Spreading speed for some cooperative systems with nonlocal diffusion and free boundaries, part 2: Precise rates of acceleration

We investigate a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and some of the equations in the system do not have a diffusion term. Such a system covers various models arising from population biology and epidemiology, including in particular a West Nile virus model [12] and an epidemic model [38], where a “spreading-vanishing” dichotomy is known to govern the long time dynamical behaviour, but the spreading rate was not well understood. We aim to develop a systematic approach to determine the spreading profile of the system. In an earlier work [13], we obtained threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading), and for the case of finite speed, we determined its value via semi-wave solutions. In the current work, we focus on the case of accelerated spreading, and obtain the precise rates of acceleration for some typical classes of kernel functions. Our results apply directly to the above mentioned concrete models.