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Zbl 0872.34047
Zhao, Tao; Kuang, Yang; Smith, H.L.
Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems. (English)
[J] Nonlinear Anal., Theory Methods Appl. 28, No.8, 1373-1394 (1997). ISSN 0362-546X

The authors consider the following delayed Gause-type predator-prey system $$\align \dot x(t)&= x(t)[g(x(t))- p(x(t))y(t)],\\ \dot y(t)&= y(t)[-\nu+ h(x(t-\tau))], \endalign$$ 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 $$\align \dot x(t)&= x(t)\Biggl[ \gamma-ax(t)- \frac{by(t)} {1+cx(t)}\Biggr],\\ \dot y(t)&= y(t)\Biggl[ -\nu+ \frac{dx(t-\tau)} {1+cx(t-\tau)}\Biggr]. \endalign$${}.
[V.Petrov (Plovdiv)]

MSC 2000:: *34K13 Periodic solutions of functional differential equations
34C25 Periodic solutions of ODE
92D25 Population dynamics

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

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