| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Behavioral Ecology Vol. 12 No. 1: 71-83
© 2001 International Society for Behavioral Ecology
State dependent behavior and the Marginal Value Theorem
Department of Organismic Biology, Ecology and Evolution, 621 Charles Young Dr. S., University of California, Los Angeles, CA 90095, USA
Address correspondence to P. Nonacs. E-mail: nonacs{at}biology.ucla.edu .
Received 23 August 1999; revised 5 May 2000; accepted 6 June 2000.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONRESULTSDISCUSSIONREFERENCES |
|---|
The Marginal Value Theorem (MVT) is the dominant paradigm in predicting patch use and numerous tests support its qualitative predictions. Quantitative tests under complex foraging situations could be expected to be more variable in their support because the MVT assumes behavior maximizes only net energy-intake rate. However across a survey of 26 studies, foragers rather consistently "erred" in staying too long in patches. Such a consistent direction to the errors suggests that the simplifying assumptions of the MVT introduce a systematic bias rather than just imprecision. Therefore, I simulated patch use as a state-dependent response to physiological state, travel cost, predation risk, prey densities, and fitness currencies other than net-rate maximization (e.g., maximizing survival, reproductive investment, or mating opportunities). State-dependent behavior consistently results in longer patch residence times than predicted by the MVT or another foraging model, the minimize µ/g rule, and these rules fail to closely approximate the best behavioral strategy over a wide range of conditions. Because patch residence times increase with state-dependent behavior, this also predicts mass regulation below maximum energy capacities without direct mass-specific costs. Finally, qualitative behavioral predictions from the MVT about giving-up densities in patches and the effects of travel costs are often inconsistent with state-dependent behavior. Thus in order to accurately predict patch exploitation patterns, the model highlights the need to: (1) consider predator behavior (sit-and-wait versus actively foraging); (2) identify activities that can occur simultaneously to foraging (i.e., mate search or parental care); and (3) specify the range of nutritional states likely in foraging animals. Future predictive models of patch use should explicitly consider these parameters.
Key words: marginal value theorem, predation risk, foraging, patch use, stochastic dynamic programming, state-dependent behavior.
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
|---|
Optimal foraging theory is an important tool for increasing our understanding of animal behavior. One optimality model that has been particularly widely used is the Marginal Value Theorem (MVT), which predicts the behavior of foraging animals collecting energy within patches. Patch depletion will eventually force the animal to move. If the animal's goal is to maximize net rate of energy intake, it should leave a patch when its foraging rate drops to the overall average intake for the entire habitat (Charnov, 1976
). The MVT
further predicts that if an animal encounters a series of patches of varying
quality, it should bias its foraging efforts such that eventually all patches
are depleted to an equal prey density. The prey density at which a forager
leaves a patch is known as the giving-up density, or GUD, and therefore
optimal foraging should result in all exploited patches having similar
GUDs.
The MVT has been extensively applied and tested. Many studies have shown
good qualitative support for MVT predictions such as animals preferring richer
food patches over poorer ones and patch residence time (PRT) correlating with
patch quality (see Stephens and Krebs,
1986: Table 9.1). However, quantitative tests of MVT predictions
have been less successful in that observed behavior often deviates measurably
from predicted behavior. One reason why quantitative predictions may be less
accurate is due to the difficulty of objectively measuring payoff rates of
various behavioral options. This is a significant problem for both the
foraging animal (which may often need to sample changing or unpredictable
environments), and for the researcher in determining and measuring the fitness
of behaviors.
Methodological problems aside, a review of quantitative tests of the MVT shows a definite pattern (Table 1). When quantitative predictions of the MVT fail it is far more common for animals to stay longer in patches than predicted. In 23 of 26 studies, quantitative observations differed from the predicted MVT optima. In most studies there was considerable variance in patch usage across subjects and across trials within subjects. Some stayed too long while others left too early. However, in 19 studies the more common or average deviation from MVT predictions was to stay too long. In two cases both leaving too early or too late were relatively equally observed either for individual animals or across sets of conditions. In the other two cases, the average deviation was in leaving patches too quickly. In another seven studies the results could not be entirely consistent with MVT quantitative predictions (e.g., animals stayed equal lengths of time in good and poor patches), but the direction in which the animals deviated could not be determined.
|
Consistent patterns in deviations from a model's predictions strongly
suggest that instead of mere imprecision resulting in random errors, something
fundamental is missing. Two major aspects of biological realism are absent
from the MVT. First, animals are probably simultaneously doing more than just
searching for food. For example, patch use patterns may often be affected by
predators, either through their physical presence or by the foragers having to
be vigilant for their appearance (Brown,
1988; Lima and Dill,
1990; Newman,
1991; Nonacs and Dill,
1990). Besides negative factors such as predation risk, foragers
may also be looking for positive benefits such as mating opportunities.
The effect of predation risk on optimal foraging strategies has been
extensively modelled, with the general result that net intake-rate
maximization is not always the best strategy. Instead, strategies that
maximize long-term survival trade off between foraging gain and exposure to
risk do better (McNamara,
1990). Consequences of such trade-offs have been predicted to
affect: (1) the amount of fat foragers were willing to accumulate
(Bednekoff and Houston, 1994;
Clark and Ekman, 1995;
Houston and McNamara, 1993;
Houston et al., 1997;
Lima, 1986;
McNamara and Houston, 1990);
(2) activity regimes over a day (Clark,
1994; Houston et al.,
1993; McNamara et al.,
1994); (3) reproductive and mating strategies
(Houston and McNamara, 1986;
McNamara and Houston, 1997)
and (4) population-level mortality rates
(Houston and McNamara, 1993;
McNamara and Houston, 1987,
1990).
Most of the above examinations of predation risk and foraging gain have
numerically solved for optimal behavior. A different analytical approach in
combining foraging gain with avoiding predation risk yields the minimize
µ/g rule (Werner et al.,
1983). Over a given time period, foragers maximize their survival
by choosing the patch that has the smallest ratio of expected predation rate
(µ) to growth rate (g), subject to the constraints that g is positive and
there is no opportunity for reproduction. Although the minimize µ/g rule
does not predict optimal behavior under conditions of strong seasonal effects
or discrete time horizons (Ludwig and
Rowe, 1990), patch choice decisions in juvenile fish have been
accurately predicted in several species
(Gilliam and Fraser, 1987;
Werner et al., 1983). However,
the extent to which the qualitative and quantitative predictions of the MVT
and minimize µ/g rules differ has not been rigorously examined.
A second aspect of biological realism lacking in the MVT is that all
animals are not the same. Foragers differ in their hunger states, energy
reserves, susceptibility to predators, and parental demands. Environments can
also vary over time in their harshness, stability, or predictability, so that
the animal's expectation of overall patch quality may be continually in flux.
Although for any animal facing any given set of circumstances, a unique MVT
prediction can be derived, in practice the model has been applied only to the
average animal in relation to the average state of the environment
(Stephens and Krebs, 1986).
When a forager's fitness depends on more than the food it collects,
state-dependent behavior may again be a better strategy than rate maximization
(Houston, 1990;
McNamara et al., 1987).
Despite the tacit consensus that natural foraging situations are much more
complex than net rate of energy-intake maximization, qualitative predictions
that follow from the MVT are still often made and tested (e.g., the studies in
Table 1). Some of these
include: (1) better food patches are used more than poorer patches; (2) giving
up densities are equalized across foraged patches, with the corollary that
unequal GUDs imply particular cognitive processes or predation risk
differences in the patches; and (3) increased patch residence time and lower
GUDs with both increased travel costs or decreased overall prey density.
However, few theoretical extensions of foraging models have directly compared
the expected fitness of foraging as predicted by the MVT versus the more
complex and realistic state-dependent models. If foraging according to the MVT
performs poorly, it can be said to have a high canonical cost and it would be
predicted that significant selection would favor behavior different from pure
net-rate maximization (McNamara and
Houston, 1986). With high canonical costs, it is possible that
some of the qualitative predictions of the MVT will also not hold.
In this article, I will derive sets of optimal behaviors for a foraging animal that is faced with simultaneous problems of survival (i.e., gaining food and avoiding predators) and provisioning young or finding mates. I will compare the relative fitness of hypothetical foragers that use either state-dependent behavior (SDB), a strategy predicted by the MVT, or a strategy predicted by the minimize µ/g rule. The goal is to identify sets of conditions under which the MVT or the minimize µ/g rule either approximate optimal behavior or do not have high canonical costs. Additionally, if the quantitative predictions significantly diverge, I will test if the qualitative predictions of the MVT are also inconsistent with more complex behavior.
Models of patch choice
My model uses Stochastic Dynamic Programming (as described in
Clark and Mangel, 2000;
Mangel and Clark, 1988) to
predict the optimal set of strategies for a foraging animal under a wide
variety of conditions and assumptions. The elements of the baseline conditions
are as follows.
Food patches
All food patches are assumed to have 20 possible hiding places (slots) for
prey items. A forager staying in a patch randomly hunts in one slot per time
period. Foragers do not systematically search through the patch, so revisits
of empty slots are possible. The patches are either rich (initially 12 prey
items scattered in the 20 slots); medium (eight prey items); and poor (four
prey items). Rich, medium and poor patches occur at equal frequency. Once in
the patch, the forager is assumed to be able to recognize and track its
overall quality, but not which slots contain food. If a forager encounters a
prey item during time t, the number of prey in the patch at time
t + 1 is decreased by one. Patches do not replenish during a
forager's visit and overall habitat quality is stable within a simulation.
Foragers traveling between patches encounter the three types randomly and have
to spend at least one time period in each encountered patch (i.e., they cannot
recognize a poor patch before entering and thus avoid it). This can be thought
of as a sampling constraint. There is only one prey type and it contains 5
units of energy. The forager expends 1 unit of energy per time period as a
basal metabolic rate (BMR), independent of what it is doing.
Predation risk
This situation is modeled as if a sit-and-wait predator is present in only
some of the patches and infrequently moves between them. Therefore the longer
a forager is in a patch without encountering a predator, the more likely it is
that the patch has no hidden predator. This decline in probability of
encountering a predator with patch residence time is set as an exponential
function, so that: pi = ae-b(t-1),
where a = 0.1 and b = 0.2. The minimum predation risk is set as
pi at t = 30, so that patches never become
totally safe. Any encounter with a predator results in death for the
forager.
Traveling between patches
Travel between patches occurs instantaneously, in that the forager does not
spend a given number of time steps in transit. Travel costs are therefore
incorporated as added energetic demands on top of BMR. Because fitness results
directly from net foraging intake, when the model reaches stationarity (see
below) traveling-cost effects are identical if considered either as an extra
energy expenditure rather than several time steps in which the animals
reserves decline due to not foraging. However, it is computationally much
easier to have a single cost rather than simultaneously considering the
effects of decisions over multiple time steps. Travel has an added predation
risk of a 0.05 probability of dying while moving between patches.
State variable
The forager's energy resource (S) is the state variable, from which its
fitness will be determined. S can have integer values from 0 to a maximum of
50. If S = 0 the individual has starved and is dead.
Fitness functions
The model finds the set of state-dependent behaviors that maximize fitness
under three different assumptions. The first function maximizes the forager's
long-term probability of survival. The terminal fitness function is positive
and equal for all energy states greater than zero (i.e., there is no benefit
for surviving at a larger size). There are no mating or reproductive
opportunities. These "Survival" conditions approximate the classic
problem of a "bird in winter," whose only goal is to survive
(e.g., Lima, 1986;
McNamara and Houston, 1990;
McNamara et al., 1994).
Maximizing fitness is thus a matter of avoiding predators and starvation. If
there is no predation risk then the optimal solution approximates the
maximization of net intake rates. An analogous situation described by this
model would be for a parasitoid that lays eggs on individual hosts, so that
instead of maximizing caloric intake rates, the animal should maximize the
number of hosts encountered over its lifetime.
The second fitness function, "Reproduction," assumes a parent caring for offspring, where fitness is maximized by collecting the most food over a lifetime to invest in offspring. Thus, the animal must avoid starvation and predation (as in the Survival model), and provision offspring. The terminal fitness function is zero for all final states in this model because animals can accrue fitness through reproduction at every time step rather than reproducing after a set period of foraging. In constructing the set of optimal behaviors, a successful foraging animal must decide between eating the prey itself or provisioning young. Once a prey item is obtained, the eat/feed decision is maximized for lifetime reproductive output identically for the SDB, MVT, and µ/g strategies. Therefore, the three strategies differ only in regards to patch use patterns and are independent of allocation strategy to self versus off-spring. The Reproduction conditions are also applicable to parasitoids that both feed and lay eggs on the same host. Such a parasitoid also faces a similar trade off in that the more it eats (and improves its condition), the less there is for its off-spring.
The third fitness function, "Mating," assumes an animal that is searching a patch for both prey and mating opportunities. Each patch contains four possible mates in the 20 slots (i.e., rich prey patches do not have more potential mates). The probability of successfully mating, if a potential mate is encountered, is a linear function of size so that large animals are more likely to be successful than are small animals. Thus under these conditions, fitness accrues as a function of mate encounter rate, foraging success, and predator avoidance. As with the Reproduction function, fitness accrues at each time step rather than at the end through a terminal fitness function. The Mating conditions are also applicable to territorial animals that must patrol for intruders. Successful guarding of territory boundaries is likely to be a function of both the territory holder's physical state and search strategy.
Running the simulations
In all cases, the models were iterated backwards for 250 time steps. An
individual in a patch has two options: it can continue to forage in that
patch, or it can travel on to the next patch. On the last day of the model
(t = 250), the stay/go decisions are numerically solved for every
combination of the forager's energy state (1 
S 50), possible number
of food items still in the patch (0 to a maximum of 12), and the level of risk
(from t = 1 to 30) in the patch. The optimal decision is the one
which gives the highest expected fitness and it is recorded in an optimal
decision matrix. The process is repeated stepwise for t = 249, which
uses the t250 decision matrix as its fitness function, and
then for t = 248, and so on.
The optimal decision matrix at t1 has reached
stationarity, where patch residence decisions are affected only by processes
occurring within the time step and are no longer affected by the initial
terminal fitness function at t = 250. Therefore, the optimal behavior
for any given set of states and options does not change from one time step to
the next (Clark and Mangel,
2000; Mangel and Clark,
1988). Given this fact, I could use t1
matrices to simulate the equilibrium behavior and success of hypothetical
populations of foragers. When simulating behavior at stationarity, the results
are independent of the assumed terminal fitness function.
In the simulations I started 20,000 foragers, with S drawn randomly from values between 10 and 40 and having just entered a randomly-determined new patch (t = 1). Each forager stays or leaves patches depending on the rule they are following (see below). Foraging success, patch type encounters and predator encounters were stochastically drawn from the expected probability distributions. For each individual, the simulation ran until it starved or was killed by a predator. There was no upper limit to the number of patches potentially encountered. Across the 20,000 simulations I calculated:
- Mean fitness in the assumed currency
- The mean and distribution of patch residence times in the three patch
types
- The mean giving-up densities at which foragers chose to leave patches for
each patch-entry state.
For comparing the overall mean GUDs of the different patch types, I simulated behavior from the decision matrices, but without predation so that foragers would not die. I started 1000 foragers at every possible state (from one to 50) in each patch type and calculated the mean GUD. In calculating the equilibrium energy reserves, I started 1000 foragers at randomly drawn states between 10 and 40, in randomly-determined patches and let them forage for 250 time steps (again with no predation to assure survival). Mean equilibrium energy state was defined as the average energy reserves over the last 50 days.
For contrast to state-dependent behavior, I repeated the simulations for foragers behaving as predicted by either the MVT (i.e., maximizing the long-term net rate of energy intake in the environment), or the minimize µ/g rule where a forager leaves a patch when the patch's ratio exceeds the expectation averaged across randomly encountered new patches.
Other models of patch choice
Two previous models also examined how patch choice might differ from MVT
predictions when there are trade-offs between foraging gain and predation risk
or other activities. Methodologically, however, there are significant
differences between these models and the one presented here. First, Brown
(1988) considered trade-offs in
terms of energy (e.g., how much foraging success is an animal willing to give
up to be safer?). Thus, the optimal predicted harvest rate at which an animal
quits a patch (H) would be the sum of the energetic cost of foraging (C),
energetic equivalent for predation risk (P), and the energetic equivalent of
missed opportunity costs (MOC) for personal maintenance, mating opportunities,
or finding alternative food sources.
If the values of P and MOC are known or biologically reasonably functions
that can be estimated, then analytical solutions give the optimal patch
exploitation pattern. In practice, however, quantitative predictions from the
H = C + P + MOC rule may be difficult to obtain
(Brown, 1992). Nevertheless, by
holding all variables except one constant, the model does predict qualitative
shifts in patch use patterns. Tests of the model in kangaroo rats
(Brown, 1988) and fox squirrels
(Brown et al., 1992) found
that, as predicted, patch use went down with increased predation risk and
missed opportunity costs. Neither result would be predicted by the MVT because
of its insensitivity to factors other than the net rate of food intake.
Brown's model, however, is based on the assumptions of the MVT. If predation risk and missed opportunities are absent or are equal across patches then both the quantitative and qualitative predictions are identical to MVT. The model does not predict behavior in currencies of fitness besides energy gain. Instead, it assumes the animal will "pay" for gaining safety or non-foraging opportunities by sacrificing energy intake rate. Although optimal behavior can be observed and measured in this context, a priori predictions about quantitative patterns of patch use are difficult to make. Therefore, the SDB model presented here is a numerical method of analyzing optimal behavior conceptualized in the Brown model under conditions where P and MOC may or may not vary across patches.
Newman (1991) used an SDB
model to predict foraging behavior under predation hazard and to see how
similar the predicted behavior might appear to an MVT model. The techniques
employed by Newman are similar to those presented here, but there are
substantial differences in the assumptions and goals of the models. Major
differences in the Newman model are: (1) only two patches are available in any
simulation, a refuge and a foraging patch; (2) predation risk does not change
within patches and there are no predation costs for travelling; (3) behavior
is modelled to the end of set time period, so the terminal fitness function
has an effect; and (4) only survival is used as a fitness currency. One model
should not be thought of as being better than the other, as they are geared
towards different problems. Newman's model predicts foraging behavior over 1
day, while the one presented here is designed more towards identifying longer
term patterns.
Sensitivity analyses
Food patches
I examined the effects on the predicted behaviors of overall habitat
quality by changing the probabilities of encountering various patch types
(Table 2). In a good habitat,
rich patches occurred 80% of the time, medium patches and poor patches each
10% of the time. In a bad habitat there are: 80% poor, 10% medium, and 10%
rich patches.
|
Predation risk
Six quantitative and qualitative changes in the predation risk functions
were examined.
- Low predation risk, where risk in the patch and traveling were each reduced
by a factor of 10.
- High risk, where risk in a patch declined inversely to the baseline
exponential function resulting in an initial slow decline in risk with patch
residence time.
- Constant risk per any time period across all patches, with a = 0.05 and b =
0. This is analogous to the predator also being an active forager moving
frequently between patches.
- A predation-risk/foraging-gain trade-off, where: a = 0.1, 0.05 or 0.025 for
rich, medium and poor patches, respectively. This simulates a situation where
a forager's predators and prey are most likely to be found in the same
place.
- Constant predation risk (b = 0) with a = 0.1, 0.05 or 0.025 for rich,
medium, and poor patches, respectively.
- No predation risk in traveling between patches.
Traveling between patches
Three quantitative changes to traveling were examined. First, travel costs
were set to zero, so that it cost the forager no extra energy above BMR to
switch patches. Second, travel costs were set high (= 4 units), so that travel
+ BMR would cost one captured prey item. Note that this would be equivalent in
time to where foragers can instantly switch patches or have to take four time
steps to move.
Marginal value theory conditions
As a reference point for all other simulations, all parameters were set to
most closely resemble the basic assumptions of the MVT. These are: (1) a
constant per time predation risk of 0.05 so that neither patch choice or
residence times have effects on mortality from predation; (2) no added
predation risk for traveling; and (3) a constant travel cost of one unit plus
BMR. Under such conditions, the state-dependent solution should be very
similar if not identical to the solution from the MVT.
|
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
|---|
There are four main outcomes from the simulations:
- Behaving as predicted by the MVT or the µ/g rule often substantially
reduces survival, mating opportunities, or off-spring production.
- Predation risk strongly influences patch use patterns if foragers can
affect the level of risk by their patch residence times.
- Qualitative predictions from the MVT concerning patch depletion patterns,
GUDs, and the effects of travel costs do not often result from state-dependent
behavior.
- Mass regulation occurs in foragers (i.e., maintaining body mass below an
obtainable maximum). This regulation results from indirect rather than direct
costs of predation risk. Fatter birds were never assumed to be directly more
at risk because of their weight. Rather, body mass is regulated through
decreasing dangerous foraging activity.
Fitness consequences
The behaviors predicted by the SDB model always have higher fitness than
behaviors predicted by the MVT and minimize µ/g strategies
(Table 3). If the foraging
situation approximates the assumptions of the MVT model, then there are small
canonical costs to deviating from state-dependent behavior. However, when
these assumptions are relaxed, state-dependent behavior does considerably
better in increasing survival, mating opportunity and reproductive success. In
all cases, the SDB strategies have higher fitness through equal or longer
patch residence times in at least one of the three patch types than predicted
by the MVT. The fact that the SDB model consistently predicts longer PRTs than
the MVT under all scenarios fits with the predominant trend in the
quantitative tests of the MVT (Table
1).
|
Not surprisingly, because the µ/g rule was derived to include both predation risk and foraging gain in patch use, behavior that minimizes µ/g is intermediate in fitness between the MVT and SDB under most simulation conditions of the Survival scenario (Table 3). Minimizing µ/g is generally a better behavioral strategy than net-rate maximization without traveling costs or risks or when risk in the patches is intermediate or constant. It is a poorer strategy whenever µ/g predicts shorter PRTs than does the MVT. However, overall the difference in success between following a µ/g strategy or a MVT strategy is small. Minimizing µ/g does not approximate state-dependent solutions significantly better than does the MVT.
Predation risk
Predation risk greatly affects foraging patterns when behavior is
state-dependent. SDB foraging reduces risk through staying longer in patches
and moving less often. SDB also predicts that within-patch risk has greater
effects than risk associated with moving between patches, in that GUDs change
more from the baseline conditions when the dynamics of risk within patches are
altered than when traveling risk is removed. Although the addition of
predation risk in these models predicts patch use patterns quite different
from the MVT, the canonical cost for non-optimal behavior is not equal across
all levels of risk. When risk levels are high or low, the MVT and µ/g rules
predict much higher GUDs than the SDB model, but all patch use patterns have
relatively similar fitnesses (Table
3).
In the above simulations there were no explicit refuge patches where the animal cannot forage but is also absolutely safe from predators. Although foragers make patches refugelike with long stays, it may be that such behavior would not be predicted with a true refuge possibility. Thus, for three sets of conditions in the Survival scenario (Table 2), I added a refuge as third option for a foraging animal (i.e., stay in current foraging patch, go to another foraging patch, or use a refuge). Under baseline and the no predation risk while traveling conditions, there is a trivial effect of the refuge. Overall, the animals spend only 0.5 and 4.9% of their time in the refuges. There is no effect on patch use patterns during time spent foraging. Under MVT conditions, however, animals spend 47.9% of their time in the refuge. If predation risk is reduced from 0.05 to 0.01 per time step, there is almost no effect on the use of the refuge with MVT conditions (animals spend 49.3% of their time in the refuge). This latter result further reinforces the finding that changes in risk with time spent in patches are more important than the absolute level of risk within a patch.
Qualitative predictions
State-dependent foraging behavior does not predict the same general results
as the MVT. First, giving-up densities are not equalized across the utilized
patches. In the simulations, the MVT predicted differences in GUD's between
the rich and medium patches in the range from 0-16% of remaining prey density.
However, state-dependent behavior produced larger mean differences in the
majority of simulations (Figures
1,2,3).
Furthermore, the expected GUDs for both the rich and medium patches were
strongly affected by energy state of the forager and the effects of state
differed across simulations. Thus, increasing a forager's state can either
increase or decrease the difference in expected GUDs as well as switch which
patch would be expected to be the more heavily exploited. When GUDs are
equalized across patches, it most often results from foragers staying much
longer than predicted by the MVT in rich and medium patches and reducing both
to very low prey densities. This effect is also state-dependent with generally
lower GUDs in all patches when foragers have high energy reserves.
|
|
|
Second, GUDs are affected by environmental quality and foraging costs, but the effects are not always in the direction predicted by the MVT. Consistent with the MVT, increasing travel costs decreased GUDs where increasing overall environmental quality increased GUDs in medium quality patches (panels E versus F and H versus G in Figures 1,2,3). However contrary to MVT predictions, GUDs always slightly increased in good patches with increased travel costs and decreased with overall environmental quality in both the Survival and Mating scenarios.
Mass regulation
By the choice of when to leave patches (and if to feed off-spring), the
forager directly controls its energy reserves, which can be considered as the
animal's body size or fat reserves. Energy held in reserve is strongly
affected by which fitness currency is assumed. Under the baseline conditions,
the equilibrium energy states (out of a maximum of 50) for the Survival,
Reproduction, and Mating scenarios are: 38.2, 8.3, and 41.5, respectively. In
the Reproductive scenario energy is invested in offspring, while in the Mating
scenario reproductive success is a positive function of size. These
differences decrease and increase the equilibrium energy state relative to the
Survival scenario. Under all three scenarios, however, the equilibrium energy
state is 15-25% lower than the equilibrium reached by animals foraging
according to the MVT. For the Survival scenario, I calculated the equilibrium
energy states for all sets of the conditions in
Table 2. Only if conditions
approximate the assumptions of the MVT, is the maximum attainable energy state
also the equilibrium energy state. For all other permutations, the equilibrium
energy states range from a low of 66% (in the Constant risk + risk trade-off
situation) to a maximum of 84% (in the Low overall risk situation) of the
equilibrium predicted by the MVT.
These results show that mass regulation below maximum levels can occur
without direct mass specific costs in increased predation risk. This is a
contested point. McNamara
(1990) found no mass
regulation without mass specific costs and Houston et al.
(1997) pointed out three
errors in formulating SDB models that can produce results that mimic the
appearance of mass regulation. These include: (1) numerical imprecision, in
that after many iterations, differences in the outcomes between optimal and
non-optimal behaviors can be rounded off to the same value, (2) choice of
inappropriate time scales such that behavioral or physiological processes are
not correctly modeled, and (3) sensitivity to the maximum energy state that
can be achieved by a forager. The first two errors were suggested as
explaining Clark's (1994)
results of mass-regulation through indirect costs. Error 1 may be present
because foraging and resting have been rounded off to the same fitness value.
Error 2 had birds with maximum energy reserves being able to survive for an
unrealistic 267 days without foraging. Since the model ran for only 80 days,
individuals with higher reserves never needed to forage. The first error was
avoided in the Survival scenario by using double rather than single precision
(see Houston et al., 1997).
The other two scenarios (Reproduction and Mating) are immune to this problem
because fitness is continually accrued at each time step rather than at the
end of some time period. The second error is avoided by modelling behavior at
a stage where the model is stationary in its patch choice predictions and
foragers at the maximum energy state still have a nonzero chance for
starving.
The models are, however, sensitive to the upper limit for energy reserves. For example, if maximum state is raised from 50 to 60, then the equilibrium state for the baseline Survival scenario increases from 38 to 46. Nevertheless, I would argue that the results are not simply a numerical artifact, but instead that the sensitivity to the upper limit illustrates an interesting point. If the models are recast such that the maximum energy state is 10 or 30, the same mass regulation phenomenon emerges. As when Smax = 50, the equilibrium mass is always approximately 75% of the given maximum energy state (Figure 4). If the simulations are begun with all foragers at Smax, their states will converge on the same lower equilibria values. Finally, foragers can be simulated using optimal strategy sets derived from higher maximum energy state models (i.e., Figure 4A; a forager behaves as if it could reach an energy reserve of 70, but its actual maximum energy state is only 10). The equilibria energy states for such strategies also remain considerably below the states reached by foragers using MVT rules, and where they differ from the calculated optimal rule, their fitness is reduced (e.g., the lifespan of foragers behaving as if Smax = 60, when Smax = 50 is, on average, 97.2% that of optimal). In summary for the models presented here, the best long term strategy is to keep mass below the maximum attainable by behaving as predicted by the MVT.
|
As the maximum energy state decreases, the degree of apparent mass
regulation decreases as well. When Smax = 10, the SDB equilibria
energy state is 89.8% that of MVT state, while with Smax = 50, the
SDB strategy is 79.7% of the MVT strategy. This suggests that the degree to
which an optimally foraging animal exhibits mass regulation is dependent on
the scale at which it usually changes its energy state. The finer the scale of
this control, the more evident mass regulation should be. Therefore, I ran the
Survival scenario again under baseline conditions, but with patches containing
one, two, or three prey items of size 20, so that the amount of food per patch
type and overall environment is the same, but it come in much bigger pieces.
With this set of conditions, the equilibrium energy states for SDB and MVT
rules were 35.6 and 39.1, respectively. Thus, the SDB strategy equilibrium was
91% that of the MVT when control of energy state was more coarsegrained. This
result is consistent with previous models on mass regulation: An increase in
environmental variability favors carrying closer to the maximum possible
levels of fat (Bednekoff and Houston,
1994; Houston and McNamara,
1993; Lima,
1986).
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
|---|
The Marginal Value Theorem predicts patch use patterns based on maximization of long term net energy-intake rates. However, quantitative predictions of the MVT differ from other theoretical models that include avoiding predation (Bednekoff and Houston, 1994
; Houston and McNamara,
1993; Houston et al.,
1993; Lima, 1986;
McNamara, 1990;
McNamara and Houston, 1990) or
reproductive behavior (Houston and
McNamara, 1986; McNamara and
Houston, 1997). Experiments with the MVT also show consistent
deviations from quantitative predictions
(Table 1). Thus, it is not
surprising that state-dependent behavior does better than
"simpler" behaviors predicted from the MVT or the µ/g rules.
What is important, however, are the wide range of conditions that predict poor
performance for foragers following those rules. Only under limited conditions
are they good "rules of thumb" that approach optimal behavior.
Furthermore, qualitative general predictions that follow from the MVT about
patch residence times and giving up densities may also often be inconsistent
with state-dependent behavior. The results affect five predictions about
foraging behavior. In the SDB model, patch residence times are longer, resulting in significantly lower GUDs than predicted by the MVT. The MVT and state-dependent strategies are more likely to predict similar behavior or to have similar fitness when food is immediately invested in reproduction, predation risk is either very low or high, foragers cannot reduce risk by moving less often between patches, foragers have low energy reserves; or food comes in large pieces. As in the majority of tests of the MVT such factors are not explicitly described, it is difficult to correlate how any of these factors affect the outcome of the tests. For example, a clear need in quantitative tests of foraging models is to examine more situations where animals are either feeding offspring or simultaneously looking for mating opportunities as these type of situations are greatly underrepresented in the existing tests of patch use (Table 1).
Decreased travel costs or an increase in overall environmental quality may result in more frequent moves between poorer patches as predicted by the MVT. However, a forager exhibiting SDB may treat the best patches as predation refuges, moving less often between them than predicted by the MVT (which does not incorporate predation risk).
The likelihood of observing equal giving-up densities across patches
depends on how closely the situation corresponds to MVT assumptions. Unequal
GUDs are an almost ubiquitous consequence of state-dependent foraging and can
result from various departures from MVT assumptions. For example, differences
in GUDs are predicted and observed across patches that differ in predation
risk (Bouskila, 1995;
Brown, 1988;
Brown et al., 1992;
Kotler, 1997). However, an
observation that GUDs differ across two patches does not mean they must differ
in predation risk or other non-foraging opportunities. If perceived predation
risk declines with time in a patch, unequal GUDs are likely to result without
any difference in initial predation risk across the patches. The better
patches will simply also function as predation refuges.
Equal GUDs are more likely with foragers that have significant energy
reserves, but quantitatively this results from staying in patches far longer
than predicted by the MVT and thus reducing them to similar, very low
densities. When patches are not reduced to similar GUDs, correlations between
initial prey density and final density result. The correlations are negative
if the forager overuses the rich patch relatively more than the other patches
and positive if vice versa. Positive correlations between initial patch
density and the GUD have been proposed to indicate that foragers are Bayesian
in their estimation of overall habitat quality
(Alonso et al., 1995;
Valone, 1991). However,
positive correlations are also predicted when conditions are similar to the
assumptions of the MVT, when there are trade-offs between risk and food, so
that the best food patches are also perceived as the most risky, or when the
foragers are simultaneously searching for food, mates, or territory intruders.
In all of these situations, animals with higher energy reserves might appear
Bayesian in their behavior, without actually being so.
Predation risk strongly affects patch use patterns, but only if there are
time-dependent or movement-dependent effects. The overall level of risk for
foraging has little effect, if it cannot be decreased by staying longer in
patches or moving less often between them. This is complementary to results
found by Newman (1991) and
Bednekoff and Houston (1994),
who also predict no effect of increasing risk per time unit of foraging
activity. In total this suggests that patch use may be greatly influenced by
how a forager expects to encounter its predators. The response to ambush or
sit-and-wait predators may be to stay longer in individual patches, but not
reduce the overall time spent foraging. Conversely, the response to actively
foraging predators may be rate-maximizing GUDs, but reduced overall foraging
and more time spent in refuges. As a hypothetical example, desert rodents may
spend more time foraging with lower GUDs per patch if they are avoiding
rattlesnakes rather than owls. Predation risk from snakes would be reduced by
staying longer in patches, and predation risk from owls would be reduced by
hiding in refuges.
Mass regulation below an attainable maximum follows from behavioral
patterns that reduce foraging activity in order to reduce encounters with
predators (similar results are found in
Clark and Ekman, 1995;
Lima, 1986; and
Newman, 1991). As such, these
conclusions are contradictory to those made by Bednekoff and Houston
(1994) and Houston et al.
(1997) who predict mass
regulation only with direct mass-specific costs. The differences between these
findings may relate to how the effect of predation risk is interpreted or how
maximum attainable weight is defined.
Bednekoff and Houston (1994)
found a large effect of mass-dependent risk and almost no effect of increasing
per unit exposure risk over four orders of magnitude. However, all their
simulations included a mass-specific predation cost, which may have masked the
effects of increased exposure risk. Thus, it is appropriate to conclude that
variation in mass-dependent costs predicts larger effects than variation in
exposure costs, but not that exposure, per se, has no effect. The latter would
only be true if the mass of a forager experiencing no risk at all were similar
to those experiencing exposure risk. Houston et al.
(1997) reject Clark and
Ekman's (1995) conclusion
about mass regulation from exposure costs based on the sensitivity of the
predicted optimal body mass to the upper constraint in maximum body mass.
However, the predicted optimum remains consistently proportional to the upper
constraint (see Houston et al., 1987:
Figure 2). Thus, as in the
models presented here, Clark and Ekman's predictions may be valid if viewed as
a proportion of attainable body mass rather than a specific value. Without
comparing Clark and Eckman's results to the expected body mass with no
predation risk, it cannot be concluded that mass regulation was truly
absent.
Mass regulation without direct cost should not be taken to imply that such
mass-specific costs do not exist. There are certainly ample reasons to believe
that carrying extra weight imposes metabolic and agility costs on animals
(e.g., Witter et al., 1994)
and in theoretical considerations such costs have always predicted large
behavioral effects (Bednekoff and Houston,
1994; Lima, 1986).
However, one cannot infer direct mass-dependent costs if foragers in a
Survival-type scenario are simply observed to "not keep reserves at the
maximum possible level" (e.g.,
Houston et al., 1997: 331).
This convenient shortcut to the difficult propostion of measuring the survival
costs of extra weight in natural situations is not justified.
The Marginal Value Theorem has served as a valuable heuristic tool in behavioral ecology, but there is too large of a disconnection between the modeling and the testing of the foraging process. Models strongly suggest that optimal behavior is not synonymous with net intake-rate maximization, but rarely explicitly examine the degree to which the MVT may fail. Experimenters may realize that the quantitative predictions of the MVT are unlikely to be upheld in the field but continue to assume that the qualitative patterns of the MVT will hold. The goal of this article is to bring these issues into focus.
A state-dependent approach to describing foraging is, by its nature,
unlikely to generate broadly applicable predictions about patch use as does
the MVT. The model presented here identifies sets of conditions where
particular types of behavior may occur, but the testing of these predictions
will be case-specific in adapting the model. The experimenter may have to know
the nature of predation risk, what is important to the forager besides its
food, the general quality of the environment that determines the forager's
access to resources, and the likely energy state the tested animal will have.
This is not as daunting a task as it may seem and including state-dependent
behavior can certainly lead to new insights. An example of this approach is
Roitberg's (1990) work with
fruit flies (Rhagoletis pomella). Female flies lay eggs on fruit
hosts and there is considerable variation in search times within trees across
females. With an SDB model, Roitberg predicted decisions as a function of: (1)
current and average patch qualities, (2) patch densities, (3) time of day, and
(4) egg load. He also varied fly behavior in terms of whether flies used a
rule of thumb that over or underestimated their rates of locating patches
("optimists" versus "pessimists"). Overall, both
strategies performed nearly as well as the optimal behavior, when the patch
estimate errors were small or moderate. However, as errors increased in
magnitude, pessimists did worse than optimists, which led Roitberg to further
predict (and confirm by observation) that the rule of thumb for optimistic
behavior is more common in natural fly populations than pessimistic behavior.
In summary, state-dependent optimality models create logical constructs
against which actual behavior can be matched to test the validity of the
assumptions (Nonacs and Dill,
1993). Models like the MVT are a reasonable starting point, but
natural complexity has to be incorporated in both the derivations of foraging
and patch use predictions and their tests.
|
ACKNOWLEDGEMENTS |
|---|
I thank Colin Clark, Tom Langen, Smadar Gilboa, Alex Kacelnik, Jonathan Newman, and several anonymous reviewers for their helpful comments during the preparation of this article. This work was partially supported by UCLA Academic Senate Grant 563828-NP-19914.
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
|---|
Alonso JC, Alonso JA, Bautista LM, Muñoz-Pulido R, 1995. Patch use in cranes: a field test of optimal foraging predictions. Anim Behav 49: 1367-1379.
Bednekoff PA, Houston AI, 1994. Optimizing fat reserves over the entire winter: a dynamic model. Oikos 71: 408-415.
Best LS, Bierzychudek P, 1982. Pollinator foraging on foxglove (Digitalis purpurea): a test of a new model. Evolution 36: 70-79.[ISI]
Bouskila A, 1995. Interactions between predation risk and competition: a field study of kangaroo rats and snakes. Ecology 76: 165-178.
Brown JS, 1988. Patch use as an indicator of habitat preference, predation risk, and competition. Behav Ecol Sociobiol 22: 77-87.
Brown JS, 1992. Patch use under predation risk. I. Models and predictions. Ann Zool Fennici 29: 301-309.
Brown JS, Morgan RA, Dow BD, 1992. Patch use under predation risk. II. A test with fox squirrels, Sciurus niger. Ann Zool Fennici 29: 311-318.
Cassini MH, Kacelnik A, Segura ET, 1990. The tale of the screaming hairy armadillo, the guinea pig and the marginal value theorem. Anim Behav 39: 1030-1050.
Cassini MH, Lichtenstein G, Ongay JP, Kacelnik A, 1993. Foraging behaviour in guinea pigs: further tests of the marginal value theorem. Behav Proc 29: 99-112.
Charnov EL, 1976. Optimal foraging, the marginal value theorem. Theor Pop Biol 9: 129-136.[ISI][Medline]
Clark CW, 1994. Antipredator behavior and the
asset-protection principle. Behav Ecol
5: 159-170.
Clark CW, Ekman J, 1995. Dominant and subordinate fattening strategies: a dynamic game. Oikos 72: 159-170.
Clark CW, Mangel M, 2000. Dynamic state variable models in ecology: Methods and applications. Oxford: Oxford University Press.
Cowie RJ, 1977. Optimal foraging in great tits (Parus major). Nature 268: 137-139.
Crowley PH, DeVries DR, Sih A, 1990. Inadvertent errors and error-constrained optimization: fallible foraging by bluegill sunfish. Behav Ecol Sociobiol 27: 135-144.
Cuthill IC, Haccou P, Kacelnik A, 1994. Starlings
(Sturnus vulgaris) exploiting patches: response to long-term changes
in travel time. Behav Ecol 5:
81-90.
Cuthill IC, Kacelnik A, Krebs JR, Haccou P, Iwasa Y, 1990. Starlings exploiting patches: the effect of recent experience on foraging decision. Anim Behav 40: 625-640.
Formanowicz DR Jr, 1984. Foraging tactics of an aquatic insect: partial consumption of prey. Anim Behav 32: 774-781.
Gaines S, 1989. Comparisons of patch-use models for wintering American Tree Sparrows. Auk 106: 118-123.
Gilliam JF, Fraser DF, 1987. Habitat selection under predation hazard: test of a model with foraging minnows. Ecology 68: 1856-1862.[ISI]
Grether GF, Palombit RA, Rodman PS, 1992. Gibbon foraging decisions and the marginal value model. Internat J Primatol 13: 1-17.
Hansen J, 1987. Tests of optimal foraging using an operant analogue. In: Foraging behavior (Kamil AC, Krebs JR, Pulliam HR, eds). New York: Plenum Press; 335-362.
Hanson J, Green L, 1989. Foraging decisions: patch choice and exploitation by pigeons. Anim Behav 37: 968-986.
Hodges CM, Wolf LL, 1981. Optimal foraging bumblebees: why is nectar left behind in flowers? Behav Ecol Sociobiol 9: 41-44.
Houston AI, 1990. The importance of state. In: Diet selection: an interdisciplinary approach to foraging behaviour (Hughes RN, ed). Oxford: Blackwell Scientific Publications; 10-31.
Houston AI, McNamara JM, 1986. The influence of mortality on the behaviour that maximizes reproductive success in a patchy environment. Oikos 47: 267-274.
Houston AI, McNamara JM, 1993. A theoretical investigation of the fat reserves and mortality levels of small birds in winter. Ornis Scand 24: 205-219.
Houston AI, McNamara JM, Hutchinson JMC, 1993. General results concerning the trade-off between gaining energy and avoiding predation. Phil Trans R Soc Lond B 241: 375-397.
Houston AI, Welton NJ, McNamara JM, 1997. Acquisition and maintenance costs in the long-term regulation of avian fat reserves. Oikos 78: 331-340.
Howell DJ, Hartl DL, 1980. Optimal foraging in glossophagine bats: when to give up. Am Nat 115: 696-704.[ISI]
Howell DJ, Hartl DL, 1982. In defense of optimal foraging by bats: a reply to Schluter. Am Nat 119: 438-439.
Hubbard SF, Cook RM, 1978. Optimal foraging by parasitoid wasps. J Anim Ecol 47: 593-604.
Kacelnik A, 1984. Central place foraging in starlings (Sturnus vulgaris). I. Patch residence time. J Anim Ecol 53: 283-299.
Kacelnik A, Todd IA, 1992. Psychological mechanisms and the Marginal Value Theorem: effect of variability in travel time on patch exploitation. Anim Behav 43: 313-322.
Kamil AC, Misthal RL, Stephens DW, 1993. Failure of
simple optimal foraging models to predict residence time when patch quality is
uncertain. Behav Ecol 4:
350-363.
Kamil AC, Yoerg SI, Clements KC, 1988. Rules to leave by: patch departure in blue jays. Anim Behav 36: 843-853.
Kotler BP, 1997. Patch use by gerbils in a risky environment: manipulating food and safety to test four models. Oikos 78: 274-282.
Kotler BP, Gross JE, Mitchell WA, 1994. Applying patch use to assess aspects of foraging behavior in Nubian ibex. J Wildl Manage 58: 299-307.
Lima SL, 1984. Downy woodpecker foraging behavior: efficient sampling behavior in simple stochastic environments. Ecology 65: 166-174.
Lima SL, 1985. Sampling behavior of starlings foraging in simple patch environments. Behav Ecol Sociobiol 16: 135-142.
Lima SL, 1986. Predation risk and unpredictable feeding conditions: determinants of body mass in birds. Ecology 67: 377-385.
Lima SL, Dill LM, 1990. Behavioral decisions made under the risk of predation: a review and prospectus. Can J Zool 68: 619-640.[ISI]
Ludwig D, Rowe L, 1990. Life-history strategies for energy gain and predator avoidance under time constraints. Am Nat 135: 686-707.
Mangel M, Clark CW, 1988. Dynamic modeling in behavioral ecology. Princeton, New Jersey: Princeton University Press.
McNamara JM, 1990. The policy which maximizes long-term survival of an animal faced with the risks of starvation and predation. Adv Appl Prob 22: 295-308.
McNamara JM, Houston AI, 1986. The common currency for behavioral decisions. Am Nat 127: 358-378.[ISI]
McNamara JM, Houston AI, 1987. Starvation and predation as factors limiting population size. Ecology 68: 1515-1519.
McNamara JM, Houston AI, 1990. The value of fat reserves and the trade-off between starvation and predation. Acta Biotheor 38: 37-61.[ISI][Medline]
McNamara JM, Houston AI, 1997. Currencies for foraging based on energy gain. Am Nat 150: 603-617.
McNamara JM, Houston AI, Lima SL, 1994. Foraging routines of small birds in winter: a theoretical investigation. J Avian Biol 25: 287-302.
McNamara JM, Mace RH, Houston AI, 1987. Optimal daily routines of singing and foraging. Behav Ecol Sociobiol 20: 399-405.
Mellgren RL, 1982. Foraging in a simulated natural environment: there's a rat loose in the lab. J Exp Anal Behav 38: 93-100.[Medline]
Munger JC, 1984. Optimal foraging? Patch use by horned lizards (Iguanidae: Phyrnosoma). Am Nat 123: 654-680.
Newman JA, 1991. Patch use under predation hazard: foraging behavior in a simple stochastic environment. Oikos 61: 29-44.
Nonacs P, Dill LM, 1990. Mortality risk vs. food quality trade-offs in a common currency: ant patch preferences. Ecology 71: 1886-1892.
Nonacs P, Dill LM, 1993. Is satisficing an alternative to optimal foraging theory? Oikos 23: 371-375.
Parker GA, 1978. Searching for mates. In: Behavioural ecology: an evolutionary approach. (Krebs JR, Davies NB, eds). Oxford: Blackwell Scientific; 214-244.
Podolsky RH, Price MV, 1990. Patch use by Dipodomys deserti (Rodentia: Heteromyidae): profitability, preference, and depletion dynamics. Oecol 83: 83-90.
Pyke GH, 1978. Optimal foraging in hummingbirds: testing the marginal value theorem. Am Zool 18: 739-752.
Roitberg BD, 1990. Optimistic and pessimistic fruit flies: evaluating fitness consequences of estimation errors. Behaviour 114: 65-82.
Roitberg BD, Prokopy RJ, 1982. Influence of intertree distance on foraging behaviour of Rhagoletis pomella in the field. Ecol Entomol 7: 437-442.
Schluter D, 1982. Optimal foraging in bats: some comments. Am Nat 119: 121-125.
Stephens DW, Krebs JR, 1986. Foraging theory. Princeton, New Jersey: Princeton University Press.
Todd IA, Kacelnik A, 1993. Psychological mechanisms and the marginal value theorem: dynamics of scalar memory for travel time. Anim Behav 46: 765-775.
Tome MW, 1988. Optimal foraging: food patch selection by ruddy ducks. Oecol 76: 27-36.
Valone TJ, 1991. Bayesian and prescient assessment: foraging with preharvest information. Anim Behav 41: 569-577.
van Alphen JJM, Gallis F, 1983. Patch time allocation and parasitization efficiency of Asorbara tabida. J Anim Ecol 52: 937-952.
Werner EE, Gilliam JF, Hall DJ, Mittelbach G, 1983. An experimental test of the effects of predation risk on habitat use in fish. Ecology 64: 1540-1548.
Witter MS, Cuthill IC, Bonner RHC, 1994. Experimental investigations of mass-dependent predation risk in the European starling, Sturnus vulgaris. Anim Behav 48: 201-222.
Ydenberg RC, 1984. Great tits and giving-up times: decision rules for leaving patches. Behaviour 90: 1-24.
CiteULike 
Connotea 
Del.icio.us What's this?
This article has been cited by other articles:
|
|
 |
|
  T. Spataro and C. Bernstein Influence of environmental conditions on patch exploitation strategies of parasitoids Behav. Ecol., July 1, 2007; 18(4): 742 - 749. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
C. Tentelier, E. Desouhant, and X. Fauvergue Habitat assessment by parasitoids: mechanisms for patch use behavior Behav. Ecol., July 1, 2006; 17(4): 515 - 521. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
B. Marsh, C. Schuck-Paim, and A. Kacelnik Energetic state during learning affects foraging choices in starlings Behav. Ecol., May 1, 2004; 15(3): 396 - 399. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. Thiel and T. S. Hoffmeister Knowing your habitat: linking patch-encounter rate and patch exploitation in parasitoids Behav. Ecol., May 1, 2004; 15(3): 419 - 425. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
O. Ovadia and H. zu Dohna The effect of intra- and interspecific aggression on patch residence time in Negev Desert gerbils: a competing risk analysis Behav. Ecol., July 1, 2003; 14(4): 583 - 591. [Abstract] [Full Text] [PDF]
|
|
| ||||||||