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Permanence and extinction for dispersal population systems. (English) Zbl 1073.34052
The system of differential equations $\dot{x}=f(t,x)$, $x\in\Bbb{R}^n$, is said to be {\it permanent} if there exists a compact set $K\subset \text{int}\,\Bbb{R}_+^n$ such that all solutions starting in $\text{int}\,\Bbb{R}_+^n$ ultimately enter and remain in $K$. The authors study the following predator-pray model in a patchy environment \align \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], \endalign where $x_i$ denotes the species $x$ in patch $i$; $d(t)>0$, $e(t)>0$, $b_i(t)>0$, $a_i(t)>0$, $D_{ij}(t)\ge 0$ 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 $x_1\to\infty$, $\phi(t,x_1)\ge 0$, $\partial\phi(t,x_1)/\partial{x_1}\le0$ and $\partial(x_1\phi(t,x_1))/\partial{x_1}\ge0$. 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 stability of ODE 34C60 Qualitative investigation and simulation of models (ODE) 92D25 Population dynamics (general)
##### Keywords:
predator-pray system; permanence; periodic solution
Full Text:
##### References:
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