Stability analysis of \(n\)-species Lotka-Volterra almost periodic competition models with grazing rates and diffusions (Q2922090)

The authors are concerned on the following system NEWLINE\[NEWLINE\frac{\partial v_i(x,t)}{\partial t}=k_i(t)\Delta v_i(x,t)+v_i(x,t)\left(c_i(t)-\sum\limits_{j=1}^{n}a_{ij}(t)v_i(x,t)\right)+f_i(t),\eqno(1)NEWLINE\]NEWLINE \(i=1,2,\cdots,n,x\in \Omega\), with Neumann boundary and initial value condition, where all coefficient functions are almost periodic functions of \(t\in \mathbb{R}\). A smooth and almost periodic function \(V(t)=(v_1(t),\cdots,v_n(t))\) on \([0,\infty)\) is called a spatial homogeneous almost periodic solution of (1) if \(V(t)\) satisfies (1). Under some conditions, it is showed that there exists a spatial homogeneous almost periodic solution of (1) which is strictly positive and global asymptotically stable. The method is upper-lower solutions, Lyapunov function and Schauder fixed point theorem.