Spreading Speed for Some Cooperative Systems with Nonlocal Diffusion and Free Boundaries, Part 3: Rate of Shifting (Journal of Nonlinear Modeling and Analysis, 2025)
This is the last part of our work series on a class of cooperativereaction-diffusion systems with free boundaries in one space dimension, wherethe diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and some of the equations in the system do not have adiffusion term. Such a system covers various models arising from populationbiology and epidemiology, including in particular a West Nile virus model [10]and some epidemic models [22, 38], where a “spreading-vanishing” dichotomyis known to govern the long time dynamical behaviour, but the spreading ratewas not well understood. In this work series, we develop a systematic approachto determine the spreading profile of the system. In Part 1 [11], we obtainedthreshold conditions on the kernel functions which decide exactly when thespreading has finite speed $c_0,$ or infinite speed (accelerated spreading), and forthe case of finite speed, we determined its value $c_0$ via semi-wave solutions. Inthis paper, for some typical classes of kernel functions, we obtain more precisedescriptions of the spreading for the finite speed case by revealing the exactrate of shifting of the spreading front from $c_0t;$ the infinite speed case is studiedseparately in Part 2 [14].