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The Maynard Smith Model of Sympatric Speciation

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The Maynard Smith Model of Sympatric Speciation ARTICLE IN PRESS Journal of Theoretical Biology 239 (2006) 172–182 www.elsevier.com/locate/yjtbi The Maynard Smith model of sympatric speciation Sergey Gavriletsa,b,Ã aDepartment of Ecology and Evolutionary Biology, University of Tennessee, Knoxville, TN 37996, USA bDepartment of Mathematics, University of Tennessee, Knoxville, TN 37996, USA Received 4 March 2005; received in revised form 23 July 2005; accepted 24 July 2005 Available online 20 October 2005 Abstract The paper entitled ‘‘Sympatric speciation,’’ which was published by John Maynard Smith in 1966, initiated the development of mathematical models aiming to identify the conditions for sympatric speciation. A part of that paper was devoted to a specific two-locus, two-allele model of sympatric speciation in a population occupying a two-niche system. Maynard Smith provided some initial numerical results on this model. Later, Dickinson and Antonovics (1973) and Caisse and Antonovics (1978) performed more extensive numerical studies on the model. Here, I report analytical results on the haploid version of the Maynard Smith model. I show how the conditions for sympatric and parapatric speciation and the levels of resulting genetic divergence and reproductive isolation are affected by the strength of disruptive selection and nonrandom mating, recombination rate, and the rates of male and female dispersal between the niches. r 2005 Elsevier Ltd. All rights reserved. Keywords: Sympatric; Speciation; Mathematical model; Maynard Smith 1. Introduction ‘‘Animal speciation and evolution’’ which forcefully argued against the importance of sympatric speciation in nature. John Maynard Smith has made a number of extremely By the mid-1960s, solid theoretical work on sympatric important contributions to evolutionary biology, many of speciation was needed to help clarify the arguments and which have also found various applications well outside conflicting evidence on sympatric speciation. evolutionary biology (see other papers in this issue). One of The first part of the paper analysed the conditions for the his earlier influential works was the paper entitled maintenance of genetic variation in a two-deme version of ‘‘Sympatric speciation’’ published by American Naturalist the Levene model (Levene, 1953) which Maynard Smith in 1966. generalized for the case of habitat choice by females. The There were two major stimulas for the paper in a few second part was devoted to the following question: given preceding years. The first was a series of laboratory that genetic variation is stably maintained under disruptive experiments with Drosophila melanogaster by Thoday and selection, can reproductive isolation evolve between the his colleagues (Millicent and Thoday, 1961; Thoday and two ‘‘extreme’’ morphs, so that no low-fitness intermedi- Boam, 1959; Thoday and Gibson, 1962) that showed rapid ates are produced even when different individuals freely emergence of very strong reproductive isolation. These encounter each other? experiments suggested that sympatric speciation can Maynard Smith discussed four possible mechanisms proceed in a straightforward way. The second was the that can lead to the evolution of reproductive isolation publication of Ernst Mayr’s (1963) monumental volume in the presence of gene flow: (1) ‘‘habitat selection,’’ by which he meant the tendency of organisms to mate in the same niche where they were born, (2) ‘‘pleiotropism,’’ ÃCorresponding author at: Department of Ecology and Evolutionary Biology, University of Tennessee, Knoxville, TN 37996, USA. Tel.: by which he meant the situations in which the alleles +1 865 974 8136; fax: +1 865 974 3067. that adapt the individuals to local conditions also E-mail address: [email protected]. control mating behavior, (3) ‘‘modifier genes,’’ by which 0022-5193/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2005.08.041 ARTICLE IN PRESS S. Gavrilets / Journal of Theoretical Biology 239 (2006) 172–182 173 he meant the establishment of selectively neutral genes to the following scheme: that cause pleiotropism in the genes underlying local AA Aa aa adaptation, and (4) ‘‘assortative mating genes,’’ by which he meant genes that cause assortative mating in niche 1 1 1 1 À s1 regardless of the genotype at the selected loci. I note that in niche 2 1 À s2 1 À s2 1: the majority of the later theoretical research on sympatric Here the coefficients s and s measure the strength of speciation has been done along these four lines first 1 2 viability selection within each niche (0 s ; s 1). That is, discussed by Maynard Smith. Maynard Smith also p 1 2p the dominant allele A is advantageous in niche 1, whereas proposed a specific model of sympatric speciation via the the recessive allele a is advantageous in niche 2. Maynard mechanism of ‘‘assortative mating genes’’ which will be the Smith assumed that surviving adults form a single focus of this paper. randomly mating population. After mating, each female In the years since 1966 sympatric speciation went lays a fraction ð1 þ HÞ=2 of her eggs in the niche where she through cycles of increased enthusiasm and skepticism was raised and lays the remaining fraction ð1 À HÞ=2 in the accompanied by a continuous growth in both empirical other niche (0 H 1). The coefficient H measures the (reviewed by Coyne and Orr, 2004) and theoretical p p degree of habitat selection by females. H ¼ 0 implies no (reviewed by Gavrilets, 2004) knowledge. Unfor- habitat selection, as in the Levene model (Levene, 1953; tunately, most of the theoretical work remains based on Gavrilets, 2004, pp. 235–240). H ¼ 1 implies complete numerical simulations which significantly limits the habitat selection. Note that, alternatively, one can think of generality of theoretical conclusions (Gavrilets, 2003). mating as taking place within the niches, with males Given the exploding growth of interests in speciation, in migrating randomly between the niches prior to mating general, and in sympatric speciation, in particular, and and females either staying in the niche where they were given the controversies that keep remain associated born or going to the other niche with probabilities ð1 þ with sympatric speciation (as is apparent from a compar- HÞ=2andð1 À HÞ=2, respectively. ison of major conclusions on sympatric speciation of The first question to ask is whether spatially hetero- Coyne and Orr, 2004 and Gavrilets, 2004 and those geneous selection maintains genetic variation in the DS of Dieckmann et al., 2004) there is a growing need locus. For simplicity, assume that the sizes of the two of a solid quantitative theory of speciation. Although populations are the same. Then, as shown by Maynard recently a number of simple models of sympatric speciation Smith (1966),ifH ¼ 0 (i.e. females lay eggs in the two have been solved analytically (Gavrilets, 2004), obtaining niches at random), genetic variation is maintained if analytical results still remains a major theoretical s2 À s1 challenge. Below I present analytical results for a À1o o1. simplified version of the model introduced by Maynard s1s2 Smith. A part of these results were briefly introduced in If H ¼ 1 (females always lay eggs in the niche where they Gavrilets (2004, Chapter 10). themselves were raised), then genetic variation is main- I will define sympatric speciation as the emergence of tained if new species from a population where mating is random 3 s2 À s1 with respect to the birthplace of the mating partners À o o3. (Gavrilets, 2003, 2004). Note that, implicitly, mating is 2 s1s2 allowed to be nonrandom with respect to, for example, These inequalities show that maintaining genetic variation genotype, phenotype, and culturally inherited traits. This requires sufficiently strong selection (i.e. large s1 and s2); definition is actually implied in most existing mathematical increasing the degree of habitat choice by females H models of sympatric speciation. increases the range of conditions resulting in stable polymorphism. At the polymorphic equilibrium the locally advanta- 2. The Maynard Smith model geous alleles (i.e. allele A in niche 1 and allele a in niche 2) have higher frequencies than the locally deleterious alleles Consider a diploid population with discrete nonoverlap- (i.e. allele a in niche 1 and allele A in niche 2). However, ping generations inhabiting two distinct niches. Assume because mating is random, offspring with locally deleter- that density-dependent factors operating indepen- ious genotypes will be constantly produced at relatively dently within each niche maintain population sizes high frequencies which will reduce the average fitness of at constant levels that do not depend on the average each population. fitness of the populations. This is the case of soft Assume next that there is also a nonrandom mating selection (e.g. Christiansen, 1975; DeMeeus et al., 1993; (NM) locus with alleles B and b controlling assortative Wallace, 1968). mating. Let the rate of recombination between the DS and 1 Let individuals differ with regard to the disruptive NM locus be r (0prp2). We will suppose that mating selection (DS) locus with alleles A and a controlling occurs according to the O’Donald model with dominance fitness (viability) in a local environment according (O’Donald, 1960, 1980; Gavrilets, 2004, pp. 287–290). That ARTICLE IN PRESS 174 S. Gavrilets / Journal of Theoretical Biology 239 (2006) 172–182 is, individuals BB and Bb have the same phenotype which selection, and weak assortative mating. The second out- is recognizably different from the phenotype of individuals come is promoted by low migration, strong selection, and bb. With probability a, an individual mates with another strong assortative mating. This outcome is interpreted as (a individual that has the same phenotype (0pap1). With step towards) speciation that is sympatric or parapatric, 1 probability 1 À a, the individual is engaged in random depending on whether male migration rate is m ¼ 2 or 1 mating. mo2. For the special case of H ¼ 1 (i.e.
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