Permanence and extinction for dispersal population systems
The system of differential equations \(\dot{x}=f(t,x)\), \(x\in\mathbb{R}^n\), is said to be \textit{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.
scientific article; zbMATH DE number 2117728