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).
The current paper {establishes} criteria for the existence of principal eigenvalues of time periodic cooperative linear nonlocal dispersal systems with Direchlet type, Neumann type or periodic type boundary conditions. It is shown that such a nonlocal dispersal system has a principal eigenvalue in the following cases: the nonlocal dispersal distance is sufficiently small; the spatial inhomogeneity satisfies a so called vanishing condition; or the spatial inhomogeneity is nearly globally homogeneous. Moreover, it is shown that the principal eigenvalue of a time periodic cooperative linear nonlo…
In this paper, the global dynamics of two-species Lotka-Volterra competition models with nonlocal dispersals is studied. Under the assumption that dispersal kernels are symmetric, we prove that except for very special situations, local stability of semi-trivial steady states implies global stability, while when both semi-trivial steady states are locally unstable, the positive steady state exists and is globally stable. Moreover, our results cover the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.
In this paper, we prove that the solution maps of a large class of nonlocal dispersal equations are $α$-contractions, where $α$ is the Kuratowski measure of noncompactness. Then we give some remarks on the spreading speeds and traveling waves for such evolution equations in a periodic habitat.
This series of two papers is devoted to the study of the principal spectral theory of nonlocal dispersal operators with almost periodic dependence and the study of the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence. In this first part of the series, we investigate the principal spectral theory of nonlocal dispersal operators from two aspects: top Lyapunov exponents and generalized principal eigenvalues. Among others, we provide various characterizations of the top Lyapunov exponents and generalized principal eigenvalues, establish the relations be…
In this paper, we study the global dynamics of a general $2\times 2$ competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper continues the work in \cite{BaiLi2017}, where competition models with symmetric nonlocal operators are considered.
In this paper, we gives a complete classification of the global dynamics of two- species Lotka-Volterra competition models with nonlocal dispersals: where K, P represent nonlocal operators, under the assumptions that the nonlo- cal operators are symmetric, the models admit two semi-trivial steady states and 0<bc<1. In particular, when both semi-trivial steady states are locally stable, it is proved that there exist infinitely many steady states and the solution with non- negative and nontrivial initial data converges to some steady state. Furthermore, we generalize these results to the case th…
In this paper, we consider the following single species model with nonlocal dispersal strategy $$ d\mathcal {L} [θ] (x,t) + θ(x,t) [m(x)- θ(x,t)]=0 , $$ where $\mathcal {L}$ denotes the nonlocal diffusion operator, and investigate how the dispersal rate of the species and the distribution of resources affect the total population. First, we show that the upper bound for the ratio between total population and total resource is $C\sqrt{d}$. Moreover, examples are constructed to indicate that this upper bound is optimal. Secondly, for a type of simplified nonlocal diffusion operator, we prove that…
This paper is devoted to a nonlocal dispersal logistic model with seasonal succession in one-dimensional bounded habitat, where the seasonal succession accounts for the effect of two different seasons. Firstly, we provide the persistence-extinction criterion for the species, which is different from that for local diffusion model. Then we show the asymptotic profile of the time-periodic positive solution as the species persists in long run.
It is known that solutions of nonlocal dispersal evolution equations do not become smoother in space as time elapses. This lack of space regularity would cause a lot of difficulties in studying transition fronts in nonlocal equations. In the present paper, we establish some general criteria concerning space regularity of transition fronts in nonlocal dispersal evolution equations with a large class of nonlinearities, which allows the applicability of various techniques for reaction-diffusion equations to nonlocal equation, and hence serves as an initial and fundamental step for further studyin…
This paper is concerned with a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Dirichlet boundary condition, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous. We introduce a basic reproduction number $R_0$ and establish threshold-type results on the global dynamic in terms of $R_0$. More specifically, we show that if the basic reproduction number is less than one, then the disease will be extinct, and if the basic reproduction number is larger than one, then the disease will persist. Particularly, our results imply tha…
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…
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 inter…
In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =u \left(λ- \int_ΩQ(x,y) |u(y)|^p dy \right) , \ \mbox{in} \ Ω, \] where $Ω\subset \mathbb{R}^{N}$, $N\geq 1$, is a bounded connected open, $p>0$, $λ$ is a real parameter, $Q:Ω\times Ω\to \mathbb{R}$ is a nonnegative function, and $L_0 : C(\overlineΩ) \to (\overlineΩ)$ is a nonlocal dispersal operator. The existence of solution is obtained via bifurcation theory.
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…
This article is concerned with a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Neumann boundary condition, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous and the total population number is constant. We first introduce the basic reproduction number $R_0$ and then discuss the existence, uniqueness and stability of steady states of the nonlocal dispersal SIS epidemic model in terms of $R_0$. In particular, we also consider the impacts of the large diffusion rates of the susceptible and infected population on the persi…
Seasonality frequently occurs in population models, and the corresponding seasonal patterns have been of great interest to scientists. This paper is concerned with traveling waves to a time-periodic bistable Lotka-Volterra competition system with nonlocal dispersal. We first establish the existence, uniqueness and stability of traveling wave solutions for this system. Then, by utilizing comparison principle and the stability property, the relationship among the bistable wave speed, the asymptotic propagation speeds of the associated monotone subsystems and the speed of upper/lower solutions is…
Variational characterizations of weighted eigenvalue and basic reproduction ratio for nonlocal dispersal systems and application
School of Mathematics(Zhuhai), Sun Yat-sen University（中山大学数学学院（珠海））；School of Mathematics and Systems Science, Guangdong Polytechnic Normal University（广东技术师范大学数学与系统科学学院）
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 as…
Propagation dynamics for nonlocal dispersal predator-prey systems in shifting habitats: A Hamilton-Jacobi approach
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；Department of Mathematics, University of Miami（迈阿密大学数学系）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal SIS (susceptible-infected-susceptible) reaction-diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, sufficient conditions for spreading and vanishing of the disease are derived.Our results s…
Asymptotic limits of the principal spectrum point of a nonlocal dispersal cooperative system and application to a two-stage structured population model
We are concerned with the persistence of both predator and prey in a diffusive predator-prey system with a climate change effect, which is modeled by a spatial-temporal heterogeneity depending on a moving variable. Moreover, we consider both the cases of nonlocal and local dispersal. In both these situations, we first prove the existence of forced waves, which are positive stationary solutions in the moving frames of the climate change, of either front or pulse type. Then we address the persistence or extinction of the prey and the predator separately in various moving frames, and achieve a co…
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 …
We study the convenience of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances - namely, a precise condition on the distribution of the resource - under which a nonlocal dispersal behavior is favored. In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete example of resources for which the equilibrium with only the local populat…
We consider a three-species predator-prey system involving two competing predators and one prey. The species diffuse with nonlocal dispersal kernels with possibly non-compact support, and they interact in a heterogeneous environment moving with a positive forced speed such that the environment is favorable to the prey in the absence of predators far ahead of the shifting boundary and it is unfavorable far behind. Such systems arise in the modeling of population dynamics under the effect of a shifting environment, such as climate change. We show on the one hand the existence of waves connecting…
This paper is concerned with a Lotka-Volterra type competition model with free boundaries in time-periodic environment. One species is assumed to adopt nonlocal dispersal and the other one adopts mixed dispersal, which is a combination of both random dispersal and nonlocal dispersal. We show that this free boundary problem with more general growth functions admits a unique solution defined for all time. A spreading-vanishing dichotomy is obtained and criteria for spreading and vanishing are provided. Moreover, under the weak competition condition we provide the long-time asymptotic behavior of…
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 dispe…
In this paper, we study the asymptotic behavior as $\varepsilon\to0^+$ of solutions $u\_\varepsilon$ to the nonlocal stationary Fisher-KPP type equation$$ \frac{1}{\varepsilon^m}\int\_{\mathbb{R}^N}J\_\varepsilon(x-y)(u\_\varepsilon(y)-u\_\varepsilon(x))\mathrm{d}y+u\_\varepsilon(x)(a(x)-u\_\varepsilon(x))=0\text{ in }\mathbb{R}^N, $$where $\varepsilon>0$ and $0\leq m<2$. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution $u\_\varepsilon$ and that $u\_\varepsilon\to a^+$ as $\varepsilon\to0^+$ where $a^+=\max\{0,a\}$. T…
This series of two papers is devoted to the study of the principal spectral theory of nonlocal dispersal operators with almost periodic dependence and the study of the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence. In the first part of the series, we investigated the principal spectral theory of nonlocal dispersal operators from two aspects: top Lyapunov exponents and generalized principal eigenvalues. Among others, we provided various characterizations of the top Lyapunov exponents and generalized principal eigenvalues, established the relations…
In this communication, we prove some important limits of the principal eigenvalue for nonlocal operator of Neumann type with respect to the parameters, which are significant in the understanding of dynamics of biological populations. We obtained a complete picture about limits of the principal eigenvalue in term of the large and small dispersal rate and dispersal range classified by "Ecological Stable Strategy" of persistence. This solves some open problems remainning in the series of work [3, 28, 29], in which we have to overcome the new difficulties comparing to [3, 28, 29] since principal e…
In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for $d\ge1$, there exist $C_0, C_1>0$, depending on the $\|m\|_{L^1}$ only, such that $$C_0\sqrt{d}<\mbox{supremum~ of~ total~ population}<C_1\sqrt{d}.$$ However, when replaced by random diffusion, a conjecture, proposed by Ni and justified in [3], indicates that in the one-dimensional case, supremum of total population…
We study the travelling wave problem J\astu - u - cu' + f (u) = 0 in R, u(-\infty) = 0, u(+\infty) = 1 with an asymmetric kernel J and a monostable nonlinearity. We prove the existence of a minimal speed, and under certain hypothesis the uniqueness of the profile for c = 0. For c = 0 we show examples of nonuniqueness.
The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore--Gibson--Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition.
In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =f(x,u)+g(x) , \ \mbox{in} \ Ω, \] where $Ω\subset \mathbb{R}^{N}$, $N\geq 1$, is a bounded connected open, $g \in C(\overlineΩ)$, $f:\overlineΩ \times \mathbb{R} \to \mathbb{R}$ are function, and $L_0 : C(\overlineΩ) \to C(\overlineΩ)$ is a nonlocal dispersal operator. Using a sub-supersolution method and the degree theory for $γ$-Condensing maps, we have obtained a result of the Ambrosetti-Prodi type, that is, we obtain a necessary condition on $g$ for the non-existence of solutions, the exi…
This paper is devoted to the investigation of the following nonlocal dispersal equation $$ u_{t}(t,x)=\frac{D}{σ^m}\left[\int_ΩJ_σ(x-y)u(t,y)dy-u(t,x)\right]+f(t,x,u(t,x)), \quad t>0,\quad x\in\overlineΩ, $$ where $Ω\subset\mathbb{R}^{N}$ is a bounded and connected domain with smooth boundary, $m\in[0,2)$, $D>0$ is the dispersal rate, $σ>0$ characterizes the dispersal range, $J_σ=\frac{1}{σ^{N}} J\left(\frac{\cdot}σ\right)$ is the scaled dispersal kernel, and $f$ is a time-periodic nonlinear function of generalized KPP type. We first study the principal spectral theory of the linear operator a…
We consider parabolic equations on bounded smooth open sets $\Om\subset \R^N$ ($N\ge 1$) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator $\mathscr{L} \coloneqq - Δ+ (-Δ)^{s}$ ($0<s<1$). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been prese…
We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval $(0,1)$ associated with the fractional Laplace operator $(-\partial_x^2)^s$, where $0<s<1$. We show that there is a control function which is localized in a non-empty open set $\mathcal{O}\subset \left(\mathbb{R}\setminus(0,1)\right)$, that is, at the exterior of the interval $(0,1)$, such that the system is null controllable at any time $T>0$ if and only if $\frac 12<s<1$.
In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $Ω$, $u=g$ in $\mathbb R^N\setminusΩ$, in non-smooth domains $Ω$. When $g$ is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right hand side, for which the boundary regularity is well understood. Here, we study the case in which $g\in C^{0,α}$, and establish the optimal Hölder regularity of $u$ up to the boundary. Our results extend previous results of Grubb for $C^\infty$ domains $Ω$.
We propose here a motivation for a mixed local/nonlocal problem with a new type of Neumann condition. Our description is based on formal expansions and approximations. In a nutshell, a biological species is supposed to diffuse either by a random walk or by a jump process, according to prescribed probabilities. If the process makes an individual exit the niche, it must come to the niche right away, by selecting the return point according to the underlying stochastic process. More precisely, if the random particle exits the domain, it is forced to immediately reenter the domain, and the new poin…
We develop a class of nonlocal delay Reaction-Diffusion (RD) models in a circular domain. Previous modeling efforts include RD population models with respect to one-dimensional unbounded domain, unbounded strip and rectangular spatial domain. However, the importance of an RD model in a symmetrical domain lies in the increasing number of empirical studies conducted with respect to symmetrical natural habitats of single species. Assuming that the single species has no directional preference to spread in the symmetrical domain, the RD model is reduced to an equation with no angular dependance. Th…
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
We study existence and multiplicity of positive radial solutions for a coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the boundary conditions are nonlocal. We use a non-standard cone to establish existence of solutions by means of fixed point index theory.
In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus Ω\\ \mathcal{N}_su(x) &= 0\;\;\mbox{in}\;\;Ω, \end{aligned} \right. \end{equation} where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is a bounded set with smooth boundary, $(-Δ)^s$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_su(x) = C_{N,s} \int_{\R^N \setminus Ω…
In this paper, we investigate the asymptotic dynamics of Fisher-KPP equations with nonlocal dispersal operator and nonlocal reaction term in time periodic and space heterogeneous media. We first show the global existence and boundedness of nonnegative solutions, and then obtain some sufficient conditions ensuring the uniform persistence. In particular, we study the existence, uniqueness and global stability of positive time periodic solutions under several different conditions.
A nonlocal-to-local transfer principle for parabolic Calderón problems
National Cheng Kung University（国立成功大学）；National Yang Ming Chiao Tung University（国立阳明交通大学）；University of Washington（华盛顿大学）；Institute for Advanced Study, the Hong Kong University of Science and Technology（香港科技大学高等研究院）
We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. In the first part, we show that if $0<s<1$, $Ω\subset\RR^N$ ($N\ge 1$) is a bounded Lipschitz domain and the parameter $δ> 0$, then there is no control function $g$ such that the following system \begin{equation*} \begin{cases} u_{tt} + (-Δ)^{s} u + δ(-Δ)^{s} u_{t}=0 & \mbox{ in }\; Ω\times(0,T),\\ u=gχ_{\mathcal O\times (0,T)} &\mbox{ in }\; (\Omc)\times (0,T) ,\\ u(\…
We study the long time behavior of solutions to the nonlocal diffusion equation $\partial_t u=J*u-u$ in an exterior one-dimensional domain, with zero Dirichlet data on the complement. In the far field scale, $ξ_1\le|x|t^{-1/2}\leξ_2$, $ξ_1,ξ_2>0$, this behavior is given by a multiple of the dipole solution for the local heat equation with a diffusivity determined by $J$. However, the proportionality constant is not the same on $\mathbb{R}_+$ and $\mathbb{R}_-$: it is given by the asymptotic first momentum of the solution on the corresponding half line, which can be computed in terms of the ini…
Principal spectral theory and variational characterizations for nonlocal coupled cooperative systems and applications
School of Mathematics, Sun Yat-sen University（中山大学数学学院）；School of Mathematics and Systems Science, Guangdong Polytechnic Normal University（广东技术师范大学数学与系统科学学院）
This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property…
We are interested in the threshold phenomena for propagation in nonlocal diffusion equations with some compactly supported initial data. In the so-called bistable and ignition cases, we provide the first quantitative estimates for such phenomena. The outcomes dramatically depend on the tails of the dispersal kernel and can take a large variety of different forms. The strategy is to combine sharp estimates of the tails of the sum of i.i.d. random variables (coming, in particular, from large deviation theory) and the construction of accurate sub-and super-solutions.
The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
