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

Sun, Jingan; Cui, Yonghong

Electron. J. Differ. Equ. 2004, Paper No. 81, 12 p. (2004).

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

Keywords:

Predator-prey system; stage structure; permanence; periodic solution; infinite delay