Dynamics in a two-species system with common dynamical resources and general competition term

Abstract

This paper deals with the following two-species competition system with common dynamical resources $\left(\begin{array}{ccc}& u_t=d_1\Delta u-{\chi }_1\nabla \cdot (u\nabla w)+f_1(u,v,w),& x\in \Omega ,t>0,\\ & v_t=d_2\Delta v-{\chi }_2\nabla \cdot (v\nabla w)+f_2(u,v,w),& x\in \Omega ,t>0,\\ & \tau w_t=d_3\Delta w-w(u+v)+\mu w(m\left(x\right)-w),& x\in \Omega ,t>0,\end{array}\right)$under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset R^n(n\ge 1)$, where $\tau \in \{0,1\}$, function $m\left(x\right)\in C\left(\bar{\Omega }\right)$, the parameters $\mu$, $d_i(i=1,2,3)$, ${\chi }_j(j=1,2)$ are positive and $f_1(u,v,w)$, $f_2(u,v,w)$ satisfy $\left(\begin{array}{cc}& f_1(u,v,w)=u\left(a_{1}w-a_{2}u-a_{3}v-a_4{\int }_{\Omega }udx-a_5{\int }_{\Omega }vdx\right),\\ & f_2(u,v,w)=v\left(b_{1}w-b_{2}v-b_{3}u-b_4{\int }_{\Omega }vdx-b_5{\int }_{\Omega }udx\right),\end{array}\right)$with $a_1,a_2,b_1,b_2>0$ and $a_3$, $a_4$, $a_5$, $b_3$, $b_4$, $b_5\in R$. First, we show that the above system admits a unique global and uniformly bounded solution in any spatial dimension for suitable large intraspecific competition parameters $a_2$ and $b_2$. Moreover, by constructing Lyapunov functionals and applying LaSalle’s invariant principle, we obtain some conditions under which the solutions converge to the coexistence steady state exponentially or to the competitive exclusion steady states algebraically as time goes to infinity under the cases where $a_i,b_i>0(i=3,4,5)$ or $a_i,b_i<0(i=3,5)$, $a_4,b_4\in R$.