Global stability and persistence in diffusive food chains (Q2762256)

This paper is devoted to study the global stability and persistence aspects of the following system of reaction-diffusion equations which may be viewed as a diffusive food chain model: NEWLINE\[NEWLINE\begin{cases} \dot u_i=d_i\Delta u_i+f_i(u), \quad & (x,t)\in \Omega\times \mathbb{R}_+,\\ {\partial u_i\over \partial \nu}=0 \quad & \text{on }\partial \Omega\times\mathbb{R}_+,\\ u_i(x,0)=u_{i0}(x)\geq 0 \quad & x\in \overline\Omega, \end{cases}\tag{1}NEWLINE\]NEWLINE where \(1\leq i\leq n\), \(0\leq d_1 \leq d_2\leq \cdots\leq d_n\), and \(u=(u_1,\dots, u_n)\). Here \(\Omega\) is a bounded connected domain in \(\mathbb{R}^N\) with smooth boundary. To this end the author uses the invariance principle of reaction-diffusion equations and the construction of the Lyapunov functional.