Boundedness and stability of a quasilinear three-species predator–prey model with competition mechanism

Abstract

In this paper, we consider the following quasilinear three-species predator–prey model with competition mechanism

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t} =\nabla \cdot \left( \phi _1\left( u\right) \nabla u\right) -\nabla \cdot \left( u \psi _1\left( u\right) \nabla w\right) + \gamma _1 u w -\theta _1 u -\mu _1 u v, \\ v_{t} =\nabla \cdot \left( \phi _2\left( v\right) \nabla v\right) -\nabla \cdot \left( v \psi _2\left( v\right) \nabla w\right) + \gamma _2 v w -\theta _2 v -\mu _2 u v, \\ w_{t} = D \Delta w - \left( u + v\right) w + \sigma w\left( 1 - w\right) , \end{array}\right. } \end{aligned}$$

in a bounded smooth domain $\Omega \subset {\mathbb {R}}^{n}\left( n\ge 2\right)$. The parameters $D, \gamma _i, \theta _i, \sigma >0$ and $\mu _i\ge 0$ are constants with $i=1,2$. The functions $\phi _i\left( s\right)$ and $\psi _i\left( s\right)$ satisfy

$$\begin{aligned} \phi _i\left( s\right) \ge d_i \left( s+1\right) ^{\alpha _i} \text { and } \psi _i\left( 0\right) =0\le \psi _i\left( s\right) \le \chi _i \left( s+1\right) ^{\beta _i-1} \end{aligned}$$

for all $s\ge 0$ with $d_i>0, \chi _i>0, \alpha _i, \beta _i\in {\mathbb {R}}\left( i=1,2\right)$. It is proved that its corresponding homogeneous Neumann initial-boundary problem possess a global bounded classical solution, provided that

$$\begin{aligned} \alpha _i>\max \left\{ \beta _i-\frac{2}{n}, -\frac{2}{n}\right\} \left( i=1,2\right) . \end{aligned}$$

Moreover, it is shown that when $\alpha _i =0, \beta _i =1\left( i=1,2\right)$, the prey-only, semi-coexistence and coexistence steady states of the above model are globally asymptotically stable under certain conditions.