Summary: We consider the following initial-boundary value problem. \[\begin{cases} u_t(x,t)=aLu-\alpha u^pf(v)\quad\text{in}\quad \Omega\times(0,\infty),\\ v_t(x,t)=bLu+cLv+Qu^pf(v)\quad \text{in}\quad\Omega\times(0,\infty),\\\frac{\partial u}{\partial N}=0,\quad\frac{\partial v}{\partial N}=0\quad\text{on}\quad\partial \Omega\times(0,\infty),\\u(x,0)=u_0(x)>0,\quad u(x,0)=u_0(x)>0\quad \text{in}\quad\Omega,\end{cases}\] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(L\) is an elliptic operator, \(p>0\), \(\alpha\ne 0\), \(a,b,c,Q\) are positive constants, \(f(s)\) is a positive and increasing function for the positive values of \(s\).
We find some conditions under which solutions of the above system either exists globally or blow up in a finite time. We also prove that if \(0<p<1\), the solutions \((u,v)\) extincts in a finite time. Some numerical results are given to illustrate our analysis.