In this talk, I will report my recent research on the threshold dynamics for a class of time-delayed reaction diffusion equations with a time-periodic shifting environment which is suitable for survival only in bounded regions. We first establish the spreading properties of solutions under the uniform asymptotic annihilation type assumptions. Then we prove the threshold dynamics of forced time-periodic waves by the abstract theory of asymptotic spectral radius. In addition, we propose some supplementary conditions for the uniqueness and stability of such time-periodic waves if it exists.
In this project, we study the propagation dynamics of a Lotka-Volterra two-species competition system in a periodic discrete habitat. Under appropriate assumptions, this lattice system with periodic initial data admits a bistable structure. We then establish the existence of the pulsating travelling front connecting two stable semi-trivial equilibria and the global stability of such a bistable wave for the wave-like initial data. We also obtin some conditions to determine the sign of the wave speed.
Spatial and temporal evolutions are very important topics in epidemiology and ecology. This thesis is devoted to the study of global dynamics of some reaction-diffusion models incorporating environmental heterogeneities. As biological invasions significantly impact ecology and human society, how invasive species' growth and spatial spread interact with the environment becomes a significant challenging problem. We start with an impulsive time-space periodic model to describe a single species with a birth pulse in the reproductive stage in Chapter 2. In-host viral infections commonly involve hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV). To explore the effects of the spread heterogeneity on the spread of within-host virus, we propose a time-delayed nonlocal reaction-diffusion model and obtain the threshold-type results in terms of the basic reproduction ratio in Chapter 3. In Chapter 4, we then explore the existence and nonexistence of traveling wave solutions for such a non-monotone system on an unbounded domain, and show that there is a minimum wave speed for traveling waves connecting the infection-free equilibrium and the endemic equilibrium. Mosquito-borne diseases are transmitted by the bite of infected mosquitoes, including Zika, West Nile, Chikungunya, dengue, and malaria. To investigate the effects of spatial and temporal heterogeneity on the spread of the Chikungunya virus, we develop a nonlocal periodic reaction-diffusion model of Chikungunya disease with periodic time delays in Chapter 5. We further establish two threshold-type results regarding the global dynamics of mosquito growth and disease transmission, respectively. A brief summary and future works are presented at the end of this thesis.
Up to now, there have been quite a few works on spreading properties of solutions for predator-prey reaction-diffusion systems. The main difficulty lies in the fact that such a system does not admit the comparison principle. It turns out that the predator and prey speies may spread in two different speeds. In this talk, I will review two methods in the study of spreading speeds. The first one is motivated by the uniform persistence idea from dynamical systems and uses the properties of the corresponding entire solutions. The second one is based on the parabolic strong maximum principle for scalar equation and on the derivation of local pointwise estimates, which are employed to compare the solutions of the predator-prey system with those of a KPP scalar equation on suitable spatio-temporal domains.
The dynamics of competitive maps and semiflows defined on the product of two cones in respective Banach spaces is studied. It is shown that exactly one of three outcomes is possible for two viable competitors. Either one or the other population becomes extinct while the surviving population approaches a steady state, or there exists a positive steady state representing the coexistence of both populations.
I will report our recent research on the spatio-temporal dynamics for a large class for time-periodic cooperative systems with either random diffusion or nonlocal dispersal. Under the assumption that the edge of the habitat is shifting and the two limiting systems have asymptotic annihilation, we establish the spreading properties of solutions and the existence, uniqueness as well as the global attractivity of the time-periodic forced wave for such a system. Our main approach is to use the theory developed for monotone evolution systems with asymptotic annihilation in [Yi and Zhao, DCDS,43(2023), 2693-2720].
In this talk, we consider two problems related to chemotaxis-Stokes models with mixed boundary conditions. The first one is regard to chemotaxis-Navier-Stokes system in a two-dimensional strip periodic domain. The nonlinearity brought by consumption term $cn^{\gamma}$ may bring some essential technical difficulties to derive some a priori estimates for global bounded solutions. By the energy analysis in domains and boundaries and applying trace interpolation inequalities, it is shown that there exists a global bounded classical solution for any $\gamma > 0$. For the second, we consider the time periodic problem to a three-dimensional chemotaxis-Stokes model with porous medium diffusion $n^m$. By using a double-level approximation method and some iterative techniques, we obtain the existence and time-space uniform boundedness of weak time periodic solutions for any $m > 1$. Moreover, the obtained periodic solutions are in fact strong periodic solutions for $m\leq 4/3$. This talk is based on the joint works with Prof. Chunhua Jin.
This research project deals with a diffusive population-toxicant model in polluted aquatic environments, with a toxicant-taxis term describing a toxicant-induced behavior change,that is, the population tends to move away from locations with high-level toxicants.The global existence of solutions is established by the techniques of the semigroup estimation and Moser iteration. Based on the properties of the principal eigenvalue for non-self-adjoint eigenvalue problems, we investigated the local and global stability of the toxin-only steady-state solution and the existence of positive steady state, which yields sufficient conditions that lead to population persistence or extinction. Finally, we use numerical simulation to study the effects of some key parameters,such as toxicant-taxis coefficient, advection rate, and effect coefficient of the toxicant on population growth, on population persistence. Both numerical and analytical results show that a weak chemotaxis effect, a small advection rate of the population, and a weak effect of the toxicant on population growth are favorable for population persistence.
In this talk, I will report our recent research on a reaction-diffusion SIS (susceptible-infected-susceptible) epidemic model with saturated incidence function and linear birth-death growth in advective environments. The related model without the linear source case has been studied in our earlier work. Our main purpose is to investigate the combined effects of varying total population, saturation infection mechanism and spatial heterogeneity on the transmission dynamics and spatial distribution of disease. The extinction and persistence of the infectious disease in terms of the basic reproduction number are established. We discuss the global attractivity of the equilibria in two special cases and explore the asymptotic profiles of the endemic equilibrium with respect to the dispersal and advection rates. Compared with our earlier results for the model without source term, our findings indicate that the linear source can enhance the persistence of an infectious disease and may provide some prospective applications in disease prevention and control.
In this talk, 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 on an ecosystem that is subject to natural boundary conditions. We determine the asymptotic behavior of the principal eigenvalue as the diffusion rate approaches zero, or the advection rate approaches infinity. 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.