Convergence of solutions of the Kolmogorov equation to travelling waves

Bramson, Maury

doi:10.1090/memo/0285

Cited by 590 publications

(926 citation statements)

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“…It has a whole family of travelling wave solutions. However, if we start from a non-negative initial condition which tends to 1 at −∞ and 0 at +∞ and decays quickly enough in space (for example one in which favoured alleles are initially confined to the negative half line), then the solution to (1) converges to the non-negative travelling wave of the smallest possible velocity, c ∞ = σ √ 2s, (Bramson, 1983). If we write the corresponding travelling wave solution as p(t, x) = p c∞ (x − c ∞ t), then, as Fisher showed, when p c∞ (z) is small it can be approximated by exp(−c ∞ z/σ 2 ).…”

Section: One Dimensional Waves Of Advancementioning

confidence: 99%

Genetic hitchhiking in spatially extended populations

Barton

Etheridge

Kelleher

et al. 2013

Theoretical Population Biology

100

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Section: One Dimensional Waves Of Advancementioning

confidence: 99%

Genetic hitchhiking in spatially extended populations

Barton

Etheridge

Kelleher

et al. 2013

Theoretical Population Biology

100

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“…(In this situation we, of course, have to replace the 'ro' at the right-hand side of (2.3) by n, where fi is the solution of f(fi) = 0.) (Various more subtle aspects of the convergence of solutions of (2.l) to travelling wave solutions are studied by, e.g., Kolmogorov et al (1937), and Bramson (1983).) Note that founder populations always have a bounded spatial support.…”

Section: (22) 2nst 2stmentioning

confidence: 99%

The velocity of spatial population expansion

1990

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Abstract. We consider the velocity with which an invading population spreads over space. For a general linear model, originally due to Diekmann and Thieme, it is shown that the asymptotic velocity of population expansion can be calculated if information is available on: (i) the net-reproduction, ~; i.e. the expected number of offspring produced by one individual throughout its life, and (ii) the (normalized) reproduction-and-dispersal kernel, {J(a, x -e); i.e. the density of newborns produced per unit of time at position x by an individual of age a born at e. By means of numerical examples we study the effect of the net-reproduction and the shape of the reproductionand-dispersal kernel on the velocity of population expansion. The reproduction-and-dispersal kernel is difficult to measure in full. This leads us to derive approximation formulas in terms of easily measurable parameters. The relation between the velocity of population expansion calculated from the general model and that from the Fisher /Skellam diffusion model is discussed. As a final step we use the model to analyse some real-life examples, thus showing how it can be put to work.

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“…When the initial value decays exponentially at z = +∞, the stability of travelling front solution with the critical speed or a noncritical speed for Fisher equation is also investigated in [7,18,24,28], where it was shown that the exponential decaying rate of the initial value determines the convergence of the solution. For more general initial value with non-exponentially spatial decay, the large time behavior of solution to Fisher equation has been recently investigated in [13].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

2013

Math. Model. Nat. Phenom.

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Abstract. This paper is concerned with the existence and stability of travelling front solutions for more general autocatalytic chemical reaction systems ut = duxx − uf (v), vt = vxx + uf (v) with d > 0 and d = 1, where f (v) has super-linear or linear degeneracy at v = 0. By applying Lyapunov-Schmidt decomposition method in some appropriate exponentially weighted spaces, we obtain the existence and continuous dependence of wave fronts with some critical speeds and with exponential spatial decay for d near 1. By applying special phase plane analysis and approximate center manifold theorem, the existence of traveling waves with algebraic spatial decay or with some lower exponential decay is also obtained for d > 0. Further, by spectral estimates and Evans function method, the wave fronts with exponential spatial decay are proved to be spectrally or linearly stable in some suitable exponentially weighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to be locally exponentially stable in some exponentially weighted spaces and Lyapunov stable in C unif (R) space, if the initial perturbation of the waves is small in both the weighted and unweighted norms; the perturbation of the waves also stays small in L1(R) norm and decays algebraically in C unif (R) norm, if the initial perturbation is in addition small in L1 norm.

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Convergence of solutions of the Kolmogorov equation to travelling waves

Cited by 590 publications

References 0 publications

Genetic hitchhiking in spatially extended populations

Genetic hitchhiking in spatially extended populations

The velocity of spatial population expansion

Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

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