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 + χv^2 \nabla \Big(\dfrac{u}{v}\Big)\right] +u(m_1-u+av), \qquad & x\inΩ, \ t>0, \\ v_t &=& d_2Δv+v(m_2-bu-v), \qquad & x\inΩ, \ 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 $(u_{\ast },v_{\ast })$ as $t\to \infty$, where $u_{\ast },v_{\ast }$ solves $u_{\ast }(m_1-u_{\ast }+a{v}_{\ast })=v_{\ast }(m_2-b{u}_{\ast }-v_{\ast })=0$ with $u_{\ast }>0$ (covering both coexistence as well as prey-extinction cases).