TY - JOUR
T1 - Spreading Speed for Some Cooperative Systems with Nonlocal Diffusion and Free Boundaries, Part 3: Rate of Shifting
AU - Du , Yihong
AU - Ni , Wenjie
JO - Journal of Nonlinear Modeling and Analysis
VL - 4
SP - 1532
EP - 1574
PY - 2025
DA - 2025/07
SN - 7
DO - http://doi.org/10.12150/jnma.2025.1532
UR - https://global-sci.org/intro/article_detail/jnma/24250.html
KW - Free boundary, nonlocal diffusion system, spreading rate.
AB - <p style="text-align: justify;">This is the last part of our work series on 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 [10]
and some epidemic models [22, 38], where a “spreading-vanishing” dichotomy
is known to govern the long time dynamical behaviour, but the spreading rate
was not well understood. In this work series, we develop a systematic approach
to determine the spreading profile of the system. In Part 1 [11], we obtained
threshold conditions on the kernel functions which decide exactly when the
spreading has finite speed $c_0,$ or infinite speed (accelerated spreading), and for
the case of finite speed, we determined its value $c_0$ via semi-wave solutions. In
this paper, for some typical classes of kernel functions, we obtain more precise
descriptions of the spreading for the finite speed case by revealing the exact
rate of shifting of the spreading front from $c_0t;$ the infinite speed case is studied
separately in Part 2 [14].</p>