Traveling wave solutions to a one prey-two competing predators model with nonlocal delay

In this paper, we investigated the traveling wave solutions to a one prey-two competing predators model with nonlocal delay. First, we analyzed the stability of the positive equilibrium by using a Lyapunov function. Then, by examining the distribution of the roots of the characteristic equation, we ascertained the critical wave speed $c^{\ast }$. Finally, employing the cross iteration method and Schauder's fixed point theorem, we proved the existence of traveling wave solutions connecting the trivial equilibrium $(0,0,0)$ with the positive equilibrium $(u^{\ast },v^{\ast },w^{\ast })$ for wave speeds $c>c^{\ast }$. The incorporation of nonlocal delay into models featuring intra-specific and inter-specific competition significantly elevates computational complexity, thereby necessitating precise analytical estimates.