Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation

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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“…by Aronson and Weinberger (1975) and others. Providing u + f (u + ) has the appropriate concave-down shape…”

Section: Prey-only Wavesmentioning

confidence: 81%

“…and, for compact initial data, it is straightforward to show that the spread rate for any level set u = u L asymptotically achieves (11) (Aronson and Weinberger, 1975). Waves that are not pulled are said to be 'pushed' (Hadeler and Rothe, 1975).…”

Section: Prey-only Wavesmentioning

confidence: 99%

How Predation can Slow, Stop or Reverse a Prey Invasion

Owen

Lewis

2001

Bulletin of Mathematical Biology

180

141

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“…[2,3,8]. In particular, the dynamics of travelling wave solutions are frequently of interest in applications.…”

Section: Introductionmentioning

confidence: 99%

“…One question addressed in [9] is precisely the effect of the cut-off function ϕ on the velocity c of the corresponding travelling fronts. In particular, for ϕ(u) = (u − ε), where denotes the Heaviside step function, it is shown in [9] that the front velocity c in (2) differs from the continuum wave speed c FKPP by a term that is logarithmic in ε, c ∼ 2 − π 2 (ln ε) 2 as ε → 0.…”

Section: Introductionmentioning

confidence: 99%

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

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“…A result of Aronson and Weinberger (1975) implies that there is a c > 0 so that starting from any initial condition v ≡ 0…”

Section: Model and Resultsmentioning

confidence: 99%

Evolution of dispersal distance

Durrett

Remenik

2011

J. Math. Biol.

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The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there is a trade off between survivability and dispersal range. Ezoe (1998) and Levin and Muller-Landau (2000) approached this problem using models that were essentially deterministic, and used calculus to find optimal dispersal parameters. Here we follow Hiebeler (2004) and use a stochastic spatial model to study the competition of different dispersal strategies. Most work on such systems is done by simulation or nonrigorous methods such as pair approximation. Here, we use machinery developed by Cox, Durrett, and Perkins (2011) to rigorously and explicitly compute evolutionarily stable strategies.

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Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation

Cited by 992 publications

References 3 publications

How Predation can Slow, Stop or Reverse a Prey Invasion

How Predation can Slow, Stop or Reverse a Prey Invasion

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

Evolution of dispersal distance

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