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* [10]Abstract
* [11]Introduction
* [12]Section snippets
* [13]References (54)
* [14]Cited by (29)

[15]Elsevier

[16]Journal of Theoretical Biology

[17]Volume 339, 21 December 2013, Pages 112-121
[18]Journal of Theoretical Biology

Behavioral refuges and predator-prey coexistence

Author links open overlay panel (BUTTON) Vlastimil Krivan ^a ^b
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[19]https://doi.org/10.1016/j.jtbi.2012.12.016[20]Get rights and content

Abstract

The effects of a behavioral refuge caused either by the predator optimal foraging
or prey adaptive antipredator behavior on the Gause predator-prey model are
studied. It is shown that both of these mechanisms promote predator-prey
coexistence either at an equilibrium, or along a limit cycle. Adaptive prey
refuge use leads to hysteresis in prey antipredator behavior which allows
predator-prey coexistence along a limit cycle. Similarly, optimal predator
foraging leads to sigmoidal functional responses with a potential to stabilize
predator-prey population dynamics at an equilibrium, or along a limit cycle.

Highlights

|> Optimal predator foraging or optimal predator avoidance by prey creates a
behavioral refuge for prey in predator-prey models. |> Such a behavioral refuge
promotes predator-prey coexistence in the Gause predator-prey model. |> Predator
avoidance by prey leads to a game that has two evolutionarily stable strategies
at current population densities. |> The existence of these ESS leads to a
hysteresis in prey behavior.

Introduction

Presence of a refuge has been known to promote predator-prey coexistence for a
long time. One of the first such experimental evidence was reported by Gause et
al. (1936) who observed in experiments with protists and yeast that when at low
densities, yeast formed into a sediment that was not accessible to protists
staying in the water column. When at low density, the yeast was effectively in a
refuge and protists and yeast densities fluctuated. Using a predator-prey model
with exponentially growing prey and decelerating functional response these
authors also showed that a refuge can promote predator-prey coexistence along a
limit cycle (for a full analysis see Krivan, 2011). For the Lotka-Volterra
predator-prey model Maynard Smith (1974) considered a refuge that protects either
a fixed number of prey, or a constant fraction of prey. He concluded that refuges
protecting a constant number of prey stabilize population dynamics more strongly
than refuges protecting a proportion of prey.

The work mentioned so far assumes passive (non-adaptive) refuge use by prey:
either a fixed number or a fixed proportion of prey stays in the refuge. However,
using a refuge leads to a trade-off, because being in a refuge increases survival
due to lower predation but decreases other components of prey fitness (e.g., food
intake or mating opportunities). It has been clearly documented that under
increasing predation risk prey reduce their activity or change their habitat
adaptively (e.g.., Sih, 1980, Sih, 1986, Lima and Dill, 1990, Peacor and Werner,
2001, Brown and Kotler, 2004). Models of adaptive refuge use (reviewed in Krivan,
1998) were also studied in the literature (e.g., Ives and Dobson, 1987, Sih,
1987, Ruxton, 1995). These models assume that prey strategy is a function of
predation risk. Krivan (1998) used a game theoretical approach to derive
evolutionarily stable prey antipredator strategy as a function of predator
density. For fitness functions based on the Lotka-Volterra population dynamics
there were only two possibilities: below a critical predator density all prey
were outside of the refuge while above the threshold they were in the refuge. The
corresponding population dynamics then had a neutrally stable equilibrium at
which either all prey were in the refuge, or out of the refuge. It was predicted
that a similar behavior can be expected in the case where the linear functional
response is replaced by the Holling type II functional response. In this paper I
will study such a model and I will show that for the Holling type II functional
response the situation is much more complex as the prey fitness depends not only
on predator abundance but also on prey abundance. This makes the prey fitness
frequency dependent, and the optimal prey strategy must be sought in the form of
an evolutionarily stable strategy (ESS Hofbauer and Sigmund, 1998).

I will also survey how predator's optimal foraging (Oaten and Murdoch, 1975,
Charnov, 1976a) can create a behavioral prey refuge. Both of these optimal
foraging models predict that at low preferred prey densities the interaction
strength between prey and predators sharply decreases because predators either
switch to an alternative prey type, or include an alternative prey type to their
diet. Such a behavior creates a refuge for the primary prey type. Although the
effects of optimal foraging were studied extensively in the literature (e.g.,
Holt, 1983, Fryxell and Lundberg, 1994, Abrams and Matsuda, 1996, Fryxell and
Lundberg, 1997, Krivan, 1997, Abrams, 1999, Krivan and Eisner, 2003, Ma et al.,
2003) analysis in this paper allows me to compare effects of adaptive refuge use
by prey with the refuge caused by predator's optimal foraging.

Section snippets

Adaptive refuge use by prey

In this paper I study the effect of refuges on the Gause predator-prey population
dynamics
[MATH: dRdt=rR-Cf(R),dCdt=(g(R)-m)C :MATH]
and its variants. Here R is prey density, C is predator density, r is the per
capita prey population growth rate, f is the functional response, g is the
numerical response, and m is the predator mortality rate. The above model is not
of the Kolmogorov type (Svirezhev and Logofet, 1983), because it assumes
unlimited exponential prey growth. When the functional response is of the

Refuges caused by predator foraging behavior

In this section I briefly review some consequences of predator optimal foraging
behavior in the context of behavioral prey refuges. I will consider the optimal
diet choice and prey switching models.

The diet choice model (Charnov, 1976b, Stephens and Krebs, 1986) assumes that
predators rank potential prey types on the basis of their profitability measured
by the ratio of energy gain over the handling time (i.e.,
[MATH: e/h :MATH]
). Here I consider two prey types, a primary prey type (R) which is more

Discussion

In this paper, effects of a refuge on the Gause predator-prey model are studied.
Two types of adaptive prey or predator behavior are studied in detail. First, I
consider the effect of adaptive refuge use by prey. In this case the prey fitness
is frequency dependent and I analyzed the corresponding prey ESS as a function of
prey and predator numbers. Due to the risk dilution effect (e.g., Foster and
Treherne, 1981) caused by the Holling type II functional response (i.e., the
decrease in

Acknowledgment

I thank two anonymous reviewers for their thoughtful suggestions. This work was
partly conducted while the author was a Sabbatical Fellow at the Mathematical
Biosciences Institute, an Institute sponsored by the National Science Foundation
under Grant DMS 0931642 and at the National Institute for Mathematical and
Biological Synthesis, an Institute sponsored by the National Science Foundation,
the US Department of Homeland Security, and the US Department of Agriculture
through NSF Award EF-0832858
(BUTTON) Recommended articles

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Cited by (29)

[34]Impact of prey refuge in a discontinuous Leslie-Gower model with harvesting and
alternative food for predators and linear functional response
2023, Mathematics and Computers in Simulation
(BUTTON) Show abstract
Since in nature there are species that take refuge, totally or
proportionally, from the predator if the prey population size exceeds a
threshold value, and become available again to the predator, with linear
predator functional response, if its population size is higher than the
threshold value, in this work we show all the changes in the dynamics
presented by two discontinuous Leslie-Gower predator-prey models, assuming
harvesting and alternative food for predators and total protection, or a
constant proportional refuge of prey from being consumed by the predator when
its population size is below the threshold value. The conditions on their
parameters to determine the dynamics of each discontinuous model are
developed by means of a bifurcation analysis, with respect to the threshold
value of the prey population size and the collection rate for predators. It
is concluded that for certain conditions on their parameters, the prey
population size could reach a convergence equal or higher than its threshold
value when considering the discontinuous model with a mechanism to protect
prey from being consumed by the predator, as opposed to the discontinuous
model with a constant proportion of prey refuges above the threshold value,
whose convergence could be lower, equal or higher than the threshold value.
*

[35]Bifurcations on a discontinuous Leslie-Grower model with harvesting and alternative
food for predators and Holling II functional response
2023, Communications in Nonlinear Science and Numerical Simulation
(BUTTON) Show abstract
This paper proposes a mathematical model that describes the interaction of
prey and predators, assuming logistic growth for both species, harvesting and
alternative food for predators and functional response of the Holling II
predator. When performing a qualitative analysis to determine conditions in
the parameters that allow the possible extinction or preservation of prey
and/or predators, a modification of the initial model is made considering
that the consumption of prey by predators is restricted if the amount of prey
is less than a critical value, whose dynamics is formulated by a planar
Filippov system. The study of the discontinuous model is carried out by
bifurcation analysis in relation to two parameters: harvesting of predators
and critical value of prey.
*

[36]Scaling from optimal behavior to population dynamics and ecosystem function
2022, Ecological Complexity
Citation Excerpt :
This highlights that the relative time-scales need to be considered before
choosing a model with instantaneous optimization. Generally, however,
behavior and population dynamics are often on decoupled time-scales (Krivan,
2013). When this decoupling occurs, incorporating behavior as an
instantaneous game is a valuable tool and our approach can be used to find
complex emergent phenomena.
(BUTTON) Show abstract
While behavioral responses of individual organisms can be predicted with
optimal foraging theory, the theory of how individual behavior feeds back to
population and ecosystem dynamics has not been fully explored. Ecological
models of trophic interactions incorporating behavior of entire populations
commonly assume either that populations act as one when making decisions,
that behavior is slowly varying or that non-linear effects are negligible in
behavioral choices at the population scale. Here, we scale from individual
optimal behavior to ecosystem structure in a classic tri-trophic chain where
both prey and predators adapt their behavior in response to food availability
and predation risk. Behavior is modeled as playing the field, with both
consumers and predators behaving optimally at every instant basing their
choices on the average population behavior. We establish uniqueness of the
Nash equilibrium, and find it numerically. By modeling the interactions as
playing the field, we can perform instantaneous optimization at the
individual level while taking the entire population into account. We find
that optimal behavior essentially removes the effect of top-down forcing at
the population level, while drastically changing the behavior. Bottom-up
forcing is found to increase populations at all trophic levels. These
phenomena both appear to be driven by an emerging constant consumption rate,
corresponding to a partial satiation. In addition, we find that a Type III
functional response arises from a Type II response for both predators and
consumers when their behavior follows the Nash equilibrium, showing that this
is a general phenomenon. Our approach is general and computationally
efficient and can be used to account for behavior in population dynamics with
fast behavioral responses.
*

[37]Asymptotic stability of delayed consumer age-structured population models with an
Allee effect
2018, Mathematical Biosciences
(BUTTON) Show abstract
In this article we study a nonlinear age-structured consumer population model
with density-dependent death and fertility rates, and time delays that model
incubation/gestation period. Density dependence we consider combines both
positive effects at low population numbers (i.e., the Allee effect) and
negative effects at high population numbers due to intra-specific competition
of consumers. The positive density-dependence is either due to an increase in
the birth rate, or due to a decrease in the mortality rate at low population
numbers. We prove that similarly to unstructured models, the Allee effect
leads to model multi-stability where, besides the locally stable extinction
equilibrium, there are up to two positive equilibria. Calculating derivatives
of the basic reproduction number at the equilibria we prove that the upper of
the two non-trivial equilibria (when it exists) is locally asymptotically
stable independently of the time delay. The smaller of the two equilibria is
always unstable. Using numerical simulations we analyze topologically
nonequivalent phase portraits of the model.
*

[38]Asymptotic stability of tri-trophic food chains sharing a common resource
2015, Mathematical Biosciences
Citation Excerpt :
Several mechanisms explaining species coexistence were proposed. These
include, but are not limited to non-equilibrium dynamics due to environmental
[3] or internal [4] fluctuations in population dynamics, relative
nonlinearity in species responses to competition [5,6], predation on
competing species [7,8], or adaptive foraging [9,10]. These mechanisms fit
into two broad categories [5]: (i) stabilizing mechanisms that tend to
increase negative intraspecific interactions relative to interspecific
interactions (density dependent mechanisms, e.g., the logistic population
growth) and (ii) equalizing mechanisms that tend to decrease average fitness
differences between species.
(BUTTON) Show abstract
One of the key results of the food web theory states that the interior
equilibrium of a tri-trophic food chain described by the Lotka-Volterra type
dynamics is globally asymptotically stable whenever it exists. This article
extends this result to food webs consisting of several food chains sharing a
common resource. A Lyapunov function for such food webs is constructed and
asymptotic stability of the interior equilibrium is proved. Numerical
simulations show that as the number of food chains increases, the real part
of the leading eigenvalue, while still negative, approaches zero. Thus the
resilience of such food webs decreases with the number of food chains in the
web.
*

[39]L-shaped prey isocline in the Gause predator-prey experiments with a prey refuge
2015, Journal of Theoretical Biology
Citation Excerpt :
Similarly, a predator isocline with a horizontal segment bounds maximal
oscillations in predator numbers. Theoretical models that predict such
isoclines can arise due to optimal prey selection by predators, or prey use
of a refuge (e.g., Rosenzweig and MacArthur, 1963; Rosenzweig, 1977; Fryxell
and Lundberg, 1994; Krivan, 1998; van Baalen et al., 2001; Brown and Kotler,
2004; Krivan, 2007, 2013). In an attempt to falsify the Lotka-Volterra
predator-prey model Gause experimented with various predator-prey systems
(Gause, 1934, 1935a; Gause et al., 1936).
(BUTTON) Show abstract
Predator and prey isoclines are estimated from data on yeast-protist
population dynamics ([40]Gause et al., 1936). Regression analysis shows that
the prey isocline is best fitted by an L-shaped function that has a vertical
and a horizontal part. The predator isocline is vertical. This shape of
isoclines corresponds with the Lotka-Volterra and the Rosenzweig-MacArthur
predator-prey models that assume a prey refuge. These results further support
the idea that a prey refuge changes the prey isocline of predator-prey models
from a horizontal to an L-shaped curve. Such a shape of the prey isocline
effectively bounds amplitude of predator-prey oscillations, thus promotes
species coexistence.

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50. https://www.elsevier.com/
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