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).
This project is dedicated to investigating a periodic predator-prey reaction-diffusion model with spatial and temporal heterogeneities. We rigorously characterize the properties of the principal eigenvalue and establish a precise relationship between the coefficient functions and the dynamics. Our results indicate that slow predator movement and short frequency of environmental periodic variations promote successful predator invasion. Conversely, reducing the predator mortality rate facilitates long-term coexistence of both populations. Additionally, we explore the asymptotic behaviors of positive periodic solutions when the diffusion coefficients are large or small, revealing the effects of diffusion on the invasion dynamics.
We investigate the spatiotemporal dynamics of an impulsive nonlocal dispersal model for species in a spatially heterogeneous environment. Since the model couples a differential equation with a recurrence relation, we reformulate the problem as a discrete-time recursive system via the solution map approach. In a bounded domain, we establish a criterion to distinguish between species persistence and extinction. In an unbounded domain, we establish the existence of a spreading speed and prove its coincidencde with the minimal wave speed for spatially periodic traveling waves in the monotone case. Numerical simulations are conducted to examine the impacts of maturation delay, spatial heterogeneity and nonlocal dispersal on the spatial distribution patterns of species.
We investigate the spreading dynamics of a time-delayed nonlocal within-host viral infection model incorporating two latent stages, which gives rise to a non-monotone semiflow. For this generalized system, we characterize the associated spreading speeds. When the basic reproduction number R_0 > 1, we first prove that for any wave speed c \geq c^* (c^* > 0), the solution converges uniformly to the infection-free equilibrium in the region |x| \geq ct as t \to +\infty. This result is established by introducing an auxiliary system as an upper control system, constructing a suitable upper solution, and applying the comparison principle. Then we employ the notion of uniform persistence to demonstrate the spreading property for c \in (-c^*, c^*) in the region |x| \leq ct as t \to +\infty. Under the assumption of equal diffusion coefficients, we further derive a stronger conclusion regarding the spreading speed for c \in (-c^*, c^*) via Lyapunov function-like arguments: every bounded solution converges uniformly to the unique positive spatially homogeneous equilibrium in the region |x| \leq ct as t \to +\infty. It turns out that c^* > 0 is the asymptotic spreading speed for this non-monotone system. Numerical simulations are conducted to calculate c^* and illustrate the long-time behavior of the solutions.
In this talk, I will report our recent research on traveling wave solutions for a reaction-diffusion-advection model of dengue fever with nonlinear incidence function. The incidence function may be non-monotonic in the variable representing the infected mosquito population. In the non-monotone case, we establish the existence of semi-wave solutions by constructing appropriate upper and lower solutions and applying Schauder's fixed point theorem. In the monotone case, we establish the existence of traveling wave solutions by constructing a Lyapunov function and applying LaSalle's invariance principle. The existence of the traveling wave with the critical speed c^* is obtained via a limiting argument. Moreover, by using the one-sided Laplace transform, we prove the nonexistence of traveling wave solutions with wave speed less that c^*.
Seasonality is common in population dynamics, since environmental conditions often change in a regular way over time. These changes may influence key biological processes, including reproduction, growth, movement, and survival, and may lead to different long-term outcomes for population systems. This thesis presents two research projects on population dynamics in seasonal environments. The main focus is a completed study of a mosquito growth model with stage structure, pair formation, and periodic developmental delay. For this model, we introduce the basic reproduction ratio as a threshold quantity for extinction and persistence and establish the corresponding dynamical results in a seasonal setting. I also briefly discuss the ongoing work on a single-species model with seasonal succession and strong Allee effect, with emphasis on bistable traveling waves.
Understanding the spatial dynamics of populations has long been a central theme in biological and ecological research, with particular emphasis on the spread and persistence of species. Accordingly, this thesis is devoted to the spatial dynamics of three classes of evolutionary systems arising in population biology. The theoretical tools employed throughout the thesis are introduced in the preliminary chapter. In Chapter 2, we first study species dynamics in time-varying domains with impulsive birth effects. To analyze the global dynamics, a hybrid population model combining continuous- and discrete-time processes in an asymptotically periodic domain is considered. In Chapter 3, we examine the long-term behavior of competing species in time-varying domains by analyzing a two-species competition reaction-diffusion model subject to homogeneous Dirichlet and Neumann boundary conditions on three different types of evolving domains. In Chapter 4, to better understand the spatial dynamics of competitive systems with spatial heterogeneity, we investigate bistable traveling waves arising from the spread of two competing species in a Lotka-Volterra competition system on a periodic discrete habitat. Finally, in concluding Chapter, we collect and discuss several open problems and potential future research directions arising from each chapter.