Global existence and stability of three species predator-prey system with prey-taxis

Gurusamy Arumugam; Gurusamy Arumugam

doi:10.3934/mbe.2023371

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8448-8475. doi: 10.3934/mbe.2023371

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Global existence and stability of three species predator-prey system with prey-taxis

Gurusamy Arumugam ^,

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong S.A.R., China

Academic Editor: Yang Kuang

Received: 31 October 2022 Revised: 21 February 2023 Accepted: 24 February 2023 Published: 02 March 2023

In this paper, we study the following initial-boundary value problem of a three species predator-prey system with prey-taxis which describes the indirect prey interactions through a shared predator, i.e.,

$ \begin{align*} \begin{cases} u_t = d\Delta u+u(1-u)- \frac{a_1uw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ v_t = \eta d\Delta v+rv(1-v)- \frac{a_4vw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ w_t = \nabla\cdot(\nabla w-\chi_1 w\nabla u-\chi_2 w\nabla v) -\mu w+ \frac{a_5uw}{1+a_2u+a_3v}+\frac{a_6vw}{1+a_2u+a_3v}, & \mbox{in}\ \ \Omega, t>0, \ \ \label{II} \end{cases} \end{align*} $

under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^n (n \geqslant 1) $ with smooth boundary, where the parameters $ d, \eta, r, \mu, \chi_1, \chi_2, a_i > 0, i = 1, \ldots, 6. $ We first establish the global existence and uniform-in-time boundedness of solutions in any dimensional bounded domain under certain conditions. Moreover, we prove the global stability of the prey-only state and coexistence steady state by using Lyapunov functionals and LaSalle's invariance principle.
- Predator-prey system,
- prey-taxis,
- local existence,
- global existence,
- global stabilization
Citation: Gurusamy Arumugam. Global existence and stability of three species predator-prey system with prey-taxis[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8448-8475. doi: 10.3934/mbe.2023371

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Abstract

In this paper, we study the following initial-boundary value problem of a three species predator-prey system with prey-taxis which describes the indirect prey interactions through a shared predator, i.e.,

$ \begin{align*} \begin{cases} u_t = d\Delta u+u(1-u)- \frac{a_1uw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ v_t = \eta d\Delta v+rv(1-v)- \frac{a_4vw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ w_t = \nabla\cdot(\nabla w-\chi_1 w\nabla u-\chi_2 w\nabla v) -\mu w+ \frac{a_5uw}{1+a_2u+a_3v}+\frac{a_6vw}{1+a_2u+a_3v}, & \mbox{in}\ \ \Omega, t>0, \ \ \label{II} \end{cases} \end{align*} $

under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^n (n \geqslant 1) $ with smooth boundary, where the parameters $ d, \eta, r, \mu, \chi_1, \chi_2, a_i > 0, i = 1, \ldots, 6. $ We first establish the global existence and uniform-in-time boundedness of solutions in any dimensional bounded domain under certain conditions. Moreover, we prove the global stability of the prey-only state and coexistence steady state by using Lyapunov functionals and LaSalle's invariance principle.

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