Genetics: Advance Online Publication, published on October 26, 2012 as 10.1534/genetics.112.144980
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2012-1023
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Evolutionary branching in a finite population:
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Deterministic branching versus stochastic branching
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Joe Yuichiro Wakano1, 2 and Yoh Iwasa3
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1. Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University,
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Kawasaki 214-8571, Japan
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2. Japan Science and Technology, PRESTO, Japan
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3. Department of Biology, Kyushu University, Fukuoka 812-8581, Japan
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Copyright 2012.
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Running title:
Evolutionary branching in a finite population
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Keywords: adaptive dynamics, genetic drift, stochastic differential equation,
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mutation-selection-drift balance, population size
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Corresponding author:
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Joe Yuichiro Wakano
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University,
1-1-1 Higashi Mita, Tama-ku, Kawasaki, Kanagawa, Japan 214-8571
Email: Joe Yuichiro Wakano <[email protected]>
Fax: +81-44-934-7660
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(to appear in Genetics)
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Abstract
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Adaptive dynamics formalism demonstrates that, in a constant environment, a
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continuous trait may first converge to a singular point followed by spontaneous
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transition from a unimodal trait distribution into a bimodal one, which is called
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"evolutionary branching". Most previous analyses of evolutionary branching have been
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conducted in an infinitely large population. Here, we study the effect of stochasticity
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caused by the finiteness of the population size on evolutionary branching. By analysing
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the dynamics of trait variance, we obtain the condition for evolutionary branching as the
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one under which trait variance explodes. Genetic drift reduces the trait variance and
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causes stochastic fluctuation. In a very small population, evolutionary branching does
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not occur. In larger populations, evolutionary branching may occur, but it occurs in two
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different manners: in deterministic branching, branching occurs quickly when the
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population reaches the singular point, whilst in stochastic branching, the population
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stays at singularity for a period before branching out. The conditions for these cases and
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the mean branching-out times are calculated in terms of population size, mutational
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effects, and selection intensity, and are confirmed by direct computer simulations of the
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individual-based model.
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1. Introduction
In addition to models using quantitative genetics (Lande 1981, Barton &
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Turelli 1991, Pomiankowski et al. 1991, Iwasa et al. 1991), adaptive dynamics
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formalism has revealed diverse behaviours of evolutionary dynamics associated with
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continuous traits in various organisms (Metz et al. 1992, Dieckmann & Law 1996,
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Geritz et al. 1997, Geritz et al. 1998). This approach allows us to analyse the evolution
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of many ecological traits for which frequency-dependent selection is prevalent without
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specifying the underlying genetic systems controlling the trait. Even in a perfectly
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constant environment, a population may show intriguing temporal behaviour. For
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example, if the trait evolves by accumulation of mutations of small magnitude and if the
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fitness is frequency dependent, a continuous trait may show convergence to a singular
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point followed by spontaneous splitting of a unimodal trait distribution into a bimodal
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(or multimodal) one, referred to as "evolutionary branching" (Metz et al. 1992, Metz et
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al. 1996, Geritz et al. 1997). Evolutionary branching is predicted to occur at an
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evolutionarily singular point that is approaching stable (or convergence stable; CS) but
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that is not an ESS, and is observed in individual-based simulations in many models for
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the evolution of ecological traits.
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In evolutionary branching, an evolving population spontaneously changes its
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trait distribution from unimodal to bimodal (or multimodal). Although evolutionary
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branching has been regarded as a model of sympatric speciation by some authors
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(Dieckmann & Doebeli 1999, Doebeli & Dieckmann 2000), the relationship between
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evolutionary branching in adaptive dynamics formalism and speciation events has been
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the focus of much debate (for review, Waxman & Gavrilets 2005).
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Most previous analyses of adaptive dynamics have assumed an infinite
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population size and that the fitness landscape acts as the deterministic driving force of
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evolution. However, in a finite population, stochastic fluctuations are unavoidable. For
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example, an individual with a lower fitness may be able to produce more offspring just
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by chance. Thus, a finite population can evolve against the selection gradient. As a
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result, stochasticity allows a population to jump from a low local peak to a higher peak.
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Hence, stochastic fluctuation can be an important factor in the outcome of evolution.
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The importance of stochasticity, or genetic drift, has been studied in detail in the context
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of quantitative population genetics (e.g., Lande 1976, Tazzyman & Iwasa 2010). On the
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other hand, in population genetics, random drift is known to decrease the genetic
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diversity of a population (e.g., Ewens 2004), which tends to slow down the action of
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natural selection. Hence, the effect of genetic drift is important to understand
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evolutionary dynamics, including evolutionary branching, in finite populations. In fact,
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Claessen et al. (2007) analysed evolutionary branching in a model including three
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species, and reported that genetic drift tends to cause a delay in evolutionary branching.
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However, a more detailed study of the relationship between evolutionary branching and
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genetic drift is needed.
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In this paper, we study the effect of stochastic fluctuation caused by the
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finiteness of population size (random genetic drift) on evolutionary branching. As an
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illustrative example, we adopt a model of a public goods game studied by Doebeli et al.
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(2004) who investigated evolutionary branching by using a corresponding
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individual-based simulation. The game model deals with a situation that is both
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biologically and socially interesting: a self-organised differentiation of players into a
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highly cooperative group and a non-cooperative group. Concerning the genetics, we in
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effect assume that the population is haploid and that the focal trait is controlled by a
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locus with many alleles. By focusing on this simplest case, we can analyse the effect of
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random genetic drift to the evolutionary outcome most clearly. We first show the results
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of individual-based simulations using different population sizes. We show that genetic
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drift is especially important when the population size is small, and that evolutionary
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branching does not necessarily occur even when the standard theory of adaptive
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dynamics predicts that it should. We also derive deterministic and stochastic versions of
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the dynamics of the trait variance. We reveal that there are two different manners in
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which evolutionary branching occurs: in deterministic branching, branching may occur
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quickly after the population reaches the singular point, whilst in stochastic branching,
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the population stays at a singularity for a period before branching out suddenly. We
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derive the conditions and mean branching-out times for these cases, and confirm our
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predictions by direct computer simulations of the individual-based model.
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2. Evolutionary Branching in a Game in a Finite Population
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2.1 Individual-based model
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As a full stochastic model of a population under natural selection, mutation and
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random genetic drift, we consider the following stochastic process. The population
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consists of N individuals reproducing asexually. Each individual has a genetically
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determined trait value z  [0,1] . At each discrete time step t, an individual i obtains a
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fecundity Fi from pairwise interactions with the other N-1 individuals. Each
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interaction is a two-player game where a payoff given to an individual i matched with
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an individual j is given by f ( zi , z j ) . Its fecundity Fi is denoted by
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Fi   f ( zi , z j ) .
(1)
j i
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Our framework is valid for any payoff function f ( zi , z j ) . As an example, we adopt a
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model by Doebeli et al. (2004) in which trait z represents the amount of investment into
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public goods shared by both players. In this game, the payoff function is represented by
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f ( zi , z j )  1  B( zi  z j )  C ( zi ) ,
(2a)
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where the benefit function is
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B ( z1  z 2 )  b1 ( z1  z 2 )  b2 ( z1  z 2 ) 2 ,
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(2b)
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and the cost function is
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C ( z )  c1 z  c2 z 2 .
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(2c)
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To keep the population size constant, the fitness of an individual with trait zi is
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denoted as
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w( zi ) 
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Fi
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(1 / N ) F j
(3)
j
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When the population size is infinitely large, Doebeli et al. (2004) has shown
z* 
c1  b1
4b2  2c2
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that
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QCS  4b2  2c2  0 and evolutionarily stable when QES  2b2  2c2  0 . The singular
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strategy is referred to as an evolutionary branching point when QCS  0 and QES  0 .
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They have observed evolutionary branching in an individual-based simulation of a large
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population (N=10,000).
the
singular
strategy
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is
convergence
stable
when
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In a finite population, the model includes random genetic drift, and the details
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of the updating rule and the order of different stages in population dynamics becomes
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important. Here, we assume a Wright-Fisher process. At each time step, fecundity is
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calculated for all N individuals. Then all individuals die and are replaced by their
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offspring which are sampled multinomially, with the probability of an individual to be
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the parent of each offspring individual proportional to the fecundity of the focal
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individual.
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After reproduction, there is a small chance of mutation  , and the mutant has
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a trait value slightly different from that of the parent. To be specific, it follows a normal
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distribution with a mean equal to the parent's and a variance  2 . Mutants are
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constrained to be between 0 and 1, which confines trait z within an interval [0,1].
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2.2 Evolutionary branching depends on the population size.
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All simulations were run with parameter values for which a unique singular
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point exists that is convergence stable and evolutionarily non-stable (i.e. the
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evolutionary branching point). Thus, evolutionary branching is expected when the
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population size is infinitely large.
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When the population size is large (N = 8,000), evolutionary branching occurs
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as soon as the trait evolves to reach the convergent stable (CS) value z* (Figure 1(a)).
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For an intermediate population size (N = 1,000), the population lingers around the CS
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value, z*, before it finally shows evolutionary branching. The waiting time until the
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evolutionary branching varies considerably among different simulation runs with the
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same parameter values (Figure 1(b) and (c)). For a small population size (N=200), we
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do not observe evolutionary branching and the population seems to stay around z*, even
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though z* is not evolutionarily stable (Figure 1(d), see also Wakano and Lehmann
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2012).
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The evolutionary branching dynamics are characterised by an abrupt increase
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or explosion of the variance of the focal phenotype (the trait variance). If all the
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parameters for the interactions are the same, a small population does not branch, an
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intermediately large population shows delayed branching, and a large population
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branches immediately. The intuitive reason for the smallness of the population size to
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suppress evolutionary branching is that genetic diversity is reduced by genetic drift
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every generation. In the following sections, we provide an approximate dynamics for
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the trait variance to understand these simulation results.
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3. Dynamics of Trait Variance
The full (individual-based) model is analytically intractable. Therefore, we
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derive a reduced simplified dynamical system here, which is tractable and gives
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predictions on the behaviour of the full model. One generation in the reduced model
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consists of two substeps.
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Substep 1: Natural selection and mutation
Let  (z ) denote a distribution of trait z, which is normalised to satisfy
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  ( z )dz  1 . When a fitness function
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sufficiently weak, the trait distribution in offspring pool after mutation, denoted by
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 (z ) , can be approximated by the following replicator equation plus a mutation term.
w(z ) is given and selection and mutation are
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(z)  (z)  w(z)  w (z) 
 2 2
2 z 2
(z) ,
(4)
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where w is the population average of fitness values. The second term on the
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right-hand side represents the effect of selection and the third term represents the effect
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of mutation. The Taylor expansion of fitness function around the average trait value z
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is denoted as
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w( z )  w( z )  w1 ( z  z ) 
w2
(z  z)2 ,
2
(5)
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where the coefficients w1 and w2 are determined by the current state z . The
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coefficient w1 represents the selection gradient. The coefficient w2 indicates
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stabilising selection if it is negative, and indicates disruptive selection if it is positive.
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The derivation of w2 from the payoff function f ( zi , z j ) is explained in Appendix A.
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For parameter values used in our simulations, we have w2  0.1 , implying that
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disruptive selection is operating. The population average and the variance of trait values
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in the offspring trait distribution  (z ) are respectively denoted as
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w2
m3 ,
2
w
w
2
2
Var [ z ]  m2  ( w1  2 )m2  w1m3  2 m4   2 ,
2
2
Av [ z ]  z  w1m2 
(6a)
(6b)
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where mk denotes the kth centralised moment of  (z ) . The above approximation is
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derived by neglecting m5 and higher order moments (see Appendix B).
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Substep 2: Random sampling (or random genetic drift)
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From the offspring pool  ( z ) , N individuals are independently sampled to
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become adults. Hence, even for a fixed  (z ) , different distributions of adult
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phenotypes are realised according to a probability distribution (probability measure)
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over all possible trait distributions. After sampling N individuals independently from
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 ( z ) , the expectation of the average trait value is identical to the average trait value in
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 (z ) . In contrast, the expectation of the trait variance is decreased by a factor (N-1)/N
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(Evens 2004). Hence, we have the following:
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E[ Av[ Z ]]  Av [ z ]
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E[Var[ Z ]] 
.
N 1
Var [ z ]
N
(7)
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Combining the two sub-steps, the expected changes in the population average
and the variance in one generation are given by
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E[Av[ Z ]]  w1m2 
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E[Var[ Z ]] 
w2
m3 ,
2
(8a)
N 1 
w

2
2
2
m2  w1m3  w1 m2  2 (m4  m2 )   2   m2 .

N 
2

(8b)
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We are particularly interested in the dynamics of the variance of the trait distribution.
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To analyse Eq. (8) further, we here assume that the trait distribution is close to a normal
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distribution whose mean and variance obey the dynamics given by Eq. (8). For normal
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distributions, m3  0 and m4  3m2 . Using these relationships, we have the following
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recursive equations for the mean and variance of the trait.
2
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z  w1m2
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m2 
(9a)
N 1
1
N 1 2
(w 2  w12 )m2 2  m2 
 .
N
N
N
(9b)
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The first, the second, and the third terms on the right-hand side of Eq. (9b) represent the
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effect of natural selection, the loss of diversity due to genetic drift, and the increase of
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diversity due to mutation, respectively. Without selection, the variance m2 converges
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to N 1 2 .
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4. Condition for the Explosion of Trait Variance
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As long as the selection gradient does not vanish ( w1  0 ), the mean trait value
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changes according to Eq. (9a), which is a standard result of population genetics (Ewens
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2004) and corresponds to the canonical equation in adaptive dynamics theory
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(Dieckmann & Law 1996, Champagnat et al. 2006). If a singular point z*, where the
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selection gradient vanishes ( w1  0 ), is convergence stable, then the system approaches
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the singular point with time (Geritz et al. 1997). Once the mean trait (i.e., the first
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moment of trait distribution) reaches the singularity, it will not change any longer.
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However, the trait variance (i.e., the second centralised moment of the distribution) may
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change with time. See Figure 2 for a schematic illustration. Here we focus on the
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recursion for the trait variance:
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m2 
1
N 1
N 1
2
w2 m2  m2 
 2
N
N
N
(10)
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which indicates that the one generational change in the trait variance is equal to the sum
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of three terms: the natural selection term, the genetic drift term and the mutation term.
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The right-hand side of Eq. (10) represents the "force" acting on the trait variance
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(Figure 3). Note that the orders of the three terms in Eq. (10) are m2 2 , m 2 , and 1,
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respectively. When the variance is small, the mutation (the third term) is dominant, and
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it increases the trait variance. As the variance increases, the genetic drift (the second
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term) increases its magnitude and it reduces the trait variance. The balance between
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these two processes would result in the equilibrium m2 *  N 2 , which represents the
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variance determined by mutation-drift balance. When the trait variance is large,
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selection (the first term) becomes dominant. Selection decreases the trait variance when
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w 2  0 (stabilising selection), but it increases the trait variance when w 2  0
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(disruptive selection).
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Combing these, we have three different cases, depending on the parameters.
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Case (i) When w2  0 , stabilising selection is at work. Then the dynamics of the trait
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variance (Eq. (10)) has a single positive equilibrium m2 * 
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which is globally stable, and the trait variance converges to this value for any initial
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value. At this equilibrium, the trait variance is maintained by mutation, which increases
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variance, and by genetic drift and stabilising selection, which decrease variance.
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Especially when the stabilising selection is weak, trait variance converges to the level
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maintained by mutation-drift balance ( m 2*  N  1 2 ) when w 2 is small.
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1  1  4( N  1) 2 w2  2
,
2( N  1) w2
1
, disruptive selection is at work, but it is not
4 ( N  1) 2
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Case (ii) When 0  w2 
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very strong. Then the dynamics of the trait variance have two positive equilibria: the
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smaller one is stable, but the larger one is unstable. If the trait variance starts from a
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small value, it converges to the smaller equilibrium, where the trait variance is
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maintained mainly by the mutation-drift balance. If the trait variance starts from a larger
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value, then disruptive selection dominates, leading to the unlimited growth of the trait
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variance.
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Case (iii) When w2 
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the trait variance no longer has a stable equilibrium. Starting from any initial value, the
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trait variance keeps increasing without limit.
2
1
, disruptive selection is strong. The dynamics of
4 ( N  1) 2
2
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Figure 4 shows a bifurcation diagram of the variance dynamics given by Eq.
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(10), with the stabilising/disruptive selection coefficient ( w2 ) as a bifurcation parameter.
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Figure 5 shows a bifurcation diagram in which the population size (N) is a bifurcation
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parameter. We have a similar bifurcation diagram where mutational effects (  2 ) serve
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as a bifurcation parameter (not shown). We regard the explosion of the trait variance in
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the dynamics Eq. (10) as evolutionary branching. Then, these bifurcation diagrams
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show that evolutionary branching occurs as a result of a saddle-node bifurcation.
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Evolutionary branching is predicted to occur at a non-ES singular point when disruptive
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selection intensity, population size, mutation rate, and mutation step size are sufficiently
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large. Specifically, deterministic dynamics Eq. (10) predict that evolutionary branching
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occurs when 4 2 ( N  1) 2 w2  1 holds.
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5.
Stochastic Calculation of the Explosion of Variance
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The deterministic dynamics of the trait variance represented by Eq. (10)
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indicates only the expectation of one generational change. There must be stochasticity
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caused by the genetic drift. Results of an individual-based model, such as those
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illustrated in Figure 1, clearly show stochasticity. To account for stochasticity, the
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deterministic difference equation Eq. (10) needs to be modified by an additional term.
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In this section, we derive the stochastic differential equation model considering the
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stochasticity of trait variance dynamics.
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The deterministic dynamics Eq. (10) can be approximated by the following
ordinary differential equation (ODE):
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d
m2  f (m2 ) ,
dt
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(11a)
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where
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( N  1) w2 x 2  x  ( N  1)  2
f ( x) 
.
N
(11b)
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If stochasticity has no correlation over generations, and if the paths can be approximated
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as continuous, then we can incorporate the Brownian motion term, and the differential
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equation Eq. (11) becomes the following stochastic differential equation (SDE):
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dM 2  f ( M 2 )dt 
2
M 2 dB ,
N
(12)
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where dB denotes Brownian motion and M 2 denotes a random variable representing
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the trait variance. f ( M 2 ) is the systematic change per generation in variable M 2 and
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is given by Eq. (11b).
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The second term on the right-hand side of Eq. (12) represents stochasticity in
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trait variance caused by random genetic drift (see Appendix C for derivation). The
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stochastic fluctuation term depends on the trait variance itself (geometric Brownian
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motion), because the amplitude of fluctuation in the trait variance in each generation of
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offspring is large when the trait variance in each parental generation is large. The
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stochastic term also depends on population size, because fluctuation is averaged out in
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large populations. Hence, random genetic drift becomes less important in larger
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populations. The system is similar to but different from the motion of a particle moved
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by "force" shown in Figure 3 under heat noise.
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5.1 Stationary distribution
The corresponding Fokker-Planck equation (or forward Kolmogorov equation)
to Eq. (12) is given by
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

1 2 2 2

p(m2;t)  
f (m2 ) p(m2 ;t)

2  m 2 p(m 2 ;t)  ,

2 m2 N
t
m2
(13)
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(Feller, 1971), where pm2 ;t dm2  Pr m2  M 2 t   m2  dm2  is the probability
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density for the trait variance m2 . The stationary distribution of the trait variance is
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~
p ( x)  C x 3e( N 1) a ( x ) ,
(14a)
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where C is a normalising constant and
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a ( x) 
w2 x 2   2
.
x
(14b)
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Figure 6 illustrates an example of stationary distribution of the trait variance, showing
356
that the system tends to show slightly smaller trait variance than the locally stable value
357
in Eq. (11), because the fluctuation is weaker when the variance is smaller (geometric
358
Brownian motion).
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Comparison with the results of simulations:
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We compared the stationary distribution of the trait variance between the
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theoretical prediction given by Eq. (14) and the observed distribution obtained from the
363
individual-based simulation (Section 2). Figure 7 shows good agreement between the
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diffusion approximation (i.e., the SDE model) and the corresponding simulation result
365
with a small population size. In particular, the approximation captures the shift of the
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mode — the peak of the distribution lies at the value smaller than N 2  0.0008 . The
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average trait variance in an interval shown in Figure 7 is close to N 2 for both
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analytic prediction (Eq. (14)) and simulation results.
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5.2. Waiting time for evolutionary branching
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In Case (ii), there are stable and unstable equilibria for the trait variance, the
372
latter being larger than the former. If the initial value of trait variance is smaller than the
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unstable equilibrium, the deterministic model Eq. (11) predicts that variance would be
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maintained indefinitely. In contrast, if the initial value is larger than the unstable
375
equilibrium, it quickly increases and grows without limit. Hence, whether trait variance
376
eventually explodes depends on the initial condition. In the presence of stochasticity,
377
however, the trait variance would not be maintained at smaller values than the unstable
378
equilibrium; instead, it would fluctuate around the stable equilibrium for a period, but
379
then eventually increase beyond the unstable equilibrium and diverge to infinity. Hence,
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under the stochastic differential equation, the trait variance is predicted to explode
381
eventually even if the initial value is small. The timing of this explosion is controlled by
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the stochasticity.
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The mean waiting time prior to explosion is a typical "first passage time"
384
problem in diffusion theory. This has also been used in calculating the mean time of
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fixation in population genetics (Ohta and Kimura 1968), and the mean time to
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extinction in conservation biology (e.g. Lande 1993; Hakoyama and Iwasa, 2000), and
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physiology (e.g. Ricciardi, 1977).
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To study the expectation of waiting time, we assume that a system has a
389
reflecting boundary at m2  l  0 and an absorbing boundary at m2  r (note that the
390
left boundary can never be achieved because the deterministic "force" is increasing the
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variance while stochastic fluctuation vanishes at m2  0 , i.e., Eq. (12) is singular at
392
m2  0 ). Let T ( ) be the expected waiting time until the system reaches the right
21
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absorbing boundary m2  r when the initial value of the trait variance is m2 (0)   .
394
To obtain T ( ) , we use the following backward Kolmogorov equation
395
d
2 d2
T ( ) 
T ( ) ,
d
N d 2
 1  f ( )
396
(15)
397
398
with the boundary condition T (r )  0 and T ' (l )  0 (see Appendix E). The solution
399
is
400
401
T ( )  N 
r

l
y
x 3 ye( N 1)( a ( x )a ( y )) dxdy
(16)
402
403
where a(x) is given by Eq. (14b). The expected waiting time is smaller when
404
population size is larger, disruptive selection is stronger, or mutational effects are larger
405
(see Appendix D).
406
407
Comparison with the results of simulations:
408
We compared the mean waiting times observed in the computer simulations of
409
the individual-based model and the prediction by the approximate stochastic differential
410
equation model given by Eq. (16). Once the system reaches a convergence stable point,
411
the trait variance m2 is expected to be close to the locally stable value N 2 (see
22
412
also our results on stationary distribution). Thus, we numerically calculate T ( N 2 )
413
and compare the corresponding result in individual-based simulation (Figure 8). We also
414
derive the probability distribution of the first passage time by using Kolmogorov's
415
backward equation in Appendix E, but the fit of the shape of the probability distribution
416
of waiting times to the distribution observed in individual-based simulations is less than
417
perfect (Figure 9).
418
As we decrease N, the waiting time increases exponentially, and it eventually
419
becomes impossible to escape from the locally stable unimodal state in a reasonable
420
time (Figure 8). This implies that evolutionary branching is unlikely to occur in a small
421
population.
422
When N is very large, genetic drift can be neglected and the system is
423
approximated by the ODE; dm2 / dt  w2 m2   2 . Thus, the waiting time can be
424
approximated by the time required to move from m2  N 2 to the right boundary
425
m2  r . This time is about 1500 for N = 1000 and 960 for N = 2000. This calculation, in
426
conjunction with Figure 8, shows that the stochastic fluctuation plays a dominant role
427
even in a population of 1000 individuals.
2
428
Our reduced model measures genetic diversity in trait distribution. An
429
alternative measure of genetic diversity is the number of segregating alleles, which
430
depends only on  and is predicted by Ewen's sampling formula when selection is
431
absent (Ewens 2004). Figure 10 shows the effect of mutation rate  and mutation step
23
432
size  2 . Simulations confirmed our prediction that only the value of  2 matters,
433
even though the number of segregating alleles in simulation with N = 1800 was about
434
50 and 150 for   0.0025 and 0.01, respectively. It is also confirmed that smaller
435
mutational effects require a longer waiting time until evolutionary branching.
436
437
6. Discussion
438
In this paper, we have studied the effect of stochastic fluctuation due to finite
439
population size on evolutionary branching. We have derived an ordinary differential
440
equation and then a stochastic differential equation for the dynamics of trait variance.
441
The analysis has revealed that there are two different manners in which evolutionary
442
branching occurs: deterministic branching and stochastic branching. In case of
443
deterministic branching, evolutionary branching occurs relatively soon after the mean
444
trait reaches the branching point. This occurs when disruptive selection intensity w2
445
exceeds a threshold value: w2  1 /(4  2 ( N  1) 2 ) , which can be approximated by
446
1 N  2 w2  2 . This condition for deterministic branching implies that the intensity of
447
random genetic drift 1 N is greater than disruptive selection intensity w 2 and the
448
mutational effects  2 (see later). If the opposite inequality holds, i.e., when
449
disruptive selection is weaker than the threshold, then a unimodal distribution is locally
450
stable in the deterministic model and stochastic fluctuation is necessary for the
451
population to undergo evolutionary branching. We call this latter type of evolutionary
24
452
branching "stochastic branching". We observed a relatively long waiting time until
453
evolutionary branching occurs in simulations. The distribution and expectation of the
454
waiting time were mathematically calculated and these predictions were consistent with
455
computer simulations of the individual-based model. Considering the several
456
assumptions we have adopted, the consistency is better than expected, although not
457
perfect. As population size or mutational effect decreases, the waiting time
458
exponentially increases. Mathematically, this is because these factors appear in the
459
exponent in the expression of the expected waiting time (Eq. 16). Although the increase
460
of the waiting time is a continuous phenomenon resulting from the underlying
461
stochasticity in individual-based simulations, we can understand evolutionary branching
462
by classifying it as either deterministic branching in which stochastic fluctuation is
463
unnecessary or to stochastic branching in which stochastic fluctuation is necessary.
464
Our analysis has revealed that evolutionary branching is almost impossible in
465
some biological situations in which the effective population size or mutational effect is
466
sufficiently small. As a population size increases, our condition for deterministic
467
evolutionary branching approaches and converges to the standard branching point
468
condition. The condition can be written as 4w2 N 2  2  1 , which can be used as an
469
approximate indicator; for example a population with size N = 108 with mutational
470
effects  2  1014 is predicted to show deterministic branching when disruptive
471
selection intensity is w2  O(1) , although such individual-based simulation is almost
25
472
impossible.
473
The condition separating deterministic and stochastic branching is understood
474
intuitively by considering the force suppressing variance (genetic drift) and the two
475
forces promoting the variance (disruptive selection and mutation). The balance between
476
them determines which type of branching occurs. The condition for deterministic
477
branching 1 N  2 w2  2 can be rewritten as (1 N ) 2  4w2   2 which means
478
(genetic drift)2 < 4 × (selection effect) × (mutation effect).
479
This implies that deterministic branching occurs when the intensity of genetic drift is
480
smaller than double the geometric average of the selection and mutation intensities. It is
481
important to note that the condition is determined by N 2 instead of N .
482
In order to mimic the full individual-based model by the deterministic and
483
stochastic equations of the trait variance, we have adopted several simplifying
484
assumptions in deriving the dynamics: i.e. trait distribution is assumed to be close to
485
normal, and the trait variance is small enough to justify series expansion. From
486
simulation results, it is clear that the trait distribution finally becomes multimodal when
487
evolutionary branching occurs. Thus, our analytic results are only valid for the initial
488
growth of the trait variance where unimodal distribution (approximated by a normal
489
distribution) becomes broader at the singular point. This initial explosion is followed by
490
highly non-linear and complex dynamics that realises a multimodal distribution.
491
Nevertheless, the agreement of our analytic prediction and simulation results suggests
26
492
that our reduced model captures the onset of evolutionary branching.
493
Based on individual-based simulations, Claessen et al. (2007) reported delayed
494
evolutionary branching and proposed two reasons: random genetic drift and extinction
495
of incipient branches. In contrast to Claessen et al. (2007), our model has no fluctuation
496
in the total population size. Thus, the delayed or absent evolutionary branching is
497
caused by demographic stochasticity— the fluctuation in the number of offspring each
498
individual produces. Our simple model has enabled us to derive the condition for
499
evolutionary branching mathematically, with random genetic drift taken into
500
consideration, which has been only verbally proposed before. In addition, our
501
mathematical analysis and simulation results have revealed that the mutation rate and
502
step size are as important as population size.
503
In this paper, to illustrate the effect of genetic drift on the evolutionary
504
branching, we used as an example a pairwise interaction game that incorporated a
505
Wright-Fisher update. We conjecture that the extension of our model to include a Moran
506
update or to a multi-player game is readily possible. In the case of including a Moran
507
update, the loss of genetic diversity (Eq.7) should be appropriately modified and also
508
the unit of time should be rescaled since N Moran steps correspond to a single
509
Wright-Fisher step. The result of our model should be easily extended to
510
resource-competition models, which have been intensively studied in adaptive dynamics
511
(e.g., Dieckmann & Doebeli 1999, Sasaki & Dieckmann 2011, Mirrahimi et al. 2012).
27
512
We have focused on the dynamic processes in which the population escapes
513
from the locally stable value of the trait variance, while keeping its trait distribution as a
514
unimodal distribution. Conceptually, if we allow all possible trait distributions, we
515
might have two locally stable states, one of which corresponds to a unimodal state and a
516
second that corresponds to a fully branched state. Stochastic branching in the present
517
model only considers the case in which a system located at the unimodal state escapes
518
from this local optimum by stochastic fluctuation (random genetic drift). However, a
519
branched state can also evolve back to the unimodal state by genetic drift; thus, a
520
stochastic analysis including both escaping from and evolving back to the unimodal
521
distribution would give us an even deeper understanding of evolutionary branching.
522
28
523
Acknowledgements
524
This work was supported by Japan Science and Technology PRESTO to J.Y.W., a
525
Grant-in-Aid for Scientific Research (B) from JSPS to Y.I., and also by the support of
526
the Environment Research and Technology Development Fund (S9) of the Ministry of
527
the Environment, Japan. We would like to thank the following people for their very
528
helpful comments: C. Hauert, Y. Kobayashi, C. Miura, W. Nakahashi, H. Ohtsuki, and A.
529
Sasaki.
530
29
531
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Wakano, J.Y., and L. Lehmann. 2012. Evolutionary and convergence stability for
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Waxman, D., and S. Gavrilets. 2005. Twenty questions on adaptive dynamics. J. Evol.
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593
33
594
Figure Legends
595
596
Figure 1. Evolutionary dynamics. (a) A simulation run with N = 8000. (b,c) two
597
different simulation runs with N = 1000 (d) A simulation run with N = 200.
598
Parameters:   0.01,   0.02, (b1 , b2 , c1 , c2 )  (6.0,1.4,4.56,1.6)
599
A schematic illustration of phenotype distribution  ( z ) and fitness
600
Figure 2.
601
function w(z ) when the mean trait value z has reached the singular strategy z*. The
602
trait variance is Var[ ( z )]  m2 so the standard deviation is
m2 .
603
604
Figure 3. "Force" plotted against the trait variance based on Eq.10. Dotted, solid, and
605
thick curves represent stabilising selection ( w2 = - 0.2), weak disruptive selection ( w2 =
606
0.2), and strong disruptive selection ( w2 = 0.6), respectively. They also correspond to
607
Case i), ii), and iii) in the main text.
608
Parameters: N=400,   0.01,   0.02
609
610
Figure 4. Stable (thick) and unstable (broken) branches of the trait variance m2 with
611
disruptive selection intensity w2 as a bifurcation parameter. Arrows indicate the
612
direction of systematic change in m2 .
613
Parameters: N=1000,   0.01,   0.02
34
614
615
Figure 5. Stable (thick) and unstable (broken) branches of the trait variance m2 with
616
population size N as a bifurcation parameter. A Log plot is shown. Arrows indicate
617
the direction of systematic change in m2 .
618
Parameters:   0.01,   0.02, (b1 , b2 , c1 , c2 )  (6.0,1.4,4.56,1.6)
619
620
Figure 6. The stationary distribution ~
p (m2 ) of the trait variance m2 (Eq.14). A
621
Log-Log plot is shown, and thus the probability density is much higher at the peak than
622
it appears. The locally stable value of the variance according to Eq.11 is m2 *  0.00081,
623
which is close to N 2 =0.0008 (dashed line). Note that a local peak lies at a value
624
smaller than 0.0008 because of geometric Brownian motion.
625
Parameters: N=200,   0.01,   0.02, (b1 , b2 , c1 , c2 )  (6.0,1.4,4.56,1.6)
626
627
Figure 7. The stationary distribution of the trait variance m2 . Curves represent an
628
analytic result (Eq. 14), while bars are a histogram of realised trait variances in a
629
simulation run for 10 6 time steps. The dashed line represents N 2 =0.0008. In the
630
simulation, the initial trait distribution was set as monomorphic at z =0.2.
631
transition from this initial state to the stationary state was negligible as it took only
632
about 5000 time steps.
633
Parameters: N=200,   0.01,   0.02, (b1 , b2 , c1 , c2 )  (6.0,1.4,4.56,1.6)
35
The
634
635
Figure 8
636
A Log-plot of waiting times for evolutionary branching for different population sizes
637
(N). The thick and thin curves represent analytic result based on a SDE model (Eq.16)
638
and that based on an ODE model (Eq.11), respectively. Each open circle represents the
639
result of a single run in individual-based simulation. The first time when the trait
640
variance exceeded 0.1 is shown. In the simulation, the initial trait distribution was set as
641
monomorphic at z =0.2 and each run consisted of 106 time steps. Simulation runs in
642
which the variance did not explode are plotted at 10 6 . Time required to reach a singular
643
point in the simulation was roughly 5000 when N = 200 and 2000 when N = 1800,
644
which is included to draw open circles. Thus, open circles are expected to lie slightly
645
above the curve.
646
Params:   0.01,   0.02, (b1 , b2 , c1 , c2 )  (6.0,1.4,4.56,1.6)
647
648
Figure 9. The distribution of waiting times. Curves represent an analytic result (see
649
Appendix E), while bars represent a histogram of the waiting time obtained in 1000
650
different runs of individual-based simulation.
651
Parameters: N = 1000,   0.01,   0.02, (b1 , b2 , c1 , c2 )  (6.0,1.4,4.56,1.6)
652
653
Figure 10.
A Log-plot of waiting times for evolutionary branching for different
36
654
population sizes and mutation parameters. The upper panel (   0.0025,   0.02 )
655
depicts a result for a smaller mutation rate than in Figure 8, while the lower panel
656
(   0.01,   0.01 ) depicts a result for a smaller mutation step size. As  2  10 6 is
657
the same, the analytic prediction curves are the same. The two simulation results are
658
also undistinguishable.
659
Parameters: (b1 , b2 , c1 , c2 )  (6.0,1.4,4.56,1.6)
660
661
662
663
37
664
Appendices
665
666
Appendix A: Fitness function and the coefficients
667
Because individuals are playing a game, their payoff, and thus their fitness
668
depend not only on an individual's own trait but also on their opponents' traits. In
669
principle, it is impossible to determine a fitness function for an individual with trait z
670
by an average trait z because more information is needed (i.e., the distribution of the
671
trait  (z ) ). However, when the trait variance Var[ ( z )]  m2 is small, fitness is
672
mainly determined by the deviation of the trait value from the average, z  z , and can
673
be approximated by
674
w( z )  w( z )  w1 ( z  z ) 
w2
(z  z)2
2
(A-1)
675
where the derivatives w1 and w2 are evaluated at a situation for which all individuals
676
(except self) carry the average trait z . In other words, Eq.(A-1) represents invasion
677
fitness. The coefficients are functions of z , so that w1  w1 ( z ) and w2  w2 ( z ) .
678
According to adaptive dynamics, a singular point z* satisfies w1 ( z*)  0 and it is
679
convergence stable if
680
a singular point is convergence stable but not evolutionarily stable, evolutionary
681
branching is known to occur.
682
d
w1 ( z )
 0 . It is evolutionarily stable if w2 ( z*)  0 . When
dz
z z*
Here we mainly focus on a case in which the average trait remains at z  z * ,
38
683
and thus we treat w1 and w2 as if they were constant coefficients. They can be
684
explicitly calculated for finite N, but when N >100 it is well approximated by the value
685
obtained by an infinite population limit. In this limit, fitness function is denoted as
w( z ) 
686
687
690
(A-2)
and thus
688
689
f ( z , z*)
,
f ( z*, z*)
w1 


1


f ( z , z*)
f ( z*, z*)  z
z  z* 
(A-3a)
w2 
 2

1
 2 f ( z , z*)


f ( z*, z*)  z
z  z* 
(A-3b)
and
691
hold. To calculate the intensity of disruptive selection, not only the sign but also the
692
magnitude is important. For the model of Doebeli et al. (2004), this becomes
693
w2 
2(b2  c2 )
f z*, z *
(A-4)
c1  b1
. We numerically confirmed that the value of w2 can be well
4b2  2c2
694
where z* 
695
approximated
696
(b1 , b2 , c1 , c2 )  (6.0,1.4,4.56,1.6)
697
magnitude of disruptive selection depends on b1 and c1 , as well as b2 and c2 . In the
by
w2  0.1
for
the
parameter
values
and for the population sizes N >100. The
39
698
vicinity of our choice of parameter values used in all figures, disruptive selection is
699
stronger for smaller values of b1 and larger values of c1 .
700
701
Appendix B: Derivation of deterministic approximation
702
When both selection and mutation are weak, their effects can be approximated
703
as additive and we obtain the replicator equation Eq.(4). By integrating Eq.(5), which is
704
the Taylor expansion of w(z) around
z   z ( z )dz ,
705
706
we have
w    ( z )w( z )dz  w( z ) 
707
708
709
710
711
712
(B-1)
w2
m2
2
(B-2)
where
mk   ( z  z ) k ( z )dz
(B-3)
is the kth centralised moment around z . Putting this into Eq.(4), we have
w
 2  2
 w

 ( z )   ( z )   2 m2  w1 ( z  z )  2 ( z  z ) 2  ( z, t ) 
 ( z ) . (B-4)
2
2 z 2
 2

Imposing zero-flux boundary conditions  / z  0 , we have
  ( z )dz  1
713
714
w2
m3
2
w
 w

2
2
  ( z )( z  z ) dz  1  22 m2 m2  w1m3  22 m4  
  ( z )( z  z )dz  w m
1
2

We denote statistical values of the offspring distribution by
40
(B-5)
Av [ z ]    ( z ) zdz
715
716
(B-6)
Var [ z ]    ( z )( z  Av [ z ]) 2 dz
then we have Av [ z ]  z  w1m2 
717
w2
m3 , which is Eq.(6a).
2
To calculate the variance in the offspring pool, we use
Var [ z ]    ( z )( z  z )  ( Av [ z ]  z )  dz
2
718
719
   ( z ) z  z  dz  ( w1m2 
2
(B-7)
w2
m3 ) 2
2
and we have
Var [ z ]  m2 
720
721
When
722
Var [ z ]  m2  ( w1 
we
w2 2
w
w
m2  w1m3  2 m4   2  ( w1m2  2 m3 ) 2 .
2
2
2
neglect
2
the
fifth
and
higher
order
terms,
(B-8)
we
have
w2
w
2
)m2  w1m3  2 m4   2 , which is Eq.(6b).
2
2
723
724
Appendix C: Derivation of SDE approximation
725
We want to approximate the dynamics by an SDE with the following form
726
dM 2  f ( M 2 )dt  g ( M 2 ) dB
(C-1)
727
where dB denotes Brownian motion and M 2 denotes a random variable representing
728
the variance in trait distribution. To obtain g (x) , we use an established result in
729
statistics that a random variable
730
731
N
Z z
X   i

v 
i 1 
2
(C-2)
obeys the chi-squared distribution where Z i 's are N independent random variables each
41
732
of which is sampled from a normal distribution with mean z and variance v 2 . The
733
variance of the chi-squared distribution is 2N. When N is large, we can approximate z in
734
Eq.(C-2) by Z . The trait variance M 2 can be written by
M2 
735
1
N
 Z
N
i 1
i Z 
2
v2
X,
N
(C-3)
736
M 2 is a random variable, due to random genetic drift. Its variance over probability
737
distribution is given by
2
 v2 
2v 4
.
Var[ M 2 ]  2 N   
N
N
738
(C-4)
739
According to our model assumptions, we approximate the variance v 2 by the trait
740
variance in offspring distribution  ( z ) . Then, we have the following approximation:
Var[ M 2 ] 
741
742
2m2
N
2
(C-5)
which yields g ( x)  2 x 2 / N and Eq.(12).
743
By a more detailed calculation, we can show
744
Var[ M 2 ] 
2( N  1)m2
,
N2
2
(C-6)
745
which results in ~
p ( x)  x  ( 3 N 2 ) /( N 1) e Na ( x ) , but for our purpose Eq.(C-5) is precise
746
enough.
747
42
748
749
Appendix D: Properties of waiting times
Here, we show that the expectation of waiting time
T  N
750
r

 l
y
x  3 ye ( N 1)( a ( x )  a ( y )) dxdy
(D-1)
751
is a decreasing function of w2 , N, and  2 . First, T is an increasing function of
752
a( x)  a( y ) , which satisfies
1 1
a ( x)  a ( y )   w2 ( y  x)   2   
x y
753
(D-2)
754
Since y>x holds in the region of integration in Eq.(D-1), the waiting time T is a
755
decreasing function of w2 and  2 . To show the dependence on N, we use
r y
d
T    x 3e ( N 1)( a ( x )a ( y )) B ( x, y ) dxdy
 l
dN
756
757
758
(D-3a)
where
B ( x, y )  y  N
( x  y )( wxy   2 )
x
(D-3b)
759
When N is large, B(x,y) has the same sign as x-y, which is negative. Thus, T is a
760
decreasing function of N for large N. In addition, since a ( x)  a( y ) is a negative
761
constant independent of N, we have
762
lim N  T  0
(D-4)
763
with all the other parameters fixed at positive values. This is true even for the vanishing
764
mutation rate (i.e. N 2  const . when N   ). However, it does not necessarily
765
go to zero if N (a ( x)  a( y )) does not go to   . For example, if Nw2 and N 2
43
766
remain constant as N   , the waiting time T can be an increasing function of N.
767
768
Appendix E: Distribution of waiting times
769
Let  (x ) be an indicator function such that  ( r )  1 and  ( x )  0 for
770
l  x  r . Then a random variable  ( M 2 (t )) is unity if absorption at m2  r has
771
occurred until time t, and it is zero otherwise. Let u ( , t )  E  ( M 2 (t )) where E
772
denotes the expectation in the stochastic process with the initial value M 2 (0)   . Then
773
the following Kolmogorov backward equation holds (Karlin & Taylor 1981)
 2u
u 1
u
 g ( ) 2
 f ( )

 2
t
774
(E-1)
775
Let Q be a random variable representing the first passage time, then it is intuitively clear
776
that
777
u( t,)  Pr[Q  t | m 2 (0)  ]
(E-2)
778
is the probability that absorption at m2  r has occurred on or before time t. The
779
probability density that absorption occurs exactly at time t is given by u / t . A similar
780
logic and calculation were adopted in a classical paper studying the fixation probability
781
in population genetics (Kimura 1957). The expected first passage time is
782
783

T ( )   t
0
u
dt .
t
(E-4)
After some calculations, we obtain
44
784
 1  f ( )
d
2 d2
T ( ) 
T ( )
d
N d 2
785
which is Eq.(15). To obtain the distribution of waiting time, we numerically solve the
786
partial differential equation Eq.(E-1) with domain   [l , r ] . We impose a reflecting
787
boundary at left and Dirichlet boundary at right u (t , r )  1 . In the numerical algorithm,
788
the initial value is set as u (0, )  0 except u (0, r )  1 . The FTCS method with small
789
dt and d values gives the numerical solution u~ (t , ) . To check the validity of this
790
result, we numerically put it into Eq.(E-4) and compared the result with that of the
791
numerically obtained value of Eq.(16). The expected waiting times obtained by the two
792
different approaches were almost the same. The distribution of the waiting time t from
793
the initial trait variance N 2 is given by u~ (t , N 2 ) , which is shown in Figure 9.
794
45
Wakano & Iwasa, Figure 1
 (z )
w(z )
m2
z
z*
Wakano & Iwasa, Figure 2
Case iii)
Case ii)
Case i)
Wakano & Iwasa, Figure 3
Case i)
no
branching
m2
0.05
0.04
Case ii)
stochastic
branching
0.03
Case iii)
deterministic
branching
w2 * 
1
4  2 N 2
0.02
0.01
-0.04 -0.02 0.00
0.02
0.04
0.06
0.08
0.10
w2
Wakano & Iwasa, Figure 4
stochastic
branching
1
deterministic
branching
0.1
m2
0.01
0.001
N *  2  2 / w2
10 -4
200
400
600
800
1000
N
Wakano & Iwasa, Figure 5
1000
~
p (m2 )
10
0.1
0.001
0.0001
0.001
0.01
0.1
m2
Wakano & Iwasa, Figure 6
frequency
m2
Wakano & Iwasa, Figure 7
Wakano & Iwasa, Figure 8
waiting time
Wakano & Iwasa, Figure 9
Wakano & Iwasa, Figure 10