Generalized traveling waves for time-dependent reaction-diffusion systems

Traveling wave solutions in general time-dependent (including time-periodic) reaction-diffusion equations and systems of equations have attracted great attention in the last two decades. The aim of this paper is to study the propagation phenomenon in a general time-heterogeneous environment. More specifically, we investigate generalized traveling wave solutions for a two-component time-dependent non-cooperative reaction-diffusion system which has applications in epidemiology and ecology. Sufficient conditions on the existence and nonexistence of generalized traveling wave solutions are established. In the susceptible-infectious epidemic model setting, generalized traveling waves describe the spatio-temporal invasion of a disease into a totally susceptible population. In the context of predator-prey systems, the generalized traveling waves describe the spatial invasion of predators introduced into a new environment where the prey population is at its carrying capacity.