On two types of moving front in quasilinear diffusion

Stokes, A. N.

doi:10.1016/0025-5564(76)90087-0

Cited by 136 publications

(158 citation statements)

References 4 publications

Supporting

Mentioning

149

Contrasting

Order By: Relevance

“…It is convenient to express the solution in terms of scaled variables. Setting (Stokes 1976). In contrast, the spatial dynamics of bistable systems depend on local dynamics averaged over all frequencies (see eq.…”

Section: One Dimension Homogeneous Environmentmentioning

confidence: 99%

“…[15]; Stokes 1976;Fife 1979b). As with Fisherian systems, the rate of spatial spread and the form of the spreading wave depend on initial conditions, but for a wide range of biologically plausible initial conditions, both the wave form and its rate of movement depend asymptotically only on the parameters of the model (Kolmogorov et al 1937;Fife 1979a;Bramson 1983).…”

Section: One Dimension Homogeneous Environmentmentioning

confidence: 99%

See 1 more Smart Citation

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

Barton

Turelli

2011

The American Naturalist

202

400

View full text Add to dashboard Cite

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

show abstract

Section: One Dimension Homogeneous Environmentmentioning

confidence: 99%

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

View full text Add to dashboard Cite

show abstract

“…For more general monostable f the propagation speed was found to be either equal to or larger than c lin and therefore one distinguishes between a linear or nonlinear selection of the propagation speed (the terminology pulled and pushed fronts, respectively, having an equivalent meaning). Aronson and Weinberger [AW75] (see also Hadeler and Rothe [HR75] and [Sto76] for other pioneering work on such matters and [vS03] for a recent review) proved that for general monostable f , in both the linear and the nonlinear selection case, the propagation speed for solutions with initial data decaying sufficiently fast is given by the minimal speed of travelling waves. They showed that monotonic travelling waves exist for all speeds c ≥ c min and none for c < c min , where c min ≥ c lin ; solutions of (1.4) with sufficiently rapidly decaying initial data propagate with speed c min .…”

mentioning

confidence: 99%

Travelling-wave analysis of a model describing tissue degradation by bacteria

Hilhorst¹,

King²,

Röger³

2007

Eur. J. Appl. Math

View full text Add to dashboard Cite

Abstract. We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the 'degradation-rate parameter' solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave-speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, nonlinear selection of the minimal speed occurs.

show abstract

“…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%

Pushed and pulled fronts in a discrete reaction–diffusion equation

King

O’Dea

2015

J Eng Math

View full text Add to dashboard Cite

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.

show abstract

On two types of moving front in quasilinear diffusion

Cited by 136 publications

References 4 publications

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

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

Travelling-wave analysis of a model describing tissue degradation by bacteria

Pushed and pulled fronts in a discrete reaction–diffusion equation

Contact Info

Product

Resources

About