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

Abstract

In this paper, we study the existence, stability, permanence, andglobal attractor of coexistence states (i.e. the densities of allthe species are positive in $\Omega$) to the following diffusivetwo-competing-prey and one-predator systems with preys competitionand predator saturation:$$\begin{cases} -\Delta\displaystyle u=u\bigg(a_1-u-b_{12}v-\frac{c_1w}{(1+\alpha_1u)(1+\beta_1w)}\bigg)& {\rm in}\ \Omega,\\\displaystyle -\Deltav=v\bigg(a_2-b_{21}u-v-\frac{c_2w}{(1+\alpha_2v)(1+\beta_2w)}\bigg)&{\rm in}\ \Omega,\\\displaystyle -\Deltaw=w\bigg(\frac{e_1u}{(1+\alpha_1u)(1+\beta_1w)}+\frac{e_2v}{(1+\alpha_2v)(1+\beta_2w)}-d\bigg)&{\rm in}\ \Omega,\\k_1\partial_\nu u+u=k_2\partial_\nuv+v=k_3\partial_\nu w+w=0 & {\rm on}\ \partial\Omega, \end{cases}$$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 $v$ aredensities of the competing preys, $w$ is the density of the predator.