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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., Comment: 24 pages

This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator $\frac{1}{\sigma^{m}}\int_{\Omega}J_{\sigma}(x-y)(u(y,t)-u(x,t))dy$ 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., Comment: 24 pages

This paper is devoted to investigating the traveling wave solutions of a nonlocal dispersal SEIR epidemic model with standard incidence. We find that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the critical wave speed. Through considering a truncate problem, combining with Schauder’s fixed-point theorem and applying a limiting argument, we prove the existence of traveling wave solutions. Meanwhile, the nonexistence of traveling wave solutions is showed by the Laplace transform method. Furthermore, the existence of traveling wave solutions with critical wave speed is also established by a delicate analysis. We also point out that both the nonlocal dispersal and coupling of system in the model bring some difficulties in the study of traveling wave solutions.

In this paper we study a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Neumann boundary condition, where the spatial movement of individuals is described by a nonlocal (convolution) diffusion operator, the transmission rate and recovery rate are spatially heterogeneous, and the total population number is constant. We first define the basic reproduction number R 0 and discuss the existence, uniqueness and stability of steady states of the nonlocal dispersal SIS epidemic model in terms of R 0 . Then we consider the impacts of the large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease. The obtained results indicate that the nonlocal movement of the susceptible or infectious individuals will enhance the persistence of the infectious disease. In particular, our analytical results suggest that the spatial heterogeneity tends to boost the spread of the infectious disease.

This paper is concerned with the existence of traveling waves for a nonlocal dispersal Kermack–McKendrick epidemic model. As we know, due to the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity, it is difficult to investigate traveling waves with the critical wave speed. Furthermore, the nonlocal dispersal and bilinear incidence bring additional difficulties to get the boundedness of traveling waves. In the present paper, we overcome these difficulties to obtain the boundedness of traveling waves by analysis technique firstly and then prove the existence of traveling waves under the critical wave speed.

This paper is concerned with the spreading or vanishing of a epidemic disease which is characterized by a diffusion SIS model with nonlocal incidence rate and double free boundaries. We get the full information about the sufficient conditions that ensure the disease spreading or vanishing, which exhibits a detailed description of the communicable mechanism of the disease. Our results imply that the nonlocal interaction may enhance the spread of the disease.

In this paper, we investigate the effects of nonlocal dispersal and spatial heterogeneity on the total biomass of species via nonlocal dispersal logistic equations. In order to make the model more relevant for real biological systems, we consider a logistic reaction term, with two parameters, \begin{document}$ r(x) $\end{document} for intrinsic growth rate and \begin{document}$ K(x) $\end{document} for carrying capacity. We first establish the existence, uniqueness and asymptotic stability of the positive steady state solution for this equation. And then we study the continuous property and asymptotic limit of the positive steady state solution with respect to the dispersal rate. Finally, the function about the total biomass of species is defined by the positive steady state solution. Our results show in a heterogeneous environment, the total biomass is always strictly greater than the total carrying capacity in the special case when the nonlocal dispersal is allowed.

It is well-known that the nonlocal dispersal operator has the advantage of capturing short-range as well as long-range factors for the dispersal of the spices by choosing the kernel function properly, and is also capable to include spatial dispersal strategies of the species beyond random (local) diffusion. This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with non-monotone incidence, which is a non-monotone system. The method of sub and super solutions combined with Schauder’s fixed-point theorem is applied to establish the existence of positive traveling waves as the wave speed is over critical speed. We further prove the existence of traveling waves with critical speed and the nonexistence of bounded positive traveling waves by the delicate analysis method. The main difficulty is to get the boundedness of traveling waves caused by the nonlocal dispersal operator.

This paper is concerned with the traveling wave solutions for a class of predator-prey model with nonlocal dispersal. By adopting the truncation method, we use Schauder's fixed-point theorem to obtain the existence of traveling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations for \begin{document}$ c>c_{*} $\end{document}, in which \begin{document}$ c_* $\end{document} is the minimal wave speed. Meanwhile, through the limiting approach and the delicate analysis, we establish the existence of traveling wave solutions with the critical speed. Finally, we show the nonexistence of traveling waves for \begin{document}$ 0 by the characteristic equation corresponding to the linearization of original system at the semi-trivial equilibrium. Throughout the whole paper, we need to overcome the difficulties brought by the nonlocal dispersal and the non-preserving of system itself.

This paper is concerned with the principal eigenvalues of some nonlocal operators. We first derive a result on the limit of certain sequences of principal eigenvalues associated with some nonlocal eigenvalue problems. Then, such a result is used to study the existence, uniqueness and asymptotic behavior of positive solutions to a nonlocal stationary problem with a parameter. Finally, the long-time behavior of the solutions of the corresponding nonlocal evolution equation and the asymptotic behavior of positive stationary solutions on parameter are discussed.

This paper is concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIR model with nonlocal delayed transmission. The threshold dynamics are determined by the so-called basic reproduction number of the corresponding reaction system and the critical wave speed. Our results imply that (i) the diffusion ability of the infected individuals can accelerate the wave speed; and (ii) the incubation period can slow down the propagation while the non-locality of interaction would speed up the spread of the disease. In particular, we remove the usual condition that the nonlocal dispersal kernel function is compacted supported.

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. This threshold dynamics are proved by Schauder’s fixed point theorem and the Laplace transform. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.

In this paper, we study the blow-up profiles of the nonlocal dispersal equation. More precisely, we prove that the positive solution of nonlocal dispersal equation has different blow-up profiles, depending on the refuge domain.

This paper is concerned with the spreading (persistence) and vanishing (extinction) of a disease which is characterized by a diffusive SIRS model with a bilinear incidence rate and free boundary. Through discussing the dynamics of a free boundary problem of an SIRS model, the spreading of a disease is described. We get the sufficient conditions which ensure the disease spreading or vanishing. In addition, the estimate of the expanding speed is also given when the free boundaries extend to the whole $$\mathbb {R}$$ .

This paper is concerned with the existence, nonexistence and minimal wave speed of traveling waves of a nonlocal dispersal delayed SIR model with constant external supplies and Holling-II incidence rate. We find that the existence and nonexistence of traveling waves of the system are not only determined by the minimal wave speed c ? , but also by the so-called basic reproduction number R 0 of the corresponding reaction system. That is, we establish the existence of traveling waves for R 0 1 and each wave speed c ? c ? , and the nonexistence for R 0 1 and any 0 < c < c ? or R 0 < 1 . We also discuss how the latency of infection and the spatial movement of the infective individuals affect the minimal wave speed. Biologically speaking, the longer the latency of infection in a vector is, the slower the disease spreads.

This paper is concerned with the existence, uniqueness and asymptotic stability of positive steady-states for a nonlocal dispersal equation arising from selection–migration models in genetics. Due to the lack of compactness and regularity of the nonlocal operators, many classical methods cannot be used directly to the nonlocal dispersal problems. This motivates us to find new techniques. We first establish a criterion on the stability and instability of steady-states. This result is effective to get a necessary condition to guarantee a positive steady-state, it also gives the uniqueness. Then we prove the existence of nontrivial solutions by the corresponding auxiliary equations and maximum principle. Finally, we consider the dynamic behavior of the initial value problem.

This paper is concerned with the traveling wave solutions of a diffusive SIR system with nonlocal delay. We obtain the existence and nonexistence of traveling wave solutions, which formulate the propagation of disease without outbreak threshold. Moreover, it is proved that at any fixed moment, the faster the disease spreads, the more the infected individuals, and the larger the recovery/remove ratio is, the less the infected individuals.

In this paper, we consider a Kermack-McKendrick epidemic model with nonlocal dispersal. We find that the existence and nonexistence of traveling wave solutions are determined by the reproduction number. To prove the existence of nontrivial traveling wave solutions, we construct an invariant cone in a bounded domain with initial functions being defined on, and apply Schauder's fixed point theorem as well as limiting argument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, the nonexistence of traveling wave solutions is obtained by Laplace transform if the speed is less than the critical velocity.

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 that the nonlocal dispersal of the infected individuals may suppress the spread of the disease even though in a high-risk domain., Comment: 21 pages

This paper is concerned with an inhomogeneous nonlocal dispersal equation. We study the limit of the re-scaled problem of this nonlocal operator and prove that the solutions of the re-scaled equation converge to a solution of the Fokker–Planck equation uniformly. We then analyze the nonlocal dispersal equation of an inhomogeneous diffusion kernel and find that the heterogeneity in the classical diffusion term coincides with the inhomogeneous kernel when the scaling parameter goes to zero.

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model with standard incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding ordinary differential model and the minimal wave speed. These threshold dynamics are proved by constructing an invariant cone and applying Schauder's fixed point theorem on this cone and the Laplace transform. The main difficulties are the lack of an occurrence of a regularizing effect and the loss of the order-preserving property of this model.