## Permanence of predator-prey system with infinite delay.(English)Zbl 1056.92050

From the paper: We consider a predator-prey system with periodic coefficients and infinite delay, in which the prey has a history that takes them through two stages, immature and mature. The system is given by: $\dot x_1=a(t)x_2-b(t)x_1- d(t)x^2_1-p(t)x_1\int^0_{-\infty}k_{12} (s)y(t-s)ds,$
$\dot x_2=c(t)x_1-f(t) x^2_2,\tag{1}$
$\dot y=y\left[-g(t)+h(t)\int^0_{-\infty}k_{21}(s)x_1(t+s) ds-q(t)\int^0_{-\infty} k_{22} (s)y(t+s)ds\right],$ where $$x_1$$ and $$x_2$$ denote the density of immature and mature populations, A(prey), respectively, and $$y$$ is the density of the predator B that preys on $$x_1$$. The coefficients are all $$\omega$$-periodic and continuous for $$t\geq 0$$, $$a(t)$$, $$b(t)$$, $$c(t)$$, $$d(t)$$ and $$f(t)$$ are all positive, $$p(t)$$, $$h(t)$$ and $$q(t)$$ are nonnegative, and $$\int^\omega_0 q(t)dt>0$$, $$\int^\omega_0g(t)dt\geq 0$$. The functions $$k_{ij}(s)$$ $$(i,j=1,2)$$ defined on $$\mathbb{R}_-=(-\infty,0]$$ are nonnegative and integrable, $$\int^0_{-\infty}k_{ij}(s)=1$$.
We provide a sufficient and necessary condition to guarantee the permanence of the system. The system has a positive periodic solution under this condition. Some known results are extended to the delay case.

### MSC:

 92D25 Population dynamics (general) 34C25 Periodic solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 92D40 Ecology
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