## Permanence and extinction for dispersal population systems.(English)Zbl 1073.34052

The system of differential equations $$\dot{x}=f(t,x)$$, $$x\in\mathbb{R}^n$$, is said to be permanent if there exists a compact set $$K\subset \text{int}\,\mathbb{R}_+^n$$ such that all solutions starting in $$\text{int}\,\mathbb{R}_+^n$$ ultimately enter and remain in $$K$$.
The authors study the following predator-pray model in a patchy environment \begin{aligned} \dot{x}_1&=x_1[b_1(t)-a_1(t)x_1-y\phi(t,x_1)]+ \sum_{j=1}^n(D_{1j}(t)x_j-D_{j1}(t)x_1),\\ \dot{x}_i&=x_i[b_i(t)-a_i(t)x_i]+ \sum_{j=1}^n(D_{ij}(t)x_j-D_{ji}(t)x_i),\;i=2,\ldots,n,\\ \dot{y}&=y[-d(t)+e(t)x_1\phi(t,x_1)-f(t)y], \end{aligned} where $$x_i$$ denotes the species $$x$$ in patch $$i$$; $$d(t)>0$$, $$e(t)>0$$, $$b_i(t)>0$$, $$a_i(t)>0$$, $$D_{ij}(t)\geq 0$$ are continuous $$\omega$$-periodic functions; $$D_{ij}(t)$$ is the dispersal coefficient of the species from patch $$j$$ to patch $$i$$, $$D_{ii}(t)\equiv 0$$; the predator functional response $$x_1\phi(t,x_1)$$ is bounded as $$x_1\to\infty$$, $$\phi(t,x_1)\geq 0$$, $$\partial\phi(t,x_1)/\partial{x_1}\leq0$$ and $$\partial(x_1\phi(t,x_1))/\partial{x_1}\geq0$$. Necessary and sufficient conditions for the permanence of the above system are presented. Sufficient conditions for the permanence of the single-species system in the absence of a predator ($$y=0$$) is also obtained. The approach is based on the well-known properties of the periodic logistic model $$\dot{z}=z(b(t)-a(t)z)$$.
The biological interpretion of the main results is given.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D25 Population dynamics (general)

### Keywords:

predator-pray system; permanence; periodic solution
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### References:

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