Boundedness and stabilization in a forager-exploiter model with competitive kinetics

Abstract

This paper considers the following forager-exploiter model \begin{eqnarray*} \left \{ \begin{array}{lll} u_t=u_{xx}-\chi_1(uw_{x})_{x} + \mu_1 u(1-u-a_1v), \quad & x\in \Omega,\quad t > 0,\\ v_t=v_{xx}-\chi_2(vu_{x})_{x} + \mu_2 v(1-v-a_2u), \quad & x\in \Omega,\quad t > 0,\\ w_t=w_{xx}-\lambda(u + v)w-\mu w + r, \quad & x\in \Omega,\quad t > 0 \end{array} \right. \end{eqnarray*} in a smooth bounded domain $\Omega\subset \mathbb{R}$ with homogeneous Neumann boundary conditions, where the parameters $\chi_{1},\chi_{2},\mu_{1},\mu_{2},a_{1},a_{2},\lambda,\mu$, and $r$ are some positive constants. It is proved that for all appropriately regular initial data, the corresponding initial-boundary value problem has a global bounded classical solution. Moreover, the large time behavior of solution will be investigated. When $0 < a_{1},a_{2} < 1$, it is asserted that forager and exploiter will approach spatially homogeneous distributions under some explicit conditions; if $0 < a_{1} < 1,a_{2}\geq1$, the exploiter will become extinct.