Global existence and boundedness of solutions to a two-dimensional forager-exploiter model with/without logistic source

Abstract

This paper is focused on the zero-flux initial–boundary value problem for a forager-exploiter model of the form $\left(\begin{array}{cccc}& u_t=\Delta u-\nabla \cdot (u\nabla w)+{\mu }_1(u-u^m),& & x\in \Omega ,t>0,\\ & v_t=\Delta v-\nabla \cdot (v\nabla u)+{\mu }_2(v-v^l),& & x\in \Omega ,t>0,\\ & w_t=\Delta w-f\left(u\right)w-g\left(v\right)w-\mu w+r(x,t),& & x\in \Omega ,t>0,\end{array}\right)$in a smoothly bounded domain $\Omega \subset R^2$, where $\mu$, ${\mu }_1$, ${\mu }_2$, $m$, $l$ are positive constants, $r(x,t)\in C^1(\overline{\Omega }\times [0,\infty ))\cap L^{\infty }(\Omega \times (0,\infty ))$ is a given nonnegative function, the functions $f,g\in C^1[0,\infty ]$ are assumed to behave essentially like $u^{\alpha }$, $v^{\beta }$ respectively, with some positive constants $\alpha$ and $\beta$. It is shown that the initial–boundary value problem possesses globally bounded classical solutions, provided that $m\ge 1$ , $l\ge 1$, $\alpha \le \frac{m}{2}$ and $\beta <\frac{l}{2}$.