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This paper is concerned with the 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 spreading speeds such as asymptotic spreading ray speeds and asymptotic spreading sets. Furthermore, we consider the dependence of the spreading speed on the dispersal rate and reaction term and prove that taking the temporal average or spatial average can decrease the spreading speed. Finally, we employ the viscosity vanishing method to establish the existence of time-space periodic traveling fronts with the critical speed in every direction under the partially temporally homogeneous case and partially nearly flat case, which solves partially the open problem raised by Rawal, Shen, and Zhang (Discrete Contin. Dyn. Syst. 35 (2015) 1609–1640). [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlocal (convolution) dispersal susceptible-infected-susceptible (SIS) epidemic model with bilinear incidence and Neumann boundary conditions. First we establish the existence and uniqueness of stationary solutions by reducing the system to a single equation. Then we study the asymptotic profiles of the endemic steady states for large and small diffusion rates to illustrate the persistence or extinction of the infectious disease. The lack of regularity of the endemic steady state makes it more difficult to obtain the limit function of the sequence of endemic steady states. We also observe the concentration phenomenon which occurs when the diffusion rate of the infected individuals tends to zero. Our analytical results demonstrate that limiting the movement of susceptible individuals is not effective in eliminating the infectious disease unless the total population size is relatively small. [ABSTRACT FROM AUTHOR]

We focus on the influences of nonlocal dispersal kernels on the spatial propagation in nonlocal dispersal cooperative systems with initial values having nonempty compact supports. It is well-known that the solution of a monostable nonlocal dispersal scalar equation spreads at a finite speed when the kernel is thin-tailed and propagates by accelerating when the kernel has a heavy tail (the fat-tailed). However, in such systems, we find that one species can propagate by accelerating although its dispersal kernel is thin-tailed, which is a new and interesting phenomenon. More precisely, we show that the spatial propagation of every species at large time is mainly determined by the tail of the maximum of their dispersal kernels, which further implies that their spatial propagation is accelerated by each other because of the cooperation relation between them. This gives us a new understanding of the cooperation relation in the spatial propagation of nonlocal dispersal cooperative systems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is concerned with the new types of entire solutions other than traveling wave solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats. We first establish the existence and properties of spatially periodic solutions connecting two steady states. Then new types of entire solutions are constructed by combining the rightward and leftward pulsating traveling fronts with different speeds and a spatially periodic solution. Finally, for a class of special heterogeneous reaction, we further establish the uniqueness of entire solutions and the continuous dependence of such an entire solution on parameters, such as wave speeds and the shifted variables. In other words, we build a five-dimensional manifold of solutions and the traveling wave solutions are on the boundary of the manifold. [ABSTRACT FROM AUTHOR]

This paper is concerned with space periodic traveling wave solutions of the following Lotka–Volterra competition system with nonlocal dispersal and space periodic dependence, { ∂ u 1 ∂ t = ∫ R N κ ( y − x ) u 1 ( t , y ) d y − u 1 ( t , x ) + u 1 ( a 1 ( x ) − b 1 ( x ) u 1 − c 1 ( x ) u 1 ) , x ∈ R N ∂ u 2 ∂ t = ∫ R N κ ( y − x ) u 2 ( t , y ) d y − u 2 ( t , x ) + u 2 ( a 2 ( x ) − b 2 ( x ) u 1 − c 2 ( x ) u 2 ) , x ∈ R N . Under suitable assumptions, the system admits two semitrivial space periodic equilibria ( u 1 ⁎ ( x ) , 0 ) and ( 0 , u 2 ⁎ ( x ) ) , where ( u 1 ⁎ ( x ) , 0 ) is linearly and globally stable and ( 0 , u 2 ⁎ ( x ) ) is linearly unstable with respect to space periodic perturbations. By sub- and supersolution techniques and comparison principals, we show that, for any given ξ ∈ S N − 1 , there exists a continuous periodic traveling wave solution of the form ( u 1 ( t , x ) , u 2 ( t , x ) ) = ( Φ 1 ( x − c t ξ , c t ξ ) , Φ 2 ( x − c t ξ , c t ξ ) ) connecting ( u 1 ⁎ ( ⋅ ) , 0 ) and ( 0 , u 2 ⁎ ( ⋅ ) ) and propagating in the direction of ξ with speed c > c ⁎ ( ξ ) , where c ⁎ ( ξ ) is the spreading speed of the system in the direction of ξ . Moreover, for c < c ⁎ ( ξ ) there is no such solution. When the wave speed c > c ⁎ ( ξ ) , we also prove the asymptotic stability and uniqueness of traveling wave solution using squeezing techniques. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the spreading speeds and traveling wave solutions of a nonlocal dispersal equation with degenerate monostable nonlinearity. We first prove that the traveling wave solution with critical minimal speed decays exponentially as , while other traveling wave solutions with do not decay exponentially as . Then the monotonicity and uniqueness (up to translation) of traveling wave solution with critical minimal speed is established. Finally, we prove that the critical minimal wave speed coincides with the asymptotic speed of spread. [Copyright &y& Elsevier]

Abstract: We consider entire solutions of nonlocal dispersal equations with bistable nonlinearity in one-dimensional spatial domain. A two-dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions is established by an increasing traveling wave front with nonzero wave speed. Furthermore, we show that such an entire solution is unique up to space–time translations and Liapunov stable. A key idea is to characterize the asymptotic behaviors of the solutions as in terms of appropriate subsolutions and supersolutions. We have to emphasize that a lack of regularizing effect occurs. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]