On the strongly competitive case in a fully parabolic two-species chemotaxis system with Lotka-Volterra competitive kinetics

Abstract

This work considers a two-species chemotaxis system with Lotka-Volterra competitive kinetic functional response term in a bounded domain with smooth boundary. We proved global bounded solutions to the system in high dimensions (n≤5) without the convexity of the domain. Moreover, by constructing appropriate Lyapunov functionals, it is proved that the solution convergences to the semi-trivial steady state in L∞(Ω) under strong competition (a1,a2≥1, where a1,a2 represent the intensity of competition between different species) if the growth coefficients of two species are appropriately large. Compared to previous work, our result removes the requirement for the convexity of the domain and proves global bounded solutions in high dimensions (n≤5). Moreover, we studied the asymptotic behavior of solutions in the case of strong competition which is obscure in the existing literature. Furthermore, the linear stability analysis is performed to find the possible patterning regimes, outside the stability parameters regime, for both semi-trivial and coexistence steady states, our numerical simulations show that non-constant steady states and spatially inhomogeneous temporal periodic patterns are all possible.