Uniform persistence for nonautonomous and random parabolic Kolmogorov systems
The authors considered the following nonautonomous parabolic Kolmogorov system: \[ \frac{\partial u_{i}}{\partial t}= \Delta u_{i}+f_{i}(t,x,u)u_{i},\quad x\in D,\;t>0, \] \[ B_{i}u_{i}=0,\quad x\in \partial D,\;t>0,\;i=1,2,3,\dots,n, \] where \(u=(u_{1},u_{2},u_{3},\dots, u_{n}),D\) is a bounded domain in \(\mathbb R^{N}\) with sufficiently smooth boundary \(\partial D,\) \(f=(f_{1},f_{2},f_{3},\dots,f_{n}):\mathbb R \times cl(D)\times [ 0,\infty )^{n}\rightarrow \mathbb R^{n}\) is a function bounded on sets of the form \(\mathbb R\times cl(D)\times B\) with bounded \(B\subset [ 0,\infty )^{n},\) uniformly Hölder continuous in \((t,x)\) and \( C^{2}\) in \(u,\) and \(B_{i}\) denotes a boundary operator either of the Dirichlet type or of the Robin type. Embedding the above system into system \(\Sigma \) of skew-product semiflow, they presented a sufficient condition for the above system to be uniformly persistent. This is proved using the notions of global attractor of \(\Sigma \) and a Morse decomposition of maximal invariant set of \(\Sigma \) in \(\partial X_{+}\times Y.\) That is, the main theorem is Theorem 3.1. Assume that \(\Sigma \) has a global attractor and that there is a Morse decomposition \(\left\{ M_{1},M_{2},\dots,M_{k}\right\} \) of the maximal compact invariant set of \(\Sigma \) in \(\partial X_{+}\times Y\) such that for any \(1\leq j\leq k,\) there is an \(1\leq i\leq n\) with the property \(\lambda _{\min }^{i}(M_{j})>0.\) Then the above system is uniformly persistent. Two sufficient conditions of another types are also given. Furthermore the strong uniform persistence of random Kolmogorov system also is investigated. As the applications strong uniform persistence in \(n\)-species random competitive system is discussed.
scientific article; zbMATH DE number 2117822
scientific article