The Wave of Advance of Advantageous Genes

Fisher, Ronald Aylmer

doi:10.1111/j.1469-1809.1937.tb02153.x

Cited by 4,357 publications

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“…Reaction-diffusion equations have been instrumental in studying spatio-temporal population dynamics ever since the seminal contributions by Fisher (1937) andSkellam (1951). As outlined in Cantrell and Cosner (2003) and Turchin (1998), reactiondiffusion equations may be derived from a detailed description of individual movement behavior in the form of a random walk model.…”

Section: Individual Dispersalmentioning

confidence: 99%

Integrodifference equations in patchy landscapes

Musgrave

Lutscher

2013

J. Math. Biol.

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In this dissertation, we study integrodifference equations in patchy landscapes. Specifically, we provide a framework for linking individual dispersal behavior with populationlevel dynamics in patchy landscapes by integrating recent advances in modeling dispersal into an integrodifference equation.First, we formulate a random-walk model in a patchy landscape with patchdependent diffusion, settling, and mortality rates. We incorporate mechanisms for individual behavior at an interface which, in general, results in the probability-density function of the random walker being discontinuous at an interface. We show that the dispersal kernel can be characterized as the Green's function of a second-order differential operator and illustrate the kind of (discontinuous) dispersal kernels that arise from our approach. We examine the dependence of obtained kernels on model parameters.Secondly, we analyze integrodifference equations in patchy landscapes equipped with discontinuous kernels. We obtain explicit formulae for the critical-domain-size problem, as well as explicit formulae for the analogous critical size of good patches on an infinite, periodic, patchy landscape. We examine the dependence of obtained formulae on individual behavior at an interface. Through numerical simulations, we observe that, if the population can persist on an infinite, periodic, patchy landscape, its spatial profile can evolve into a discontinuous traveling periodic wave. We derive a dispersion relation for the speed of the wave and illustrate how interface behavior ii

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Section: Individual Dispersalmentioning

confidence: 99%

Integrodifference equations in patchy landscapes

Musgrave

Lutscher

2013

J. Math. Biol.

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“…Diffusion approximations describing this property have been made for the two-deme model (Barton and Rouhani, 1991). Also, a travelling wave approximation (Fisher, 1937) has been used to describe the conditions permissible for the establishment and spread, despite underdominance, of an allele in continuous habitats with a higher homozygous genotype fitness (Barton, 1979;Piálek and Barton, 1997;Soboleva et al, 2003). We do not mean to imply that underdominance is commonly responsible for the maintenance of polymorphism in natural structured populations.…”

Section: Overviewmentioning

confidence: 99%

Using underdominance to bi-stably transform local populations

Altrock

Traulsen

Reeves

et al. 2010

Journal of Theoretical Biology

106

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Underdominance refers to natural selection against individuals with a heterozygous genotype. Here, we analyse a single-locus underdominant system of two large local populations that exchange individuals at a certain migration rate. The system can be characterized by fixed points in the joint allele frequency space. We address the conditions under which underdominance can be applied to transform a local population that is receiving wildtype immigrants from another population. In a single population, underdominance has the benefit of complete removal of genetically modified alleles (reversibility) and coexistence is not stable. The two population system that exchanges migrants can result in internal stable states, where coexistence is maintained, but with additional release of wildtype individuals the system can be reversed to a fully wildtype state. This property is critically controlled by the migration rate. We approximate the critical minimum frequency required to result in a stable population transformation. We also concentrate on the destabilizing effects of fitness and migration rate asymmetry. Practical implications of our results are discussed in the context of utilizing underdominance to genetically modify wild populations. This is of importance especially for genetic pest management strategies, where locally stable and potentially reversible transformations of populations of disease vector species are of interest.

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“…When m * > 0, (8) is called Fisher's (1937) equation for the advance of an advantageous gene, or the KPP equation, after Kolmogoroff, Petrovsky, and Piscounov (1937). 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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The Wave of Advance of Advantageous Genes

Cited by 4,357 publications

References 0 publications

Integrodifference equations in patchy landscapes

Integrodifference equations in patchy landscapes

Using underdominance to bi-stably transform local populations

Evolution of dispersal distance

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