Stability and bifurcation for a kind of nonlinear delayed differential equations

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Abstract

In this paper, a kind of nonlinear delayed differential equations which can describe a two species predator–prey system with stage structure is discussed. The stability of the equilibria of the system and the existence of Hopf bifurcations are given. And the stability and the direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem.

Introduction

Linear and nonlinear delayed differential equations have been extensively applied in many areas such as biology, network, robot engineering, control of signal and so on. The models in the above areas are almost similar in form which have one or many delays. In this paper, we will study a kind of nonlinear delayed differential equations which can describe a two species predator–prey system with stage structure. The kind of model have been studied in [7], [13], [14], but the stage structure of species has been ignored in many literature. In the natural world, however, there are many species whose individual members have a life history which takes them through two stages: immature and mature in [1], [2], [4], [6], [8], [9], [10], [12]. In particular, we have in mind mammalian populations and some amphibious animals, which exhibit these two stages. In these models, the age to maturity is represented by a time delay, leading to systems of retarded functional differential equations. Cannibalism has been observed in a great variety of species, including a number of fish species. Cannibalism models of various types have also been investigated in [5]. For general models population growth one can see [11].

In this paper, we consider a kind of nonlinear delayed differential equations taken in the following form:dX1(t)dt=α0X2(t)-γX1(t)-αX2(t-τ),dX2(t)dt=αX2(t-τ)-βX22(t)-a1X2(t)Y(t),dY(t)dt=Y(t)(-r1+a2X2(t)-bY(t)),X1(0)>0,Y(0)>0,X2(t)=φ(t)0,-τt0,where X1(t),X2(t) represent respectively the immature and mature prey populations densities; Y(t), represents the density of predator population; a1>0, is the transformation coefficient of mature predator population; α represents the immatures who were born at time t  τ and survive at time t (with the immature death rate γ), and τ represents the transformation of immatures to matures; γ>0, is the death rate of the immature prey population; β>0, represents the mature death and overcrowding rate. The model is derived under the following assumptions:

  • [H1]

    The birth rate of the immature prey is proportional to the existing mature population with α>0; the death rate of the immature prey is proportional to the existing immature population with γ>0; the death rate of the mature is proportional to square of itself with a proportionality constant β>0.

  • [H2]

    The predator only depend on the prey, that is, the predator will die out with the absence of prey. We also assume that r1>0,a2>0,b>0.

Section snippets

Stability and Hopf bifurcation

Note that the quantities u2(x,t) and v(x,t) of the system (1.1) are independent of the quantity u1(x,t), so we may only consider the following subsystem to be easy to yield the dynamic behaviors of the system (1.1):du1dt=αu1(x,t-τ)-βu12(x,t)-a1u1(x,t)u2(x,t),du2dt=u2(x,t)(-r1+a2u1(x,t)-bu2(x,t)).If a2α>βr1, then the system (2.1) has a unique positive equilibrium E(c1,c2) withc1=bα+a1r1a1a2+bβ,c2=a2α-βr1bβ+a1a2.Normalizing the delay τ by the time scaling tt/τ, then the system (2.1) is

Stability of Hopf bifurcation

From Theorem 2.2, we know that Hopf bifurcation will occurs if 2p+α2>q2, p>mα and Δ>0 are satisfied. Next, based on the normal form method and the center manifold theory by Hassard et al. in [3], we study the direction of the bifurcation and stability of the bifurcating periodic solutions.

For convenience, we introduce a new parameter θ=τ-τn(τn+orτn-) and rewrite (2.2) asu˙(t)=Lθut+F(ut,θ),where u(t)=(u1(t),u2(t))T and Lθ:CR2, F:C×R+R2 are given byLθ(φ)=(τn+θ)-c1βφ1(0)+a1φ2(0)+αφ1(-1)-αφ1(0)c2

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    On the one hand, we can decrease amplitude to restrain the vibration behavior of system; on the other hand, we can obtain a desirable effect via increasing amplitude. Therefore, Hopf bifurcation control and amplitude control with various objectives have been implemented in mechanics, biology, and economy [5–9]. The rest of this paper is organized as follows.

Supported by the Natural Science Foundation of China (No. 10671154).

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