I will report our recent research on the propagation dynamics for a large class for nonautonomous cooperative systems with nonlocal dispersal in a time-periodic shifting environment. Under the assumption that each of the two limiting systems has both leftward and rightward spreading speeds, we establish the spreading properties of solutions for such a system by appealing to the theory developed for monotone evolution systems with asymptotic translation invariance. Due to the lack of compactness, the existence of the time-periodic forced wave is obtained with the help of the Kuratowski measure of noncompactness. We further prove the uniqueness of the forced wave by using the sliding technique and its global attractivity via the monotone semiflows approach.
In this talk, I will discuss my recent research in two distinct projects. The completed project is about the propagation dynamics for a class of integro-difference equations in a shifting habitat. We first use a classical transformation to convert this question into a nonmonotone equation with a new kernel function. In two directions of the spatial variable, the resulting equation has two limiting equations admitting spatial translation invariance. Under the hypothesis that each of these two limiting equations has both leftward and rightward spreading speeds, we establish the spreading properties of solutions and the existence of nontrivial forced waves for the original equation by appealing to the abstract theory of nonmonotone semiflows. We also show the stability of the forced waves under certain conditions. The ongoing project is related to a class of periodic time-delayed reaction-diffusion equations in a shifting environment. I will introduce the analytical approach and present our expected results for such an equatuon.
This talk is concerned with the propagation dynamics for nonlocal diffusion systems with monostable and time-space periodic nonlinearity. In the cooperative case, we characterize the spreading speed in terms of principle eigenvalues and tail of dispersal kernels. More specifically, when the dispersal kernels are all light-tailed, we obtain the existence and variational characterization of the linear spreading speed; while the species with heavy-tailed kernel can accelerate the spatial propagation of the other species, and the accelerated spreading rate can be determined by the tail of the maximum of dispersal kernels. In the noncooperative case, by constructing two auxiliary cooperative systems, we establish the existence of the minimal wave speed for semi- transition-waves and its coincidence with the spreading speed when the dispersal kernels are all light-tailed; while as a consequence of acceleration propagation, the semi-transition-waves do not exist if one species has heavy-tailed dispersal kernel.
In this talk, I will report two research projects in my Ph.D. thesis. The first one deals with a population toxin model with chemotaxis effect in an open advective environment. We first obtain the existence and boundedness of solutions by using semi-group estimation and Moser iteration. Based on a detailed analysis of the principal eigenvalues for non-self-adjoint eigenvalue problems, we establish the stability of the toxin-only steady-state solution, which yields sufficient conditions for population persistence or extinction. The second project is devoted to the study of invasion dynamics of Aedes aegypti mosquitos by a reaction-diffusion-advection system in closed advective environments with spatial heterogeneity. We first establish the threshold dynamics in terms of the basic reproduction number R0. Using some properties of R0 with respect to the diffusion rate D and the advection rate q, we obtain detailed dynamic classifications of the system in the D-q plane.
In this talk, I will report recent research on the asymptotic behavior of solutions for some nonlinear diffusion problems. Specifically, in the first part, I will consider the Cauchy problem of one dimensional porous medium equation with a certain class of bistable nonlinearity. In the second part, I will consider a free boundary problem for diffusion equation with the same nonlinearity as above. A complete classification of asymptotic behavior of bounded solutions is given for each of these two models. Furthermore, when spreading happens, it is proved that the spreading speed can be uniquely determined from the related travelling-wave problem. My proof is mainly based on the construction of super-/sub-solutions and the phase plane analysis.
In this talk, I will report our recent research on qualitative analysis for two reaction-diffusion-advection SIS epidemic models. In the first part, we consider a reaction-diffusion SIS epidemic model with standard incidence infection mechanism and linear source in advective heterogeneous environments. We derive the threshold-type dynamics in terms of the basic reproduction number R0 and prove the global asymptotic stability of the endemic equilibrium in a special case. We mainly investigate the effects of linear source, advection and diffusion on asymptotic profiles of the endemic equilibrium. In the second part, we analyze a reaction-diffusion SIS epidemic model governed by the saturated incidence infection mechanism in advective environments. We derive a variational expression of the basic reproduction number R0 and establish the global dynamics of the system in terms of R0. We also explore qualitative properties of the basic reproduction number and investigate the spatial distribution of the individuals with respect to the dispersal and advection.
To investigate the combined effects of drug resistance, seasonality and vector-bias, we formulate a periodic two-strain reaction-diffusion model. It is a competitive system for sensitive and resistant strains, but the single-strain subsystem is cooperative. We derive the basic reproduction number R_i and the invasion reproduction number \hat{R}_i for strain i=1,2, and establish the transmission dynamics in terms of these four quantities. More precisely, (i) if R_1<=1 and R_2<= 1, then the disease is extinct; (ii) if R_1>1>=R_2 (R_2>1>=R_1), then the sensitive (resistant) strains are persistent, while the resistant (sensitive) strains die out; (iii) if \hat{R}_1>1 and \hat{R}_2>1, then two strains are coexistent and periodic oscillation phenomenon is observed. We also study the asymptotic behavior of the basic reproduction number R_0 = max{R_1, R_2} for our model regarding small and large diffusion coefficients. Numerically, we demonstrate the outcome of competition for two strains in different cases.
In this talk, I will report our recent research on a general time-periodic stoichiometric ODE model of the algal growth. The model system has singularities induced by the zero nitrogen concentration. For such a system, we first show that there exists a threshold value λ0, which is exactly the principal eigenvalue of a nonlinear eigenvalue problem associated with a homogeneous of degree one system and get the formula to calculate the value of $\lambda _0$ by introducing an auxiliary system. Furthermore, we establish the global dynamics of the model system in terms of $\lambda _0$. In particular, we obtain the uniqueness of the positive periodic solution when $\lambda _0 > 0$. Finally, we carry out simulations to illustrate the analytic results and make a brief discussion.
In this talk, I will report our recent research on principal eigenvalues for nonlocal dispersal cooperative systems. We establish the necessary and sufficient conditions for the existence of the principal eigenvalue, and investigate the limiting profiles of the spectral bound for the nonlocal dispersal problem as the dispersal rates go to zero and infinity, respectively. As an application, we consider a time-periodic nonlocal dispersal susceptible-infected-susceptible epidemic model with Neumann boundary conditions. We define its basic reproduction ratio R0 and obtain the existence, uniqueness and stability of steady states in terms of R0. Finally, we discuss the impacts of small and large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease.
I will report our recent research on the global dynamics of a time-delayed nonlocal reaction-diffusion model of within-host viral infections. We introduce the basic reproduction number R0 and show that the infection-free steady state is globally asymptotically stable as R0<=1, while the disease is uniformly persistent as R0>1. In the case where all coefficients and reaction terms are spatially homogeneous, we obtain an explicit formula for R0 and the global attractivity of the positive constant steady state. Numerically, we illustrate the analytical results and conduct the sensitivity analysis. We also investigate the influence of parameters on the model and explore the impact of drugs on curtailing the virus' spread.
In this project, we study traveling waves for a time-delayed nonlocal reaction-diffusion model of within-host viral infections.
Firstly, we establish the existence of semi-traveling waves that converge to an unstable infection-free equilibrium as the moving coordinate goes to
$-\infty$, provided that the wave speed $c > c^*$ for some positive number $c^*$ and the basic reproduction number $\mathcal R_0>1$. Then we
construct a Lyapunov functional to show that the semi-travelling waves converge to an endemic equilibrium as the moving coordinate goes to $+\infty$,
and use a limiting argument to obtain the existence of the traveling wave connecting these two equilibria for $c=c^*$ and $\mathcal R_0>1$.
We further employ a Laplace transform technique to prove the non-existence of bounded semi-traveling waves when $0 < c