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This paper is concerned with the propagation dynamics of the time periodic Lotka-Volterra competition systems with nonlocal dispersal in a shifting habitat. We first obtain three types of time-periodic forced waves connecting the extinction state to the co-existence state, itself and the semi-trivial state, which describe the conversion from the state of two aboriginal co-existent competing species, two invading alien competitors, and a saturated aboriginal competitor with another invading alien competitor to the extinction state, respectively. This provides a comprehensive explanation of the point-wise extinction dynamics of these two competing species under such a time-periodic worsening habitat. Then, we establish the spreading properties of the associated Cauchy problem depending on the range of the shifting speed. More specifically, we give a complete description on the threshold values for the extinction as well as persistence (by moving with asymptotic speed). Our results reveal the possibility that a competitively weaker species with a much faster spreading speed can drive a competitively stronger species with a slower spreading speed to extinction. The discussion in this paper applies to both cases of weak competition and strong-weak competition. In particular, we need to point out that some combined effects of nonlocal dispersal, two-variable coupling and time-periodic shifting heterogeneity in this system pose extra difficulties in mathematical treatment, which are dealt with by introducing new approaches. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the propagation dynamics of a Lotka-Volterra two-species competition system in a periodic discrete habitat. Under appropriate assumptions, we show that a semi-trivial equilibrium is globally stable for the spatially periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for the spatially periodic rightward traveling waves. We further obtain sufficient conditions for the linear determinacy of the rightward spreading speed. Finally, we apply these results to a specific model of two-species competition and conduct numerical simulations for the spreading speed and traveling waves. [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]

Existence, uniqueness and stability of forced pulsating waves as well as gap formation of reaction diffusion models in a shifting habitat are challenging problems in the field of propagation dynamics. In this paper, we design to tackle these problems for a time periodic Lotka-Volterra competition system. By using alternatively-coupling upper-lower solution method, we establish the existence of forced pulsating traveling wave, as long as the shifting speed falls in a finite interval where the end points are obtained from KPP-Fisher speeds. The asymptotic behaviors of the forced pulsating traveling wave are derived and this helps us show that this wave is unique. Moreover in the Lyapunov sense, the stability of the forced pulsating wave is also studied. Finally, when the forced speed is beyond the finite interval, gap formation is investigated theoretically and illustrated further by numerical simulations. [ABSTRACT FROM AUTHOR]

A reaction-diffusion model which is called the field-road model was introduced by Berestycki, Roquejoffre and Rossi [9] to describe biological invasion with fast diffusion on a line. In this paper, we investigate this model in a heterogeneous landscape and establish the existence of the asymptotic spreading speed c ⁎ as well as its coincidence with the minimal wave speed of pulsating fronts along the road. We start with a truncated problem with an imposed Dirichlet boundary condition. We prove the existence of spreading speed c R ⁎ which coincides with the minimal speed of pulsating fronts for the truncated problem in the direction of the road. The arguments combine the dynamical system method with PDE's approach. Finally, we turn back to the original problem in the half-plane via generalized principal eigenvalue approach as well as an asymptotic method. [ABSTRACT FROM AUTHOR]

Spreading speed of spatio-temporal nonlinear dynamical system can sometimes be determined either by its corresponding linear system with an explicit speed formula, or by the complicated nonlinear system itself with the existence of a pushed wavefront. In this paper, the spreading speed (the minimal speed of wavefronts) for a Lotka-Volterra competition model in spatially periodic habitats is investigated. We establish new results on the linear and nonlinear selections in terms of the spatio-periodic coefficient functions. In the case of nonlinear selection, lower and upper bound estimates of the minimal speed are provided. [ABSTRACT FROM AUTHOR]

We study an integro-difference equation that describes the spatial dynamics of a species in a shifting habitat. The growth function is nondecreasing in density and space for a given time, and shifts at a constant speed c. The spreading speeds for the model were previously studied. The contribution of the current paper is to provide sharp conditions for existence of forced traveling waves with speed c. We show the existence of traveling waves with zero value at ∞ or −∞ for c in different value ranges determined by the spreading speeds. We also show the existence of a traveling wave with any speed c for the case that the species can grow everywhere. Our results demonstrate the existence of different types of traveling waves with the same speed. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat. We first show that this equation admits a periodic forced wave with the speed at which the habitat is shifting by using the monotone iteration method combined with a pair of generalized super- and sub-solutions. Then we establish the nonexistence, uniqueness and global exponential stability of periodic forced waves by applying the sliding technique and the comparison argument. Finally, we obtain the spreading properties for a large class of solutions. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to the study of pulsating fronts and pulsating front-like entire solutions for a reaction–advection–diffusion model of two competing species in a periodic habitat. Under certain assumptions, the competition system admits a leftward and a rightward pulsating fronts in the bistable case. In this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts. Some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the x -axis, which turn out to be unique and Liapunov stable 2-dimensional manifolds of solutions, furthermore, the leftward and rightward pulsating fronts are on the boundary of these 2-dimensional manifolds. The others behave as the two pulsating fronts propagating from one side of the x -axis, the faster one then invades the slower one as t → + ∞. These kinds of pulsating front-like entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat. [ABSTRACT FROM AUTHOR]

We consider a time-periodic competition system with a free boundary, which can be used to describe two species that are struggling on the free boundary to obtain their own habitats and the final competition result is the regional partition of two species. In [15] , we have studied the (classical) traveling wave solutions of the temporal homogeneous problems. In this paper, we consider the time-periodic case and study periodic traveling wave solutions of the problem. More precisely, we will prove the existence and uniqueness of the periodic traveling wave solution, and show that the speed of the periodic traveling wave is uniquely determined by and monotonically dependent on the time-periodic terms in the system. Moreover, we will present some sufficient conditions to guarantee that the wave travels with a positive speed. [ABSTRACT FROM AUTHOR]