A Time-Periodic Parabolic Eigenvalue Problem on Finite Networks and Its Applications

In this paper, we investigate the eigenvalue problem of a time-periodic parabolic operator on a finite network. The network under consideration can support various types of flows, such as water, wind, or traffic. Our focus is to determine the asymptotic behavior of the principal eigenvalue as the diffusion rate approaches zero, or the advection rate approaches infinity, under reasonable boundary conditions that can be derived from ecosystems. Our results demonstrate that such asymptotics is primarily influenced by the boundary conditions at the upstream and downstream vertices of the network, rather than the geometric structure of the finite network itself provided that it is simple and connected. We then apply our results to a single-species population model and two SIS epidemic systems on networks and reveal the substantial impact of the diffusion and advection rates as well as the boundary conditions on the long-time dynamics of the population and the transmission of infectious diseases.