Traveling Waves in a Diffusive Single-Species Model with a Weak Spatiotemporal Memory Kernel

Abstract

We study traveling wave solutions in a nonlinear reaction-diffusion model incorporating a weak spatiotemporal distributed memory kernel. The model describes a single-species population whose movement is influenced by both random diffusion and memory-based dispersal, with the latter expressed as a convolution term involving a temporal weighting function and a spatial Green’s function. This framework captures the gradual decay of spatial memory and its effect on dispersal dynamics. Using perturbation expansions, operator theory, and the Banach fixed-point theorem, we establish the existence of traveling wavefronts connecting equilibrium states in two parameter regimes: (i) a small memory-based diffusion coefficient and (ii) a large wave speed. The analysis addresses significant challenges arising from the nonlocal, nonlinear memory term by employing integral equation representations and precise estimates. Numerical simulations illustrate how the memory diffusion coefficient and mean delay influence wave speed, front shape, and population distribution. The results provide a rigorous characterization of wave propagation in systems with weak distributed memory, offering a unified approach applicable to models in population dynamics, biological invasion, and other spatiotemporal processes with memory effects.