Global existence and stabilization in a diffusive predator-prey model with population flux by attractive transition

TL;DR

This paper proves global existence, boundedness, and stabilization of solutions in a diffusive predator-prey model with population flux driven by attractive transition, in two and three dimensions.

Contribution

It establishes the first rigorous results on global solutions and long-term behavior for this complex predator-prey system with population flux.

Findings

Solutions converge to a steady state as time approaches infinity.

Global classical solutions exist in 2D, weak solutions in 3D.

The steady state depends on model parameters and can represent coexistence or prey extinction.

Abstract

The diffusive Lotka-Volterra predator-prey model \begin{eqnarray*} \left\{ \begin{array}{rcll} u_t &=& \nabla\cdot \left[ d_1\nabla u + \chi v^2 \nabla \Big(\dfrac{u}{v}\Big)\right] +u(m_1-u+av), \qquad & x\in\Omega, \ t>0, \\ v_t &=& d_2\Delta v+v(m_2-bu-v), \qquad & x\in\Omega, \ t>0, \end{array} \right. \end{eqnarray*} is considered in a bounded domain $\Omega\subset\mathbb{R}^n$, $n \in\{2,3\}$, under Neumann boundary condition, where $d_1, d_2, m_1, \chi, a, b$ are positive constants and $m_2$ is a real constant. The purpose of this paper is to establish global existence and boundedness of classical solutions in the case $n=2$ and global existence of weak solutions in the case $n=3$ as well as show long-time stabilization. More precisely, we prove that the solutions $(u(\cdot,t), v(\cdot,t))$ converge to the constant steady state $(u_*, v_*)$ as $t \to \infty$, where $u_*, v_*$ solves $u_*(m_1-u_*+av_*)=v_*(m_2-bu_*-v_*)=0$ with $u_* > 0$ (covering both coexistence as well as prey-extinction cases).