From stochastic, individual-based models to the canonical equation of adaptive dynamics – in one step
We consider a model for Darwinian evolution in an asexual population with a large but non-constant populations size characterized by a natural birth rate, a logistic death rate modelling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population () size, rare mutations (), and small mutational effects (), proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, e.g. by Champagnat and Méléard, we take the three limits simultaneously, i.e. and , tend to zero with , subject to conditions that ensure that the time-scale of birth and death events remains separated from that of successful mutational events. This slows down the dynamics of the microscopic system and leads to serious technical difficulties that requires the use of completely different methods. In particular, we cannot use the law of large numbers on the diverging time needed for fixation to approximate the stochastic system with the corresponding deterministic one. To solve this problem we develop a ”stochastic Euler scheme” based on coupling arguments that allows to control the time evolution of the stochastic system over time-scales that diverge with .
Key words and phrases:adaptive dynamics, canonical equation, large population limit, mutation-selection individual-based model
In this paper we study a microscopic model for evolution in a population characterized by a birth rate with a probability of mutation at each event and a logistic death rate, which has been studied in many works before [6, 7, 8, 9, 13]. More precisely, it is a model for an asexual population in which each individual’s ability to survive and to reproduce is a function of a one-dimensional phenotypic trait, such as body size, the age at maturity, or the rate of food intake. The evolution acts on the trait distribution and is the consequence of three basic mechanisms: heredity, mutation and selection. Heredity passes the traits trough generations, mutation drives the variation of the trait values in the population, and selection acts on individuals with different traits and is a consequence of competition between the individuals for limited resources or area.
The model is a generic stochastic individual-based model and belongs to the models of adaptive dynamics. In general, adaptive dynamic models aim to study the interplay between ecology (viewed as driving selection) and evolution, more precisely, the interplay between the three basic mechanisms mentioned above. It tries to develop general tools to study the long time evolution of a wide variety of ecological scenarios [10, 11, 21]. These tools are based on the assumption of separation of ecological and evolutionary time scales and on the notion of invasion fitness [19, 20]. While the biological theory of adaptive dynamics is based on partly heuristic derivations, various aspects of the theory have been derived rigorously over the last years in the context of stochastic, individual-based models [6, 7, 8, 9, 15, 16]. All of them concern the limit when the population size, , tends to infinity. They either study the separation of ecological and evolutionary time scales based on a limit of rare mutations, , combined with a limit of large population [6, 9], the limit of small mutation effects, , [7, 9, 15], the stationary behavior of the system , or the links between individual-based and infinite-population models . A important concept in the theory of adaptive dynamics is the canonical equation of adaptive dynamics (CEAD), introduced by U. Dieckmann and R. Law . It is an ODE that describes the evolution in time of the expected trait value in a monomorphic population. The heuristics leading to the CEAD are based on the biological assumptions of large population and rare mutations with small effects and the assumption that no two different traits can coexist. (Note that we write sometimes mutation steps instead of effects.) There are mathematically rigorous papers that show that the limit of large population combined with rare mutations leads to a jump process, the Trait Substitution Sequence, , and that this jump process converges, in the limit of small mutation steps, to the CEAD, . Since these two limits are applied separately and on different time scales, they give no clue about how the biological parameters (population size , probability of mutations and size of mutation steps ) should compare to ensure that the CEAD approximation of the individual-based model is correct.
The purpose of the present paper is to analyse the situation when the limits of large population size, , rare mutations, , and small mutation steps, , are taken simultaneously. We consider populations with monomorphic initial condition, meaning that at time zero the population consists only of individuals with the same trait. Then we identify a time-scale where evolution can be described as a succession of mutant invasions. To prove convergence to the CEAD, we show that if a mutation occurs, the individuals holding this mutant trait can either die out or invade the resident population on this time scale, where invasion means that the mutant trait supersedes the resident trait i.e. the individuals with the resident trait become extinct after some time. This implies that the population stays essentially monomorphic with a trait that evolves in time. We will impose conditions on the mutation rates that imply a separation of ecological and evolutionary time scales in the sense that an invading mutant population converges to its ecological equilibrium before a new invading (successful) mutant appears. In order to avoid too restrictive hypothesis on the mutation rates, we do, however, allow non-invading (unsuccessful) mutation events during this time, in contrast to all earlier works.
We will see that the combination of the three limits simultaneously, entails some considerable technical
difficulties. The fact that the mutants have only a -dependent small evolutionary advantage
decelerates the dynamics of the microscopic process such that the time of any
macroscopic change between resident and mutant diverges with .
This makes it impossible to use a law of large numbers as in 
to approximate the stochastic system with the corresponding deterministic system during the time of invasion.
Showing that the stochastic system
still follows in an appropriate sense the
corresponding competition Lotka-Volterra system (with -dependent coefficients)
requires a completely new approach. Developing this approach, which can be seen
as a rigorous ”stochastic Euler-scheme”, is the main novelty of the present paper. The proof requires methods, based on
with discrete time Markov chains combined some standard potential theory arguments for the ”exit from a domain problem”
in a moderate deviations regime, as well
as comparison and convergence results of branching processes.
Note that since the result of  is already different from
classical time scales separations results (cf. ), our result differs from them a fortiori.
Thus, our result can be seen as a rigorous justification of the biologically motivated, heuristic assumptions which lead to
The remainder of this paper is organised as follows. In Section 2 and 3 we introduce the model and give an overview on previous related results. In Section 4 we state our results and give a detailed outline of the proof. Full details of the proof are presented in the Section 6, 7 and 8. In the appendix we state and prove several elementary facts that are used throughout the proof.
2. The individual-based model
In this section we introduce the model we analyze.
We consider a population of a single asexual species that is composed of a finite number of individuals, each of them characterized by a one-dimensional phenotypic trait.
The microscopic model is an individual-based model with non-linear density-dependence, which has already been studied in ecological or evolutionary contexts by many authors [8, 6, 9, 13].
The trait space is assumed to be a compact interval of . We introduce the following biological parameters:
is the rate of birth of an individual with trait .
is the rate of natural death of an individual with trait .
is a parameter which scales the population size.
is the competition kernel which models the competition pressure felt by an individual with trait from an individual with trait .
with is the probability that a mutation occurs at birth from an individual with trait , where is a scaling parameter.
is the mutation law of the mutational jump . If the mutant is born from an individual with trait , then the mutant trait is given by , where is a parameter scaling the size of mutation and is a random variable with law . We restrict for simplicity the setting to mutation measures with support included in .
The three scaling parameters of the model are the population size, controlled by the scaling parameter , the mutation probability, controlled by the scaling parameter , the mutation size, controlled by the scaling parameter . The novelty of our approach is that we consider the case where all these parameters tend to their limit jointly, more precisely that both and are functions of and tend to zero as tends to infinity (subject to certain constraints).
At any time we consider a finite number, , of individuals, each of them having a trait value . It is convenient to represent the population state at time by the rescaled point measure, , which depends on , and
Let denote the integral of a measurable function with respect to the measure . Then and for any , the positive number is called the density of trait at time . With this notation, an individual with trait in the population dies due to age or competition with rate
Let denote the set of finite nonnegative measures on , equipped with the weak topology, and define
Similar as in , we obtain that the population process, , is a -valued Markov process with infinitesimal generator, , defined for any bounded measurable function from to and for all by
The first and second terms are linear (in ) and describe the births (without and with mutation), but the third term is non-linear and describes the deaths due to age or competition. The density-dependent non-linearity of the third term models the competition in the population, and hence drives the selection process.
We will use the following assumptions on the parameters of the model:
, and are measurable functions, and there exist such that
For all , , and there exists such that , .
The support of is a subset of and uniformly bounded for all . This means that there exists an such that
, where for any .
Assumptions (i) and (iii) allow to deduce the existence and uniqueness in law of a process on with infinitesimal generator (cf. ). Note that Assumption (iii) differs from the assumptions in  because we restrict the setting to mutation measures with support included in and that it ensures that a mutant trait remains in . Assumption (ii) prevents the population from exploding or becoming extinct too fast. Since is compact, Assumption (iv) ensures that the derivatives of the functions and are uniformly Lipschitz-continuous.
3. Some notation and previous results
We start with a theorem, due to N. Fournier and S. Méléard, which describes the behavior of the populations process, for fixed and , when .
Theorem 3.1 (Theorem 5.3 in ).
Fix and .
Let Assumption 1 hold and assume in addition that
the initial conditions converge for in law and for the weak topology on
to some deterministic finite measure
and that .
Then for all , the sequence , generated by , converges for in law, in , to a deterministic continuous function . This measure-valued function is the unique solution, satisfying , of the integro-differential equation written in its weak form: for all bounded and measurable functions, ,
Without mutation one obtains a convergence to the competitive system of Lotka-Volterra equations defined below (see ).
Corollary 3.2 (The special case and is n-morphic).
If the same assumptions as in the theorem above with hold and if in addition , then is given by , where is the solution of the competitive system of Lotka-Volterra equations defined below.
For any , we denote by the competitive system of Lotka-Volterra equations defined by
Next, we introduce the notation of coexisting traits and of invasion fitness (see ).
We say that the distinct traits and coexist if the system admits an unique non-trivial equilibrium, named , which is locally strictly stable in the sense that the eigenvalues of the Jacobian matrix of the system at are all strictly negative.
The invasion of a single mutant trait in a monomorphic population which is close to its equilibrium is governed by its initial growth rate. Therefore, it is convenient to define the fitness of a mutant trait by its initial growth rate.
If the resident population has the trait , then we call the following function invasion fitness of the mutant trait
The unique strictly stable equilibrium of is , and hence for all .
There is a relation between coexistence and invasion fitness (cf. ).
There is coexistence in the system if and only if
The following convergence result from  describes the limit behavior of the populations process, for fixed , when and . More precisely, it says that the rescaled individual-based process converges in the sense of finite dimensional distributions to the ”trait substitution sequence” (TSS), if one assumes in addition to Assumption 1 the following ”Invasion implies fixation” condition.
Given any , Lebesgue almost any satisfies one of the following conditions: (i) or (ii) and .
Note that by Proposition 3.6, this means that either a mutant cannot invade, or cannot coexist with the resident.
Theorem 3.7 (Corollary 1 in ).
Fix also and let be a sequence of -valued random variables such that converges for in law to and is bounded in for some . Consider the processes generated by with monomorphic initial state .
Then the sequence of the rescaled processes converges in the sense of finite dimensional distributions to the measure-valued process
where the -valued Markov jump process has initial state and infinitesimal generator
Here we write if when . Note that,
for any , the convergence does not hold in law for the Skorokhod topology on , for any topology such that the total mass function is continuous, because the total mass of the limit process is a discontinuous function.
The main part of the proof of this theorem is the study of the invasion of a mutant trait
that has just appeared in a monomorphic population with trait .
The invasion can be divided into three steps.
Firstly, as long as the mutant population size
is smaller than a fixed small , the resident population size
stays close to . Therefore, can be approximated by a linear branching process with birth rate and death rate
until it goes extinct or reaches .
has reached , for large , is close to the solution of
with initial state , which reaches the -neighborhood of in finite time. This is a consequence of Corollary 3.2.
Finally, once is close to and
is small, can be approximated by a subcritical process, which becomes extinct a.s. .
The time of the first and third step are proportional to , whereas the time of the second step is bounded.
Thus, the second inequality in (3.5) guarantees that, with high probability,
the three steps of invasion are completed before a new mutation occurs.
Without Assumption 2 it is possible to construct the ”polymorphic evolution sequence” (PES) under additional assumptions on the -morphic logistic system. This is done in . Finally, in , the convergence of the TSS with small mutation steps scaled by to the ”canonical equation of adaptive dynamics” (CEAD) is proved. We indicate the dependence of the TSS of the previous Theorem on with the notation .
Theorem 3.8 (Remark 4.2 in ).
If Assumption 1 is satisfied and the family of initial states of the rescaled TSS, , is bounded in and converges to a random variable as , then, for each , the rescaled TSS converges when , in the Skorohod topology on , to the process with initial state and with deterministic sample paths, unique solution of the ordinary differential equation, known as CEAD:
where denotes the partial derivative of with respect to the first variable .
Note that this result does not imply that, applying to the individual-based model first the limits and afterwards the limit yields its convergence to the CEAD. One problem of theses two successive limits is, for example, that the first convergence holds on a finite time interval, the second requires to look at the Trait Substitution Sequence on a time interval which diverges. Moreover, as already mentioned these two limits give no clue about how , and should be compared to ensure that the CEAD approximation is correct.
4. The main result
In this section, we present the main result of this paper, namely the convergence to the canonical equation of adaptive dynamics in one step. The time scale on which we control the population process is and corresponds to the combination of the two time scales of Theorem 3.7 and 3.8. Since we combine the limits we have to modify the assumptions to obtain the convergence. We use in this section the notations and definitions introduced in Section 3.
For all , .
Assumption 3 implies that either : or : . Therefore coexistence of two traits is not possible. Without loss of generality we can assume that, , . In fact, a weaker assumption is sufficient, see Remark 3.(iii).
Fix and let be a sequence of -valued random variables such that
converges in law, as , to the positive constant and
is bounded in , for some .
For each , let be the process generated by with monomorphic initial state . Then, for all , the sequence of rescaled processes, , converges in probability, as , with respect to the Skorokhod topology on to the measure-valued process , where is given as a solution of the canonical equation of adaptive dynamics,
with initial condition .
If for , then (4.3) is , i.e. the process stops.
We can prove convergence for a stronger topology. Namely, let us equip , the vector space of signed finite Borel-measures on , with the following Kantorovich-Rubinstein norm:
where is the space of Lipschitz continuous functions from to with Lipschitz norm one (cf.  p. 191). Then, for all , we will prove that
By Proposition 9.1 this implies convergence in probability with respect to the Skorokhod topology.
The main result of the paper actually holds under weaker assumptions. More precisely, Assumption 3 can be replaced by
Assumption 3’. The initial state has a.s. (deterministic) support with satisfying
The reason is that since is continuous, the Assumption 3 (a) is satisfied locally and since is Lipschitz-continuous, the CEAD never reaches an evolutionary singularity (i.e. a value such that ) in finite time. In particular, for a fixed , the CEAD only visits traits in some interval of where . By modifying the parameters of the model out of in such a way that everywhere in , we can apply Thm. 4.1 to this modified process and deduce that has support included in for with high probability, and hence coincides on this time interval.
The condition allows mutation events during an invasion phase of a mutant trait, see below, but ensures that there is no ”successful” mutational event during this phase.
The fluctuations of the resident population are of order , therefore ensures that the sign of the initial growth rate is not influenced by the fluctuations of the population size. We will see later that if a mutant trait appears in a monomorphic population with trait , its initial growth rate is since .
is the time the resident population stays with high probability in a -neighborhood of an attractive domain. This can be seen as a moderate derivation result. Thus the condition ensures that the resident population is still in this neighborhood when a mutant occurs.
Note that the that we use in the proof of the theorem and in the main idea below will not depend on , but it will converge to zero in the end of the proof of Theorem 4.1. The constant introduced below is going to be fixed all the time. It depends only the parameters of the model, in particular not on and .
The conditions about family of initial states imply that and therefore, since , the family of random variable is uniformly integrable (cf.  Lem. 1).
4.1. The main idea and the structure of the proof of Theorem 4.1
Under the conditions of the theorem the evolution of the population will be described as
a succession of mutant invasions.
We prove that, on the timescale of this result, coexistence of two traits cannot occur,
namely, when a mutant trait invades the population, the resident trait (i.e the trait that gave birth to the mutant trait) dies out.
We say the mutant trait fixates in the population. Note that this does not prevent coexistence with other mutant traits that do not invade.
In order to analyze the invasion of a mutant we divide the time until a mutant trait has fixated in the population into two phases.
The two invasion phases: (compare with Figure 1) First, we prove that, as long as all mutant densities are smaller than , the resident density stays in an -neighborhood of . Note that, because the mutations are rare and the population size is large, the monomorphic initial population has time to stabilize in an -neighborhood of this equilibrium before the first mutation occurs. (The time of stabilization is of order and the time where the first mutant occurs is of order ). This allows us to approximate the density of one mutant trait by a branching process with birth rate and death rate such that we can compute the probability that the density of the mutant trait reaches , which is of order , as well as the time it takes to reach this level or to die out. Therefore, the process needs mutation events until there appears a mutant subpopulation which reaches a size . Such a mutant is called successful mutant and its trait will be the next resident trait. (In fact, we can calculate the distribution of the successful mutant trait only on an event with probability , but we show that on an event of probability , this distribution has support in . Therefore, the exact value of the mutant trait is unknown with probability , but the difference of the possible values is only of order .) We prove in this step also that there are never too many different mutants alive at the same time. From all this we deduce that the subpopulation of the successful mutant reaches the density , before a different successful mutant appears. Note that we cannot use large deviation results our time scale as used in  to prove this step. Instead, we use some standard potential theory and coupling arguments to obtain estimates of moderate deviations needed to prove that a successful mutant will appear before the resident density exists a -neighborhood of its equilibrium.
Second, we want to prove that if a mutant population with trait reaches the size , it will increase to an -neighborhood of its equilibrium density . Simultaneously, the density of the resident trait decreases to and finally dies out. Since the fitness advantage of the mutant trait is only of order , the dynamics of the population process and the corresponding deterministic system are very slow. (Even if we would start at a macroscopic density , the deterministic system needs a time of order to reach an -neighborhood of its equilibrium density). Thus we can not apply the law of large numbers for density-dependent population processes (see Chap. 11 of ) on our time scales which was used in  and  to approximate the population process by the solution of the corresponding competition Lotka-Volterra system. This is the main difficulty, which requires entirely new techniques. The method we develop to handle this situation can be seen as a rigorous stochastic ”Euler-Scheme”. Nevertheless, the proof contains an idea which is strongly connected with the properties of the deterministic dynamical system. Namely, the deterministic system of equations for the case has an invariant manifold of fix points with a vector field independent of pointing towards this manifold. Turning on a small , we therefore expect the stochastic system to stay close to this invariant manyfold and to move along it with a speed of order . With this method we are able prove that, in fact, the mutant density reaches the -neighborhood of and the resident trait dies out. Note that it is possible that a unsuccessful mutant is alive a this time. Therefore, we prove that after the resident trait has died out, there is a time where the population consists only of one trait, namely the one that had fixed, before the next successful mutant occurs. We will divide this phase into several steps. A more detailed outline of the structure of the proof is given in Section 7.
Note that Figure 1 is only a rough draft and not a ”real” simulation.
If and are two random variables on , we write if we can construct a random variable, on the probability space as , such that , and that for all .
If and are two measures in , then we write if:
(i) and (ii)
Note that (i) and (ii) imply that, for all that are monotone increasing and for all ,
Convergence: Given , with the results of the two invasion phases, we will define for all two measure-valued processes, and , in , such that, for all ,
and, for all and ,
for some function such that when . This implies (4.5) and therefore the theorem.
More precisely, let be the random time of the th invasion phase, i.e. the first time after such that a mutant density is larger than , and let be the trait of the th successful mutant. Knowing the random variables and , we are able to approximate and : After the (-1)th invasion phase (of the process ), we define two random times, and , and two random variables and in , such that
Thus we define and through
for some appropriate masses and . In fact, will be approximately for and approximately for . We will prove that the times and are (approximately) exponentially distributed with parameters of order , and that the difference of is of order . The processes and will be constructed by slightly modifying the two processes and in order to make them Markovian. This will imply by standard arguments from  that the processes and converge to when , where is the solution of the canonical equation of adaptive dynamics.
All the remaining sections are devoted to the proof of the Theorem 4.1.
5. An augmented process and some elementary properties
In the proof of Theorem 4.1 we need to construct a augmented process that keeps track of part of the history of the population. In particular, we record the number of mutations that occurred before .
Let denote the set of finite non-negative point measures on rescaled by . We write , where and . The augmented process, , is a continuous time stochastic process with state space . The label of an individual with trait denotes that there were mutational events in the population before the trait appeared for the first time in the population. As in , we give a path-wise description of .
Let and let be the number of individuals holding a mutation of label . Then we rewrite as follows,
In fact, the will be equal in our situation, because the only variation in the trait value is driven by mutational events. We need to define three functions. First, is defined as
Second, us given in terms of by
i.e., if , then . Third, is defined as follows: if , then
Let be a abstract probability space. On this space, we define the following independent random elements:
a -valued random variable (the random initial trait),
a sequence of independent Poisson point measures, , on with intensity measure ,
a sequence of independent Poisson point measures, , on with intensity measure ,
a Poisson point measures, , on with intensity measure .
Let and , then we consider the process defined by the following equation