# American Institute of Mathematical Sciences

July  2022, 27(7): 3913-3931. doi: 10.3934/dcdsb.2021211

## Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms

 Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília- DF, Brazil

* Corresponding author: Willian Cintra

Received  March 2021 Revised  May 2021 Published  July 2022 Early access  August 2021

Fund Project: The first author was partially supported by the project CAPES/Brazil - PrInt $n^o$ $88887.466484/2019-00$. The second author was partially supported by the project CAPES/Brazil - PrInt $n^o$ $88887.466484/2019 - 00$ and CNPq/Brazil with the grant $311562/2020 - 5$. The third autor is supported by the project CAPES/Brazil Proc. $n^o$ $2788/2015-02$

In this paper, we present results about existence and non-existence of coexistence states for a reaction-diffusion predator-prey model with the two species living in a bounded region with inhospitable boundary and Holling type II functional response. The predator is a specialist and presents self-diffusion and cross-diffusion behavior. We show the existence of coexistence states by combining global bifurcation theory with the method of sub- and supersolutions.

Citation: Willian Cintra, Carlos Alberto dos Santos, Jiazheng Zhou. Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3913-3931. doi: 10.3934/dcdsb.2021211
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##### References:
The region of coexistence of (1): on the left side, we have the general setting; the dark part represents the region of non-existence of coexistence states, and the area with dashed line represents the region of coexistence state. On the right, the portion with dashed lines represents the region of coexistence states in the simplest case $d \equiv d(0)$, $P\equiv 1$ and $R \equiv 1$
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