Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems. (English) Zbl 0872.34047

Zhao, Tao; Kuang, Yang; Smith, H. L.

Nonlinear Anal., Theory Methods Appl. 28, No. 8, 1373-1394 (1997).

The authors consider the following delayed Gause-type predator-prey system \[ \begin{aligned} \dot x(t)&= x(t)[g(x(t))- p(x(t))y(t)],\\ \dot y(t)&= y(t)[-\nu+ h(x(t-\tau))], \end{aligned} \] where \(x(t),y(t)\) denote the population density of prey and predator at time \(t\), respectively, the positive constant \(\nu\) stands for the death rate of predator \(y\) in the absence of prey \(x\), and \(\tau\) is the recover-time. The main goal of the paper is to establish global existence of nonconstant periodic solutions. This is done by proving the existence of nontrivial fixed points of an appropriate map. In the last section of the paper the authors apply their results to the following delayed Lotka-Volterra predator-prey system \[ \begin{aligned} \dot x(t)&= x(t)\Biggl[ \gamma-ax(t)- \frac{by(t)} {1+cx(t)}\Biggr],\\ \dot y(t)&= y(t)\Biggl[ -\nu+ \frac{dx(t-\tau)} {1+cx(t-\tau)}\Biggr]. \end{aligned} \] {}.

Reviewer: V.Petrov (Plovdiv)

MSC:

34K13	Periodic solutions to functional-differential equations
34C25	Periodic solutions to ordinary differential equations
92D25	Population dynamics (general)

Keywords:

delayed Gause-type predator-prey system; nonconstant periodic solutions; delayed Lotka-Volterra predator-prey system

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