Probability of fixation under weak selection: A branching process unifying approach

https://doi.org/10.1016/j.tpb.2006.01.002Get rights and content

Abstract

We link two-allele population models by Haldane and Fisher with Kimura's diffusion approximations of the Wright–Fisher model, by considering continuous-state branching (CB) processes which are either independent (model I) or conditioned to have constant sum (model II). Recent works by the author allow us to further include logistic density-dependence (model III), which is ubiquitous in ecology.

In all models, each allele (mutant or resident) is then characterized by a triple demographic trait: intrinsic growth rate r, reproduction variance σ and competition sensitivity c. Generally, the fixation probability u of the mutant depends on its initial proportion p, the total initial population size z, and the six demographic traits. Under weak selection, we can linearize u in all models thanks to the same master formulau=p+p(1-p){grsr+gσsσ+gcsc}+o(sr,sσ,sc),where sr=r-r, sσ=σ-σ and sc=c-c are selection coefficients, and gr, gσ, gc are invasibility coefficients ( refers to the mutant traits), which are positive and do not depend on p. In particular, increased reproduction variance is always deleterious. We prove that in all three modelsgσ=1σ,and gr=z/σ for small initial population sizes z.

In model II, gr=z/σ for all z, and we display invasion isoclines of the ‘mean vs variance’ type. A slight departure from the isocline is shown to be more beneficial to alleles with low σ than with high r.

In model III, gc increases with z like ln(z)/c, and gr(z) converges to a finite limit L>K/σ, where K=r/c is the carrying capacity. For r>0 the growth invasibility is above z/σ when z<K, and below z/σ when z>K, showing that classical models I and II underestimate the fixation probabilities in growing populations, and overestimate them in declining populations.

Introduction

Since the pioneering works of Fisher, Haldane and Wright in the modelling of evolution, mathematical models for biological invasions, with particular emphasis on the probability of fixation of a mutant allele, have repeatedly received attention from biologists and mathematicians. Although much progress in this direction has been made since the late 1940s (see in particular Malécot, 1948, Kimura, 1964, Ewens, 1979), a common belief outside the population geneticists’ community is that this field was mapped out by the 1930s. While such a vision is certainly an overstatement, it holds an element of truth in the sense that the work of the three authors, including even early works such as Fisher (1922), Haldane (1924) and Wright (1931), still remain the standing foundations of population genetics. For this reason, major references in diffusion theory and branching processes applied to population genetics are sometimes 70 or 80 years old.

Here, we start with two celebrated models—one by Haldane, 1927, Haldane, 1932 and Fisher, 1922, Fisher, 1930 based on the branching process in its early form and one by Wright, 1931, Wright, 1945, Fisher, 1922, Fisher, 1930 and Kimura, 1957, Kimura, 1962 relying on diffusion theory—that gave rise to widely known analytical results as far as fixation probabilities are concerned. The first mathematical model consists of a pair of independent branching processes, and the second one is a diffusion approximation of the so-called Wright–Fisher model (featuring fixed population size).

In this paper, we provide a unified way of presenting these two models in terms of modern branching diffusions, and we introduce a similar third model, which improves the first two by allowing density-dependence in a natural way, à la Quételet–Verhulst (logistic growth model), thus making the model ecologically more relevant. Our purpose is then to give new results about fixation probabilities in each of these three models (sometimes under weak selection). The comparison and discussion of these results will then be facilitated by the unified framework we have introduced.

The simplest stochastic model used in population dynamics is the well-known Bienaymé–Galton–Watson process or branching process, abbreviated as BGW-process (Athreya and Ney, 1972, Jagers, 1975). In this time-discrete, integer-valued (asexual) process, cells die or split into a random number of daughter cells who behave in the same way as, and independently of, their mother cell. The distribution of the number of replacing cells per capita at each generation (0 when the cell dies) is called the offspring distribution. The fate of the branching process depends on the mean m of the offspring distribution. When m>1 (supercritical regime), the survival probability is positive. When m<1 (subcritical regime) or m=1 (critical regime), extinction occurs eventually with probability 1.

The model considered in Haldane, 1927, Haldane, 1932 and Fisher, 1922, Fisher, 1930 is based on the BGW-process. Consider two populations whose dynamics are driven by two independent BGW-processes. The resident dynamics are driven by a critical (mean 1) process, and the mutant population by a supercritical (mean 1+s) process starting with a single individual. The real number s>0 is called the relative fitness of the (beneficial) mutant allele. In this setting, recalling that critical BGW-processes go extinct with probability 1, Haldane defines the fixation of the mutant as non-extinction of the supercritical process. A famous result is the equivalent of the fixation probability as s0+, when the offspring distributions are Poisson. Using the standard notation u for this probability,u2sass0+.Among the objections that can be raised against this model, we will retain only two. First, the fixation probability does not depend on the initial proportion of the mutant allele in the population and it vanishes in the neutral limit (s=0) (whereas it should converge to this initial proportion). Indeed, this approximation is actually only valid for a mutant allele with selection coefficient s (small but) large enough so that the inverse of the initial population size 1/N is negligible as compared to s (smooth genetic drift compared to the action of selection). Second, when fixation occurs in this model, the fate of the mutant population size is to go to . This well-known property of BGW-processes is of little relevance for natural populations modelling (although large population rescalings can eliminate this effect; see e.g. Champagnat, 2006).

At variance with population dynamics, most popular models in population genetics assume constant population size. The so-called Wright–Fisher model consists in a diploid two-allele population with constant population size 2N, where a homozygous wild-type genotype, a heterozygous genotype and a homozygous mutant genotype are sampled in the ratios 1:1+sh:1+s, where s is the selection coefficient of the mutant allele and h is the dominance coefficient. When there is no dominance (h=12), a celebrated result by Moran (1959/1960), Kimura, 1957, Kimura, 1962 and Gillespie (1974) based on the diffusion approximation of this model isu=1-e-2spNe1-e-2sNe,where p is the initial fraction of mutants and Ne is the effective population size. Indeed, an alternative assumption is to assert that natural populations behave ‘as if’ they had constant size Ne, where Ne is called the (variance) effective size, and is supposed to account for the stochasticity measured in the real population (stochasticity decreases when Ne increases).

A severe shortcoming of this assumption is to ignore ecological interactions responsible for the size fluctuations occurring in natural populations. In particular, it should be of general interest to investigate the fate of mutant alleles in a background population which grows or declines (Fisher, 1958, Kimura and Ohta, 1974b, Otto and Whitlock, 1997). However, one could note that the assumption of fixed population size forces the allele with smallest real (even positive) growth rate, to actually be modelled by a declining population.

It is well-known that branching populations that do not go extinct grow exponentially, thus making the BGW-process as ecologically irrelevant as the Wright–Fisher model. Indeed, negative density-dependence prevents natural populations from growing exponentially without limit in time. The oldest way of modelling negative density-dependence goes back to Quételet, 1835, Verhulst, 1838 and is called the logistic growth model. It allows the population size to converge to a finite limit, and is, therefore, the simplest model in population dynamics resembling those population genetics’ models featuring constant size. In this model, the population size Zt at time t solves the following ordinary differential equation (ODE):dZt=rZtdt-cZt2dt,where competition is enclosed in the negative quadratic term. In this model, the population size is not fixed (although it converges to K:=r/c), but no stochasticity is permitted.

The logistic branching process (Lambert, 2005), abbreviated as LB-process, is the stochastic analogue of the logistic growth model, and is defined as a continuous time branching process with extra deaths due to competition pressure. In a continuous time branching process, individuals give birth at constant rate b and die at constant rate d. In the LB-process, individuals give birth as in the pure branching case, but can die either naturally (at constant rate d) or by competition pressure (at rate c(N-1) when the total population has size N, proportionally to the number of extant conspecifics). The total death rate is thus the sum of the total natural death rate dN (linear) and the total competition death rate cN(N-1) (quadratic). A nice feature of this model is that it allows the population size N to fluctuate randomly through time, and when d=0, to oscillate eternally around a mean value equal to (b/c)/(1-exp(-b/c)). In particular, there is no need to consider any effective population size Ne, since the census size itself is stochastic.

The aim of this paper is to compare three models of competing alleles (regarding the fixation probability of the mutant allele): Haldane and Fisher's model (two independent branching processes, model I), the Wright–Fisher model (constant population size, model II), and an ecological model (two coupled logistic branching processes, model III). To be able to do this comparison properly, we introduce in the next subsection a unifying framework based on the diffusion approximations of branching processes.

A diffusion (Xt,t0) is a random, Markovian, continuous function of the real time t. The law of a diffusion is characterized by its infinitesimal generator G (Rogers and Williams, 1994). The infinitesimal generator of a diffusion is a functional operator that maps twice continuously differentiable real functions into continuous real functions. There are always continuous mappings a and b (b is nonnegative) such that the image Gf by G of any twice differentiable function f is given byGf=af+12bf.Moreover, denoting by x the starting point of the diffusion (X0=x), we also have Gf(x)=limt01t(Ex(f(Xt))-f(x)),so that G characterizes the behaviour of the diffusion X at small times. When X is a d-dimensional diffusion, the generator is of the form Gf=i=1daifxi+12i,j=1dbij2fxixj,where the a's and b's are real functions on Rd (b is symmetric nonnegative).

When population geneticists began using diffusion theory, diffusions were mainly known through their generator as in (4). Thanks to Itô's stochastic integration (integration against the paths of the Brownian motion), however, their pathwise definition was also made possible via stochastic differential equations (SDE, see Revuz and Yor, 1999, for a comprehensive account on that subject). Both viewpoints are interchangeable, but the SDE formalism sometimes allows for a quicker understanding. More specifically, the diffusion with generator G given by (4) is equivalently defined as the solution to the following SDE: dXt=a(Xt)dt+b(Xt)dBt,where B is the standard Brownian motion.

For a diffusion modelling the dynamics of the size of a population, the state 0 is called an absorbing barrier. To compute fixation probabilities in the sequel we will use the following general result. The probability u that a diffusion is tricked at an absorbing barrier x0 solves Gu=0 (where u is taken as a function of the starting point) with boundary condition u(x0)=1 (and u(x1)=0 at any other absorbing barrier x1).

A CB-process is a continuous time, real-valued branching process. Those CB-processes which are continuous functions of time are then called branching diffusions1 and are strong solutions of SDEs of the following type (Lamperti, 1967):dZt=rZtdt+σZtdBt.The real number r is termed the Malthusian intrinsic growth rate, and the positive real number σ is the so-called Gaussian coefficient, or reproduction variance.

As in the discrete setting, a CB-process can either be subcritical (r<0), critical (r=0), or supercritical (r>0). When r=0, Z is the celebrated Feller diffusion (Feller, 1951). When r0, extinction occurs in finite time with probability 1. When r>0, either extinction occurs or Zt grows exponentially as t increases. To compute the survival probability u, recall from diffusion theory that u, as a function of the starting point x, solves 2ru+σu=0, with u(0)=1 and u()=0. Then it is elementary to see that u=1-exp(-2rx/σ).There are several ways of looking at CB-processes (the interested reader is referred to Haccou et al., 2005, p. 83) which eschew natural criticisms regarding diffusions (Gillespie, 1989).

The first way is to check (we will not do it here) that CB-processes are the only diffusions that satisfy the very same additive property as BGW-processes. In words, a CB-process starting from the positive real number z=x+y is the sum of two independent CB-processes starting, respectively, from x and y. Thus, a CB-process can be seen as the total population size summed over several lineages evolving independently (just as BGW-processes).

Another viewpoint is to consider CB-processes as continuous approximations of discrete dynamics, exactly in the same sense as what happens when σ=0. Indeed, the solutions of dZt=rZtdt are straightforward exponentials which are used ubiquitously in population dynamics in lieu of the solutions of Nt+1=(1+R)Nt. The same reasons that motivate these approximations in the deterministic setting apply to the stochastic setting. Let us mention two of them. The first reason is technical. Not only are mathematical computations facilitated when handling continuous objects, but the parameterization of the branching process is much simpler in the continuous case (two parameters r and σ) than in the discrete case (offspring distribution). The second one is phenomenological. Discrete dynamics can be seen as embedded in the continuous dynamics, that is, as a sample of the continuous dynamics at discrete times, which yields nice correspondences between both settings. In the deterministic setting, replace r with ln(1+R), and notice that Nt is then exactly equal to Zt, for t integer. In the stochastic setting, recall that the extinction probability for a discrete branching population starting from n individuals is of the form qn. Now for a continuous population the extinction probability is 1-u=exp(-2rx/σ), so that the extinction probabilities for discrete and continuous populations coincide for x integer, provided r/σ=-ln(q)/2.

Observe that we have not mentioned so far any assumption of large population. But of course, a third and popular way to look at CB-processes is to consider them by way of the angle that led to their discovery, namely, as limits of BGW-processes rescaled in time and state-space (Jirina, 1958). More specifically, consider any (nice) integer-valued branching process N(t) starting from N0=az and let Z(t)=N(at)/a. Then, as a, Z converges to a CB-process starting from z. This rescaling means that CB-processes can be obtained as the density of a population censused over a large area a, but it does not mean that CB-processes can only be used to model large populations. Though, we must specify that when we take the limit z0 in the sequel, we think of a large population with small density z=N0/a1N0.

From now on, ‘model I’ will refer to the case of two independent CB-processes. Strictly speaking, in terms of the parameters of these processes, the continuous counterpart of Haldane and Fisher's model should have r1>r2=0 (the resident undergoes critical dynamics and the mutant supercritical dynamics), and σ1=σ2. However, it will not be necessary to meet such requirements.

To compare branching processes and the Wright–Fisher model, we need to introduce the diffusion approximation of the latter. However, in a particular case of the discrete setting, there is already a well-known link between them. Start with a discrete n-allele diploid population of constant size N, and assume that the dynamics of the 2N individual genes (at a given locus) are those of the neutral Wright–Fisher model. Then observe that the number of copies of each gene left to the next generation follows the multinomial distribution with total number of trials 2N, and success probabilities 1/2N. In addition, the only dependence between the n random quantities of each allele is to have fixed sum 2N. In this case, we remind the reader (see Haccou et al., 2005) that the underlying allele genealogy is that of n independent critical BGW processes, all with Poisson (1) distributed offspring numbers, conditioned on having constant sum 2N through time.

This property translates to the continuous setting in a more general fashion (Proposition 2.1). Set X=X(1)/(X(1)+X(2)), where X(1),X(2) are two independent branching diffusions with growth rates r1,r2, respectively, and both with the same reproduction variance σ (this last restriction is not required in the general result stated in Proposition 2.1). Then conditional on Xt(1)+Xt(2)=z for all times t, (Xt,t0) is a diffusion with generator A, whereAf(p)=(r1-r2)p(1-p)f(p)+σ2zp(1-p)f(p).This result was first mentioned in Gillespie, 1974, Gillespie, 1975. Next, let us turn to the Wright–Fisher diffusion approximation. Wright, 1931, Wright, 1945 followed Fisher, 1922, Fisher, 1930 who had used, for the first time, Kolmogorov forward equations (also called Fokker–Planck equations) to approximate the dynamics of the BGW-process with Poisson offspring. The proper use of general diffusion theory was later made by Feller (1951), Kimura, 1957, Kimura, 1962, Moran (1961), Gillespie (1974), Ewens (1979) and others. When there is no dominance (h=12), the dynamics of the fraction of mutants through time in the Wright–Fisher model can be approximated by the diffusion with generator G (see e.g. Ewens, 1979), whereGf(p)=s2p(1-p)f(p)+12Nep(1-p)f(p).As a consequence, the diffusion approximation of the Wright–Fisher model (7) has exactly the same distribution as a pair of independent CB-processes, with growth rates r1, r2, respectively, and identical reproduction variance σ, conditioned on their sum remaining constant equal to z (6), provided that s=2(r1-r2) and σ=z/Ne. This result justifies that we consider CB-processes as the starting point for unifying classical models in population genetics and population dynamics.

The reproduction variance σ can then be interpreted in several manners. It can be seen as the variance of the offspring produced per time unit and per capita. Since it is a diffusion coefficient, it is a powerful surrogate for demographic stochasticity, allowing us to quantify genetic drift. It can evolve as a life history trait in natural populations, if regarded as the lack of survival investment and parental care (which reduce stochastic causes of mortality). To use more standard terminology, σ is also the ratio of the initial census size over the (variance) effective population size Ne.

Throughout this paper, we will use the term ‘model II’ to refer to the case of two CB-processes conditioned upon their sum being constant.

Real-valued LB-processes are defined as solutions of SDEs of the following type (Lambert, 2005):dZt=rZtdt-cZt2dt+σZtdBt,where c is the so-called competition rate, or competition sensitivity coefficient. When c=0, the LB-process is a mere CB-process, and when σ=0 it reduces to the logistic growth model. When c0 and σ0, the LB-process is a diffusion that cannot go to and so goes extinct with probability 1, regardless of the value of r. However, when r is large, the extinction time can be very long, and its expectation actually increases with r as exp(r2/σc) (unpublished work).

Since it only requires the one extra parameter c, we claim that the LB-process is a most parsimonious and natural density-dependent generalization of branching processes. Etheridge (2004) mentioned this process for the first time as ‘Feller diffusion with logistic growth’.

Model I is a toy-model for randomly fluctuating populations. One step further, model II is a straightforward way of sticking to this scheme while coping with model I's shortcoming of letting population sizes go to . A minimal refinement in this direction is to continue with branching diffusions, but instead of artificially conditioning the total population size to be constant, to introduce density-dependence into the model. One parsimonious way of doing it is to consider two LB-processes where density-dependence is expressed w.r.t. the overall population size. This model will be referred to as ‘model III’. Since density-dependence in the LB-process impedes the population size from going to , model III features randomly fluctuating total population size with negatively correlated mutant and resident populations, where neither of them can increase indefinitely.

In the previous subsections, we explained how to situate Haldane–Fisher's and Wright–Fisher–Kimura's models in the single framework of modern, continuous-state branching processes. In the first case, allele populations’ dynamics are driven by independent branching processes, and in the second case by branching processes with fixed sum. By considering general CB-processes right from the start, we will give direct proofs of old results and provide new interesting formulae for the fixation probability of the mutant gene (when there is no dominance). To improve these two models, we will thirdly consider coupled logistic branching processes as a more natural way of modelling density-dependence than conditioning upon the total population being constant. Let us point out that works on branching processes in questions of fixations of mutant alleles are still common today (though in the discrete setting), e.g. Eshel (1981), Athreya (1992), Haccou and Iwasa (1996), Lange and Fan (1997), Johnson and Gerrish (2002), and De Oliveira and Campos (2004). For further results in fixation probabilities for discrete branching populations and applications to adaptive dynamics, we refer the reader to Champagnat and Lambert (2006).

In Section 2, we give formal definitions of the three models in the unifying CB-process framework. In Section 2.1, we introduce notation for two independent CB-processes modelling, respectively, the wild-type and mutant dynamics. In Section 2.2, we condition these independent diffusions to have constant sum, and prove that after conditioning, the proportion of mutant alleles satisfies a certain SDE, with Wright–Fisher diffusion approximation as a special case (Proposition 2.1). In Section 2.3, we rigorously define the coupled logistic branching processes as CB-processes with added quadratic regulatory mechanism expressed w.r.t. the overall population size.

In Section 3, we consider for each of these three models the fixation probability u of the mutant allele in the population. To correct for the aforementioned shortcomings of Haldane and Fisher's model, we propose a different and more general definition of fixation, which applies to all kinds of models (although it would have to be slightly changed for models including mutation or migration).

Definition 1.1

Fixation of the allele A occurs if the lineages of all other alleles present at time 0 at that locus eventually go extinct, and do so before the lineage of A does, if it does at all.

As is often done in population genetics and most evolutionary setups, and for the sake of mathematical treatment as well, we will generally consider that the demographic traits of the mutant are ‘close’ to those of the wild-type, as will be specified in Section 3.0. This case will be referred to as nearly neutral (or weak selection) setting. Let s denote generically the difference between one triple demographic trait of the mutant (growth rate, reproduction variance or competition sensitivity) and that of the resident. We will focus on what we call the invasibility coefficient g of the resident (z;r,σ,c), where z is the initial total population size, and g can be defined implicitly as u=u0+p(1-p)gs+o(s),where u0 is the so-called neutral fixation probability and p the initial proportion of the mutant allele in the population. In particular, we prove that g does not depend on p (or s), and that u0=p in the vast majority of models, called fair models (Definition 3.1).

In Section 3.1 (model I), we display new results for the fixation probability according to Definition 1.1, and provide nice approximations of the invasibility coefficients gr and gσ. In Section 3.2 (model II), we are able to compute u explicitly (not solely when selection is weak). We thus recover the analogue of (2), and also display new formulae when the reproduction variances of each allele are different (Proposition 3.2). In the nearly neutral setting, the result coincides exactly with that of model I when the resident population underwent critical dynamics (see Eqs. (15) and (17)). We also display a mean–variance trade-off, in the sense that reduced growth rate can be compensated by reduced reproduction variance, therefore inducing situations in which different alleles may have different trait values but still be neutral to each other. We call this situation neutral polymorphism, and further show that a small departure from that trade-off is advantageous for those alleles that initially had small reproduction variance and growth rate, rather than high ones.

The mathematical task in Section 3.3 (model III) is more challenging, but still, we obtain the invasibility coefficients gr, gσ and gc as solutions in z of second-order linear differential equations (Theorem 3.3). We are able to show that gσ=1/σ and to display fine equivalents of gr and gc (Theorem 3.5). Particularly for large z, gc(z)ln(z)/c and gr(z) approaches a finite limit L, which has L>r/(σc). The comparison of growth invasibilities in all three models is interesting. Whereas gr is equal to z/σ in the classical models I and II, in model III gr is above z/σ for z<r/c and below z/σ for z>r/c. This proves that classical models tend to underestimate fixation probabilities in growing resident populations (z<r/c) and overestimate them in declining resident populations (z>r/c). Furthermore, the sensitivity of the fixation probability to the value of the mutant advantage (in either r or c) is at its highest for an intermediate population size z, which is lower than the carrying capacity r/c.

Finally, Section 4 is devoted to discussing and commenting on the model and the results.

Section snippets

The models

In this section, we set up notation and give formal definitions of models I–III. We also clearly specify for each model how to define fixation in accordance with Definition 1.1.

Neutral and nearly neutral settings

In this subsection, we set up the analytical framework corresponding to weak selection, i.e. appropriate for approximations of u when the difference between the mutant and the resident is small but nonzero (nearly neutral setting). Recall that r1,σ1,c1 (r2,σ2,c2) are, respectively, the growth rate, reproduction variance and competition sensitivity of the mutant (resident) allele. In models I and II, c1=c2=0. However, for the sake of generality and conciseness, we will continue characterizing

Comments on the model(s)

The continuous branching (CB) process framework unifies Haldane and Fisher's branching approach, Wright–Fisher–Kimura's diffusion methods, as well as Quételet and Verhulst's so-called logistic process. In this framework, a population is characterized by (at most) three demographic parameters, the growth rate r, the reproduction variance σ>0 and the competition sensitivity c0. See Section 1.3.2 for a discussion on the use of CB-processes in population genetics.

The direct translation of

Acknowledgment

This work was partially supported by a grant from the (French) Ministère de l’Enseignement Supérieur et de la Recherche (ACI IMPB63).

References (47)

  • W.J. Ewens

    Mathematical Population Genetics

    (1979)
  • Feller, W., 1951. Diffusion processes in genetics. In: Proceedings of the Second Berkeley Symposium on Mathematical...
  • R.A. Fisher

    On the dominance ratio

    Proc. R. Soc. Edinburgh

    (1922)
  • R.A. Fisher

    The distribution of gene ratios for rare mutations

    Proc. R. Soc. Edinburgh

    (1930)
  • R.A. Fisher

    The Genetical Theory of Natural Selection

    (1958)
  • S. Gavrilets

    Fitness Landscapes and the Origin of Species. Monographs in Population Biology

    (2004)
  • J.H. Gillespie

    Natural selection for within-generation variance in offspring number

    Genetics

    (1974)
  • J.H. Gillespie

    Natural selection for within-generation variance in offspring number II. Discrete haploid models

    Genetics

    (1975)
  • J.H. Gillespie

    Natural selection for variance in offspring number: a new evolutionary principle

    Am. Nat.

    (1977)
  • Gillespie, J.H., 1989. When not to use diffusion processes in population genetics. In: Feldman, M.W. (Ed.),...
  • D.R. Grey

    Asymptotic behaviour of continuous-time, continuous state-space branching processes

    J. Appl. Probab.

    (1974)
  • P. Haccou et al.

    Branching Processes. Variation, Growth, and Extinction of Populations. Series: Cambridge Studies in Adaptive Dynamics (No. 5)

    (2005)
  • J.B.S. Haldane

    A mathematical theory of natural and artificial selection. Part I

    Trans. Cambridge Philos. Soc.

    (1924)
  • Cited by (71)

    • Quasi-neutral evolution in populations under small demographic fluctuations

      2022, Journal of Theoretical Biology
      Citation Excerpt :

      Fixation probabilities in populations that change size deterministically (and where the population size dynamics are often separated from the evolutionary dynamics) have been extensively studied in the population genetics (Kimura et al., 1974; Uecker et al., 2011; Waxman, 2011). Analytical expressions for fixation probabilities under stochastically varying populations were derived only recently by using diffusion approximations (Lambert, 2005; Lambert, 2006) or by analyzing the Kolmogorov forward equation (Champagnat et al., 2007). A two species system with stochastically varying population size and density dependent birth rates was investigated in Parsons and Quince (2007a), Parsons and Quince (2007b) while the system with density dependent death rates was studied in Parsons et al. (2010).

    • Disentangling eco-evolutionary effects on trait fixation

      2018, Theoretical Population Biology
      Citation Excerpt :

      However, most theoretical studies focus on a constant population size, neglecting potential ecological effects on the population dynamics. Just recently the interaction of evolutionary and population dynamics has gained more attention (Lambert, 2006; Champagnat and Lambert, 2007; Parsons and Quince, 2007a, b; Parsons et al., 2010; Uecker and Hermisson, 2011; Constable and McKane, 2015; Parsons and Rogers, 2017). Classical equations of ecology like the Lotka–Volterra dynamics have to be re-evaluated when finite populations are considered (Gokhale et al., 2013; Papkou et al., 2016).

    View all citing articles on Scopus
    View full text