Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type

Abstract

This paper deals with the two-species chemotaxis-competition system $u_t=d_1\Delta u-{\chi }_1\nabla \cdot (u\nabla w)+{\mu }_{1}u(1-u-a_{1}v)$, $v_t=d_2\Delta v-{\chi }_2\nabla \cdot (v\nabla w)+{\mu }_{2}v(1-a_{2}u-v)$, $0=d_3\Delta w+\alpha u+\beta v-\gamma w$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary, $n\ge 2$; ${\chi }_i$ and ${\mu }_i$ are constants satisfying some conditions. The above system was studied in the cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$, and it was proved that global existence and asymptotic stability hold when $\frac{{\chi }_i}{{\mu }_i}$ are small. However, the conditions in the above two cases strongly depend on $a_1,a_2$, and have not been obtained in the case that $a_1,a_2\ge 1$. Moreover, convergence rates in the cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all $a_1,a_2>0$ which covers the case that $a_1,a_2\ge 1$, and lead to convergence rates for solutions of the above system in the cases that $a_1,a_2\in (0,1)$ and $a_1\ge 1>a_2$.

21 pages

Cited by in corpus (2)

• Two-species competition model with chemotaxis: well-posedness, stability and dynamics
• Can chemotactic effects lead to blow-up or not in two-species chemotaxis-competition models?