Wave train selection behind invasion fronts in reaction–diffusion predator–prey models

Merchant, Sandra M.; Nagata, Wayne

doi:10.1016/j.physd.2010.04.014

Cited by 16 publications

(14 citation statements)

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“…For partial differential equations, families of PTWs typically subdivide into solutions that are stable and unstable as solutions of the original model equations. Wave stability is a key issue in applications where a given set of boundary and initial conditions selects a particular member of the PTW family [42,43]. As parameters are varied, the selected PTW can lose stability, often heralding the onset of spatiotemporal chaos [35,44,45].…”

Section: Introduction Many Natural Populations Exhibit Long-term Oscmentioning

confidence: 99%

Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Sherratt¹

2014

SIAM J. Appl. Dyn. Syst.

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Periodic traveling waves (wavetrains) have been extensively studied for reaction-diffusion equations. One important motivation for this work has been the identification of periodic traveling wave patterns in spatiotemporal data sets in ecology. However, for many ecological populations, diffusion is no more than a rough phenomenological representation of dispersal, and spatial convolution with a dispersal kernel is more realistic. This paper concerns periodic traveling wave solutions of differential equations with nonlocal dispersal terms, and with local dynamics of lambda-omega form. These kinetics include the normal form near a standard supercritical Hopf bifurcation and are therefore significant for a wide range of applications. For general dispersal kernels, an explicit family of periodic traveling wave solutions is derived, as well as the condition for waves to be stable to perturbations of arbitrarily small wavenumber. Three specific kernels are then considered in detail: Laplace, Gaussian, and top hat. For Laplace and Gaussian kernels, it is shown that stability to perturbations of arbitrarily small wavenumber implies stability, a result that also applies for reaction-diffusion equations with lambda-omega kinetics. However, for the top hat kernel it is shown that periodic traveling waves may be stable to perturbations with small wavenumber but not to those with larger wavenumber. The wave family for the top hat kernel also shows significant qualitative differences from those for the Laplace and Gaussian kernels, and for reaction-diffusion equations with the same kinetics.Motivated by the form of PTW solutions of (1.1), I look for solutions of (2.1) with the form r = R, θ = αx + βt, where R ≥ 0, α ≥ 0, and β are constants, givingNote that (2.2) is only valid for even kernels K(.). Figure 1 shows various examples of such PTW solutions, when K(.) is the Laplace kernel (defined in (3.1) below). Figure 1. Examples of PTW solutions of (1.4) for the Laplace kernel (3.1) with a = 1, plotted as a function of space x. The functions λ(.) and ω(.) are given by (1.2) with λ1 = 1; the values of λ0 and α are as shown in the figure. Note that the values of ω0 and ω1 do not affect the PTW as a function of space. The left-and right-hand columns illustrate wave families for λ(0) < 1 and λ(0) > 1, respectively. In both cases, waves for small α have large wavelength and high amplitude, approaching a spatially homogeneous oscillation as α → 0 + . When λ(0) < 1, the wave amplitude is zero at a finite value of α, which is 2 for the value of λ0 used in the left-hand column of the figure. In contrast when λ(0) > 1 there is no upper bound on the values of α giving waves, and the wavelength → 0 with the amplitude remaining nonzero as α → ∞.Since λ(.) is strictly decreasing, (2.2) has a (unique) solution for R and β if and only if C 0 ∈ [1 − λ(0), 1]. Intuitive understanding of this is helped by considering the cases α = 0 and Downloaded 11/06/14 to 137.195.26.108. Redistribution subject to SIAM license or copyright; see PERIODIC WAVES...

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Section: Introduction Many Natural Populations Exhibit Long-term Oscmentioning

confidence: 99%

Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Sherratt¹

2014

SIAM J. Appl. Dyn. Syst.

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“…In contrast, a solution is unstable if some small perturbation grows over time. To investigate the stability of PTWs in the CGLE, we consider (21) and linearize about the PTW solution given by (r, θ) = (r 0 , 1 − r 2 0 x + (α − β)r 2 0 + β t), yielding a set of equations for small perturbationsr,θ,r…”

Section: Without Loss Of Generality Choosementioning

confidence: 99%

“…Ecologists, in general, would be interested in largeamplitude PTWs that have a more significant impact on the surrounding ecosystem. This is the natural direction for future work on PTW selectionlittle has been done in this area, with the exception of recent work by Merchant and Nagata [21,22], who developed a new method of PTW prediction that retains accuracy further from the Hopf bifurcation by assuming the existence of a front between the spatially homogeneous steady state and the selected PTW. • Nonlocal dispersal: For many natural populations, diffusion is considered to be a crude representation of movement, failing to capture instances of rare long-distance dispersal events.…”

Section: Examplementioning

confidence: 99%

Periodic Traveling Waves Generated by Invasion in Cyclic Predator--Prey Systems: The Effect of Unequal Dispersal

Bennett¹,

Sherratt²

2017

SIAM J. Appl. Math.

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Periodic traveling waves (wavetrains) have been an invaluable tool in the understanding of spatiotemporal oscillations observed in ecological data sets. Various mechanisms are known to trigger this behavior, but here we focus on invasion, resulting in a predator-prey-type interaction. Previous work has focused on the normal form reduction of PDE models to the well-understood λ-ω equations near a Hopf bifurcation, though this is valid only when assuming an equal rate of dispersion for both predators and prey-an unrealistic assumption for many ecosystems. By relaxing this constraint, we obtain the complex Ginzburg-Landau normal form equation, which has a one-parameter family of periodic traveling wave solutions, parametrized by the amplitude. We derive a formula for the wave amplitude selected by invasion before investigating the stability of the solutions. This gives us a complete description of small-amplitude periodic traveling waves in the governing model ecosystem.

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“…A particular focus of recent research has been the spatial distribution of cyclic populations, with field studies documenting periodic traveling waves (PTWs) in a number of natural populations including voles [3,4], moths [5], and red grouse [6] (see [7] for additional examples). Spatially extended oscillatory systems have a family of PTW solutions [8], and the initial and boundary conditions select one member of the family [11,12,13,14]. Solution of this wave selection problem is crucial for a thorough understanding of the PTWs seen in the field.…”

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confidence: 99%

“…There is an extensive mathematical literature on PTW generation [15,18,17,12,16,19,13,14], but it concerns almost exclusively reaction-diffusion equations. Although such equations are widely used in ecological modeling (see, for example, [20]), their realism is limited by the use of diffusion to represent dispersal.…”

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confidence: 99%

How Does Nonlocal Dispersal Affect the Selection and Stability of Periodic Traveling Waves?

Sherratt¹

2018

SIAM J. Appl. Math.

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In ecology a number of spatiotemporal datasets on cyclic populations reveal periodic traveling waves of abundance. This calls for studies of periodic traveling wave solutions of ecologically realistic mathematical models. For many species, such models must include long-range dispersal. However, mathematical theory on periodic traveling waves is almost entirely restricted to reactiondiffusion equations, which assume purely local dispersal. I study integrodifferential equation models in which dispersal is represented via a convolution. The dispersal kernel is assumed to be of either Gaussian or Laplace form; in either case it contains a parameter scaling the width of the kernel. I show that as this parameter tends to zero, the integrodifferential equation asymptotically approaches a reaction-diffusion model. I exploit this limit to determine the effect of a small degree of nonlocality in dispersal on periodic traveling wave properties and on the selection of a periodic traveling wave solution by localized perturbation of an unstable steady state. My analysis concerns equations of "λω" type, which are the normal form of a large class of oscillatory systems close to a Hopf bifurcation point. I finish the paper by showing how my results can be used to determine the effect of nonlocal dispersal on spatiotemporal dynamics in a predator-prey system.

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Wave train selection behind invasion fronts in reaction–diffusion predator–prey models

Cited by 16 publications

References 31 publications

Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Periodic Traveling Waves Generated by Invasion in Cyclic Predator--Prey Systems: The Effect of Unequal Dispersal

How Does Nonlocal Dispersal Affect the Selection and Stability of Periodic Traveling Waves?

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