Travelling wavefronts in the diffusive single species model with Allee effect and distributed delay

Abstract

In this paper, we consider the diffusive single species model with Allee effect and distributed delay time. Special attention is paid to the existence of travelling wavefront solutions. First, we shall show that such fronts exist when the convolution kernel assumes the strong generic delay kernel and the delay is sufficiently small. Then, in the non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times, we shall show that such fronts still exist for the weak generic delay kernel and small delay. The approach used in this paper is the geometric singular perturbation theory.

Introduction

Gopalsamy and Ladas [1] introduced the following delayed Lotka–Volterra type single species population growth model

$$\text{x}\text{̇}\text{(t)=x(t)[a+bx(t−τ)−cx}{}^2\text{(t−τ)]}$$

where a, b, c are real constants and a, c>0, τ∈[0,∞). This equation has been widely studied by several investigators in recent years ([1], [2], [3], [4] and their references). But there has been relatively little investigation of the consequences of allowing spatial dispersal in this model. In fact, any individuals always move around in space in the realistic world and if a model is to take into account such moving then the model must have diffusion term. It is our aim in this paper to study a diffusive version of (1.1) and to deal with the question of the existence of travelling wavefronts for this delay reaction–diffusion equation. In recent years, there has a great deal of interest in travelling wave solutions of time-delay reaction–diffusion equations ([5], [6], [7], [8], [9], [10], [11], [12], [13] and references therein). The paper of Schaaf [5] is the pioneering work. But the work of Schaaf applies only to a scalar delayed reaction–diffusion equation where the non-linearity either is of the Hodgin–Huxley type or satisfies the quasi-monotonicity condition. A general method for establishing existence of such solutions in reaction–diffusion equations with discrete delays was developed by Wu and Zou [8], using the monotone iteration and the upper–lower solution technique. Unfortunately, their approach does not apply to the case of distributed delay. But for some particular distributed delay, we can recast the travelling wave equations into a finite-dimensional system of non-delay ODEs by the linear chain technique and then dynamic systems theory, specifically geometric singular perturbation theory [14], will be applied to this system for the case when the delay τ is small. Using this theory, it will be shown that the evolution (in the wave variable) takes place on a two-dimensional invariant manifold which will be computed explicitly. The implicit function theorem is used to establish existence of a heteroclinic connection in this manifold, which is travelling fronts of corresponding equation.

In the present paper, we will consider the following more general version of (1.1):

$$\text{∂}\text{u}\text{∂}\text{t}\text{=}\text{∂}{}^2\text{u}\text{∂}\text{x}{}^2\text{+ru[1−a}{}_1\text{(f∗u)−a}{}_2\text{(f∗u)}{}^2\text{],}$$

where the parameter D, r, a₁, a₂ are positive constants. In what follows, we shall study travelling wave fronts of (1.2) when the convolution $\text{f∗u}$ is taken to be different cases.