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.


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


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