Large densities in a competitive two-species chemotaxis system in the non-symmetric case

Abstract

This paper deals with the two-species chemotaxis system with Lotka-Volterra competitive kinetics, \begin{align*} \begin{cases} u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1 - u - a_1 v), & x\inΩ,\ t>0,\\ v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1 - a_2 u - v), & x\inΩ,\ t>0,\\ 0 = d_3 Δw + αu + βv - γw, & x\inΩ,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions and suitable initial conditions, where $\Omega \subset {\mathbb{R}}^n$ $(n\in \mathbb{N})$ is a bounded domain with smooth boundary, $d_1,d_2,d_3,{\chi }_1,{\chi }_2,{\mu }_1,{\mu }_2>0$, $a_1,a_2\ge 0$ and $\alpha ,\beta ,\gamma >0$. Under largeness conditions on ${\chi }_1$ and ${\chi }_2$, we show that for suitably regular initial data, any thresholds of the population density can be surpassed, which extends the previous results to the non-symmetric case. The paper contains a well-posedness result for the hyperbolic-elliptic limit system with $d_1=d_2=0$.