Traveling wavefronts of a diffusive competing pioneer and climax system with delays

In this paper, we study a delayed diffusion–reaction system for competing pioneer and climax species. We establish the existence criterion of traveling wavefronts connecting the pioneer-invasion-only equilibrium and the co-invasion equilibrium. A transformation of variables makes the monotone iteration scheme and the Schauder fixed point theorem is applicable. By constructing a pair of admissible upper and lower solutions, we show that the system does support such co-invasion wavefronts for c≥c∗c≥c∗, and c∗c∗ is the minimal wave speed, where c∗>0c∗>0 can be decided by a single characteristic equation which is a quadratic algebraic equation. We also discuss the asymptotic behavior of the wave tail for the traveling wavefronts.