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

Jun Zhou



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:

-\Delta u=u\bigg(a_1-u-b_{12}v-\frac{c_1w}{(1+\alpha_1u)(1+\beta_1w)}\bigg) & {\rm in}\ \Omega,
-\Delta v=v\bigg(a_2-b_{21}u-v-\frac{c_2w}{(1+\alpha_2v)(1+\beta_2w)}\bigg) &{\rm in}\ \Omega,
-\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,

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.


Coexistence; Predator-prey model; fix point index;extinction; permanence; global attractor; stability

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