arXiv subjects

Wan-Tong Li

Publications and source records attributed to Wan-Tong Li.

At least 19 recordsLinked to original sources

arXiv2026

A Hamilton-Jacobi Approach to Time-Delayed Nonlocal Diffusion Models in Shifting Habitats

This paper is concerned with the spatial propagation dynamics of time-delayed nonlocal diffusion equations in shifting habitats. We employ the theory of viscosity solutions for Hamilton-Jacobi equations to provide a complete classification of the spreading speeds. In particular, we derive variational formulas that explicitly characterize how these speeds depend on the decay rate of the initial data, the habitat shifting speed, and the maturation delay across three regimes: locally determined, nonlocally selected, and locked. We reveal a distinct directional asymmetry in the nonlocal selection mechanism and a novel delay-insensitivity phenomenon. We also establish the threshold conditions under which the habitat locking effect eliminates the classical decelerating role of time delay.

math.AP↗

arXiv2026

Propagation Dynamics for Multidimensional Nonlocal Diffusion Equations: A General Freidlin-G\"artner Formula

In this paper, we establish a unified geometric description for the propagation behavior of multidimensional nonlocal diffusion equations. By extending the classical Freidlin-G\"artner framework to general asymmetric, nonlocal operators, our theory naturally captures biased propagation--a regime where the intrinsic spreading set may exclude the origin. A key consequence is the representation of the spreading set as a Minkowski sum, which holds for both bounded and unbounded initial supports. Within this framework, we derive uniform spreading estimates and prove the local Hausdorff convergence of level sets in arbitrary dimensions. Our work therefore not only recovers known isotropic results but also provides a complete characterization of the biased case. Moreover, the methods developed here are readily adaptable to a broader class of diffusion problems featuring general operators and nonlinearities.

math.AP↗

arXiv2026

Propagation dynamics for nonlocal dispersal predator-prey systems in shifting habitats: A Hamilton-Jacobi approach

This paper is concerned with the spreading speeds of nonlocal dispersal predator-prey systems in shifting habitats under general initial conditions. By employing geometric optics techniques and theory of viscosity solutions, we reformulate the problem into the study of Hamilton-Jacobi equations. Through a detailed analysis of the structure of viscosity solutions, we provide a complete classification of explicit formulas for the spreading speed of the prey population, especially in cases where it invades the habitat more rapidly than predators, yielding two fundamentally distinct ``nonlocal determinacy'' results derived by different mechanisms. We also obtain an upper bound for spreading speed of the predators, incorporating the decay rate of the initial data and the speed of shifting habitats. These findings demonstrate that there are complex connections among spreading speeds, habitat shifting speed and initial conditions, and emphasize the significance of nonlocal dispersal in determining the propagation dynamics of predator-prey systems.

math.AP↗

arXiv2026

Principal spectral theory and asymptotic analysis for time-periodic cooperative systems with temporally nonlocal dispersal

This paper investigates the principal spectral theory and the asymptotic behavior of the principal spectrum point for a class of time-periodic cooperative systems with nonlocal dispersal operators, incorporating both coupled and uncoupled nonlocal terms. By applying the theory of resolvent positive operators and their perturbations, we first establish criteria for the existence of the principal eigenvalue. We then construct sequences of smooth upper and lower approximating matrix-valued functions, each of whose corresponding operators satisfies the principal eigenvalue existence condition. This approximation framework allows the principal spectrum point to effectively substitute for the principal eigenvalue in characterizing the global dynamics of the nonlinear system. Moreover, it facilitates the study of the asymptotic behavior of the principal spectrum point with respect to parameters under fairly general assumptions. Subsequently, for systems with both coupled and uncoupled nonlocal terms, we analyze the asymptotic behavior of the principal spectrum point in terms of the dispersal rate, dispersal range, and frequency. Finally, we illustrate the applicability of our theoretical results through a Zika virus model and a stem cell model.

math.AP↗

arXiv2026

Dynamics of a nonlocal epidemic model with a new free boundary condition, part 1: Spreading-vanishing dichotomy

This paper investigates the long-time dynamics of a nonlocal epidemic model with free boundaries, where a pathogen with density $u(t,x)$ and the infected humans with density $v(t,x)$ evolve according to a reaction-diffusion system with nonlocal diffusion over a one dimensional interval $[g(t), h(t)]$, which represents the epidemic region expanding through its boundaries $x=g(t)$ and $x=h(t)$, known as free boundaries. Such a model with free boundary conditions based on those of Cao et al. \cite{fb27} was considered by several works. Inspired by recent works of Feng et al. \cite{fb20} and Long et al. \cite{fb5}, we propose a new free boundary condition, where the expansion rate of the epidemic region, determined by $h'(t)$ and $g'(t)$, is proportional to a linear combination of the outward flux of the pathogen $u$ through the range boundary (as in \cite{fb27}) and the weighted total population of infected individuals $v$ within the region (as in \cite{fb5}). We prove that the system under this new free boundary condition is well-posed, and its long-time dynamical behavior is characterized by a spreading-vanishing dichotomy. Moreover, we obtain sharp criteria for this dichotomy, including a sharp threshold in terms of the initial data $(u_0,v_0)$; and by studying a related eigenvalue problem, we also find a sharp threshold in terms of the diffusion rate, which complements related results in Nguyen and Vo \cite{fb7}. This is Part $1$ of a two part series. In Part $2$, we will determine the spreading speed of the model when spreading occurs, and for some typical classes of kernel functions, we will obtain the precise rates of accelerated spreading.

math.AP↗

arXiv2025

Uniqueness and stability of monostable pulsating fronts for multi-dimensional reaction-diffusion-advection systems in periodic media

In this paper, we consider the phenomenon of monostable pulsating fronts for multi-dimensional reaction-diffusion-advection systems in periodic media. Recent results have addressed the existence of pulsating fronts and the linear determinacy of spreading speed (Du, Li and Shen, \textit{J. Funct. Anal.} \textbf{282} (2022) 109415). In the present paper, we investigate the uniqueness and stability of monostable pulsating fronts with nonzero speed. We first derive precise asymptotic behaviors of these fronts as they approach the unstable limiting state. Utilizing these properties, we then prove the uniqueness modulo translation of pulsating fronts with nonzero speed. Furthermore, we show that these pulsating fronts are globally asymptotically stable for solutions of the Cauchy problem with front-like initial data. In particular, we establish the uniqueness and global stability of the critical pulsating front in such systems. These results are subsequently applied to a two-species competition system.

math.AP↗

arXiv2020

Spatial Propagation in Nonlocal Dispersal Fisher-KPP Equations

In this paper we focus on three problems about the spreading speeds of nonlocal dispersal Fisher-KPP equations. First, we study the signs of spreading speeds and find that they are determined by the asymmetry level of the nonlocal dispersal and $f'(0)$, where $f$ is the reaction function. This indicates that asymmetric dispersal can influence the spatial dynamics in three aspects: it can determine the spatial propagation directions of solutions, influence the stability of equilibrium states, and affect the monotone property of solutions. Second, we give an improved proof of the spreading speed result by constructing new lower solutions and using the new "forward-backward spreading" method. Third, we establish the relationship between spreading speed and exponentially decaying initial data. Our result demonstrates that when dispersal is symmetric, spreading speed decreases along with the increase of the exponentially decaying rate. In addition, the results on the signs of spreading speeds are applied to two special cases where we present more details of the influence of asymmetric dispersal.

math.AP↗

arXiv2020

Spatial Propagation in an Epidemic Model with Nonlocal Diffusion: the Influences of Initial Data and Dispersals

This paper studies an epidemic model with nonlocal dispersals. We focus on the influences of initial data and nonlocal dispersals on its spatial propagation. Here the initial data stand for the spatial concentrations of infectious agent and infectious human population when the epidemic breaks out and the nonlocal dispersals mean their diffusion strategies. Two types of initial data decaying to zero exponentially or faster are considered. For the first type, we show that the spreading speeds are two constants whose signs change with the number of elements in some set. Moreover, we find an interesting phenomenon: the asymmetry of nonlocal dispersals can influence the propagating directions of solutions and the stability of steady states. For the second type, we show that the spreading speed is decreasing with respect to the exponentially decaying rate of initial data, and further, its minimum value coincides with the spreading speed for the first type. In addition, we give some results about the nonexistence of traveling wave solutions and the monotone property of solutions. Finally, some applications are presented to illustrate the theoretical results.

math.AP↗

arXiv2020

Curved fronts of bistable reaction-diffusion equations in spatially periodic media

In this paper, curved fronts are constructed for spatially periodic bistable reaction-diffusion equations under the a priori assumption that there exist pulsating fronts in every direction. Some sufficient and some necessary conditions of the existence of curved fronts are given. Furthermore, the curved front is proved to be unique and stable. Finally, a curved front with varying interfaces is also constructed. Despite the effect of the spatial heterogeneity, the result shows the existence of curved fronts for spatially periodic bistable reaction-diffusion equations which is known for the homogeneous case.

math.AP↗

arXiv2020

Propagation Phenomena for Nonlocal Dispersal Equations in Exterior Domains

This paper is concerned with the spatial propagation of nonlocal dispersal equations with bistable or multistable nonlinearity in exterior domains. We obtain the existence and uniqueness of an entire solution which behaves like a planar wave front as time goes to negative infinity. In particular, some disturbances on the profile of the entire solution happen as the entire solution comes to the interior domain. But the disturbances disappear as the entire solution is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile and continue to propagate in the same direction as time goes to positive infinity for compact convex interior domain. Our work generalizes the local (Laplace) diffusion results obtained by Berestycki et al. (2009) to the nonlocal dispersal setting by using new known Liouville results and Lipschitz continuity of entire solutions due to Li et al. (2010).

math.AP↗

arXiv2019

Asymptotic Behaviors for Nonlocal Diffusion Equations about the Dispersal Spread

This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator $\frac{1}{σ^{m}}\int_ΩJ_σ(x-y)(u(y,t)-u(x,t))dy$ and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalised principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter $m\in[0,\infty)$. However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter $m$ is in a different range. In particular, for the case $m=0$, we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition.

math.AP↗

arXiv2019

The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries

We consider an epidemic model with nonlocal diffusion and free boundaries, which describes the evolution of an infectious agents with nonlocal diffusion and the infected humans without diffusion, where humans get infected by the agents, and infected humans in return contribute to the growth of the agents. The model can be viewed as a nonlocal version of the free boundary model studied by Ahn, Beak and Lin \cite{ABL2016}, with its origin tracing back to Capasso et al. \cite{CP1979, CM1981}. We prove that the problem has a unique solution defined for all $t>0$, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are also obtained, which reveal significant differences from the local diffusion model in \cite{ABL2016}. Depending on the choice of the kernel function in the nonlocal diffusion operator, it is expected that the nonlocal model here may have accelerated spreading, which would contrast sharply to the model of \cite{ABL2016}, where the spreading has finite speed whenever spreading happens \cite{ZLN2019}.

math.AP↗

arXiv2019

The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

This paper is mainly concerned with the generalised principal eigenvalue for time-periodic nonlocal dispersal operators. We first establish the equivalence between two different characterisations of the generalised principal eigenvalue. We further investigate the dependence of the generalised principal eigenvalue on the frequency, the dispersal rate and the dispersal spread. Finally, these qualitative results for time-periodic linear operators are applied to time-periodic nonlinear KPP equations with nonlocal dispersal, focusing on the effects of the frequency, the dispersal rate and the dispersal spread on the existence and stability of positive time-periodic solutions to nonlinear equations.

math.AP↗

arXiv2019

A Lotka-Volterra competition model with nonlocal diffusion and free boundaries

This paper is concerned with a nonlocal diffusion Lotka-Volterra type competition model that consisting of a native species and an invasive species in a one-dimensional habitat with free boundaries. We prove the well-posedness of the system and get a spreading-vanishing dichotomy for the invasive species. We also provide some sufficient conditions to ensure spreading success or spreading failure for the case that the invasive species is an inferior competitor or a superior competitor, respectively.

math.DS↗

arXiv2018

Pulsating Fronts for a Bistable Lotka-Volterra Competition System with Advection in a Periodic Habitat

This paper is concerned with the following Lotka-Volterra competition system with advection in a periodic habitat \begin{equation*} \begin{cases} \frac{\partial u_1}{\partial t} =d_1(x)\frac{\partial^2 u_1}{\partial x^2}-a_1(x)\frac{\partial u_1}{\partial x}+u_1\left(b_1(x)-a_{11}(x)u_1-a_{12}(x)u_2\right),\\ \frac{\partial u_2}{\partial t} =d_2(x)\frac{\partial^2 u_2}{\partial x^2}-a_2(x)\frac{\partial u_2}{\partial x}+u_2\left(b_2(x)-a_{21}(x)u_1-a_{22}(x)u_2\right), \end{cases} t>0,~x\in\Bbb R, \end{equation*} where $d_i(\cdot)$, $a_i(\cdot)$, $b_i(\cdot)$, $a_{ij}(\cdot)$ $(i,j=1,2)$ are $L$-periodic functions in $C^ν(\Bbb{R})$ with some $ν\in(0,1)$. Under certain assumptions, the system admits two periodic locally stable steady states $(u_1^*(x),0)$ and $(0,u_2^*(x))$. In this work, we first establish the existence of the pulsating front $U(x,x+ct)=(U_1(x,x+ct),U_2(x,x+ct))$ connecting two periodic solutions $(0,u_2^*(x))$ and $(u_1^*(x),0)$ at infinities. By using a dynamical method, we confirm further that the pulsating front is asymptotically stable for front-like initial values. As a consequence of the global asymptotically stability, we finally show that the pulsating front is unique up to translation.

math.AP↗

arXiv2018

Asymptotic Behavior of Traveling Fronts and Entire Solutions for a Periodic Bistable Competition-Diffusion System

This paper is concerned with a time periodic competition-diffusion system \begin{equation*} \begin{cases} {u_t}={u_{xx}}+u(r_1(t)-a_1(t)u-b_1(t)v),\quad t>0,~x\in \mathbb R, {v_t}=d{v_{xx}}+v(r_2(t)-a_2(t)u-b_2(t)v),\quad t>0,~x\in \mathbb R, \end{cases} \end{equation*} where $u(t,x)$ and $v(t,x)$ denote the densities of two competing species, $d>0$ is some constant, $r_i(t),a_i(t)$ and $b_i(t)$ are $T-$periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang [J. Differential Equations, 255 (2013), 2402-2435] that this system admits a periodic traveling front connecting two \textbf{stable} semi-trivial $T-$periodic solutions $(p(t),0)$ and $(0,q(t))$ associated to the corresponding kinetic system. Assume further that the wave speed is non-zero, we investigate the asymptotic behavior of the periodic \textbf{bistable} traveling front at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then give some key estimates. Finally, by applying super- and subsolutions technique as well as the comparison principle, we establish the existence and various qualitative properties of \emph{entire solutions} defined for all time and whole space.

math.AP↗

arXiv2017

Spatial Dynamics of a Nonlocal Dispersal Population Model in a Shifting Environment

This paper is concerned with spatial spreading dynamics of a nonlocal dispersal population model in a shifting environment where the favorable region is shrinking. It is shown that the species will become extinct in the habitat once the speed of the shifting habitat edge $c>c^*(\infty)$, however if $c<c^*(\infty)$, the species will persist and spread along the shifting habitat at an asymptotic spreading speed $c^*(\infty)$, where $c^*(\infty)$ is determined by the nonlocal dispersal kernel, diffusion rate and the maximum linearized growth rate. Moreover, we demonstrate that for any given speed of the shifting habitat edge, this model admits a nondecreasing traveling wave with the wave speed at which the habitat is shifting, which indicates that the extinction wave phenomenon does happen in such a shifting environment.

math.AP↗

arXiv2017

Entire Solution in an Ignition Nonlocal Dispersal Equation: Asymmetric Kernel

This paper mainly focus on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function $J$ may not be symmetric here. The asymmetry of $J$ has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of $f'(0)=0$, we can no longer use the common method which mainly depending on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, thus we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds.

math.AP↗