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

The system of differential equations $\stackrel{˙}{x}=f\left(t,x\right)$, $x\in {ℝ}^{n}$, is said to be permanent if there exists a compact set $K\subset \text{int}\phantom{\rule{0.166667em}{0ex}}{ℝ}_{+}^{n}$ such that all solutions starting in $\text{int}\phantom{\rule{0.166667em}{0ex}}{ℝ}_{+}^{n}$ ultimately enter and remain in $K$.

The authors study the following predator-pray model in a patchy environment

$\begin{array}{cc}\hfill {\stackrel{˙}{x}}_{1}& ={x}_{1}\left[{b}_{1}\left(t\right)-{a}_{1}\left(t\right){x}_{1}-y\varphi \left(t,{x}_{1}\right)\right]+\sum _{j=1}^{n}\left({D}_{1j}\left(t\right){x}_{j}-{D}_{j1}\left(t\right){x}_{1}\right),\hfill \\ \hfill {\stackrel{˙}{x}}_{i}& ={x}_{i}\left[{b}_{i}\left(t\right)-{a}_{i}\left(t\right){x}_{i}\right]+\sum _{j=1}^{n}\left({D}_{ij}\left(t\right){x}_{j}-{D}_{ji}\left(t\right){x}_{i}\right),\phantom{\rule{4pt}{0ex}}i=2,...,n,\hfill \\ \hfill \stackrel{˙}{y}& =y\left[-d\left(t\right)+e\left(t\right){x}_{1}\varphi \left(t,{x}_{1}\right)-f\left(t\right)y\right],\hfill \end{array}$

where ${x}_{i}$ denotes the species $x$ in patch $i$; $d\left(t\right)>0$, $e\left(t\right)>0$, ${b}_{i}\left(t\right)>0$, ${a}_{i}\left(t\right)>0$, ${D}_{ij}\left(t\right)\ge 0$ are continuous $\omega$-periodic functions; ${D}_{ij}\left(t\right)$ is the dispersal coefficient of the species from patch $j$ to patch $i$, ${D}_{ii}\left(t\right)\equiv 0$; the predator functional response ${x}_{1}\varphi \left(t,{x}_{1}\right)$ is bounded as ${x}_{1}\to \infty$, $\varphi \left(t,{x}_{1}\right)\ge 0$, $\partial \varphi \left(t,{x}_{1}\right)/\partial {x}_{1}\le 0$ and $\partial \left({x}_{1}\varphi \left(t,{x}_{1}\right)\right)/\partial {x}_{1}\ge 0$. 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 $\stackrel{˙}{z}=z\left(b\left(t\right)-a\left(t\right)z\right)$.

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