Spreading speeds in a nonlocal delayed competition system without monotonicity

This paper studies the asymptotic spreading in a reaction-diffusion competition system with nonlocal delays. Owing to the nonlocal delays present in intraspecific competition terms, this system fails to satisfy the classical comparison principle applicable to competition systems. Under the weak competition assumption, we investigate two distinct invasion processes, both of which result in the eventual coexistence of the two competitors in the sense of the compact open topology. In the first scenario, one is the native, while the other is the invader that satisfies the appropriate decaying initial conditions. The spreading speed of the invader, along with certain convergence results, is presented. Particularly, when the delayed intraspecific competition is relatively weak, the invasion speed is determined by the corresponding linearized problem at the semitrivial steady state. In the second scenario, we estimate the spreading dynamics in the context where both species act as invaders that satisfy the appropriate decaying initial conditions. Our results indicate that two invaders can exhibit distinct invasion capacities, a finding that differs from the well-investigated traveling wave solutions.