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

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).