Generation of periodic travelling waves in cyclic populations by hostile boundaries

Sherratt, Jonathan A.

doi:10.1098/rspa.2012.0756

Cited by 4 publications

(8 citation statements)

References 74 publications

(130 reference statements)

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“…PTW (wavetrain) solutions of reaction-diffusion equations have been studied since the 1970s because of their relevance to spiral waves and target patterns in oscillatory chemical reactions [12]. The discovery of their ecological importance in the 1990s was parallelled by renewed mathematical interest, and the literature on PTWs in reaction-diffusion models for interacting populations is now extensive (see [13,14,15] for recent examples). 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.…”

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 particular, they demonstrate that for smaller values of r 0 , waves can destabilize for larger values of k-this is sometimes known as a "Hopf"-type instability, illustrated in Figure 5. Figure 5(a) demonstrates, in parameter space, how Eckhaus curves given by (31) can differ significantly from the true stability boundary, which we calculate numerically. By setting r 0 = 0.72, we select PTWs with amplitude small enough for destabilization via Hopf instability.…”

Section: Stability Of Ptws In the Cgle For Certain Parameter Valuesmentioning

confidence: 99%

“…Plots to illustrate a Hopf-type instability in the CGLE in terms of ecological parameters relating to (32) defined in section 7, with δ = −0.68 (note this is an ecologically insignificant parameter choice). In (a), we can see the difference in the Eckhaus curve (dashed line), calculated using (31), and the true stability line (solid line), which is calculated numerically and separates unstable (shaded) and stable (white) regions of the a-b parameter plane. (b)-(d) are plots of essential spectra for different parameter values.…”

Section: Stability Of Ptws In the Cgle For Certain Parameter Valuesmentioning

confidence: 99%

“…For other methods of wave generation, PTWs could destabilize via Hopf instability, and one would have to calculate stability numerically (see section 5)-we give an example of this for (32) in Figure 5(a) for a fictional method of wave generation that generates PTWs of amplitude r 0 = 0.72. Figures 6(a)-(c) use (31) to calculate stable and unstable regions of the parameter plane. In addition, we calculate absolutely unstable regions (where spatiotemporal irregularity is observed immediately behind invasion) and convectively unstable regions (where PTWs are observed but then destabilize after some time) by calculating branch points in the absolute spectrum.…”

Section: Examplementioning

confidence: 99%

“…In this paper, we focus on predator-prey interactions. There are a number of ecological situations one could consider in a predator-prey system that would generate PTWs, including heterogeneous habitats [4,13], hostile boundaries [31], migration driven by pursuit and evasion [3], and an invasion of alien species [23]. In this paper, we consider the latter of these mechanisms.…”

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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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Large scale patterns in mussel beds: stripes or spots?

Bennett

Sherratt

2018

J. Math. Biol.

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An aerial view of an intertidal mussel bed often reveals large scale striped patterns aligned perpendicular to the direction of the tide; dense bands of mussels alternate periodically with near bare sediment. Experimental work led to the formulation of a set of coupled partial differential equations modelling a mussel-algae interaction, which proved pivotal in explaining the phenomenon. The key class of model solutions to consider are one-dimensional periodic travelling waves (wavetrains) that encapsulate the abundance of peak and trough mussel densities observed in practice. These solutions may, or may not, be stable to small perturbations, and previous work has focused on determining the ecologically relevant (stable) wavetrain solutions in terms of model parameters. The aim of this paper is to extend this analysis to two space dimensions by considering the full stripe pattern solution in order to study the effect of transverse two-dimensional perturbations-a more true to life problem. Using numerical continuation techniques, we find that some striped patterns that were previously deemed stable via the consideration of the associated wavetrain solution, are in fact unstable to transverse two-dimensional perturbations; and numerical simulation of the model shows that they break up to form regular spotted patterns. In particular, we show that break up of stripes into spots is a consequence of low tidal flow rates. Our consideration of random algal movement via a dispersal term allows us to show that a higher algal dispersal rate facilitates the formation of stripes at lower flow rates, but also encourages their break up into spots. We identify a novel hysteresis effect in mussel beds that is a consequence of transverse perturbations.

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Generation of periodic travelling waves in cyclic populations by hostile boundaries

Cited by 4 publications

References 74 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

Large scale patterns in mussel beds: stripes or spots?

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