Weak traveling wave solutions to a two-species competitive chemotaxis system incorporating Holling-Ⅱ functional responses

This work focuses on analyzing a two-species competitive chemotaxis model with Holling-Ⅱ functional responses. First, the stability of the positive equilibrium point $E^{\ast }$ is analyzed by applying the continuous Fourier transform and the Routh-Hurwitz criterion. Second, proof by contradiction and the generalized principal eigenvalue argument are employed to prove the non-existence of weak traveling wave solutions for sufficiently small wave speeds $c$. Finally, the existence of weak traveling wave solutions is proved via the upper and lower solutions method and Schauder's fixed-point theorem for all sufficiently large $c$.