Ecology, 87(8), 2006, pp. 2094–2102
Ó 2006 by the the Ecological Society of America
OPTIMAL MOVEMENT STRATEGIES FOR SOCIAL FORAGERS IN
UNPREDICTABLE ENVIRONMENTS
PENELOPE A. HANCOCK1
AND
E. J. MILNER-GULLAND
Department of Environmental Science and Technology, Manor House, Silwood Park Campus, Imperial College London, Ascot,
Berkshire SL5 7PY UK
Abstract. Spatial movement models often base movement decision rules on traditional
optimal foraging theories, including ideal free distribution (IFD) theory, more recently
generalized as density-dependent habitat selection (DDHS) theory, and the marginal value
theorem (MVT). Thus optimal patch departure times are predicted on the basis of the densitydependent resource level in the patch. Recently, alternatives to density as a habitat selection
criterion, such as individual knowledge of the resource distribution, conspecific attraction, and
site fidelity, have been recognized as important influences on movement behavior in
environments with an uncertain resource distribution. For foraging processes incorporating
these influences, it is not clear whether simple optimal foraging theories provide a reasonable
approximation to animal behavior or whether they may be misleading. This study compares
patch departure strategies predicted by DDHS theory and the MVT with evolutionarily
optimal patch departure strategies for a wide range of foraging scenarios. The level of
accuracy with which individuals can navigate toward local food sources is varied, and
individual tendency for conspecific attraction or repulsion is optimized over a continuous
spectrum. We find that DDHS theory and the MVT accurately predict the evolutionarily
optimal patch departure strategy for foragers with high navigational accuracy for a wide range
of resource distributions. As navigational accuracy is reduced, the patch departure strategy
cannot be accurately predicted by these theories for environments with a heterogeneous
resource distribution. In these situations, social forces improve foraging success and have a
strong influence on optimal patch departure strategies, causing individuals to stay longer in
patches than the optimal foraging theories predict.
Key words: density-dependent habitat selection; foraging theory; genetic algorithm; ideal free
distribution; individual-based model; marginal value theorem; optimal patch use; spatial movement models.
INTRODUCTION
An understanding of animal movement processes is
important to many ecological questions, including
ecosystem management (Fryxell et al. 2004), assessing
the viability of endangered species (Ruckelshaus et al.
1997), and predicting animal disease outbreaks (Morgan
et al. 2004). Given the difficulty of sampling animal
locations and movements at large spatial scales, spatially
explicit simulation models are widely used to investigate
questions relating to animal movement behavior (Zollner and Lima 1999, Rands et al. 2003, 2004, Fryxell et
al. 2005). Applications of these models range from
simulating theories about social behaviors that drive
animal movement, such as the selfish herd hypothesis
(Reluga and Viscido 2005), to incorporating resourcebased drivers for animal movement based on experimentally parameterized herbivore grazing system models (Fryxell et al. 2004).
Manuscript received 18 October 2005; revised 24 January
2006; accepted 27 January 2006. Corresponding Editor: B. P.
Kotler.
1 E-mail: [email protected]
Theories of density-dependent habitat selection recognize the importance of population density in influencing the habitats animals choose to occupy (Rosenzweig
1991, Jonzén et al. 2004). Animals sacrifice habitat
quality for the sake of occupying a lower-density
environment (Rosenzweig 1991), and any advantages
of grouping behavior, such as collective decision-making
or predator avoidance, come with a density-dependent
cost. Density-dependent habitat selection as a driver for
movement behavior has its basis in ideal free distribution (IFD) theory. Fretwell and Lucas (1969) show that
under a set of ‘‘ideal’’ conditions, the IFD (whereby the
number of individuals in a patch is directly proportional
to the patch’s resources) is an evolutionarily stable
spatial distribution. The model assumptions allow the
effect of density dependence on the spatial distribution
of animals to be considered in isolation from more
realistic foraging constraints and include unlimited
movement ability and full knowledge of the global
distribution of per capita food availability. More
generally, density-dependent habitat selection (DDHS)
theory extends IFD theory to predict an evolutionarily
stable strategy of fitness equalization across all patches,
where fitness can be any negative function of density
(Possingham 1992, Jonzén et al. 2004).
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MOVEMENT STRATEGIES FOR SOCIAL FORAGERS
In situations in which the population density or
resource distribution is continually changing, individuals must make spatial movement decisions in response to
the changing IFD. The question of the optimal time to
depart from a patch and move to another is thus central
to understanding animal movement. In accordance with
the arguments of Fretwell and Lucas (1969) and Jonzén
et al. (2004), DDHS theory predicts that individuals will
leave a patch when their fitness drops below the average
fitness across all patches. Foraging models usually derive
patch departure rules according to the marginal value
theorem (MVT; Charnov 1976), which specifies that
individuals will leave a patch when their uptake rate is
less than the mean uptake rate across all patches
(Bernstein et al. 1991, Beauchamp et al. 1997, Tyler
and Hargrove 1997, Beecham and Farnsworth 1998,
Ward et al. 2000, Fryxell et al. 2004). The MVT assumes
that the forager knows the resource level of all patches
and rarely or never revisits patches and that the rate of
food intake in any patch decreases monotonically with
the time spent in the patch. In a system in which average
food uptake rate is a measure of fitness, the MVT patch
departure strategy is the same as that which arises from
DDHS theory. This patch departure strategy has been
widely tested empirically and agrees qualitatively with
the behavior of a number of foragers (Vivas and Saether
1987, Distel et al. 1995, Bonser et al. 1998).
While the MVT and DDHS theory are derived from
different conceptual frameworks, they both incorporate
restrictive sets of assumptions that limit the extent to
which they can explain the behavior of real foragers.
Many factors influence foraging behavior that are not
encompassed by DDHS theory or the MVT, including
uncertainty associated with individual knowledge (Clark
and Mangel 2000), stochasticity in the resource supply,
and varying spatial scales of individual knowledge,
perceptual ability, and animal movement (Tyler and
Hargrove 1997, Koops and Abraham 2003). Recently,
stochastic versions of DDHS theory have been developed for a two-patch system (Jonzén et al. 2004). In
general, however, as foraging models become increasingly complex it is not clear whether simple optimal
foraging theories provide a reasonable approximation to
animal behavior or whether they may be misleading
(Bernstein et al. 1991, Tyler and Hargrove 1997, Ward et
al. 2000). Thus theoretical patch departure studies
emphasize the need for further examination of the
applicability of the MVT to a wider range of abstract
problems and to the reality of animal behavior (Clark
and Mangel 1984, Bernstein et al. 1991).
Recent studies seek alternatives to density as a habitat
selection criterion and focus on the importance of
information-gathering mechanisms such as conspecific
attraction and fidelity to memorized locations for
foragers in uncertain environments (Dall et al. 2005,
Gautestad and Mysterud 2005). Movement biases based
on these mechanisms violate the assumptions of movement models conforming to the predictions of DDHS
2095
theory (Gautestad and Mysterud 2005); however, very
few studies have investigated the effect of these processes
on the accuracy of the predictions of DDHS theory. The
role of conspecific attraction in increasing preference for
patches with higher population density and generating
erratic changes in the spatial distribution of animals is
noted by Fretwell and Lucas (1969). Beauchamp et al.
(1997) modeled conspecific attraction behavior using a
producer–scrounger model and showed that the incorporation of these social foraging tactics prevented
foragers from reaching the IFD. However, both these
models assume the existence of conspecific attraction
without incorporating an underlying mechanism or
motivation for this behavior.
We present a generalized foraging model varying the
level of information available to the individual about the
resource distribution. Conspecific attraction and repulsion behavior is incorporated using the model presented
in Hancock et al. (2006). In this model, individuals can
use information derived from the movement directions
of conspecifics to reduce their own directional error in
navigating to surrounding food sources. We generalize
the ideas of Beauchamp et al. (1997) by optimizing
conspecific attraction and repulsion behavior over a
continuous spectrum of strategies and determining the
effect of these social behaviors on optimal patch
departure strategies.
Spatially explicit individual-based models with statedependent foraging rules are used to model the foraging
process with a high degree of flexibility, and genetic
algorithms are used to find optimal patch departure
strategies. Other studies have examined the effect of
relaxing the assumptions of IFD theory to incorporate
greater realism into the foraging process on the ability of
the population to reach an IFD (Tyler and Hargrove
1997, Ranta 1999), but none have computed optimal
patch departure strategies for non-ideal foragers. We
investigate the degree to which DDHS theory accurately
predicts the optimal patch departure strategy for
foragers whose knowledge of the local resource distribution is limited in spatial accuracy and extent.
METHODS
Nonsocial movement
An individual-based coupled map lattice model was
developed to simulate spatial movement behavior in
state-dependent foragers in the absence of social forces.
Each cell in the lattice is assigned a certain resource
level, and each individual has a certain spatial location,
reserve level, and age. Energy accumulation and
expenditure is modeled using a simple state-based model
incorporating temporally structured reproductive activity. Individual reserve levels are assumed to depend
stochastically on food supply and population density, as
follows:
j
b
Rjkþ1 ¼ Rjk þ Xi;k
ð1Þ
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PENELOPE A. HANCOCK AND E. J. MILNER-GULLAND
Ecology, Vol. 87, No. 8
FIG. 1. Cells in the sensing range Sd, where the cell number is equal to the direction d. Arrows point to values of d that
correspond to arrow direction.
where Rjk is the reserve level of the jth individual at time
k. Reserves are metabolized at a constant rate b. Reserve
levels increase according to the stochastic consumption
of food, modeled by realizations of a normal random
j
; N(ci,k, rR), with mean ci,k and standard
variable Xi;k
deviation rR, where ci,k is the per capita resource supply
in cell i:
ci;k ¼
aFi;k
:
Pi;k
ð2Þ
Fi,k is the resource supply and Pi,k is the population
abundance in cell i and time k, and the competition
coefficient a is a constant. Negative realizations of the
random variable Xkj are set to zero. At equilibrium the
mean reserve level for the population is constant over
time, and thus b is equal to ci,k.
Fecundity and mortality depend directly on the
individual’s reserve level. Death occurs if the reserve
level falls below a threshold RM. Individuals older than
age aT reproduce if their reserve level is above a second
threshold RT. An individual produces a litter containing
two offspring every time it reproduces, and there is a
minimum time tL between litters. This model is a simple
energy resource/flow model similar to those developed
by Fielding (2004), Rands et al. (2004), and Hancock et
al. (2005). Model structure and parameter values are the
same as those used in Hancock et al. (2005), because this
model was shown to be robust to parameter variation.
Initially 15 000 individuals with randomly assigned
reserve levels and ages are randomly placed in cells on a
50 3 50 lattice. Each animal can make a decision to stay
in its current cell or move to one of eight neighboring
cells three times in each time step, giving a maximum
movement distance of three cells per time step. This
allows the individual to move to anywhere within its
perceptual range (which is defined below) in a given time
step. All animals in the population make one movement
decision before the next round of decisions, and the
order in which animals in the population make movement decisions is randomized for each time step.
Movement destination is decided on the basis of the
perceived resource level in each direction, which is
detected within a perceptual range. The direction d
moved by an animal in cell i is given by the direction that
has the maximum value of
U1 sd
d ¼ 1; . . . ; 8
ð3Þ
where sd is the relative resource amount that the
individual senses is in direction d and U1 is a random
number generated from a uniform distribution ranging
from 0 to 1. The ability of an individual to sense a food
source declines exponentially with its distance from the
source; thus sd is given by
X
sd0 ¼
ehq Fq;k :
ð4Þ
sd ¼ ðsd0 Þz
q2Sd
Sd is the set of cells that contribute to the relative
resource amount sensed in direction d, reflecting
decreasing directional precision with distance (Fig. 1).
The distance between cell i and cell q is given by hq. The
exponent z, termed the ‘‘sensing power,’’ dampens the
variation in the amount of resource sensed in the
direction d. The higher the value of z, the stronger the
ability of the individual to sense the difference between
the resource level in each direction and the greater its
chance of moving in the direction that has the highest
resource level. We assume that all individuals have the
same z value, although stochasticity in individual
sensing ability is incorporated via Eq. 3. By varying z
between simulations, we can vary the level of uncertainty in knowledge of surrounding food supplies on a
continuous spectrum.
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MOVEMENT STRATEGIES FOR SOCIAL FORAGERS
2097
TABLE 1. Spatial distribution of resources for environment types with varying degrees of
heterogeneity in the resource distribution.
Environment type
No.
active
cells
Resource
level in
active cells
Resource
level in
inactive cells
High patchiness
Medium/high patchiness
Medium/low patchiness
Low patchiness
30
60
200
0
2000
1000
200
...
10
10
20
l ¼ 32 For the low patchiness environment all cells have the same mean resource level and a level
of variability x ¼ 0.2 (see Methods: Nonsocial movement).
An individual decides to leave its current cell and
move to a neighboring cell if ci,k is less than its value of
an evolvable patch departure threshold parameter cE .
Values of cE are evolved using a genetic algorithm in a
manner similar to that adopted by Fielding (2004) and
Reluga and Viscido (2005). In our case, cE values are
assigned randomly among individuals in the initial
population within an experimentally determined range
of [0,10]. Offspring inherit their parent’s cE value plus
random variation ranging from 0% to 5% of the parental
value. For each offspring, there is a probability of
mutation of 0.001, in which case the value of cE is
randomly assigned a value within the initial range.
According to density-dependent habitat selection
theory, the optimal value of cE is equal to the mean
food intake rate across the entire lattice, ci.k. We
compare evolved optimal cE values with this theoretical
optimum. To give an indication of the similarity of the
spatial distribution with the ideal free distribution, we
also compare evolved cE values with the uptake rate for
each individual if the population was distributed in an
ideal free manner, termed cIFD.
The evolutionarily optimal patch departure strategy
was computed for four different simulated environment
types designed to represent varying levels of environmental heterogeneity (Table 1). For high, medium/high,
and medium/low patchiness environments, a certain
number of cells, termed ‘‘active’’ cells, have a rich
resource supply, and the remaining ‘‘inactive’’ cells have
a low resource level. For the low patchiness environment, all cells are termed inactive and have the same
mean resource level li,k, with random variation among
the cells such that the resource amount in each cell Fi,k ¼
li,k þ ei,k where ei,k ; N(0, xli,k) and where x is the level
of variability. The mean total amount of resource for the
lattice is constant throughout time and is the same for
each environment type. At every time step the resource
supply matrix is reset so that the patches are repositioned according to the uniform random distribution.
Animals must therefore continually search for food. Fi,k
is constant throughout the kth time step, and thus the
model addresses the effect of density-dependent reduction in uptake rate in isolation from resource depletion.
As the value of cE depends on the mean uptake rate,
we set the mean uptake rate to bci,k, multiplying ci,k by a
factor of b, considering b values of 1/3, 1/2, 1, and 2. We
then compared evolved cE values across these different
uptake rates. The effect of increasing the mean uptake
rate is to decrease the total abundance, and therefore the
mean density, of the population.
Modeling social behavior
To determine whether foraging success for this system
can be improved through conspecific attraction and
repulsion and how these behaviors influence patch
departure strategies, we incorporated the social foraging
model of Hancock et al. (2006) into the above model
framework. This involved adding a conspecific interaction term to Eq. 3 to give
U1 sd þ U2 pnd
d ¼ 1; . . . ; 8
ð5Þ
where U2 is a random number generated from a uniform
distribution ranging from 0 to 1, nd is the number of
individuals in cell i that have already moved in direction d
in the current time step, and p is a continuous, evolvable
parameter (termed the ‘‘pull’’ parameter). The pull
parameter determines the level of conspecific attraction
or repulsion. If sd is similar in all directions, the individual
is more likely to move in the direction taken by the largest
proportion of the individuals that have already moved if
p is positive and less likely if p is negative.
The parameter p is coevolved with the patch departure
threshold parameter cE using a real-coded genetic
algorithm with discrete crossover recombination (Herrera and Lozano 2000). This involves combining the
parameter values of the reproducing individual and
those of a ‘‘partner’’ randomly chosen from the
population of other reproducing individuals (those with
Rjk . RT and age . aT) to give the offspring’s set of
parameter values. For each evolvable parameter, the
offspring’s parameter value is randomly chosen from the
set fxi, yig, where xi is the value of the evolvable
parameter from the first parent and yi is that from the
second parent. Random variation ranging from 0 to 5%
of the parental value is then added to the offspring’s
parameter value for each evolvable parameter. For cE,
the values assigned to the initial population and the
probability of mutation are set in the same manner as
described above. For the pull parameter p, the mutation
probability is the same, and the initial populations are
randomly assigned values within an experimentally
determined range of [100, 100].
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PENELOPE A. HANCOCK AND E. J. MILNER-GULLAND
Ecology, Vol. 87, No. 8
FIG. 2. Histograms showing distributions of evolved patch departure threshold (cE) values for (A) high-, (B) medium/high-, (C)
medium/low-, and (D) low-patchiness environments and mean uptake rates of bci,k (where ci,k represents the per capita resource
supply). The value of b is varied from 1/3 to 2, with 1/3 giving the highest population density and 2 giving the lowest population
density. Long-dashed lines represent the mean uptake rate bci,k, and short-dashed lines represent the mean uptake rate if the spatial
distribution was ideal free distribution (IFD; cIFD).
RESULTS
The algorithm was run for 20 000 iterations, which
was sufficient for convergence of the mean value of cE
for all cases considered and also of the parameter p for
simulations including conspecific attraction. We first
examined a case in which individuals had high directional food-sensing accuracy, setting the sensing power z
such that .90% of movement decisions were in the
direction of the highest resource supply. A value of z ¼
0.5 was found to be sufficient for this purpose for all
environment types.
For many scenarios, a well-defined optimum was
obtained (Fig. 2). In all cases, the modal value of cE
agrees closely with the mean uptake rate bci,k, indicating
that fitness equalization is an optimal strategy for this
system regardless of environmental heterogeneity, population density, or variability in food patch location.
Close agreement between the optimal cE and the uptake
rate for the ideal free distribution, cIFD, occurs if the
population density is high or environmental heterogeneity is low. For the low patchiness environment (Fig.
2D) the evolved distribution of cE values is normal with
a negative skew and a modal value slightly less than
cIFD. The negative skew indicates frequency dependence
in the optimal strategy. Some individuals benefit from
waiting longer before departing a patch given that the
majority of the population will depart earlier and the
uptake rate of remaining individuals will increase. For
this environment type the mean resource level is the
same for each cell, so individual uptake rate is a function
of local population density alone and is thus more likely
to be influenced by frequency dependence.
As population density decreases, the discrepancy
between the optimal cE and cIFD increases, particularly
for more heterogeneous environments, indicating that
the ideal free distribution is not obtained. For the highpatchiness environment at lowest population density
(Fig. 2A, top), the evolutionarily optimal cE value
indicates that individuals are moving primarily to
distribute themselves evenly among the inactive cells.
The modal optimal cE value for this scenario is 4.0,
slightly less than half the food amount in the inactive
cells. Thus an individual leaves an inactive cell if it
contains more than two other foragers, which corresponds to an even distribution across the grid given
that the mean number of foragers per cell for this
scenario is 2.3. Thus for low population density and
high environmental heterogeneity, the distribution
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MOVEMENT STRATEGIES FOR SOCIAL FORAGERS
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FIG. 3. Abundance vs. mean uptake rate b (as a proportion of mean per capita resource supply ci,k) for different patch
departure strategies in (A) high-, (B) medium/high-, (C) medium/low-, and (D) low-patchiness environments. Lines show the
evolutionarily optimal strategy (squares), the evolved patch departure threshold (cE) fixed at 1/3 of the modal evolutionarily
optimal value (circles), and cE fixed at three times the modal evolutionarily optimal value (triangles).
reflects local rather than regional resource availability
because a lower proportion of the population will
discover food patches when the population density is
low. As population density increases or environmental
heterogeneity decreases, the results show a gradual
shift in the scale of the environmental driver for
movement behavior, from local to regional resource
levels. Thus cE converges toward the value predicted
by the IFD.
Fitness consequences of suboptimal gamma values
Variation in total population size for cE values fixed
three times higher and three times lower than the modal
evolutionarily optimal value demonstrates the fitness
consequences of adopting suboptimal patch departure
strategies (Fig. 3). For all environment types, including
those for which the IFD is not reached, a patch
departure threshold lower than cE (i.e., moving at a
per capita food supply level lower than optimal) is
significantly more detrimental than one higher than cE
(i.e., moving before resources are as low as cE). The
evolutionarily optimal cE represents a threshold strategy, below which the rate of loss of fitness increases
markedly. As environmental heterogeneity increases and
population density decreases, it becomes more detrimental to stay in a patch longer than the optimal patch
departure threshold, particularly for the highly heterogeneous environments. For high and medium/high
patchiness environments, the population goes extinct
for the lowest population density scenario if cE is three
times lower than optimal.
The effect of uncertainty
To examine the case in which individuals are uncertain
about which direction has the highest resource supply,
we reduced the sensing ability so that ;50% of movement decisions were in the direction of the highest food
supply. This corresponded to z ¼ 0.08 for the high- and
medium/high-patchiness environments and z ¼ 0.1 for
the medium- and low-patchiness environments. For the
low- and medium/low-patchiness environments, the cE
distributions barely changed with the decrease in sensing
ability, because movement behavior is driven by local
variations in population density rather than the finding
of rich food patches. For the same reason, there is also
little change in the optimal distribution for the more
heterogeneous environments at the two lowest population density levels (Fig. 4). For the higher density levels (b
¼ 1 and b ¼ 2), the cE distributions are flat in comparison
to those for high sensing power simulations. This
indicates that an optimal patch departure strategy does
not exist in situations in which the majority of the
regional resource supply is unpredictably located in a
relatively small number of rich food patches. Situations
in which foraging success is highly dependent on chance
maintain a high variability in uptake rates and prevent
convergence toward an optimal patch departure strategy.
The effect of social interactions
Social forces were incorporated into the above
uncertain forager simulations. The only scenarios for
which the pull parameter p converged to a nonzero value
were for high- and medium/high-patchiness environments with a b value of 1/2, the value corresponding to
the highest population density scenario. These are the
situations in which social forces would be expected to
evolve as factors in foraging decision-making. They are
also the scenarios for which lowering the sensing power
made the greatest change to the optimal patch departure
strategy, giving a flat distribution of cE values (Fig. 4).
For these cases the majority of the individuals (.86%)
have positive p parameters, i.e., in the presence of
uncertainty about food source locations, following
others becomes worthwhile (the many wrongs principle;
Hancock et al. 2006). The inclusion of conspecific
attraction forces restored the presence of an optimal
cE value (Fig. 5). However, cE is tightly distributed at a
value smaller than the mean uptake rate. This indicates
that individuals are staying longer than is optimal
according to density-dependent habitat selection theory.
DISCUSSION
The prediction of DDHS theory and the MVT was
found to agree well with the evolved optimal patch
departure strategy for simulations in which key assumptions of these theories were violated, including the
requirement of a deterministic environment with a
continuous rate of resource uptake and the need for
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FIG. 4. Histograms showing distributions of evolved patch departure threshold (cE) values for simulations with individual
uncertainty regarding the local resource distribution for (A) high-, (B) medium/high-, (C) medium/low-, and (D) low-patchiness
environments and mean uptake rates of bci,k (where ci,k represents the per capita resource supply). The value of b is varied from 1/3
to 2, with 1/3 giving the highest population density and 2 giving the lowest population density. Long-dashed lines represent the
mean uptake rate bci,k, and short-dashed lines represent the mean uptake rate if the spatial distribution was ideal free distribution
(IFD; cIFD).
individuals to possess perfect knowledge of the global
resource distribution. Many foraging models incorporate these theories, although there is often uncertainty
about the applicability of abstract theories to realistic
foraging scenarios that violate the theoretical assumptions (Bernstein et al. 1991, Koops and Abraham 2003,
Fryxell et al. 2004). We have shown that, provided
individuals have accurate knowledge of the resource
distribution in the local area encompassing their movement range, the predictions of DDHS theory and the
MVT were accurate for all levels of environmental
heterogeneity and population density considered. This
study therefore shows that DDHS theory and the MVT
are more robust than previously thought and provides a
framework for testing the applicability of theoretical
predictions to unpredictable environments.
Using genetic algorithm optimization allows optimal
foraging strategies to be calculated for a wide range of
foraging systems with a continuous spectrum of
uncertainties, population densities, and environment
types. By examining the effect of variation in these
variables on the optimal solution, we can identify those
that have a predominant influence on animal movement
strategies. Firstly, the results show that regional
population density influences the scale at which resource
distribution affects the optimal patch departure strategy.
As the total abundance in the region increases, the
optimal movement strategy becomes a response to the
regional resource distribution rather than the local
resource distribution and the population approaches
the ideal free distribution. This phenomenon is an
example of regional scale population dynamics influencing local scale individual behavior. A related finding was
obtained by Tyler and Hargrove (1997), who demonstrated that population size was an important determinant of agreement between the simulated distribution
and the ideal free distribution when resource distributions were spatially discontinuous.
Of all the variables examined, the only one that
caused the model to depart from the predictions of
DDHS theory was the accuracy of individual knowledge
of the local resource distribution. In heterogeneous
environments where there is considerable uncertainty
associated with individual knowledge of surrounding
food sources and high population density, foraging
strategy is improved considerably by incorporating
social forces, allowing the use of information derived
from the behavior of conspecifics. Without social forces,
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MOVEMENT STRATEGIES FOR SOCIAL FORAGERS
2101
FIG. 5. Histograms showing distributions of evolved patch departure threshold values (cE) and the pull parameter ( p) for
simulations with social interactions for (A) high- and (B) medium/high-patchiness environments, a mean uptake rate of bci,k (where
ci,k represents the per capita resource supply), and b ¼ 1/2 (the value corresponding to the highest population density scenario).
an optimal patch departure strategy cannot evolve under
these conditions. The results indicate that individual
food intake becomes more evenly spread among
members of the population when social forces are
incorporated, allowing an optimal patch departure
strategy to be much more clearly defined in comparison
to simulations of the equivalent scenario without social
forces.
Interestingly, when social forces are incorporated, the
optimal strategy has a clearly defined peak at an uptake
rate lower than the mean uptake rate, indicating that it is
optimal for individuals to stay longer in a patch than
DDHS theory predicts. It is likely that this is related to
the way in which information is gained from moving
conspecifics, i.e., individuals are waiting until others
have moved from the patch in order to improve their
estimate of the movement direction containing the
highest resource levels. Thus when social interactions
are considered, the value of a patch includes the benefit
of information derived from the departures of others.
This result is consistent with quantitative experimental studies of patch departure strategies, which commonly find that animals stay longer in patches than
predicted (WallisDeVries et al. 1999, Nonacs 2001). This
strategy is contradictory to the predictions of the
simulations conforming to the predictions of DDHS
theory, which indicated that waiting in a patch longer
than the evolutionarily optimal strategy is considerably
more detrimental than leaving earlier. Leaving earlier is
rational if there is no specific mechanism providing an
advantage to waiting longer in a patch because it allows
individuals to depart for a wider range of uptake rates,
including the mean uptake rate. However the use of
group navigation to overcome uncertainty is a mechanism by which waiting longer in a patch than predicted
by DDHS theory is optimal. This conclusion extends the
findings of Beauchamp et al. (1997) by showing that, if
conspecific attraction and repulsion forces are allowed
to evolve, these behaviors can bring the spatial
distribution of animals closer to the IFD.
We have used a simulation model and genetic
algorithm to demonstrate that conspecific attraction
and density dependence are fundamental determinants
of the spatial distribution of animals. We have adopted a
flexible modeling framework that allows theoretical
rules of thumb to be compared to evolutionarily optimal
strategies for a wide range of foraging scenarios. The
predictions of DDHS theory and the MVT were
accurate for the majority of the scenarios. The
exceptions were scenarios with populations of individuals with an uncertain knowledge of the local resource
distribution in heterogeneous environments where the
location of rich food patches is important to foraging
success. In these cases, it is important for individuals to
gather available information, including socially acquired
information, to reduce their uncertainty. This can
2102
PENELOPE A. HANCOCK AND E. J. MILNER-GULLAND
become the overriding mechanism driving movement
(Dall et al. 2005). In these situations, patch departure
rules based on density-dependent habitat selection or the
marginal value theorem are not appropriate (Gautestad
and Mysterud 2005). The effects of density-dependent
drivers and information-gathering mechanisms must be
considered in combination.
ACKNOWLEDGMENTS
Thanks to Hugh Possingham, Sigrun Eliassen, and Niclas
Jonzén for providing helpful comments on the manuscript. This
work was funded by the Leverhulme Trust.
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