Global Existence and Boundedness of a Two-Competing-Species Chemotaxis Model

Abstract

In this paper, we consider the following fully parabolic two-competing-species chemotaxis model $$ \left\{\begin{array}{ll} \displaystyle u_{1t}=\Delta{u_{1}}-\chi \nabla\cdot(u_{1}\nabla{v_{1}})+\mu_{1}u_{1}(1-u_{1}-e_{1}u_{2}),&x\in\Omega,~ t>0,\\ \displaystyle u_{2t}=\Delta{u_{2}}-\xi\nabla\cdot(u_{2}\nabla{v_{2}})+\mu_{2}u_{2}(1-e_{2}u_{1}-u_{2}),&x\in\Omega,~t>0,\\ \displaystyle v_{1t}=\Delta{v_{1}}+u_{1}- v_{1},&x\in\Omega,~ t>0, \\ \displaystyle v_{2t}=\Delta{v_{2}}+u_{2}- v_{2},&x\in\Omega,~ t>0 \end{array}\right. $$ under homogeneous Neumann boundary conditions, where Ω ⊂ ℝn  (n≥3) is a convex bounded domain with smooth boundary. Relying on a comparison principle, we show that the problem possesses a unique global bounded solution if μ1 and μ2 are large enough.