Pattern formation of a spatial predator–prey system

Abstract

There are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors. To investigate how these migration (directed movement) and diffusion (random movement) affect predator–prey systems, we have studied the spatiotemporal complexity in a predator–prey system with Holling–Tanner form. A theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated. At the same time, we present the properties of pattern solutions. The results of numerical simulations show that migration has prominent effect on the pattern formation of the population, i.e., changing Turing pattern into traveling pattern. This study suggests that modelling by migration and diffusion in predator–prey systems can account for the dynamical complexity of ecosystems.

Introduction

Investigation of spatial patterns in the distribution of organisms is a central issue in ecology [18]. This issue is usually treated in terms of reaction–diffusion models under a basic assumption that the motion of population is random and isotropic, i.e., without any preferred direction. Mathematically speaking, in a rather general case the dynamics of the given populations u (prey), and v (predator) can be described by the following equation [1], [2], [5], [11], [19], [20], [31], [32], [35], [36], [41]:

$$\frac{\partial u}{\partial t}=N(u\text{,}v)+D_u{\nabla }^{2}u\text{,}$$

$$\frac{\partial v}{\partial t}=P(u\text{,}v)+D_v{\nabla }^{2}v\text{,}$$

where ${\nabla }^2={\partial }^2/\partial x^2$ or (${\partial }^2/\partial x^2+{\partial }^2/\partial y^2$) is the usual Laplacian operator in 1 or 2-dimensional space and $D_u$, $D_v$ are prey and predator diffusion coefficients, respectively. $N(u\text{,}v)$ and $P(u\text{,}v)$ are associated with local processes such as birth, death and predation.

However, another type of dynamics can be given when the individuals exhibit a correlated motion towards certain direction. The origin of this motion may have different mechanism, and the typical motion is migration, which is corresponding to the case that the correlated motion caused by purely environmental factors such as wind in case of seeds or pollen spreading or water current in case of plankton communities [13], [24], [25]. Allowing for the above migration and diffusion, the spatiotemporal dynamics of a given population is then described by the following equation:

$$\frac{\partial u}{\partial t}+c_u\nabla u=N(u\text{,}v)+D_u{\nabla }^{2}u\text{,}$$

$$\frac{\partial v}{\partial t}+c_v\nabla v=P(u\text{,}v)+D_v{\nabla }^{2}v\text{,}$$

where $\nabla =\partial /\partial x$ (or $\partial /\partial y$), which means that the individuals move in one direction. Here, $c_u$ and $c_v$ are migration speed.

The purpose of this paper is to investigate the effects of migration in a predator–prey system with self diffusion. To determine those features in the dynamics that are specifically caused by migration terms, we choose nonlinearity in the local kinetics, taking the form of a non-dimensional Holling–Tanner system.

The paper is organized as follows. In Section 2, we give a Holling–Tanner model with both migration and diffusion whose parameters are rational biological determine the dynamics of the system. We analyze the dynamics of the model, and derive the dispersal relation with respect to model parameters in Section 3. In Section 4, by performing simulations, we illustrate the emergence of travelling pattern. Moreover, wavelength and pattern speed are calculated. Properties of pattern solutions are revealed by using AUTO in Section 5. Finally, some conclusions and discussions are given.