Convergence to pushed fronts

Rothe, Franz

doi:10.1216/rmj-1981-11-4-617

Cited by 79 publications

(94 citation statements)

References 9 publications

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“…then spread rate for (9) is (Hadeler and Rothe, 1975;Rothe, 1981;Lewis and Kareiva, 1993). The sufficient conditions given above (10) guarantee wave speed c + = 2 √ −εak only for the values of a yielding f (0) = 1, (a ≤ −1) whereas this speed is also correct for −1 < a ≤ −1/2.…”

Section: Prey-only Wavesmentioning

confidence: 98%

How Predation can Slow, Stop or Reverse a Prey Invasion

Owen

Lewis

2001

Bulletin of Mathematical Biology

180

141

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Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, [238][239][240][241][242][243][244][245][246][247][248][249][250][251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predatorprey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.

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Section: Prey-only Wavesmentioning

confidence: 98%

How Predation can Slow, Stop or Reverse a Prey Invasion

Owen

Lewis

2001

Bulletin of Mathematical Biology

180

141

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“…Hence, the direction of wave motion depends on dynamics averaged over all frequencies. In contrast, Fisherian variants essentially always spread, and their wave speed depends only on dynamics at the leading edge, via , the rate of increase f (0) near (Stokes 1976;Rothe 1981 (Mollison 1977;Kot et al 1996), whereas bistable waves are much less sensitive to non-Gaussian dispersal (Wang et al 2002). Stokes (1976) refers to these qualitatively different regimes as "pulled" versus "pushed" waves, respectively.…”

Section: One Dimension Homogeneous Environmentmentioning

confidence: 99%

Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects

Barton

Turelli

2011

The American Naturalist

202

400

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. abstract: Unlike unconditionally advantageous "Fisherian" variants that tend to spread throughout a species range once introduced anywhere, "bistable" variants, such as chromosome translocations, have two alternative stable frequencies, absence and (near) fixation. Analogous to populations with Allee effects, bistable variants tend to increase locally only once they become sufficiently common, and their spread depends on their rate of increase averaged over all frequencies. Several proposed manipulations of insect populations, such as using Wolbachia or "engineered underdominance" to suppress vector-borne diseases, produce bistable rather than Fisherian dynamics. We synthesize and extend theoretical analyses concerning three features of their spatial behavior: rate of spread, conditions to initiate spread from a localized introduction, and wave stopping caused by variation in population densities or dispersal rates. Unlike Fisherian variants, bistable variants tend to spread spatially only for particular parameter combinations and initial conditions. Wave initiation requires introduction over an extended region, while subsequent spatial spread is slower than for Fisherian waves and can easily be halted by local spatial inhomogeneities. We present several new results, including robust sufficient conditions to initiate (and stop) spread, using a one-parameter cubic approximation applicable to several models. The results have both basic and applied implications. The University of Chicago Press and

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“…While results characterising the transition between pushed and pulled fronts in the continuum version of the equation studied herein are well established (see e.g. Hadeler and Rothe [20], Stokes [21] and Rothe [27]), and the linearly-selected speed of travelling waves in discrete systems is straightforward to obtain, we are not aware of previous detailed results for pushed waves in discrete equations.…”

Section: Discussionmentioning

confidence: 98%

“…For reasons of notational convention, we set µ = ε in this section, and consider the behaviour of propagating fronts in (26), (27) for 0 < ε ≪ 1. It should be clear how the nonlinearity f (u; a) could be generalised; we shall again limit ourselves to the cubic form defined by (2).…”

Section: Naïve Time-dependent Analysismentioning

confidence: 99%

Pushed and pulled fronts in a discrete reaction–diffusion equation

King

O’Dea

2015

J Eng Math

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We consider the propagation of wave fronts connecting unstable and stable uniform solutions to a discrete reaction-diffusion equation on a one-dimensional integer lattice. The dependence of the wavespeed on the coupling strength µ between lattice points and on a detuning parameter (a) appearing in a nonlinear forcing is investigated thoroughly. Via asymptotic and numerical studies, the speed both of 'pulled' fronts (whereby the wavespeed can be characterised by the linear behaviour at the leading edge of the wave) and of 'pushed' fronts (for which the nonlinear dynamics of the entire front determine the wavespeed) is investigated in detail. The asymptotic and numerical techniques employed complement each other in highlighting the transition between pushed and pulled fronts under variations of µ and a.

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Convergence to pushed fronts

Cited by 79 publications

References 9 publications

How Predation can Slow, Stop or Reverse a Prey Invasion

How Predation can Slow, Stop or Reverse a Prey Invasion

Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects

Pushed and pulled fronts in a discrete reaction–diffusion equation

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