Predator density-dependent prey dispersal in a patchy environment with a refuge for the prey

BIOLOGICAL MODELLING   Predator density-dependent prey dispersal in a patchy environment with a refuge for the prey     K. Dao DucI, *; P. AugerI, II; T. Nguyen-HuuI, II IIXXI, ENS Lyon,46 allée d'Italie, 69364 Lyon cedex 07, France IIIRD UR Geodes, Centre IRD de l'Ile de France, 32, Av. Henri Varagnat, 93143 Bondy cedex, France     ABSTRACT In this article, we examine a two-patch predator–prey model which incorporates a refuge for the prey. We suppose that prey migration is dependent on predator density, according to a general function. We consider two different time scales in the dynamics of the model, a fast one describing patch to patch migration, and a slow one involving local prey and predator interaction. We take advantage of the time scales to reduce the dimension of the model by use of methods of aggregation of variables, and thereby examine the effect of predator density-dependent migration of prey on the stability of the predator–prey system. We establish a simple criterion of viability, namely, the existence of a positive and globally stable equilibrium, and show that density dependence has beneficial effects on both species by providing larger equilibrium densities.     Introduction Predator–prey theory has long been and remains a dominant and important theme in ecology and mathematical ecology, for which many problems remain open.1 Considered since the first Lotka-Volterra model as a classical application of mathematics in biology, models based on differential equations for interactions between species, thanks to analytical techniques and computerization, have become progressively more complex. They increasingly give a more realistic description of ecological systems, and thereby have improved our understanding of the dynamic relationship between prey and predator. The hiding behaviour of prey in particular has been incorporated as an important ingredient of predator–prey systems2 and its consequences on stability have been studied in several models. The traditional way in which refuge has been introduced is via a 'snapshot' approach, requiring that a constant proportion or number of prey cannot be killed by the predators.3 Some early theoretical work suggests that the use of refuges by prey, according to this approach, has a stabilizing effect on predator–prey dynamics,4–6 whereas other models show no such simple pattern.7,8 More recently, several studies have taken into account the dynamic nature of the refuge, and more generally, the importance of spatial heterogeneity,9,10 using patchy environment in their models.11–13 The behavioural aspect of this kind of migration focuses on its possible modalities. The density dependence of dispersal has been studied in many papers, whereas the dependence of predator density in prey migration13,14 is relatively less studied than the dependence of prey density in predator migration.15–18 Spatial heterogeneity leads to the consideration of two different types of dynamics – local interactions between species, on the one hand, and migrations from patch to patch on the other. In some cases, there exist two different time scales (for instance, a fast time scale corresponding to individual processes like migration, and a slow one for demographic changes (see ref. 14); it is then possible to reduce, via results provided by geometrical singular perturbation (GSP) theory, the dimension of the mathematical model to obtain a reduced or 'aggregated' model which can be handled analytically).19–21 It has been shown11 by using these methods that the refuge has a stabilizing effect on the equilibrium for a simple Lotka-Volterra model with refuge and density-independent migration. The purpose of this article is to examine the impact of predator density on the migration of prey in such a model. We add here the idea that, to survive, prey has to search for resources outside its refuge and so exposes itself to predation. We first describe the predator–prey model, comprising a set of three ordinary differential equations governing the local dynamics of prey and predator population densities. These dynamics present two time scales, which enables us to use aggregation of variables methods, based on perturbation techniques and on application of a centre manifold theorem of Fenichel26 to reduce the model to an aggregated one that consists of two equations. To evaluate the impact of density dependence in general, our model also uses a general predator density-dependent function for prey migration. We then study this model and its equilibrium points, and find a simple criterion of stability for a positive equilibrium, depending on various parameters and on the density-dependent migration function. The stability analysis of the non-trivial equilibrium point so found is then considered with a discussion of the results and their ecological interpretations.   The predator–prey model The model considers two patches, 1 and 2. The prey can move on both patches whereas the predator remains on patch 1. Patch 2, therefore, is a refuge for the prey. Let us denote ni(t) as the density of the prey at time t on patch i (i = 1, 2) and p(t) the density of the predator at time t on patch 1. We assume that there are two different time scales of the associated dynamics. Migrations are considered to be fast compared to predator–prey interactions. In the prey equations, the dynamics on patch 1 (conversely 2) is represented by a positive (conversely negative) term describing the natural growth (conversely mortality, as there are no resources in the refuge) and a negative term representing prey killed by predators on patch 1. For the predator, we consider a constant natural mortality rate and assume that growth is proportional to the density of prey captured. The complete system, composed of a set of three ordinary differential equations, is described as follows: The term r1 > 0 represents the intrinsic growth rate of the prey population in patch 1. Terms r2 and µ are natural mortality rate for prey in patch 2 and for predator in patch 1, respectively. The predation rates are given by a and b. The parameter k represents the prey migration rate from patch 2 to patch 1; the prey migration rate from patch 1 to patch 2 is assumed to be predator-density dependent with the function (p), which we suppose to be positive and to increase with p. In other words, the more predators found on a patch, the more prey tends to leave the patch. We suppose (0) > 0 because there is a natural migration from the resource patch to the refuge even if there is no predator. Let us define: n(t) = n1 + n2 (t), which is the total prey density. Our reduction method is based on classical aggregation methods.21 We now consider the model as an ε-perturbation of the non-perturbed problem obtained for ε = 0, which presents the following fast equilibrium: with For each value of n and p, this equilibrium is hyperbolically stabl (...truncated)