Spatial heterogeneity in behavioral responses and human mobility can jointly shape epidemic dynamics, yet their interaction has received limited attention due to analytical challenges. Here we develop a reaction-diffusion model coupling imitation-based behavioral switching with disease transmission, where the switching process is driven by differences in strategy payoffs. We define the basic reproduction number and characterize the threshold dynamics of the system. A key analytical step is to establish the ultimate boundedness of solutions in the presence of nonlinear behavioral switching terms. Furthermore, we analyze the asymptotic profiles of endemic equilibria under three limiting diffusion regimes. Our numerical results show that the effect of behavioral switching on long-term prevalence is threshold dependent and that spatial heterogeneity in behavior change can reduce disease prevalence under appropriate parameter regimes. Interestingly, increasing the mean switching rate enhances the protective effect of behavioral switching, whereas increasing the mean perceived threshold weakens this effect and may even reverse it for sufficiently large infected diffusion. In addition, the interaction between human mobility and behavioral heterogeneity can lead to non-monotone pattern in disease prevalence.
We focus on the long-term dynamics of an impulsive nonlocal dispersal system with nonlocal pulses on time-varying domains, including the asymptotically fixed, asymptotically time-periodic, and asymptotically unbounded cases. We first establish the well-posedness of the impulsive system and reduce its dynamics to a nonautonomous discrete-time process. For the asymptotically bounded cases, we characterize the threshold dynamics of the original impulsive system by combining the spectral properties and global dynamics of the limiting system with the theory of internally chain transitive sets. The asymptotically unbounded case presents an additional difficulty due to the interplay between time-dependent nonlocal dispersal and nonlocal impulsive effects, for which the limiting dynamics alone are insufficient to derive the required estimates. To overcome this difficulty, we develop an incremental subsolution method, in which a parametrized family of subsolutions is advanced through successive positive increments. Using this construction, we establish the global dynamics of positive solutions.
In this talk, I will report our recent research on the global dynamics of reaction-diffusion competition models with seasonal succession. We first analyze a seasonal single-species growth model and establish its threshold dynamics. As a key ingredient, we investigate the principal eigenvalue of a periodic eigenvalue problem with coefficients that are discontinuous in time and derive several useful properties. We then apply the theory of abstract competitive systems to characterize the global dynamics of the two-species competition model. In particular, explicit threshold criteria are obtained in terms of unfavorable-season mortality, seasonal duration, spatial resource distribution, and diffusion rates. These results reveal how seasonal succession and spatial dispersal jointly affect species extinction, persistence, and coexistence. Numerical simulations are also presented to illustrate the theoretical findings.
The purpose of this project is to study the propagation dynamics for a class of reaction-diffusion models of single species growth with seasonal succession and strong Allee effect. We first give a complete classification of the global dynamics for the spatially homogeneous system of such a model by using the Poincare (period) map approach. For the spatial model, we then establish the existence, uniqueness and global stability with phase shift of the monotone periodic traveling wave connecting two stable and spatially homogeneous periodic solutions. We further find sufficient conditions to determine the sign of the bistable wave speed in the cases of general and cubic nonlinearities. In particular, we show that under appropriate conditions, the variation of the good season proportion can reverse the propagation direction of the bistable waves.