Zhigui Lin

Spreading-Vanishing Dichotomy in the Diffusive Logistic Model with a Free Boundary

In this paper we investigate a diffusive logistic model with a free boundary in one space dimension. We aim to use the dynamics of such a problem to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. We prove a spreading-vanishing dichotomy for this model, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or it fails to establish and dies out in the long run. Sharp criteria for spreading and vanishing are given. Moreover, we show that when spreading occurs, for large time, the expanding front moves at a constant speed. This spreading speed is uniquely determined by an elliptic problem induced from the original model.

A free boundary problem for a predator?prey model

This article is concerned with a system of semilinear parabolic equations with a free boundary, which arises in a predator–prey ecological model. The conditions for the existence and uniqueness of a classical solution are obtained. The evolution of the free boundary problem is studied. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time.

Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries

In this paper, we consider a Fisher-KPP equation with an advection term and two free boundaries, which models the behavior of an invasive species in one dimension space. When spreading happens (that is, the solution converges to a positive constant), we use phase plane analysis and upper/lower solutions to prove that the rightward and leftward asymptotic spreading speeds exist, both are positive constants. Moreover, one of them is bigger and the other is smaller than the spreading speed in the corresponding problem without advection term.

Long time behavior of solutions of a diffusion-advection logistic model with free boundaries ✩

Abstract In this paper, we study the long behavior of solutions of a diffusion–advection logistic model with free boundaries in one dimensional space when the influence of advection is small. We give a spreading–vanishing dichotomy for this model, that is, the solution either converges to 1 locally uniformly in R , or to 0 uniformly in its occupying domain. Moreover, by introducing a parameter σ in the initial data, we exhibit the sharp threshold between vanishing and spreading, that is, there exists a value σ ∗ such that spreading happens when σ > σ ∗ , vanishing happens when σ ≤ σ ∗ .

Spreading and vanishing in a West Nile virus model with expanding fronts

We study a simplified version of a West Nile virus (WNv) model discussed by Lewis et al. (2006), which was considered as a first approximation for the spatial spread of WNv. The basic reproduction number R0 for the non-spatial epidemic model is defined and a threshold parameter R0D for the corresponding problem with null Dirichlet boundary condition is introduced. We consider a free boundary problem with a coupled system, which describes the diffusion of birds by a PDE and the movement of mosquitoes by an ODE. The risk index R0F(t) associated with the disease in spatial setting is represented. Sufficient conditions for the WNv to eradicate or to spread are given. The asymptotic behavior of the solution to the system when the spreading occurs is considered. It is shown that the initial number of infected populations, the diffusion rate of birds and the length of initial habitat exhibit important impacts on the vanishing or spreading of the virus. Numerical simulations are presented to illustrate the analytical results.

Free boundary models for mosquito range movement driven by climate warming

As vectors, mosquitoes transmit numerous mosquito-borne diseases. Among the many factors affecting the distribution and density of mosquitoes, climate change and warming have been increasingly recognized as major ones. In this paper, we make use of three diffusive logistic models with free boundary in one space dimension to explore the impact of climate warming on the movement of mosquito range. First, a general model incorporating temperature change with location and time is introduced. In order to gain insights of the model, a simplified version of the model with the change of temperature depending only on location is analyzed theoretically, for which the dynamical behavior is completely determined and presented. The general model can be modified into a more realistic one of seasonal succession type, to take into account of the seasonal changes of mosquito movements during each year, where the general model applies only for the time period of the warm seasons of the year, and during the cold season, the mosquito range is fixed and the population is assumed to be in a hibernating status. For both the general model and the seasonal succession model, our numerical simulations indicate that the long-time dynamical behavior is qualitatively similar to the simplified model, and the effect of climate warming on the movement of mosquitoes can be easily captured. Moreover, our analysis reveals that hibernating enhances the chances of survival and successful spreading of the mosquitoes, but it slows down the spreading speed.

Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment

This paper is concerned with a strongly-coupled elliptic system, which describes a West Nile virus (WNv) model with cross-diffusion in a heterogeneous environment. The basic reproduction number is introduced through the next generation infection operator and some related eigenvalue problems. The existence of coexistence states is presented by using a method of upper and lower solutions. The true positive solutions are obtained by monotone iterative schemes. Our results show that a cross-diffusive WNv model possesses at least one coexistence solution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough, while if the basic reproduction number is less than or equal to one, the model has no positive solution. To illustrate the impact of cross-diffusion and environmental heterogeneity on the transmission of WNv, some numerical simulations are given.

Asymptotic periodicity in a diffusive West Nile virus model in a heterogeneous environment

This paper is concerned with a diffusive West Nile virus model (WNv) in a heterogeneous environment. The basic reproduction number R0 for spatially homogeneous model is first introduced. We then de...