In this talk, I will report our recent research on a time-space periodic population growth model with impulsive birth. We first discuss some eigenvalue problems and obtain a threshold-type result. Then we establish the existence of the spreading speeds and their coincidence with the minimal speeds of time-space periodic traveling waves in the monotone and non-monotone cases, respectively. Further, we study the global dynamics for this model in a bounded spatial domain. We will also give numerical simulations to interpret the obtained analytical results.
This talk is concerned with the propagation dynamics of a large class of time-space periodic and partially degenerate reaction-diffusion systems with time delay and monostable nonlinearity. In the cooperative case, based on the theory of principal eigenvalues for linear and partially degenerate systems with time delay, we establish the existence of spreading speeds and its coincidence with the minimal wave speed of time-space periodic traveling waves. In the noncooperative case, we introduce the definition of transition semi-waves and prove the existence and equality of the spreading speed and the minimal wave speed of transition semi-waves by constructing two auxiliary cooperative systems and using the comparison arguments. To overcome the difficulty arising from the lower regularity of solutions for partially degenerate systems, some new prior estimate is established to prove the existence of the transition semi-waves.
The purpose of this project is to study the propagation dynamics of an n-species cooperative system \begin{equation}\nonumber \partial_t u=D(t) \partial_{xx} u-\alpha (t)\partial_x u + g(t,x-\beta (t),u),\quad x\in\mathbb{R},t>0. \end{equation} We firstly establish the spreading properties of such a system that resembles that of the limiting system as well as the existence of forced time periodic waves by appealing to the abstract theory of monotone semiflows. Under certain conditions, we further show the uniqueness of forced waves by the sliding method and it attracts other solutions in $L^\infty$- sense according to the tail behavior of initial values by applying the dynamical systems approach. An application is given finally to illustrate the effectiveness of our theoretical result.
In this talk, I will report our recent study on 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, such an 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 for a class of nonmonotone semiflows. We also obtain the stability of forced waves under certain conditions.
In this talk, I will report our recent research on a Lotka-Volterra two-species competition system in a periodic discrete habitat. Under appropriate conditions, we find sufficient conditions for the spatially periodic initial value problem to admit a bistable structure. Then we establish the existence of the traveling wave connecting two stable semitrivial spatially periodic steady states.
Statistics of COVID-19 cases shows that the epidemic situation in many countries fluctuates seasonally. In this talk, I will report our recent research on a COVID-19 epidemic model with seasonality. We first propose a COVID-19 epidemic model with seasonality and define the basic reproduction number R0 for the disease transmission. It is proved that the disease-free equilibrium is globally asymptotically stable when R0<1, while the disease is uniformly persistent and there exists at least one positive periodic solution when R0>1. Numerically, we observe that there is a globally asymptotically stable positive periodic solution in the case of R0>1. Further, we conduct a case study of the COVID-19 transmission in USA by using statistical data.
This talk is concerned with propagation dynamics of nonlocal dispersal monostable equations in time-space periodic habitats. We first show that such an equation admits a single spreading speed in every direction under certain conditions and then give several spreading properties in terms of the spreading speeds such as spreading ray speeds and spreading sets. Furthermore, we consider the dependence of the spreading speeds on the dispersal rate and reaction term and prove that taking the temporal average or spatial average can decreases the spreading speeds. Finally, we employ the viscosity vanishing method to establish the existence of time-space periodic traveling front with critical speed in every direction under the partially temporally homogeneous case and partially nearly flat case.
Climate changes caused by global warming, industrialization and overdevelopment have led to the shifting of habitats for biological species and seriously threatened and destroyed the ecological environment and biological diversity. It is very important to study the the effects of the climate change on the population dynamics. In this talk, I will report some recent results on the propagation dynamics for the competitive systems in (time-periodic) shifting habitats.
The basic reproduction number R0 is an important concept in population biology. As a threshold quantity for population dynamics, it is unquestionably one of the most valuable mathematical ideas brought to theoretical ecology and epidemiology. In this talk, I will report our recent research on the linear stability and R0 for autonomous and cooperative functional differential equations (FDEs). In particular, we will give a general formula of R0 for autonomous and compartmental FDE models, where both the production (infection) and internal transition have time delays. We also extend this theory to abstract autonomous FDEs on an ordered Banach space so that it can be applied to time-delayed reaction-diffusion systems. As an illustrative example, we establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of R0.
In this talk, we report some results on the free boundary problems for delayed diffusive epidemic models, which describe the positive feedback interaction between the infective human population and the concentration of bacteria or viruses in the environment. The global dynamics of solutions and asymptotic spreading speeds of free boundaries will be presented. We discuss the spreading and vanishing phenomena of bacteria in terms of basic reproduction number. When spreading occurs, the asymptotic spreading speeds of free boundaries are uniquely determined by the corresponding semi-wave problem with time delay(s). We introduce two ways to establish the existence of semi-wave solution.