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This paper is concerned with the propagation phenomenon of a three species predation-competition system with nonlocal dispersal and climate change effects. That is to say, there is not only competition for food, but also competitive interaction between two preys. The growth rate of each prey is nondecreasing along the $ x $-axis and shifts rightward at a speed $ s $. We mainly consider the population dynamics for three cases: (ⅰ) the predator spreads faster than the two preys, (ⅱ) the predator spreads between two preys and (ⅲ) the predator with slower speed spreads behind the two preys, including multiple layers with different speeds to achieve a complete picture. Due to lack of comparison principle in the prey-predator system and compactness of the nonlocal operator, we give some priori estimates of the solutions to obtain the persistence for three species, which depends on improving the regularity of the dispersal kernels and proposing some parameter conditions of the considered system. [ABSTRACT FROM AUTHOR]

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]

In this paper, we study the spreading properties of a nonlocal dispersal reducible cooperation model under a shifting habitat. By considering the corresponding upper and lower control systems, we find that the left and right expansion fronts of the solutions with exponentially unbounded kernels propagate with finite speed and infinite speed, respectively. This is quite different from the case of exponentially bounded kernels (Wang and Li, 2020), in which the expansion fronts of the solutions always spread to the left and right with finite speeds. That is to say, we have found a new phenomenon that linear extension and accelerating propagation appear at the same time for such a system. [ABSTRACT FROM AUTHOR]

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 c > c*, in which c* 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 0 < c < c* 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. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on the entire solutions of nonlocal dispersal equations with bistable nonlinearity. Under certain assumptions of wave speed, firstly constructing appropriate super- and sub-solutions and applying corresponding comparison principle, we established the existence and related properties of entire solutions formed by the collision of three and four traveling wave solutions. Then by introducing the definition of terminated sequence, it is proved that there has no entire solutions formed by k traveling wave solutions that collide with each other as long as k ≥ 5. Finally, based on the classical weighted energy approach, we obtain the global exponentially stability of the entire solutions in some weighted space. [ABSTRACT FROM AUTHOR]

This paper is concerned with the travelling waves for a class of non-local dispersal non-cooperative system, which can model the prey-predator and disease-transmission mechanism. By the Schauder's fixed-point theorem, we first establish the existence of travelling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations, whose bounds are deduced from a precise analysis. Further, we characterize the minimal wave speed of travelling waves and obtain the non-existence of travelling waves with slow speed. Finally, we apply the general results to an epidemic model with bilinear incidence for its propagation dynamics. [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 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, r(x) for intrinsic growth rate and K(x) 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. [ABSTRACT FROM AUTHOR]

We investigate the existence and qualitative properties of conical traveling fronts for the bistable nonlocal equations. Based on the existence result of pyramidal traveling fronts (Li et al., 2021), we construct a sequence of pyramidal traveling fronts and take its limit. However, it is not easy to show that the limit function is a real conical traveling front, and the proofs of its qualitative properties are quite different from those of classical equations, although some of the ideas used are similar. [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 entire solutions of the monostable equation with nonlocal dispersal, i.e., $u_{t}=J*u-u+f(u)$. Here the kernel $J$ is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds $c^{*}$ and $\hat{c}^{*}$, where $c^*$ and $\hat{c}^{*}$ are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel $J$ is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively. [ABSTRACT FROM AUTHOR]

This paper is concerned with the traveling waves for a three-species competitive system with nonlocal dispersal. It has been shown by Dong, Li and Wang (DCDS 37 (2017) 6291-6318) that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first investigate the asymptotic behavior of traveling waves at negative infinity by a modified version of Ikehara's Theorem. Then we prove the uniqueness of traveling waves by applying the stronger comparison principle and the sliding method. Finally, we establish the exponential stability of traveling waves with large speeds by the weighted-energy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as x → -∞, but can be arbitrarily large in other locations. [ABSTRACT FROM AUTHOR]

This paper is concerned with entire solutions for a two-dimensional periodic lattice dynamical system with nonlocal dispersal. In the bistable case, by applying comparison principle and constructing appropriate upper- and lowersolutions, two different types of entire solutions are constructed. The first type behaves like a monostable front merges with a bistable front and one chases another from the same side; while the other type can be represented by two monostable fronts merge and converge to a single bistable front. In the monostable case, we first establish the existence and properties of spatially periodic solutions which connect two steady states. Then new types of entire solutions are constructed by mixing a heteroclinic orbit of the spatially averaged ordinary differential equations with traveling wave fronts with different speeds. Further, for a class of special heterogeneous reaction function, we establish the uniqueness and continuous dependence of the entire solution on parameters, such as wave speeds and shifted variables. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is concerned with traveling wave solutions of a nonlocal dispersal Susceptible–Infective–Removal–Healing (for short SIRH ) model with relapse. It is found that the existence and nonexistence of traveling waves of the system are not only determined by the critical wave speed c ∗ , but also by the basic reproduction number R 0 of the corresponding system of ordinary differential equations. More precisely, we use Schauder’s fixed-point theorem to obtain the existence of traveling waves for R 0 > 1 and c > c ∗ , and the nonexistence of traveling waves for R 0 > 1 and 0 < c < c ∗ . Some numerical simulations and discussions are also provided to illustrate our analytical results. [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]

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. [ABSTRACT FROM AUTHOR]

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. [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 entire solutions of the Fisher-KPP equation with nonlocal dispersal, i.e., , which is a one-dimensional nonlocal version of the Fisher-KPP equation describing the spatial spread of a mutant in a given population and the dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modeled by a convolution operator. Here the entire solutions are defined in the whole space and for all time . A comparison principle is employed to establish the existence of entire solutions by combining two traveling wave solutions with different speeds and coming from both ends of the real axis and some spatially independent solutions. The main difficulty is that a lack of regularizing effect occurs. This is probably the first time the existence of entire solutions of reaction equations with nonlocal dispersal has been studied. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter in equation. For , the solution decays to zero as ; while for , the solution converges to the unique positive stationary solution as . In addition, we show that the solution blows up under some conditions. [Copyright &y& Elsevier]

Recently, the spreading speeds and periodic traveling wave solutions for a general class of partially degenerate nonlocal dispersal cooperative systems in time and space periodic habitats have been investigated. In this paper, we continue to study the stability and convergence rate for such time and space periodic traveling waves. We show that if the initial function perturbation is uniformly bounded with respect to a weighted maximum norm, the periodic traveling wave solution is globally exponentially stable when the wave speed is greater than the spreading speed. [ABSTRACT FROM AUTHOR]

This paper is concerned with propagation phenomena of a general class of partially degenerate nonlocal dispersal cooperative systems in time and space periodic habitats. We first show that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions. Next, we prove that if the wave speed is greater than the spreading speed, there exists a time and space periodic traveling wave solution connecting the stable positive time and space periodic steady state and 0. Finally, we apply these results to two species partially degenerate competition systems and a partially degenerate epidemic model with nonlocal dispersal in time and space periodic habitats. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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]