The evolution of dispersal

Hutson, V.; Martínez, Salomé; Mischaikow, Konstantin; Vickers, G.T.

doi:10.1007/s00285-003-0210-1

Cited by 349 publications

(306 citation statements)

References 5 publications

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“…A similar derivation for a space-jump process is found in [18], however, the birth-jump process is not included there. We first discretize time into disjoint intervals of length t that cover the non-negative real line and discretize space into disjoint patches of size x that cover the whole real line.…”

Section: Derivation From a Random Walkmentioning

confidence: 89%

“…It should be noted that for β = 0, the equation coincides with Equation (5) in [18]. We cancel x, divide by t, and consider the limit as x, t → 0.…”

Section: Derivation From a Random Walkmentioning

confidence: 99%

“…To develop the critical domain size, we work with the compact form of Equation (17), namely the nonlinear Fisher-KPP Equation (18). We rewrite Equation (18) as…”

Section: Critical Domain Size From Linearization At Zeromentioning

confidence: 99%

“…These models arise from what we call birth-jump processes and their corresponding birth-jump partial differential equations; PDEs. The birth-jump processes are generalizations of position-jump processes as discussed, for example, in [18,31] on the one hand and of reaction-diffusion models on the other hand [36].…”

Section: Introductionmentioning

confidence: 99%

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Birth-jump processes and application to forest fire spotting

Hillen

Greese

Martín

et al. 2014

Journal of Biological Dynamics

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Birth-jump models are designed to describe population models for which growth and spatial spread cannot be decoupled. A birth-jump model is a nonlinear integro-differential equation. We present two different derivations of this equation, one based on a random walk approach and the other based on a two-compartmental reaction-diffusion model. In the case that the redistribution kernels are highly concentrated, we show that the integro-differential equation can be approximated by a reaction-diffusion equation, in which the proliferation rate contributes to both the diffusion term and the reaction term. We completely solve the corresponding critical domain size problem and the minimal wave speed problem. Birth-jump models can be applied in many areas in mathematical biology. We highlight an application of our results in the context of forest fire spread through spotting. We show that spotting increases the invasion speed of a forest fire front.

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Section: Derivation From a Random Walkmentioning

confidence: 89%

“…It should be noted that for β = 0, the equation coincides with Equation (5) in [18]. We cancel x, divide by t, and consider the limit as x, t → 0.…”

Section: Derivation From a Random Walkmentioning

confidence: 99%

“…To develop the critical domain size, we work with the compact form of Equation (17), namely the nonlinear Fisher-KPP Equation (18). We rewrite Equation (18) as…”

Section: Critical Domain Size From Linearization At Zeromentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Birth-jump processes and application to forest fire spotting

Hillen

Greese

Martín

et al. 2014

Journal of Biological Dynamics

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“…An integral dispersal operator, arising in the biological context, has been considered in [11,9,10,15]. One assumes for example that…”

Section: Basic Assumptionsmentioning

confidence: 99%

Non-Local dispersal and bistability

Hutson

Grinfeld

2006

Eur. J. Appl. Math.

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The scalar initial value problemis a model for dispersal. Here u represents the density at point x of a compact spatial region Ω ∈ R n and time t, and u(·) is a function of t with values in some function space B. D is a bounded linear operator and f (u) is a bistable nonlinearity for the associated ODE u t = f (u). Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with D = ∆ is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case D = ∆). We develop a technique for proving that indeed convergence does hold for small ρ and show by constructing a counter-example that this result does not hold in general for all ρ.

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Optimizing the use of suppression zones for containment of invasive species

Lampert

Liebhold

2023

Ecological Applications

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Despite efforts to prevent their establishment, many invasive species continue to spread and threaten food production, human health, and natural biodiversity. Slowing the spread of established species is often a preferred strategy; however, it is also expensive and necessitates treatment over large areas. Therefore, it is critical to examine how to distribute management efforts over space cost‐effectively. Here we consider a continuous‐space bioeconomic model and we develop a novel algorithm to find the most cost‐effective allocation of treatment efforts throughout a landscape. We show that the optimal strategy often comprises eradication in the yet‐uninvaded area, and under certain conditions, it also comprises maintaining a “suppression zone,” an area between the invaded and the uninvaded areas, where treatment reduces the invading population but without eliminating it. We examine how the optimal strategy depends on the demographic characteristics of the species and reveal general criteria for deciding when a suppression zone is cost effective.

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The evolution of dispersal

Cited by 349 publications

References 5 publications

Birth-jump processes and application to forest fire spotting

Birth-jump processes and application to forest fire spotting

Non-Local dispersal and bistability

Optimizing the use of suppression zones for containment of invasive species

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