Dispersal evolution and resource matching in a spatially and temporally variable environmentby Robin Aguilée, Pierre de Villemereuil, Jean-Michel Guillon

Journal of Theoretical Biology

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Dispersal evolution and resource matching in a spatially and temporally variable environment

Robin Aguilée a,1, Pierre de Villemereuil b,2, Jean-Michel Guillon c,n a Université Toulouse 3 Paul Sabatier, CNRS, ENFA; UMR 5174 EDB (Laboratoire Evolution & Diversité Biologique); Toulouse, France b Laboratoire d'Écologie Alpine (UMR CNRS 5553), Université Joseph Fourier, Grenoble, France c Laboratoire Ecologie, Systématique et Evolution, Université Paris-XI, UMR 8079, CNRS, AgroParisTech, F-91405 Orsay, France

H I G H L I G H T S  We model the evolution of dispersal rates in a temporally variable environment.  We develop asexual and sexual life-cycle models that we solve analytically.  Resource matching is predicted in expectation before habitat quality variation.  The individuals' distribution undermatches resources after habitat quality variation.  The overall flow of individuals matches the overall flow of resources between patches. a r t i c l e i n f o

Article history:

Received 8 July 2014

Received in revised form 28 October 2014

Accepted 17 January 2015

Available online 29 January 2015

Keywords:

Convergence stable strategy

Habitat dependent dispersal

Ideal free distribution

Individuals' distribution

Individuals' flow a b s t r a c t

Metapopulations may consist of patches of different quality, and are often disturbed by extrinsic processes causing variation of patch quality. The persistence of such metapopulations then depends on the species' dispersal strategy. In a temporally constant environment, the evolution of dispersal rates follows the resource matching rule, i.e. at the evolutionarily stable dispersal strategy the number of competitors in each patch matches the resource availability in each patch. Here, we investigate how the distribution of individuals resulting from convergence stable dispersal strategies would match the distribution of resources in an environment which is temporally variable due to extrinsic disturbance.

We develop an analytically tractable asexual model with two qualities of patches. We show that convergence stable dispersal rates are such that resource matching is predicted in expectation before habitat quality variation, and that the distribution of individuals undermatches resources after habitat quality variation. The overall flow of individuals between patches matches the overall flow of resources between patches resulting from environmental variation. We show that these conclusions can be generalized to organisms with sexual reproduction, and to a metapopulation with three qualities of patches when there is no mutational correlation between dispersal rates. & 2015 Elsevier Ltd. All rights reserved. 1. Introduction

Many natural populations occupy a spatially fragmented landscape and may be satisfactorily described as metapopulations, i.e. as arrays of subpopulations connected by dispersal. The persistence of metapopulations depends both on the rate of disturbance and succession – an extrinsic variable – and on the species' dispersal properties (Levin and Paine, 1974). Dispersal may thus be viewed as an adaptation to ephemeral habitats: dispersal may allow tracking favorable environments (Recer et al., 1987), or, if tracking is not possible, may be a bet-hedging strategy (Philippi and Seger, 1989; McPeek and Holt, 1992). Analyzing the distribution of individuals across space and time is another way to look at dispersal strategies which has been fruitful for decades.

Fretwell and Lucas (1969) introduced the concept of ideal free distribution to predict the distribution of organisms competing for resources in patchy, heterogeneous landscapes. Their seminal work assumed that competitors are equal in food acquisition ability, move between patches at no cost, and have perfect information of the resource supply and competitors' distribution.

In these conditions, the number of competitors in each patch is predicted to match the resource availability in each patch. More generally, the ideal free distribution is the one such that an

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Journal of Theoretical Biology http://dx.doi.org/10.1016/j.jtbi.2015.01.018 0022-5193/& 2015 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ33 1 69 15 57 01.

E-mail addresses: robin.aguilee@univ-tlse3.fr (R. Aguilée), bonamy@horus.ens.fr (P. de Villemereuil), jean-michel.guillon@u-psud.fr (J.-M. Guillon). 1 Tel.: þ33 5 61 55 64 39. 2 Tel.: þ33 4 76 63 53 37.

Journal of Theoretical Biology 370 (2015) 184–196 individual could not attain higher fitness by relocating to another patch. This distribution has been shown to be evolutionarily stable when fitness is a negative function of density (Cressman and

Křivan, 2006; Křivan et al., 2008). However, experiments often report undermatching, i.e. a lack of individuals in the more rewarding patches, and an excess in the less rewarding patches (Kennedy and Gray, 1993). This may be the result of deviations from the initial hypotheses, such as imperfect knowledge of patch quality or unequal competitive abilities (Abrahams, 1986; Houston and McNamara, 1988). Undermatching is also the outcome of most experiments when the resource supply rate varies within patches (Recer et al., 1987; Earn and Johnstone 1997; but see

Hakoyama, 2003).

Spatial and temporal variability of the environment may be satisfactorily described as Markovian process, where the probability to reach a given state at the next time step only depends on the state at its present time. For example, a Markovian process has been used to describe the states of vegetation in a forest with tree replacement (Wagooner and Stephens, 1970; Horn, 1975), and to describe disturbed environments submitted to fires (Callaway and

Davis, 1993; Hibbard et al., 2003). Theoretical studies also have consistently applied a Markovian process to variable environments, e.g. to model environments subject to climate disturbance (Casagrandi and Gatto, 2002; Tuljapurkar et al., 2003). In a metapopulation, Olivieri et al., (1995) described a general stochastic process of patch extinction and succession with a Markov chain at stationarity (see also Valverde and Silvertown, 1997). At stationarity, a Markov chain has a convenient property: the proportions of time spent in the different states are constant over time. In addition, a Markov chain allows to characterize the environmental noise with its color (Vasseur and Yodzis, 2004), since a colored noise can be interpreted as a continuous limit of a discrete