A geometric approach in the study of traveling waves for some classes of non-monotone reaction–diffusion systems

In this paper we further extend a recently developed method to investigate the existence of traveling waves solutions and their minimum wave speed for non-monotone reaction–diffusion systems. Our approach consists of two steps. First we develop a geometrical shooting argument, with the aid of the theorem of homotopy invariance on the fundamental group, to obtain the positive semi-traveling wave solutions for a large class of reaction–diffusion systems, including the models of predator–prey interaction (for both predator-independent/dependent functional responses), the models of combustion, Belousov–Zhabotinskii reaction, SI-type of disease transmission, and the model of biological flow reactor in chemostat. Next, we apply the results obtained from the first step to some models, such as the Beddinton–DeAngelis model and the model of biolocal flow reactor, to show the convergence of these semi-traveling wave solutions to an interior equilibrium point by the construction of a Lyapunov-type function, or the convergence of semi-traveling waves to another boundary equilibrium point by the further analysis of the asymptotical behavior of semi-traveling wave solutions.