We consider a stochastic individual-based model for the evolution of a haploid, asexually reproducing population. The space of possible traits is given by the vertices of a (possibly directed) finite graph $G=(V,E)$. The evolution of the population is driven by births, deaths, competition, and mutations along the edges of $G$. We are interested in the large population limit under a mutation rate $μ_K$ given by a negative power of the carrying capacity $K$ of the system: $μ_K=K^{-1/α},α>0$. This results in several mutant traits being present at the same time and competing for invading the resident population. We describe the time evolution of the orders of magnitude of each sub-population on the $\log K$ time scale, as $K$ tends to infinity. Using techniques developed in [Champagnat, Méléard, Tran, 2019] we show that these are piecewise affine continuous functions, whose slopes are given by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. This work generalises [Kraut, Bovier, 2019] to the stochastic setting, and Theorem 3.2 of [Bovier, Coquille, Smadi, 2018] to any finite mutation graph. We illustrate our theorem by a series of examples describing surprising phenomena arising from the geometry of the graph and/or the rate of mutations.
We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the microscopic, stochastic description of a population of discrete individuals characterized by one or several adaptive traits. The population is modelled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trai…
We consider a fitness-structured population model with competition and migration between nearest neighbors. Under a combination of large population and rare migration limits we are particularly interested in the asymptotic behavior of the total population partition on supporting trait sites. For the population without mutation on a finite trait space we obtain the equilibrium configuration and characterize the right time scale for fixation. For the model with mutation on an infinite trait space a jump process-trait substitution tree model is established on a rarer mutation time scale against t…
We consider an interacting particle Markov process for Darwinian evolution in an asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate, and a probability $μ$ of mutation at each birth event. We introduce a renormalization parameter $K$ scaling the size of the population, which leads, when $K\to+\infty$, to a deterministic dynamics for the density of individuals holding a given trait. By combining in a non-standard way the limits of large population ($K\to+\infty$) and of small mutations ($μ\to 0$), we prove that a time scal…
We consider a continuous time stochastic individual based model for a population structured only by an inherited vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and we view the process in the timescale of mutations. Using averaging techniques due to Kurtz (1992), we give a new proof of the convergence of the individual based model to the trait substitution sequence of Metz et al. (1992) first worked out by Dieckman and Law (1996) and rigorously proved by Champagnat (2006): rigging the model …
How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. Population dynamics are driven by births and deaths, mutations at birth, and competition between individuals. Trait values influence ecological processes (demographic events, competition), and competition generates selection on trait variation, thus closing the eco-evolutionary feedback loop. The demographic effec…
We propose a type-dependent branching model with mutation and competition for modeling phylogenies of a virus population. The competition kernel depends for any two virus particles on the particles' types, the total mass of the population as well as genetic information available through the number of nucleotide substitutions separating the virus particles. We consider the evolving phylogenies of this individual based model in the huge population, short reproduction time and frequent mutation regime, and show tightness in the state space of marked metric measure spaces. Due to heterogeneity in …
This paper introduces a stochastic adaptive dynamics model for the interplay of several crucial traits and mechanisms in bacterial evolution, namely dormancy, horizontal gene transfer (HGT), mutation and competition. In particular, it combines the recent model of Champagnat, Méléard and Tran (2021) involving HGT with the model for competition-induced dormancy of Blath and Tóbiás (2020). Our main result is a convergence theorem which describes the evolution of the different traits in the population on a `doubly logarithmic scale' as piece-wise affine functions. Interestingly, even for a relativ…
We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. …
We consider a stochastic model of population dynamics where each individual is characterised by a trait in {0,1,...,L} and has a natural reproduction rate, a logistic death rate due to age or competition and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness. We focus on the limit of large population and rare mutations at several speeds. In particular, when th…
We consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth- and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [Méléard and Tran, EJP, 2012], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present artic…
We present a model for the dynamics of a population of bacteria with a continuum of traits, who compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait with possible mutations. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We consider a stochastic individual-based pure jump process taking values in the space of point measures, and whose jump events describe the individual reproduction, transfer and death mechanisms. In a large population scale, the stochastic process is proved …
In this paper we consider two continuous-mass population models as analogues of logistic branching random walks, one is supported on a finite trait space and the other one is supported on an infinite trait space. For the first model with nearest-neighbor competition and migration, we justify a well-described evolutionary path to the short-term equilibrium on a slow migration time scale. For the second one with an additional evolutionary mechanism-mutation, a jump process-trait substitution tree model is established under a combination of rare mutation and slow migration limits. The transition …
We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mutation steps. When considering the evolution of the population in a long time scale, the contribution of small sub-populations may strongly influence t…
In this work we model the dynamics of a population that evolves as a continuous time branching process with a trait structure and ecological interactions in form of mutations and competition between individuals. We generalize existing microscopic models by allowing individuals to have multiple offspring at a reproduction event. Furthermore, we allow the reproduction law to be influenced both by the trait type of the parent as well as by the mutant trait type. We look for tractable large population approximations. More precisely, under some natural assumption on the branching and mutation mecha…
We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framew…
Understanding the evolution of binary traits, which affects the birth and survival of species and also the rate of molecular evolution, remains challenging. A typical example is the evolution of mating systems in plant species. In this work, we present a probabilistic modeling framework for binary trait, random species trees, in which the number of species and their traits are represented by a two-type, continuous time Markov branching process. We develop our model by considering the impact of mating systems on dN/dS, the ratio of nonsynonymous to synonymous substitutions. A methodology is int…
We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the ca…
We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence, is called polymorphic evolution…
We consider a trait-structured population subject to mutation, birth and competition of logistic type, where the number of coexisting types may fluctuate. Applying a limit of rare mutations to this population while keeping the population size finite leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. The probability of fixation of a mutant is interpreted as a fitness landscape that depends on the current state of the population. It was in adaptive dynamics that this kind of model was first invente…
Horizontal transfer (HT) of heritable information or `traits' (carried by genetic elements, endosymbionts, or culture) is widespread among living organisms. Yet current ecological and evolutionary theory addressing HT is limited. We present a modeling framework for the dynamics of two populations that compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We capture this feedback with a stochastic individual-based model, from whic…
We consider a model of a population of fixed size N in which each individual gets replaced at rate one and each individual experiences a mutation at rate μ. We calculate the asymptotic distribution of the time that it takes before there is an individual in the population with m mutations. Several different behaviors are possible, depending on how μchanges with N. These results have applications to the problem of determining the waiting time for regulatory sequences to appear and to models of cancer development.
Understanding how stochastic and non-linear deterministic processes interact is a major challenge in population dynamics theory. After a short review, we introduce a stochastic individual-centered particle model to describe the evolution in continuous time of a population with (continuous) age and trait structures. The individuals reproduce asexually, age, interact and die. The 'trait' is an individual heritable property (d-dimensional vector) that may influence birth and death rates and interactions between individuals, and vary by mutation. In a large population limit, the random process con…
We consider an individual-based spatially structured population for Darwinian evolution in an asexual population. The individuals move randomly on a bounded continuous space according to a reflected brownian motion. The dynamics involves also a birth rate, a density-dependent logistic death rate and a probability of mutation at each birth event. We study the convergence of the microscopic process when the population size grows to $+\infty$ and the mutation probability decreases to $0$. We prove a convergence towards a jump process that jumps in the infinite dimensional space of the stable spat…
Motivated by models of cancer formation in which cells need to acquire $k$ mutations to become cancerous, we consider a spatial population model in which the population is represented by the $d$-dimensional torus of side length $L$. Initially, no sites have mutations, but sites with $i-1$ mutations acquire an $i$th mutation at rate $μ_i$ per unit area. Mutations spread to neighboring sites at rate $α$, so that $t$ time units after a mutation, the region of individuals that have acquired the mutation will be a ball of radius $αt$. We calculate, for some ranges of the parameter values, the asymp…
Horizontal gene transfer consists in exchanging genetic materials between microorganisms during their lives. This is a major mechanism of bacterial evolution and is believed to be of main importance in antibiotics resistance. We consider a stochastic model for the evolution of a discrete population structured by a trait taking finitely many values, with density-dependent competition. Traits are vertically inherited unless a mutation occurs, and can also be horizontally transferred by unilateral conjugation with frequency dependent rate. Our goal is to analyze the trade-off between natural evol…
We study the adaptive dynamics of predator-prey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexistence and the trait of the resident prey type converges to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. In this case, depending on the parameters being varied, we see (i) the number of coexisting predators remains tight and the di…
We consider an infinite-dimensional system of stochastic differential equations describing the evolution of type frequencies in a large population. The type of an individual is the number of deleterious mutations it carries, where fitness of individuals carrying k mutations is decreased by αk for some α>0. Along the individual lines of descent, new mutations accumulate at rate λper generation, and each of these mutations has a probability γper generation to disappear. While the case γ=0 is known as (the Fleming-Viot version of) Muller's ratchet, the case γ>0 is associated with compensatory mut…
We present an individual-based model of phenotypic trait evolution in two-sex populations, which includes semi-random mating of individuals of the opposite sex, natural death and intra-specific competition. By passing the number of individuals to infinity, we derive the macroscopic system of nonlinear differential equations describing the evolution of trait distributions in male and female subpopulations. We study solutions, give criteria for persistence or extinction, and state theorem on asymptotic stability, which we apply later to particular examples of trait inheritance.
Consider a population where individuals give birth at constant rate during their lifetimes to i.i.d. copies of themselves. Individuals bear clonally inherited types, but (neutral) mutations may happen at the birth events. The smallest subtree containing the genealogy of all the extant individuals at a fixed time τ, is called the coalescent point process. We enrich this process with the history of the mutations that appeared over time, and call it the marked coalescent point process. With the help of limit theorems for Lévy processes with marked jumps established in a previous work (arXiv:1305.…
A superprocess limit for an interacting birth-death particle system modelling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to e…
We consider a general, neutral, dynamical model of biodiversity. Individuals have i.i.d. lifetime durations, which are not necessarily exponentially distributed, and each individual gives birth independently at constant rate λ. We assume that types are clonally inherited. We consider two classes of speciation models in this setting. In the immigration model, new individuals of an entirely new species singly enter the population at constant rate μ(e.g., from the mainland into the island). In the mutation model, each individual independently experiences point mutations in its germ line, at const…
We reconsider the deterministic haploid mutation-selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation's solution (and, in particular, of its equilibrium) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error thresho…
We consider a (sub) critical Galton-Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. The approach combines an extension of Harris representation of Galton-Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given.
We investigate the interplay between two fundamental mechanisms of microbial population dynamics and evolution called dormancy and horizontal gene transfer. The corresponding traits come in many guises and are ubiquitous in microbial communities, affecting their dynamics in important ways. Recently, they have each moved (separately) into the focus of stochastic individual-based modelling (Billiard et al. 2016, 2018; Champagnat, Méléard and Tran, 2021; Blath and Tóbiás 2020). Here, we examine their combined effects in a unified model. Indeed, we consider the (idealized) scenario of two sub-popu…
We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows u…
From stochastic individual-based models to free-boundary Hamilton-Jacobi equations
Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Ecole Polytechnique（巴黎综合理工学院）；Institut polytechnique de Paris（巴黎理工学院）；Univ Toulouse（图卢兹大学）；INSA Toulouse（图卢兹国立应用科学学院）；IMT（图卢兹数学研究所）；Univ. Lille（里尔大学）
Microbial dormancy is an evolutionary trait that has emerged independently at various positions across the tree of life. It describes the ability of a microorganism to switch to a metabolically inactive state that can withstand unfavorable conditions. However, maintaining such a trait requires additional resources that could otherwise be used to increase e.g. reproductive rates. In this paper, we aim for gaining a basic understanding under which conditions maintaining a seed bank of dormant individuals provides a "fitness advantage" when facing resource limitations and competition for resource…
We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits…
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on the trait of the mother and on the number of descendants. In this article, we explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait along the spine by giving its associated infinitesimal generator. We prove a Many-to-One formula and a Many-to-O…
We consider a branching model in discrete time where each individual has a trait in some general state space. Both the reproduction law and the trait inherited by the offsprings may depend on the trait of the mother and the environment. We study the long time behavior of the population and the ancestral lineage of typical individuals under general assumptions, which we specify for applications to some models motivated by biology. Our results focus on the growth rate, the trait distribution among the population for large time, so as local densities and the position of extremal individuals. The …
Some population is made of n individuals that can be of p possible species (or types). The update of the species abundance occupancies is from a Moran mutational model designed by Karlin and McGregor in 1967. We first study the equilibrium species counts as a function of n, p and the total mutation probability $ν$ before considering various asymptotic regimes on n, p and $ν$. Running title: KMG Model with Mutations.
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineag…
We consider an individual based model of phenotypic evolution in hermaphroditic populations which includes random and assortative mating of individuals. By increasing the number of individuals to infinity we obtain a nonlinear transport equation, which describes the evolution of distribution of phenotypic traits. Existence of an one-dimensional attractor is proved and the formula for the density of phenotypic traits in the limiting (asymptotic) population is derived in some particular case.
We consider a population of haploid individuals reproducing sexually, i.e. for which the genome of each individual is a random mixture of the genome of its two parents. We assume that initially one individual carries a mutation at one locus, and that individuals carrying this mutation have an advantage regarding genome transmission. Our aim is to study the long time effect of this mutation on the genetic composition of the population, when population size is large.
We consider a branching population where individuals live and reproduce independently. Their lifetimes are i.i.d. and they give birth at a constant rate b. The genealogical tree spanned by this process is called a splitting tree, and the population counting process is a homogeneous, binary Crump-Mode-Jagers process. We suppose that mutations affect individuals independently at a constant rate $\θ$ during their lifetimes, under the infinite-alleles assumption: each new mutation gives a new type, called allele, to his carrier. We study the allele frequency spectrum which is the numbers A(k, t) o…
We study a mutation-selection model with a fluctuating environment. More precisely, individuals in a large population are assumed to have a modifier locus determining the mutation rate $u \in [0,\vartheta]$ at a second locus with types $v \in [0,1]$. In addition, the environment fluctuates, meaning that individual types change their fitness at some high rate. Fitness only depends on the type of the second locus. We obtain general limit results for the evolution of the allele frequency distribution for rapidly fluctuating environments. As an application, we make use of the resulting Fleming-Vio…
We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $μ_N$, and each beneficial mutation increases the individual's fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual's fitness to give birth. Under certain conditions on the parameters $μ_N$ and $s_N$, we obtain rigorous results for the rate at which mutations accumulate in the population and the distribution of the fitnesses of individuals in the population at a given tim…
We consider a population model where individuals behave independently from each other and whose genealogy is described by a chronological tree called splitting tree. The individuals have i.i.d. (non-exponential) lifetime durations and give birth at constant rate to clonal or mutant children in an infinitely many alleles model with neutral mutations. First, to study the allelic partition of the population, we are interested in its frequency spectrum, which, at a fixed time, describes the number of alleles carried by a given number of individuals and with a given age. We compute the expected val…
We investigate a simple quantitative genetics model subjet to a gradual environmental change from the viewpoint of the phylogenies of the living individuals. We aim to understand better how the past traits of their ancestors are shaped by the adaptation to the varying environment. The individuals are characterized by a one-dimensional trait. The dynamics -- births and deaths -- depend on a time-changing mortality rate that shifts the optimal trait to the right at constant speed. The population size is regulated by a nonlinear non-local logistic competition term. The macroscopic behaviour can b…
