Spreading Speeds and Linear Determinacy for Two Species Competition Systems with Nonlocal Dispersal in Periodic Habitats

Kong, Liang; Rawal, Nar; Shen, Wenxian

doi:10.1051/mmnp/201510609

Cited by 25 publications

(25 citation statements)

References 45 publications

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“…Throughout this paper, we assume (H0). The following results on the persistence, coexistence, and extinction in (1.1) have been proved in literature (see, for example, [10], [18], [21]).…”

Section: Introductionmentioning

confidence: 82%

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Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems

Kong¹,

Nguyen²,

Shen³

2019

Communications on Pure &Amp; Applied Analysis

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The current paper is devoted to the study of two species competition systems of the form u t (t,is a smooth non-negative convolution kernel supported on an interval centered at the origin),, and a i , b i , and c i (i = 1, 2) are spatially homogeneous when |x| ≫ 1, that is,, and |x| ≫ 1. Such a system can be viewed as a time periodic competition system subject to certain localized spatial variations. We, in particular, study the effects of localized spatial variations on the uniform persistence and spreading speeds of the system. Among others, it is proved that any localized spatial variation does not affect the uniform persistence of the system, does not slow down the spreading speeds of the system, and under some linear determinant condition, does not speed up the spreading speeds.

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“…Throughout this paper, we assume (H0). The following results on the persistence, coexistence, and extinction in (1.1) have been proved in literature (see, for example, [10], [18], [21]).…”

Section: Introductionmentioning

confidence: 82%

“…See [1,4,5,6,7,8,10,12,13,14,15,19,24,25], etc. for the case that Au = u xx and see [2,6,9,18,20,21], etc. for the case Au(t, x) = R κ(y − x)u(t, y)dy − u(t, x).…”

Section: Introductionmentioning

confidence: 99%

Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems

Kong¹,

Nguyen²,

Shen³

2019

Communications on Pure &Amp; Applied Analysis

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“…For monotone semiflows of bistable type, Fang and Zhao [9] interpreted the bistability from a monotone dynamical system point of view to find a link between the monostable subsystems and bistable system itself, which is used to establish the existence of bistable wavefronts. We refer to [21,6,34,35] for nonlocal dispersal equations, and two survey papers [43,15] for more references. There are also quite a few investigations on time-periodic fronts of reaction-diffusion equations, see, e.g., [1,2,10,26,46,47,45] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…For a reaction-diffusion competition model with seasonal succession, Ma and Zhao [26] studied the existence of single spreading speed and its linear determinacy, and showed that the spreading speed coincides with the minimal wave speed of time-periodic traveling waves. More recently, Kong, Rawal and Shen [21] proposed a competition model with nonlocal dispersal in a time and space periodic habitats, and investigated the spreading speed and its linear determinacy. For traveling waves in a time-delayed reaction-diffusion competition model with nonlocal terms, we refer to Gourley and Ruan [13].…”

Section: Introductionmentioning

confidence: 99%

Traveling waves and spreading speeds for time–space periodic monotone systems

Fang

Xiao

Zhao

2017

Journal of Functional Analysis

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The theory of traveling waves and spreading speeds is developed for timespace periodic monotone semiflows with monostable structure. By using traveling waves of the associated Poincaré maps in a strong sense, we establish the existence of time-space periodic traveling waves and spreading speeds. We then apply these abstract results to a two species competition reactionadvection-diffusion model. It turns out that the minimal wave speed exists and coincides with the single spreading speed for such a system no matter whether the spreading speed is linearly determinate. We also obtain a set of sufficient conditions for the spreading speed to be linearly determinate.

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“…Behavior of solutions near the stability boundary is studied using the techniques of weakly nonlinear analysis. The spreading speed in the system of competition of species with nonlocal dispersion is estimated in [26]. Pattern formation in the system of competition of species with nonlocal consumption is studied in [10].…”

Section: About This Issuementioning

confidence: 99%

Preface to the Issue Nonlocal Reaction-Diffusion Equations

Alfaro

Apreutesei

Davidson

et al. 2015

Math. Model. Nat. Phenom.

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Nonlocal reaction-diffusion equations in population dynamicsNonlocal reaction-diffusion equations are intensively studied during the last decade in relation with problems in population dynamics and other applications. In comparison with traditional reaction-diffusion equations they possess new mathematical properties and richer nonlinear dynamics. Many studies are devoted to the nonlocal reaction-diffusion equationwherewhich describes the distribution of population density in the case of nonlocal consumption of resources.Here k is a positive integer, k = 1 corresponds to asexual and k = 2 to sexual reproduction. If the kernel φ of the integral is replaced by the δ-function, then we obtain conventional reaction-diffusion equation.In the case k = 1, it is the logistic equation with the reproduction term au(1 − u) proportional to the population density u and to available resources (1 − u). In the case of nonlocal consumption of resources, the integral J(u) describes consumption of resources at the space point x by individuals located in some area around this point. The function φ(x − y) determines the efficiency of such consumption. Introduction of nonlocal consumption of resources changes the properties of solutions of this equation. Consider for certainty the case where k = 1 and σ = 0. Suppose that ∞ −∞ φ(y)dy = 1. Then u = 1 is a stationary solution of this equation. It is stable in the case of the local equation but it can lose its stability for the nonlocal equation. If it becomes unstable, then a periodic in space stationary solution bifurcates from it [13], [19], [21]. This simple result of the linear stability analysis has important consequences from the mathematical point of view and for the applications.

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Spreading Speeds and Linear Determinacy for Two Species Competition Systems with Nonlocal Dispersal in Periodic Habitats

Cited by 25 publications

References 45 publications

Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems

Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems

Traveling waves and spreading speeds for time–space periodic monotone systems

Preface to the Issue Nonlocal Reaction-Diffusion Equations

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