and their environment are described by the parameter values
presented in Table 1. At time
t = 0 both consumers should forage and if a confrontation over food
occurs, then A should act as hawk and B as a dove.
Figure 2A shows that temporal
partitioning is extremely unlikely to be optimal over any part of the FP. In
fact, it is nearly always optimal for both consumers to forage if hungry, and
A should act as a hawk and B as a dove. There are a few exceptions to this
rule. If time is starting to run out and B has been unlucky at finding food,
then it may be optimal for B to adopt the hawk strategy. If at the same time A
is still hungry but has little food left to find, it may be optimal for A to
act as a dove; otherwise, A should continue to adopt the hawk strategy, in
which case some confrontations may end in the death of one of the consumers
(see Figure 2A, t =
60).
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). The first
value represents the probability a consumer forages, and the second value
represents the probability a consumer is aggressive if it is involved in a
confrontation over food. The upper pair is the evolutionarily stable strategy
for consumer A (i = A), and the lower pair is the evolutionarily
stable strategy for consumer B (i = B). (B) Expected probability of
survival for both consumers when (1) both adopt the optimal foraging strategy,
(2) both forage if hungry and play dove, (3) both forage if hungry and play
hawk, (4) both forage if hungry, A plays hawk, B plays dove, (5) both forage
if hungry, A plays dove, B plays hawk (6) B only forages when A has finished
foraging, and (7) A only forages when B has finished foraging.The table in Figure 2B shows that consumer A is advantaged by the possibility of confrontations when both consumers adopt the optimal strategies. Like the previous example, consumers can only adopt the optimal strategy if they know the state of their competitor. Again, I examined the expected rewards (i.e., probability of survival from one FP to the next) when much simpler foraging strategies are implemented, which assume no knowledge of competitor's state. The six alternative strategies I considered are (1) both forage if hungry and always act as a dove, (2) both forage if hungry and always act as a hawk, (3) both forage if hungry, A as a hawk and B as a dove, (4) both forage if hungry, A as a dove and B as a hawk, (5) B always lets A forage alone, and (6) A always lets B forage alone. Both consumers do better if they adopt the optimal strategy unless their competitor is always submissive (i.e., always gives way or always plays dove). In this example the gain in having knowledge of competitor's state is negligible because the optimal strategy is nearly state independent, and the most likely outcome is one where both consumer's optimal strategy does not change over time.
I now examine how food abundance can change the optimal strategies. Figure 3 shows results when food abundance at the start of the FP is high (F = 16). In this case, B should initially give way to A. When food is abundant, temporal partitioning is often optimal during the early part of the FP, and B typically gives way to A because of the asymmetry assumption. The most likely outcome when the optimal strategy is adopted is that B continues to allow A to forage alone until A finds its food requirement, then B begins to forage and eventually finds its own requirement. However, if A is unsuccessful at finding food early on, then it may be optimal for B to begin to co-feed because the cost to B if it continues to wait is high. If co-feeding occurs and B by chance quickly finds and consumes food, then it may be optimal for B to again allow A to forage alone. Figure 3A shows that when food is abundant, the probability that both foragers choose to act as a hawk is extremely low and would only occur late in the FP if both consumers were extremely unlucky at finding food and their expectation of finding food is low.
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Figure 3B shows expected rewards for both consumers when the optimal strategy and the six above-mentioned state-independent strategies are adopted. Increased food abundance always increases the expected reward, regardless of the foraging strategy. Like the previous example, the optimal strategy gives similar results when compared with a state-independent strategy; however, in this case the state-independent strategy is different—consumer B always gives way to A.
In the two previous examples it was assumed that, if during a time step a
consumer found food, then there was only a 20% chance that its competitor
would notice and have the option to initiate a confrontation (i.e., =
0.2). Figure 4 shows results
when confrontation opportunities are much more common ( = 0.8). The
potential for greater numbers of confrontations typically favors consumer A
because it is more likely to adopt the hawk strategy, and hence steal food
from B. The increased cost of co-feeding to B is apparent when
Figure 4A is compared with
Figure 2A. Initially it is
optimal for both consumers to co-feed, A as a hawk and B as a dove. However,
if B is lucky and finds food quickly, then it may be optimal for B to give way
to A (Figure 4A). Previously,
when confrontation opportunities were low, it was extremely rare for temporal
partitioning to be optimal (Figure
2A). Now, consumer A often prefers that B does not wait because it
will lose opportunities to steal food. The result is that early on during the
FP it is sometimes optimal for A to not forage with certainty because it
encourages B to forage (see Figure
4A; t = 0, 20). Like the previous examples, as time goes
on, the optimal strategy is more likely to be to co-feed if hungry and adopt
the hawk behavior.
|
Unlike the two previous examples, the expected rewards when the optimal strategy is adopted are now significantly different from all the state-independent strategies (Figure 4B). One reason for this difference is due to confrontations being relatively common. Now, the cost of losing food, or benefit of gaining food via confrontations, is more significant. For consumer B, its optimal strategy typically changes from co-feeding as a dove, to giving way, to co-feeding again, first as a dove then maybe later as a hawk. B gains benefits if it feeds early because of high food abundance, and it also benefits by feeding late because A is likely to have found some food, making loss of food from confrontations less likely. In this example B's optimal strategy is sensitive to A's initial success at finding food.
|
DISCUSSION |
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|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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The model has shown that if an individual experiences negative effects when co-feeding, then it may increase its chance of survival by exhibiting temporal partitioning. Negative effects may include reduced food encounter rates or increased predation risks. Negative effects may also occur because of interference from dominant competitors, who may steal food or fight and cause injury or death. In this model death may come from three sources: predators, fights with competitors, or starvation. Variation among individuals may make them differently susceptible to each source. A good foraging strategy is one where food is found quickly, encounters with dominant competitors are unlikely, and the risk of not finding one's food requirement before the end of the FP is low. An individual should forage if the benefits of foraging are greater than the costs, and here it has been shown that these benefits and costs can be highly time and state dependent. Good foraging strategies can range from following a very simple rule (e.g., always feed if hungry and always act as a dove during a confrontation), to complicated rules based on the time of day and on how much food both competitors have consumed.
Food abundance has been shown to be an important environmental variable
that affects the costs and benefits of co-feeding. At the start of the FP,
food encounter rates are highest, and the benefits of foraging early on are
high. If food is abundant, then the cost of having food stolen by a competitor
may be low. When food abundance is low, food encounter rates are low, which
increases exposure time to predators and increases the likelihood of
starvation. Dominant consumers (i.e., those who are likely to escalate and win
confrontations) are typically advantaged by co-feeding because of
opportunities for stealing food, and their best strategy is often to begin
feeding at the start of the FP and continue feeding until their food
requirement is met. The optimal foraging strategy for less dominant consumers
is often more complicated. If, on one hand, the probability of confrontation
each time step, , is low, then it may be optimal to forage early on,
taking advantage of the high food encounter rates and accepting occasional
losses of food to competitors. On the other hand, if confrontation rates are
high, then the cost of having food constantly stolen may be great because it
greatly prolongs exposure time to predators with little reward. In this case
it may be better to initially not forage and wait for dominants to cease
feeding or to resume feeding when the risk of starvation is greater than the
risk of predation and mortality during confrontations. In some cases it may be
optimal to forage initially, then give way, then forage again
(Figure 4). When food abundance
is low, the best strategy for all consumers may simply be to forage if hungry
(Figure 2).
The model highlights how the optimal foraging decision of an individual may be strongly dependent on the state of the individual and also on the state of its competitor. The hungrier a consumer is, the higher the cost it pays when waiting or playing dove, and this cost increases as time goes on. Thus, hungry consumers are less likely to partition time or play dove, particularly toward the end of the FP. Because foraging is a game, the state of both consumers can dramatically influence each consumer's optimal decision. For example, it may be optimal for a consumer who is normally dominant during confrontations to suddenly be submissive and play dove. This change in strategy may occur when the dominant consumer is not hungry but its competitor is (Figure 2A, t = 40). In this case the normally submissive consumer will play hawk because of hunger, and the probability of death during a hawk—hawk contest is not worth the risk for the normally dominant consumer.
Here the optimal foraging strategy has been evaluated assuming both
competitors always know the state of the other (i.e., how much they want to
eat), and they also know the state of the environment (i.e., how much resource
remains to found). Although such detailed information is unlikely to be
available to a consumer (Houston and
McNamara, 1999), in many cases it may be possible for consumers to
make estimates based on previous encounters and recent foraging success
(Olsson and Holmgren, 1999).
For this study, I have chosen parameter values that could describe a situation
where the two competitors compete over a number of consecutive FPs, which may
occur when consumers tend to form long-lasting groups. In such cases consumers
may learn each other's feeding requirements and competitive abilities. The
advantage in having detailed knowledge about the state of competitors varies.
If food is very scarce or very abundant, then the gains of knowing each
other's state may be low (Figures
2 and
3); however, gains may be high
when food abundance is intermediate and confrontations over food are common
(Figure 4).
The model also highlights the potentially dramatic effect of stochasticity on foraging dynamics. Because the optimal foraging strategy is state and time-dependent, even if the optimal strategy is adopted by both consumers each FP, the actual foraging dynamics observed each FP may be quite different. However, the final result is nearly always the same: both consumers survive the FP and both find their food requirement.
An advantage of the model presented here is that it is relatively simple to
investigate behavioral responses due to intrinsic variation in feeding ability
and susceptibility to interference among individuals. Such variation has been
suggested to be an important determinant of population dynamics
(Caldow et al., 1999). Under
certain conditions two Nash equilibria exist, and here I assume that only one
is ever adopted. Assuming asymmetries for who is most likely to give way and
who is most likely to act as a hawk often strongly influences the optimal
foraging decision. In general, differences among consumers, such as feeding
abilities (ai and bi) and competitive
weights (wi), have to be sufficiently great before they
change the effects of asymmetries.
I now compare and contrast the predictions of the model with those
predicted by previous modeling studies of foraging. Schoener
(1974a) used a model to
predict when an individual should partition its foraging activities in time.
The model considered only a single consumer and assumed that its objective was
to maximize its net energy intake rate. Unless feeding resulted in a greater
loss of energy than waiting, the model predicted that feeding should occur,
suggesting that diel feeding activities should in general be broad (i.e.,
forage whenever food abundance is sufficiently high). However, this result is
dependent on the assumption that the consumer is a rate maximizer. Schoener
(1974a) suggested that,
alternatively, if the consumer had a fixed daily caloric requirement, then the
opposite might occur (i.e., the consumer should specialize in its time of
feeding). This alternative suggestion is consistent with the findings
presented here, even though interactions between two consumers are explicitly
considered. Here a consumer benefits by restricting its period of foraging
because it minimizes its risk of predation. Dominant consumers will often
restrict their time exposed to predators if they forage at the start of the
FP; however, submissive consumers may restrict their time if they feed later
when dominants have ceased foraging.
Case and Gilpin (1974)
presented one of the first modeling studies on the evolution of aggression. In
their model populations, consumers competed for common resources and
populations could choose to adopt aggressive behavior toward each other, which
was expressed as an interference competition term in their governing equations
of population dynamics. They found that aggression was only likely to evolve
when its costs were low and effects were high. An assumption made in their
model is that aggression is an all-or-nothing characteristic of a population.
Here, aggression is allowed to be expressed depending on the state of the
consumer and its environment. When co-feeding, it is often optimal for one
consumer to act as a hawk and the other as a dove, and it is never optimal for
both to act as doves. This result is consistent with predictions from the
standard hawk—dove model when the value of food is less than the cost of
fighting (Maynard Smith,
1982). Aggression (i.e., hawk plays hawk) is rarely observed when
consumers adopt the optimal strategy, and if it does occur, it is often when
both consumers are hungry and time is running out (i.e., when the value of
food is high because the cost of death due to fighting or predation is less
than the risk of starvation).
Houston and McNamara (1988)
presented a foraging model where an infinite population of consumers compete
for food and often partake in contests of the hawk—dove type. Consumers
have a state variable that represents the animal's level of energy reserves.
The evolutionarily stable strategy was found to be relatively simple; play
hawk if reserves are below a threshold, which increased slightly as the FP
came to a close. Thus, like the model here, the optimal foraging strategy
depended on both consumer state and time, with hungry consumers more likely to
play hawk and more likely to play hawk as time goes on. The model of Houston
and McNamara (1988) differs
from the one presented here; they assumed that consumers always forage,
consumers are intrinsically identical except for their energy reserves, there
are no diminishing returns, and there is no predation. Another important
difference is that Houston and McNamara
(1988) assumed that when there
are two Nash equilibria, the strategies adopted are probabilistic, whereas I
assumed one of the equilibria is always adopted (i.e., consumers exhibit a
consistent dominance). Here it has been shown that these factors can also
influence the optimal foraging strategy.
An interesting outcome of the model is that early on, the dominant consumer
may increase its food encounter rate if it chooses not to forage with
certainty because it may entice the submissive consumer to forage and hence
provide opportunities for stealing food
(Figure 4A; t = 0,
20). A similar result has been observed in a predator—prey model
developed by Brown et al.
(2001). They found that it is
often evolutionarily stable for predators (e.g., owls) not to feed with
certainty throughout the night because refraining from feeding extends the
period of time that the environment is favorable enough for some prey (e.g.,
gerbils) to forage.
Few natural communities have been studied in sufficient detail to allow
clear comparisons with the model's predictions; however, one notable exception
is the competition for seeds by two species of desert gerbils (Gerbillus
allenbyi and G. pyramidum). Kotler et al.
(1993) and Ziv et al.
(1993) both showed that these
two species coexist and exhibit temporal partitioning of foraging activities.
When sympatric, G. pyramidum uses the early part of the night and
G. allenbyi uses the later part of the night; however, both forage
during the early part when allopatric. In addition, G. pyramidum is
dominant during aggressive encounters. It has been suggested that interference
may be a key factor for understanding temporal partitioning in this system
(Ziv et al., 1993). The
results presented here are consistent with the temporal activity patterns
observed by Kotler et al.
(1993) and Ziv et al.
(1993) and support the
suggestion that interference competition is an important factor.
The prediction that aggression among foragers is more likely to be observed
at lower resource abundances is consistent with observations made on a number
of communities, including those involving birds (e.g.,
Armstrong, 1991;
Dolman, 1995) and fish
(Ryer and Olla, 1996). It has
also consistent with other modeling studies of aggression
(Houston and McNamara, 1988;
Sirot, 2000). Although not
presented here, model simulations show that the relationship between
aggression and predation risk may be more complicated. When predation rate,
mi, is high, consumers benefit greatly if they only forage
over a short period of time. For submissive consumers, short foraging times
may be more likely if they initially give way to dominants, more so than when
predation rate is low. However, later on their optimal foraging strategy
typically changes so that they then forage aggressively, thereby defending all
food found. Thus the model predicts that when predation risk is high,
submissive consumers will attempt to provide few opportunities for
confrontations but when they do occur they will typically result in mutual
aggression (i.e., hawk plays hawk). Similarly, Houston and McNamara
(1988) found that with their
foraging model, when the environment became increasingly worse, consumers were
more likely to adopt the hawk strategy. Experimental manipulations involving
fish (Martel and Dill, 1993)
and gerbils (Kotler et al.,
1993) suggest that aggression levels often decrease when predation
risk is increased. This result may be partly due to fights attracting
predators, which I have not attempted to model here.
In most natural systems it is difficult to identify the state of a consumer (e.g., energy reserves) when feeding. It is often much simpler to identify changes in foraging activity and behavior over time. Predictions from the model that can be compared with this type of data include (1) temporal partitioning occurs when food is abundant, but less segregation occurs as food abundance is reduced; (2) dominant competitors always forage at the start of the FP and less dominant competitors partition time; (3) mutual aggression is typically avoided, and a ranking of competitor dominance typically determines who steals from whom; and (4) if mutual aggression does occur, then it is most likely to be observed late in the FP. The assumption that each consumer seeks a fixed caloric quota implies that (5) provided food abundance is not too low, fixed populations consume the same amount of food each FP, and the period of foraging activity decreases as food abundance increases. If consumer state can be identified, then the following prediction can also be compared: (6) the more consumers have eaten, the more likely they are to partition time and less likely to be aggressive.
An important assumption made in the model is that there are only two
consumers. The reason for this assumption is that it keeps the state space of
the system small enough so that the optimal foraging strategy for all
consumers can be exactly evaluated. However, the two-consumer model still
provides insights about when the costs and benefits of co-feeding and
aggression are important, regardless of group size. The general predictions
presented here are also valid for situations where there are many consumers.
For example, when food abundance is high, the most dominant consumers in a
group are predicted to forage first, and less dominant consumers should appear
as their dominant competitors cease feeding. Increased group size will make
the timing of foraging for the less dominant consumers more complicated
because now they have to evaluate the costs of losing food to dominant
consumers and the benefits of stealing food from submissive consumers.
Regardless of group size, consumers are still more likely play hawk when they
have eaten little food and when the FP is coming to a close. The parameter
, which represents the probability that an encounter with food is
noticed by others, allows predictions related to consumer density. If consumer
density is high (i.e., is close to 1), then submissive consumers are
predicted to be more likely to wait, particularly early on; however, when
consumer density is low, the cost of having food stolen may be low, which may
make co-feeding optimal (cf. Figures
2 and
4).
The current model may be most applicable to situations where food is not a
major limiting factor. If all consumers are the same species and the
population is strongly regulated by food abundance, then over time food
abundance may be reduced to the point where it becomes highly valuable, in
which case the model predicts that temporal partitioning is unlikely to be an
optimal strategy for an individual. Brown
(1989) has shown that two
species may coexist on a pulsed resource if there is a trade-off between
foraging and maintenance efficiency. The species with the higher maintenance
efficiency can exploit resources when they are abundant and wait when resource
abundance is low. The species with the higher foraging efficiency can persist
by continuing to exploit resources when they are low. The model of Brown
(1989) does not include
interference competition and predicts that both species should begin feeding
at the start of the FP and each species should cease foraging when resource
abundance drops below a species-specific threshold. Results presented here
suggest that coexistence may be possible in the presence of interference
competition if species exhibit the maintenance-foraging efficiency trade-off
and the species that waits early on is the one with the higher foraging
efficiency (see also Kotler et al.,
1993; Vance,
1984).
APPENDIX A
Calculation of the optimal strategies for aggression
()
Let
 |
|
APPENDIX B
Calculation of the optimal strategies for foraging
(
)
Let
 |
)
is calculated in the same way as in Appendix A. For case a (see
Table A1), we assume both
consumers choose to forage; in other words,
()
= (1, 1); the alternative NE is neither forages, which is clearly unrealistic.
For case b (see Table A1),
either consumer A or consumer B foraging alone is an NE. In this case I assume
an asymmetry where consumer A is the one who forages and B waits; in other
words,
()
= (1, 0). |
ACKNOWLEDGEMENTS |
|---|
Many thanks to A.M. de Roos, J.S. Brown, F.R. Adler, D.F. Westneat, an anonymous reviewer, and the theoretical ecology group at the University of Amsterdam for their helpful comments. This work was supported in part by a grant awarded to A.M. de Roos by the Netherlands Organization for Scientific Research (NWO); the National Center for Ecological Analysis and Synthesis, a center funded by the National Science Foundation (grant DEB-0072909); and the University of California (Santa Barbara).
|
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