Dynamical behavior of a harvest single species model on growing habitat (Q2925720)

The authors are concerned with the topic of domain growth effect on the long time behavior of the following harvest single species logistic model with an isotropic domain growth NEWLINE\[NEWLINE\begin{aligned} u_t=d\Delta u+ru(1-\frac{u}{K})-hu,\;\; x\in \Omega,t>0;\,\\ u(x,t)=0,\;\;x\in \partial\Omega(0),t>0,\\ u(x,0)=u_0(x),\;\;x\in \Omega.\end{aligned}NEWLINE\]NEWLINE Here, the domain is \(\Omega(t)\subset\mathbb{R}^n\), being a simply connected bounded growing domain with its growing boundary \(\partial\Omega(t)\). They first transformed the model into an equivalent form via \(v(y,t)=u(\rho(t)y,t)\). Assuming that \(\rho(t)=\frac{\exp(kt)}{1+\frac{1}{m}(\exp(kt)-1)}\), achieved the global stability of the solutions. They showed that \(\lim\limits_{t\to \infty}v(y,t)=0\) as \(r\leq \frac{d}{m^2}\lambda_1+h\), and \(\lim\limits_{t\to \infty}v(y,t)=v^*(y)\) as \(r>\frac{d}{m^2}\lambda_1+h\), where \(v^*(y)\) is the nonnegative steady state, and \(\lambda_1\) is the principal eigenvalue of \(-\Delta \phi=\lambda\phi,\;x\in \Omega\) with Dirichlet boundary condition. Numerical simulations were also consistence with the analytical results.