Stability of a class of delay fisher model with diffusion (Q2763917)

The authors study the following model of fishery: NEWLINE\[NEWLINE\begin{multlined} \bigl(1/N(x,t)\bigr) \partial N(x,t)/ \partial t=a+bN(x,t) +f\bigl( N(x,t-\sigma)\bigr) +dN(x,t)-E(x,t),\\ \bigl(1/E(x,t)\bigr) E(x,t)= k\bigl(pN(x,t)-c \bigr), \end{multlined}\tag{1}NEWLINE\]NEWLINE where \(x=(x_1,x_2,x_3) \in\Omega= \{x\in\mathbb{R}^3, |x|<m\}\), \(\Delta\) denotes the three dimensional Laplace operator, \(N(x,t)\) is the fish population, \(N(x,t)(a+bN(x,t)+ f(N(x,t-\sigma))\) is the natural growth rate, \(f\in C(\mathbb{R}^+,\mathbb{R})\)), \((x,t)\) is a measure of the fishing effort, \(p\) is the price of one fish, \(c\) is the cost per unit of effort, \(k\) is a positive constant, \(d\Delta N(x,t)\) is the diffusion of the population. The system (1) is considered together with the conditions: NEWLINE\[NEWLINEN(x,t)=\varphi (x,t),\;(x,t)\in \Omega\times [-\sigma,0],\;\varphi(x,0)>0,\;E(x,0)>0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\varphi\in C(\Omega \times[-\sigma,0], \mathbb{R}^+),\;\partial N(x,t)/ \partial n=0,\;(x,t)\in \partial\Omega \times[-\sigma, \infty).NEWLINE\]NEWLINE If \(a+bcp+ f(cp)>0\), then the system (1) has a positive equilibrium NEWLINE\[NEWLINEN^*=cp,\quad E^*=a+bcp+f(cp).NEWLINE\]NEWLINE Some conditions in which every positive solution \(N(x,t)\) has the property \(\lim_{t\to\infty} \|N(x,t)-N^*\|=0\), are also given.