Global generalized solutions to a three species predator-prey model with prey-taxis

Xin Wang; Ruijing Li; Yu Shi

doi:10.3934/dcdsb.2022031

Article Contents

2022, Volume 27, Issue 12: 7021-7042. Doi: 10.3934/dcdsb.2022031

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Global generalized solutions to a three species predator-prey model with prey-taxis

a.
School of Science, Qingdao University of Technology, Qingdao, 266033, China
b.
School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China
c.
School of Science, Wuhan University of Technology, Wuhan 430070, China

^* Corresponding author: Ruijing Li
^* Corresponding author: Ruijing Li

Received: October 2021

Revised: January 2022

Early access: February 2022

Published: December 2022

This work was supported by the National Natural Science Foundation of China (Grant No. 11901112)

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Abstract

In this paper, we study the following three species predator-prey model with prey-taxis:

$ \left\{ \begin{array}{lll} u_t = d_1\Delta u+\chi_1\nabla\cdot(u\nabla v)+r_1u(1-u-kv-b_1w), &\quad x\in \Omega, t>0, \\ v_t = d_2\Delta v+r_2v(1-hu-v-b_2w), &\quad x\in \Omega, t>0, \\ w_t = d_3\Delta w-\chi_2\nabla\cdot(w\nabla u)-\chi_3\nabla\cdot(w\nabla v)\\ \ \ \ \ \ \ \ +r_3w(-1+au+av-w), &\quad x\in \Omega, t>0. \end{array}\right. $

We prove that if (1.7) and (1.6) hold, the model ($ \ast $) admits at least one global generalized solution in any dimension.

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Mathematics Subject Classification: Primary: 35A01, 35K57; Secondary: 35Q92, 92C17.

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