Boundedness in a high-dimensional forager–exploiter model with nonlinear resource consumption by two species

Abstract

We investigate a forager–exploiter model in a high-dimensional smooth bounded domain with zero-flux Neumann boundary condition:

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{cc} {\displaystyle u_{t}=\Delta u-\chi _{1}\nabla \cdot (u\nabla w),}&{}{}\quad \,\,\,\,x\in \Omega ,\,t>0,\\ {\displaystyle v_{t}=\Delta v-\chi _{2}\nabla \cdot (v\nabla u),}&{}{}\quad \,\,\,\,x\in \Omega ,\,t>0,\\ w_{t}=\mathrm {d}\Delta w-\frac{u+v}{(1+u+v)^{\gamma }}w-\mu w+r(x,t),&{}{}\quad \,\,\,\,x\in \Omega ,\,t>0.\end{array}\right. \end{aligned} \end{aligned}$$

This model characterizes the social interactions between the two species, foragers and exploiters, denoted by u and v, searching for the same food resource w. The positive taxis effects $\chi _{1}$ and $\chi _{2}$ reflect doubly tactic modelling hypothesis that the foragers chase food resource directly, while the exploiters follow after them. The spatio-temporal dynamics of food resource include its reaction-diffusion at rate d, natural reduction at rate $\mu$, renewed production at rate r and especially its nonlinear consumption by the two species. For a positive constant $\gamma$ weighing the nonlinear sensitivity of resource consumption rate, we find a sufficient condition such that the system possesses a unique nonnegative global bounded classical solution.