The study of global stability of a diffusive Michaelis-Menten and Tanner predator-prey model

Abstract

In this paper, we consider a parabolic-parabolic predator-prey model of Michaelis-Menten and Tanner functional response with random diffusion:
\begin{equation}
\begin{aligned}
&u_{t}=d_{1}\Delta u+au-bu^{2}-\frac{\delta uv}{\alpha u+v},\\
&v_{t}=d_{2}\Delta v+rv-\gamma\frac{v^{2}}{u}\nonumber
\end{aligned}
\end{equation}
with $d_{1},d_{2},a,b,r,\alpha,\gamma,\delta>0$ under the no-flux boundary condition in a smooth bounded domain $\Omega\subset\mathbb{R}^{n}\left(n=1,2,3\right)$. By applying a new method, we establish much improved global asymptotic stability of the unique positive equilibrium solution than works in literature. We also show the result can be extended to more general type of systems with heterogeneous environment.