Asymptotic dynamics in a two-species chemotaxis model with non-local terms

In this study, we consider an extended attraction two species chemotaxis system of parabolic-parabolic-elliptic type with nonlocal terms under homogeneous Neuman boundary conditions in a bounded domain \begin{document} $Ω \subset \mathbb{R}^n(n≥1)$ \end{document} with smooth boundary. We first prove the global existence of non-negative classical solutions for various explicit parameter regions. Next, under some further explicit conditions on the coefficients and on the chemotaxis sensitivities, we show that the system has a unique positive constant steady state solution which is globally asymptotically stable. Finally, we also find some explicit conditions on the coefficient and on the chemotaxis sensitivities for which the phenomenon of competitive exclusion occurs in the sense that as time goes to infinity, one of the species dies out and the other reaches its carrying capacity. The method of eventual comparison is used to study the asymptotic behavior.