## Topological Methods in Nonlinear Analysis

### Coexistence states of diffusive predator-prey systems with preys competition and predator saturation

Jun Zhou

#### Abstract

In this paper, we study the existence, stability, permanence, and global attractor of coexistence states (i.e. the densities of all the species are positive in $\Omega$) to the following diffusive two-competing-prey and one-predator systems with preys competition and predator saturation: $$\begin{cases} -\Delta \displaystyle u=u\bigg(a_1-u-b_{12}v-\frac{c_1w}{(1+\alpha_1u)(1+\beta_1w)}\bigg) & {\rm in}\ \Omega,\\ \displaystyle -\Delta v=v\bigg(a_2-b_{21}u-v-\frac{c_2w}{(1+\alpha_2v)(1+\beta_2w)}\bigg) &{\rm in}\ \Omega,\\ \displaystyle -\Delta w=w\bigg(\frac{e_1u}{(1+\alpha_1u)(1+\beta_1w)}+\frac{e_2v}{(1+\alpha_2v)(1+\beta_2w)}-d\bigg) &{\rm in}\ \Omega,\\ k_1\partial_\nu u+u=k_2\partial_\nu v+v=k_3\partial_\nu w+w=0 & {\rm on}\ \partial\Omega, \end{cases}$$ where $k_i\geq 0$ $(i=1,2,3)$ and all the other parameters are positive, $\nu$ is the outward unit rector on $\partial\Omega$, $u$ and $v$ are densities of the competing preys, $w$ is the density of the predator.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 45, Number 2 (2015), 509-550.

Dates
First available in Project Euclid: 30 March 2016

https://projecteuclid.org/euclid.tmna/1459343994

Digital Object Identifier
doi:10.12775/TMNA.2015.025

Mathematical Reviews number (MathSciNet)
MR3408834

Zentralblatt MATH identifier
1371.35104

#### Citation

Zhou, Jun. Coexistence states of diffusive predator-prey systems with preys competition and predator saturation. Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 509--550. doi:10.12775/TMNA.2015.025. https://projecteuclid.org/euclid.tmna/1459343994

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