Research on the uniqueness of traveling wave solutions for non-cooperative reaction-diffusion systems remains limited. In this talk, we will address the uniqueness of the decay exponent of traveling wave solutions in two classes of models: SI-type infectious disease reaction-diffusion systems and predator-prey reaction-diffusion systems. Specifically, we focus on the behavior of these waves at the invaded equilibrium-namely, the disease-free equilibrium in epidemiological models and the prey-only equilibrium in ecological models. Both continuous (Laplacian) diffusion and discrete diffusion are considered. Our analysis relies on the bilateral Laplace transform as a key analytical tool. By establishing the uniqueness of the decay exponent, we aim to shed light on the broader question of uniqueness for traveling wave solutions in such non-cooperative settings.
In order to investigate the effects of heterogeneous shifting environments on the dynamical behaviors for an invading species with long-distance free diffusion and birth pulse hybrid, we focus on an impulsive nonlocal diffusive system with spatiotemporal shifting heterogeneities, in which both scenarios of monotone and non-monotone reproductive terms are considered. The evolution of this hybrid system can be transformed into a discrete-time recursion composed of a continuous noncompact semi-flow and a discrete-time map. Firstly, we establish the spreading properties including the upward convergence and asymptotic annihilation of the recursive system. Secondly, we obtain the existence, uniqueness and nonexistence of forced waves of the saturation or extinction type in terms of the shifting speeds. Finally, we conduct numerical simulations to illustrate our analytical results and demonstrate rich dynamics driving by the interplay between the shifting speed of the habitat and the speed at which species spread in the optimal habitat. This work may be an early attempt to study spatiotemporal propagation patterns and traveling waves of impulsive nonlocal diffusive systems with shifting heterogeneities, in which we need to overcome some nontrivial difficulties induced by the lack of compactness, translation invariance and continuity in time simultaneously. This talk is based on a joint work with Profs. Jia-Bing Wang (CUG) and Xiao-Qiang Zhao (MUN).
A general modeling framework is proposed to investigate the effect of behavior change on disease transmission dynamics. Specifically, based on the classic SIS model, we considered two different behavioral patterns: the first pattern (Model I) characterizes behavior change through an imitation process, where individuals exclusively adopt the behavior associated with higher payoff, while the second one (Model II) assumes that the switching pattern of behavior change only relies on the fraction of individuals currently intending to change their decisions, leading to a non-smooth co-evolution model. We initially investigated the global dynamics of the first model in terms of the basic reproduction number R0 and demonstrated that the unique disease-free equilibrium is globally asymptotically stable if R0<1 and two endemic equilibria are globally asymptotically stable under different conditions, respectively. Then for the non-smooth model, the dynamics of two subsystems, the nature of equilibria and the global properties were examined. Furthermore, we numerically explored the effectiveness of human behavior change on the disease transmission. Compared to the classic SIS model, we observed that a lower prevalence threshold or a faster rate of behavior change leads to a more pronounced reduction in both daily infections and peak prevalence, with earlier epidemic peak. Interestingly, Model II exhibits a lower daily infection fraction and a reduced epidemic peak compared to Model I. Our results reveal that adopting the second behavioral pattern is more effective in reducing disease prevalence, thereby explaining the free-rider effect. Also, behavior change does not necessarily lead to oscillations, which is related to the modeling approach of epidemics. This talk is based on a joint work with Prof. Yanni Xiao (XJTU).