Search

This paper is concerned with the propagation phenomena for a nonlocal dispersal Lotka–Volterra competition model with shifting habitats. It is assumed that the growth rate of each species is nondecreasing along the x-axis, positive near ∞ and nonpositive near - ∞ , and shifting rightward with a speed c > 0 . In the case where both species coexist near ∞ , we established three types of forced waves connecting the origin, respectively to the coexistence state with any forced speed c; to itself with forced speed c > c ∗ (∞) ; and to a semi-trivial steady state with forced speed c > c ¯ (∞) , where c ∗ (∞) and c ¯ (∞) are two positive numbers. In the case where one species is competitively stronger near ∞ , we also obtain the existence and nonexistence of forced waves connecting the origin to the semi-trivial steady state. Our results show the existence of multiple types of forced waves with the same forced speed. The mathematical proofs involve integral equations and Schauder's fixed point theorem, and heavily rely on the construction of various upper-lower solutions, which adds new techniques to deal with the "shifting environments" problem. [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 the spatial propagation of bistable nonlocal dispersal equations in exterior domains. We first obtain the existence and uniqueness of an entire solution which behaves like a planar traveling wave front for large negative time. Then, when the entire solution comes to the interior domain, the profile of the front will be disturbed. However, the disturbance is local in space for finite time, which means the disturbance disappears as its location is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile uniformly in space and continue to propagate in the same direction for large positive time provided that the interior domain is compact and convex. 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). [ABSTRACT FROM AUTHOR]

This paper is devoted to considering entire solutions of the following two-species Lotka–Volterra strong competition system with nonlocal dispersals 0.1 u t = ∫ R k (x - y) u (y , t) d y - u + u 1 - u - a v , x ∈ R , t ∈ R , v t = d ∫ R k (x - y) v (y , t) d y - d v + r v 1 - v - b u , x ∈ R , t ∈ R , where a > 1 , b > 1 , d, and r are positive constants. By constructing appropriate super- and sub-solutions, and with the aid of corresponding comparison principle, we established the existence and related qualitative properties of entire solutions originating from three and four traveling wave solutions. Meanwhile, we considered the non-existence of entire solutions originating from more than seven traveling fronts by introducing the definition of non-extendable (terminated) sequence. Compared to the known works for Lotka–Volterra competition system with classical diffusions, we have to overcome many difficulties due to the appearance of nonlocal dispersal operators in current paper. [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 focus on the propagation phenomena of a bistable man–environment epidemic model with nonlocal dispersals, where there exists a positive feedback interaction between the concentration of infectious agent and infectious human population. The monostable and bistable traveling wave solutions and three-wave entire solutions are studied. First, by applying and developing the known results for monostable case, we give a summary of the existence and asymptotic behavior of all monostable traveling wave solutions in two different monostable intervals and further find some relationship between them. The existence of bistable traveling wave solutions is obtained by introducing the results about monotone semiflows with weak compactness. Second, we give twelve types of three-wave entire solutions, which contain all possibilities of three-wave entire solutions originating from three traveling wave solutions with different nonzero wave speeds, by constructing new auxiliary functions and super- and sub-solutions for every type. We also show that these entire solutions are globally Lipschitz continuous with respect to spatial variable. In addition, the nonexistence result of entire solutions originating from more than four traveling wave solutions is obtained. [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 devoted to the study of propagation dynamics for a time periodic nonlocal dispersal model with stage structure. In the case where the birth rate function is monotone, we establish the existence of the spreading speed and its coincidence with the minimal wave speed for monotone periodic traveling waves by appealing to the theory developed for monotone semiflows. In the case where the birth rate function is non-monotone, we first obtain the spreading properties by the squeezing technique combined with some known results for the monotone case, and then investigate the existence of periodic traveling waves by using the asymptotic fixed point theorem due to the lack of parabolic estimates and compactness. Finally, we apply the general results to the Nicholson's blowflies model for its spatial 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]

This paper mainly focuses on the entire solutions of a nonlocal dispersal equation with asymmetric kernel and bistable nonlinearity. Compared with symmetric case, the asymmetry of the dispersal kernel function makes more diverse types of entire solutions since it can affect the sign of the wave speeds and the symmetry of the corresponding nonincreasing and nondecreasing traveling waves. We divide the bistable case into two monostable cases by restricting the range of the variable, and obtain some merging-front entire solutions which behave as the coupling of monostable and bistable waves. Before this, we characterize the classification of the wave speeds so that the entire solutions can be constructed more clearly. Especially, we investigate the influence of the asymmetry of the kernel on the minimal and maximal wave speeds. [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 the spatial dynamics of a nonlocal dispersal population model in a shifting environment where the favorable region is shrinking. It is shown that the species becomes extinct in the habitat if the speed of the shifting habitat edge c>c∗(∞), while the species persists and spreads along the shifting habitat at an asymptotic speed c∗(∞) if c, where c∗(∞) 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, the model system 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. [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]

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space. [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]

This paper is concerned with the nonlinear stability of traveling wavefronts for a single species population model with nonlocal dispersal and age structure. By using the weighted energy method together with the comparison principle, we prove that the traveling wavefront is exponentially stable, when the initial perturbation around the wavefronts decays exponentially at –∞, but it can be arbitrarily large in other locations. In particular, our result implies that the time delay is harmless for stability of traveling wavefronts of the model. [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]

In this paper, we study a Lotka–Volterra cooperative system with nonlocal dispersal under worsening habitats. By constructing appropriate vector super-/subsolutions combined with the monotone iteration scheme, we obtain the existence of bounded and positive forced waves connecting zero equilibrium to the coexistence state of the limiting system corresponding to the most favorable resource with the wave speed at which the habitat is worsening. Compared with the existence result of traveling wave to the homogeneous system, we find that the wave phenomenon is more likely to occur in such shifting environment. Here, we remove the common assumption that the dispersal kernels are compactly supported and symmetric. Further, we investigate the tail behavior of the forced waves, especially for the rate of convergence to the extinction state, and derive the long-time dynamics of two species. Our result shows that weak interspecies cooperation and nonlocal diffusion pattern cannot prevent species from disappearing eventually under such a worsening habitat. [ABSTRACT FROM AUTHOR]

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]

This paper is concerned with the propagation phenomenon of a three species predator–prey system in a shifting environment with nonlocal dispersal. Assuming that the growth rate of each prey is moving to the right at a speed of s and the habitat is suitable for each prey to survive before the shifting edge, while the habitat behind the boundary is not suitable for survival. For any given speed of the shifting habitat edge, we obtain the existence of two different types of forced waves. On the one hand, we demonstrate the existence of front type forced waves connecting coexistence state to the origin at any forced velocity. On the other hand, we obtain the existence and nonexistence of mixed front-pulse type forced waves which connect semi-trivial states to the origin and only one prey state to the origin at super-critical speeds. Particularly, we also demonstrate the existence of these waves with critical speeds. [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]