Stability and Hopf bifurcation for a delay competition diffusion system

doi:10.1016/S0960-0779(02)00068-1

Chaos, Solitons & Fractals

Volume 14, Issue 8, November 2002, Pages 1201-1225

https://doi.org/10.1016/S0960-0779(02)00068-1 Get rights and content

Abstract

This paper investigates the stability and Hopf bifurcation of a delay competition diffusion system. Firstly we discuss the existence and stability of the corresponding steady state solutions. Secondly our purpose is to give more detail information about the Hopf bifurcation of this system. We derive the basis of the eigenfunction subspace and then convert the existence of periodic solutions to the study of the existence of the implicit function. Finally, we analyze the stability of the periodic solutions by reducing the original system on the center manifold.

Introduction

Situation involving a negative delayed feedback control occur in many mathematical models of biological systems with spatial diffusion. For a review of these models, see [6], [14]. For a set of different species interacting with each other in an ecological community, perhaps the simplest and probably the most important question from a practical point of view is whether all the species in the system survive in the long term. Therefore, the periodic phenomena of biological system are often discussed, for example, the time delay is considered as a parameter and questions of stability of equilibria and periodicity are discussed in [2], [5]. For this reason, in the present paper we consider the spatially non-homogeneous periodic solutions, that is, the so-called Hopf bifurcation which arises from the spatially non-homogeneous steady state solution, of the following autonomous competition diffusion equations with delay: $U_{t} =U_{xx} +kU(t,x)[1−b_{1} U(t−r,x)−c_{1} V(t−r,x)],$ $V_{t} =V_{xx} +kV(t,x)[1−b_{2} U(t−r,x)−c_{2} V(t−r,x)], t>0, 0<x< π,$ $U(t,0)=U(t, π)=V(t,0)=V(t, π)=0, t⩾0,$ $(U,V)=(φ_{1},φ_{2}), −r⩽t⩽0, 0⩽x⩽ π,$ with $b_{1} /b_{2} >1>c_{1} /c_{2} .$

Systems of type (1.1) have been extensively studied in recent years, see [1], [2], [3], [9], [11], [12], [13]. The closely related problems with Neumann boundary conditions have also been studied by Zhou and Hussein [11], as we know in these cases there always exist spatially homogeneous stationary solutions therefore the stability and bifurcation clearly are treated. However, under Dirichlet boundary conditions such homogeneous stationary solutions do not exist in the problem (1.1). We will present a detailed analysis of the occurrence of the Hopf bifurcation arising from the spatially non-homogeneous steady state of system (1.1). Our first consideration here is to study the existence and stability of non-trivial spatially non-homogeneous steady state solutions, and then a remaining interesting question is whether an increase in delay r will destabilize the steady state (U_k(x),V_k(x)) and lead to the occurrence of periodic solutions. Finally, the stability of the bifurcation periodic solution will be considered.

In this paper we get the conclusions as follows. The asymptotic behavior of the delay competition diffusion system is determined by the coefficients b_i,c_i, and the time delay r. This means that there is a threshold r_k₀>0, the steady state (U_k(x),V_k(x)) will attract the solutions of the initial boundary value problem if 0⩽r<r_k₀, and the periodic oscillator (U_k,r(x),V_k,r(x))=U_k,r(x)+w₁(x,t/(1+β)),V_k,r(x)+w₂(x,t/(1+β)) is an attractor as r>r_k₀.

The conclusion shows the persistent dynamics of this kind of systems and the method in this paper will be useful for other ecological systems or physical systems. Specially the study of periodicity can be developed by this approach as J.K. Hale mentioned in [3].

This paper is organized as follows. In Section 2, we present the existence of steady state solutions. Section 3 contains the eigenvalue problem. In Section 4 we describe the stability of the positive equilibrium. In Section 5 we study the Hopf bifurcation which arises from the positive equilibrium as the delay r goes through a critical point r₀, and determine a formula that establishes the stability of the bifurcation periodic solutions. Finally some numerical simulations are given to demonstrate the asymptotic behavior of the species.

Section snippets

The existence of steady state solutions

Consider the following steady state problem: $U″(x)+kU(1−b_{1} U−c_{1} V)=0, 0<x< π,$ $V″(x)+kV(1−b_{2} U−c_{2} V)=0, 0<x< π,$ $U(0)=U(π)=V(0)=V(π)=0,$ $U>0, V>0, 0<x< π .$ Here k is restricted in some neighborhood of 1. Suppose that the solution (U_k(x),V_k(x)) of (2.1) has the following expressions: $U_{k} (x)=(k−1)α(sin x+(k−1)ξ(x)), 〈ξ, sin x〉=0,$ $V_{k} (x)=(k−1)β(sin x+(k−1)η(x)), 〈η, sin x〉=0,$ where $〈g(x), sin x〉=∫_{0}^{π} g(x) sin x d x.$ Substitute (2.2) into (2.1) we get $(D^{2} +1)ξ+ sin x+(k−1)ξ−k[b_{1} α(sin x+(k−1)ξ)^{2} +c_{1} β(sin x+(k−1)ξ)(sin x+(k−1)η)]=0,$ $(D^{2} +1)η+ sin$

Eigenvalue problem

Let $1<k⩽k^{∗}$ and (U_k(x),V_k(x)) be the positive equilibrium of the system (1.1) described in Section 2. The linearized system of (1.1) around (U_k(x),V_k(x)) has the form $u_{t} =u_{xx} +ku−k(b_{1} U_{k} +c_{1} V_{k})u−kU_{k} (b_{1} u(t−r,x)+c_{1} v(t−r,x)),$ $v_{t} =v_{xx} +kv−k(b_{2} U_{k} +c_{2} V_{k})v−kV_{k} (b_{2} u(t−r,x)+c_{2} v(t−r,x)), t>0, 0<x< π,$ $u(t,0)=u(t, π)=v(t,0)=v(t, π)=0, t⩾0,$ $(u,v)=(φ_{1},φ_{2}), (t,x)∈[−r,0]×[0, π],$ where $(φ_{1},φ_{2})∈C([−r,0],X×X), X=L^{2} (0, π).$ Introduce the operator A(k):D(A(k))→X×X defined by $A(k)=(D^{2} +k)−k b_{1} U_{k} +c_{1} V_{k} 00 b_{2} U_{k} +c_{2} V_{k},$ where D²=∂²/∂x², with domain D(A(k

Stability of the positive equilibrium

In this section we study the stability of the positive equilibrium (U_k,V_k) with $k∈(1,k^{∗}]$ fixed, and the delay r considered as a parameter. To describe stability of (U_k,V_k) it suffices to investigate how the eigenvalue λ=iν varies as the delay r passing through $r_{k_{n}}, n=0,1,…$

First we have to solve the adjoint problem of (3.4) as $A(k)− i ν_{k} − e^{−iθ_{k}} k b_{1} U_{k} b_{2} V_{k} c_{1} U_{k} c_{2} V_{k} y_{k}^{∗} z_{k}^{∗} =0.$ Similarly we let $y_{k}^{∗} = sin x+(k−1)δ_{k}^{∗}, z_{k}^{∗} =(L_{k}^{∗} + i M_{k}^{∗}) sin x+(k−1)χ_{k}^{∗} .$ After the same argument, we obtain, there is a continuously

Hopf bifurcation

In this section we study the Hopf bifurcation which arises from the positive equilibrium (U_k(x),V_k(x)) as the delay r crosses r_k₀=r₀.

The generic Hopf bifurcation theorem is well established [11]. Our purpose here is to give more detailed information about the Hopf bifurcation for the delay competition diffusion system (1.1). In fact we are going to prove the following theorem.

Theorem 5.1

For each fixed $k∈(1,k^{∗}]$ , Hopf bifurcation occurs as the delay increasingly passes through r₀. Specifically for r₀, there

Numerical results

In this section we give some numerical result for the asymptotic behavior in the time delay logistic equation on a one-dimensional spatial domain $Ω=(0,1)$ : $u_{t} =u_{xx} +ku(t,x)[1−u(t−r,x)], t>0, 0<x<1,$ $u(t,0)=u(t,1)=0, t⩾0,$ $u(t,x)=0.1(1+t/r) sin (π x), −r⩽t⩽0, 0⩽x⩽1.$ By discreting system (6.1) into a finite difference system, numerical solution is then solved through the forward elimination and backward substitution.

When $Ω=(0,1)$ , the principal eigenvalue of the Laplacian operator under Dirichlet boundary

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^☆: The project supported by National Natural Science Foundation of China.

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Stability and Hopf bifurcation for a delay competition diffusion system☆

Abstract

Introduction

Section snippets

The existence of steady state solutions

Eigenvalue problem

Stability of the positive equilibrium

Hopf bifurcation

Numerical results

J. Differential Equations

Nonlinear Anal. TMA

Nonlinear Anal. TMA

J. Math. Anal. Appl.

Math. Biosci.

Interaction of spatial diffusion and delay in models of genetic control by repression

J. Math. Biol.

Theory of functional differential equations