Asymptotic behavior in a doubly tactic resource consumption model with proliferation

Abstract

This paper is concerned with the doubly tactic model

$$\begin{aligned} \left\{ \begin{array}{ll} u_t=\Delta u-\chi _u\nabla \cdot (u\nabla w)+uw,&{} x\in \Omega , t>0, \\ v_t=\Delta v-\chi _v\nabla \cdot (v\nabla u)+vw, &{}x\in \Omega , t>0, \\ w_t=\Delta w-\lambda (u+v)w-\mu w,&{} x\in \Omega , t>0, \end{array}\right. \end{aligned}$$

in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^N$ ($N\ge 1$) with positive parameters $\chi _u, \chi _v, \lambda$ and nonnegative parameter $\mu$, for the spatiotemporal evolution of forager–exploiter groups u and v, which simultaneously consume a common nutrient w and proliferate. It is shown that for all suitably regular small initial data, the corresponding Neumann initial-boundary value problem possesses a globally classical solution, which approaches spatially homogeneous profiles at an exponential rate.