Global Dynamics of Reaction-Diffusion Systems with a Time-Varying Domain

SIAM Journal on Applied Mathematics, Volume 84, Issue 4, Page 1742-1765, August 2024.
Abstract. This paper is devoted to the study of the global dynamics for a large class of reaction-diffusion systems with a time-varying domain. By appealing to the theories of asymptotically autonomous and periodic semiflows, we establish the threshold-type results on the long-time behavior of solutions for such a system in the cases of asymptotically bounded and periodic domains, respectively. To investigate the model system in the case of asymptotically unbounded domain, we first prove the global attractivity for nonautonomous reaction-diffusion systems with asymptotically vanishing diffusion coefficients via the method of sub- and supersolutions and then use the comparison arguments to obtain the threshold dynamics. We also apply these analytical results to a reaction-diffusion model of dengue fever transmission to investigate the effect of time-varying domain on the basic reproduction number. It turns out that the basic reproduction numbers with dengue fever transmission for the asymptotically bounded and unbounded domains are always less than that for the spatially homogeneous case, and under appropriate conditions, the basic reproduction numbers for asymptotically bounded and periodic domains are larger than or equal to that for the stationary bounded domain.