The evolution of slow dispersal rates: a reaction diffusion model

Dockery, Jack D.; Hutson, V.; Mischaikow, Konstantin; Pernarowski, Mark

doi:10.1007/s002850050120

Cited by 371 publications

(312 citation statements)

References 26 publications

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“…Hence, if the distribution of the resident is completely homogeneous, then the selection gradient will be 0. Second, since both Ð A *2 r dx and Ð j∇A * r j 2 dx are always positive, selection will always be for lower diffusivity, which is in line with the findings of earlier investigations by Hastings (1983) and Dockery et al (1998).…”

Section: Perturbation-based Methods For Calculating Selection Across Hsupporting

confidence: 77%

Determining Selection across Heterogeneous Landscapes: A Perturbation-Based Method and Its Application to Modeling Evolution in Space

Wickman

Diehl

Blasius

et al. 2017

The American Naturalist

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Online enhancements: appendixes.abstract: Spatial structure can decisively influence the way evolutionary processes unfold. To date, several methods have been used to study evolution in spatial systems, including population genetics, quantitative genetics, moment-closure approximations, and individualbased models. Here we extend the study of spatial evolutionary dynamics to eco-evolutionary models based on reaction-diffusion equations and adaptive dynamics. Specifically, we derive expressions for the strength of directional and stabilizing/disruptive selection that apply both in continuous space and to metacommunities with symmetrical dispersal between patches. For directional selection on a quantitative trait, this yields a way to integrate local directional selection across space and determine whether the trait value will increase or decrease. The robustness of this prediction is validated against quantitative genetics. For stabilizing/disruptive selection, we show that spatial heterogeneity always contributes to disruptive selection and hence always promotes evolutionary branching. The expression for directional selection is numerically very efficient and hence lends itself to simulation studies of evolutionary community assembly. We illustrate the application and utility of the expressions for this purpose with two examples of the evolution of resource utilization. Finally, we outline the domain of applicability of reaction-diffusion equations as a modeling framework and discuss their limitations.

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Section: Perturbation-based Methods For Calculating Selection Across Hsupporting

confidence: 77%

Determining Selection across Heterogeneous Landscapes: A Perturbation-Based Method and Its Application to Modeling Evolution in Space

Wickman

Diehl

Blasius

et al. 2017

The American Naturalist

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“…Dispersal strategies can vary from unconditional strategies in which the probability of dispersing from a patch is independent of the local environmental conditions to conditional strategies in which the likelihood of dispersing depends on local environmental factors. Understanding how natural selection acts on these different modes and strategies of dispersal has been the focus of much theoretical work [2,5,8,10,12,15,16,17]. For instance, using coupled ordinary differential equation models for populations passively dispersing between two patches, Holt [8] showed that slower dispersing populations could always invade equilibria determined by faster dispersing populations.…”

mentioning

confidence: 99%

“…For instance, using coupled ordinary differential equation models for populations passively dispersing between two patches, Holt [8] showed that slower dispersing populations could always invade equilibria determined by faster dispersing populations. Hastings [5] and Dockery et al [2] considered evolution of dispersal in continuous space using reaction diffusion equations. Dockery et al proved that for two competing populations differing only in their diffusion constant, the population with the larger diffusion constant is excluded.…”

mentioning

confidence: 99%

On the Evolution of Dispersal in Patchy Landscapes

Kirkland¹,

Li²,

Schreiber³

2006

SIAM J. Appl. Math.

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Abstract. To better understand the evolution of dispersal in spatially heterogeneous landscapes, we study difference equation models of populations that reproduce and disperse in a landscape consisting of k patches. The connectivity of the patches and costs of dispersal are determined by a k × k column substochastic matrix S, where S ij represents the fraction of dispersing individuals from patch j that end up in patch i. Given S, a dispersal strategy is a k×1 vector whose ith entry gives the probability p i that individuals disperse from patch i. If all of the p i 's are the same, then the dispersal strategy is called unconditional; otherwise it is called conditional. For two competing populations of unconditional dispersers, we prove that the slower dispersing population (i.e., the population with the smaller dispersal probability) displaces the faster dispersing population. Alternatively, for populations of conditional dispersers without any dispersal costs (i.e., S is column stochastic and all patches can support a population), we prove that there is a one parameter family of strategies that resists invasion attempts by all other strategies.

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“…It is well-known that the problem d∆θ + θ(r(x) − θ) = 0 in Ω, ∂ ν θ = 0 on ∂Ω, has a unique positive solution (see, e.g., [3]), which is denoted by θ d,r . Then the following remarkable result is established by Hastings [13] and Dockery et al [10]. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 88%

“…For reaction-diffusion model, Hastings [13] and Dockery et al [10] showed that, for two competing species with different (random) dispersal rate but otherwise identical in a heterogeneous environments, the slower diffuser always wins. To be more precise, consider the following Lotka-Volterra competition-diffusion system ( [10])…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Lotka-Volterra diffusion-advection competition system with heterogeneity vs homogeneity

Xu¹,

Jiang²

2017

J. Nonlinear Sci. Appl.

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This paper mainly studies the dynamics of a Lotka-Volterra reaction-diffusion-advection model for two competing species which disperse by both random diffusion and advection along environmental gradient. In this model, the species are assumed to be identical except spatial variation: one lives in the heterogeneity environment, the other lives in the homogeneity environment. The main results of this paper are two fold: (i) The species living in homogeneous environment can never wipe out their competitor; (ii) Explore the condition on the diffusion and advection rates for exclusion and coexistence. It is proved that for fixed dispersal rates, when the strength of the advection is sufficiently strong, the two competitive species coexist. This is a remarkable different result with that obtained by He and Ni recently for corresponding systems without advection [X.

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The evolution of slow dispersal rates: a reaction diffusion model

Cited by 371 publications

References 26 publications

Determining Selection across Heterogeneous Landscapes: A Perturbation-Based Method and Its Application to Modeling Evolution in Space

Determining Selection across Heterogeneous Landscapes: A Perturbation-Based Method and Its Application to Modeling Evolution in Space

On the Evolution of Dispersal in Patchy Landscapes

Dynamics of Lotka-Volterra diffusion-advection competition system with heterogeneity vs homogeneity

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