The interplay of two mutations in a population of varying size: A stochastic eco-evolutionary model for clonal interference

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The interplay of two mutations in a population of varying size: A stochastic eco-evolutionary model for clonal interference Sylvain Billiard a , Charline Smadi b,c,∗ a Unit´e Evo-Eco-Pal´eo, UMR CNRS 8198, Universit´e des Sciences et Technologies Lille, Cit´e scientifique, 59655

Villeneuve d’Ascq Cedex-France, France b Irstea, UR LISC, Laboratoire d’Ing´enierie des Syst`emes Complexes, 9 avenue Blaise Pascal-CS 20085, 63178

Aubi`ere, France c Department of Statistics, University of Oxford, 1 South Park Road, Oxford OX1 3TG, UK

Received 11 May 2015; received in revised form 4 May 2016; accepted 27 June 2016

Abstract Clonal interference, competition between multiple beneficial mutations, has a major role in adaptation of asexual populations. We provide a simple stochastic model of clonal interference taking into account a wide variety of competitive interactions. The population evolves as a three-type birth-and-death process with type dependent competitions. This allows us to relax the classical assumption of transitivity between mutations, and to predict genetic patterns, such as coexistence of several mutants or emergence of Rock–Paper–Scissors cycles, which were not explained by existing models. In addition, we call into questions some classical preconceived ideas about fixation time and fixation probability of competing mutations. c 2016 Elsevier B.V. All rights reserved. ⃝

MSC: 92D25; 60J80; 60J27; 92D15; 92D10 Keywords: Eco-evolution; Clonal interference; Birth and death processes; Lotka–Volterra systems; Couplings; Population genetics

∗ Corresponding author at: Irstea, UR LISC, Laboratoire d’Ing´enierie des Syst`emes Complexes, 9 avenue Blaise Pascal-CS 20085, 63178 Aubi`ere, France. E-mail addresses: [email protected] (S. Billiard), [email protected] (C. Smadi).

http://dx.doi.org/10.1016/j.spa.2016.06.024 c 2016 Elsevier B.V. All rights reserved. 0304-4149/⃝

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Introduction From the works of the ‘great trinity’ of Fisher [20,21], Wright [47] and Haldane [24] the questions of fixation probability and fixation time of new beneficial mutations have been widely studied. These are indeed fundamental questions if we aim at understanding how and how fast a population can adapt to a changing environment, the dynamics of genetic diversity, or the long term behaviour of ecological systems. The first models of adaptation in asexual populations postulated that the beneficial mutations were rare enough for populations to evolve sequentially by rapid fixations of a positively selected mutation alternating with long periods with no mutation (see [40] for a review of these models). However, various empirical evidences [17,32,33] show that in large asexual populations, several mutations can co-occur, which especially can lead to a competition between beneficial mutations. This phenomenon is known as clonal interference [23], and has a major importance for adaptation of asexual populations, such as bacteria and other prokaryotes, yeasts and other fungi, or cancers. Consequently in recent years there has been a growing interest in developing experimental studies and theoretical models to analyse clonal interference [2,23,17,8,16,31,32]. Most models investigating how clonal interference might affect the probability and time of fixation of beneficial mutations, and thus adaptation, made two important but limiting assumptions: first population sizes are constant and independent of the fitness of the individuals, and second, fitnesses only depend on the type of the mutations and not on the state of the population, i.e. fitnesses are assumed transitive. However, as emphasized by Nowak and Sigmund [39], there is a reciprocal feedback between adaptation and environmental changes because of ecological interactions between individuals, especially competition. Mutation with the highest fitness invades the population, but its fitness might depend on the density of the other mutations present in the population and might thus change during the course of adaptation. Thus, fitnesses are not necessarily transitive, i.e. selection can be frequency dependent, and phenomena such as cyclical dynamics or stable coexistence can occur. Interestingly, non-transitivity of fitnesses has been documented empirically in asexual populations [41,30]. In this paper we aim at providing and studying a simple stochastic model taking into account the wide variety of competitive interactions which can be found in nature. We show that type dependent competitive interactions are able to generate ecological patterns which are observed but not explained by conventional models. In particular we relax the classical assumption of transitivity between the different mutations (see the discussion in Section 2.1) and call into questions some classical preconceived ideas about clonal interference (see Propositions 3, 5 and 6). To model precisely the interactions between individuals we extend the model introduced in [9] where the author only considered the occurrence of one mutation. The population dynamics, described in Section 1, is a multitype birth and death Markov process with density-dependent competition. Individuals are characterized by ecological parameters depending on their genetic type and governing their growth rate and competition with other individuals. As a consequence, the “fitness” of an individual depends on the population state and is not an intrinsic characteristic of individuals. We reflect the carrying capacity of the underlying environment by a scaling parameter K ∈ N and state results in the limit for large K . Such an eco-evolutionary approach has been introduced by Metz and coauthors [34] and has been made rigorous in the probabilistic setting in the seminal paper of Fournier and M´el´eard [22]. Then it has been developed by Champagnat, M´el´eard and coauthors (see [9,11,10] and references therein) for the haploid asexual case, by Smadi and coauthors [45,5] for the haploid sexual case, and by Collet, M´el´eard and Metz [12] and Coron and coauthors [14,13] for the diploid sexual

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case. Most of these works make an ‘invasion implies fixation’ assumption, which means that if a mutant population hits a non negligible fraction of the population size, it replaces the resident population. In other words mutations cannot coexist. To the contrary, [11] specifically study the case of the fate of mutants when there is a locally stable coexisting equilibrium. They show that a multidimensional Lotka–Volterra is a good approximation of the process when the population is large, and that the stochastic process with mutations converges to a succession of polymorphic states (they called ‘Polymorphic Evolutionary Sequence’), under the assumption of time scale separation. The main novelty of our work is that we do not a priori assume that an equilibrium is reached between the resident and first mutant populations when the second mutation appears (no time scale separation), and we do not make any assumption on the three-dimensional limiting Lotka–Volterra system in large population. This work is the first one to study the impact of type dependent competition on clonal interference. By using couplings with birth and death processes without competition and comparison with deterministic systems, we are able to provide a complete description of the possible population dynamics when two mutations are in competition. We show that a large variety of behaviours can be observed, but that in all cases, the dynamics consists in an alternation of long stochastic phases (with a duration of order log K ) and short phases (with a duration of order 1) which can be approximated by deterministic processes. In the case of Rock–Paper–Scissors dynamics, we can even have a non bounded number of such alternations (see Proposition 7). The deterministic approximations will be either two-dimensional or threedimensional Lotka–Volterra competitive systems. Unlike the two-dimensional case where the final state is completely determined by the signs of the invasion fitnesses (defined in (1.6) and (1.17)), the long time behaviour of a three dimensional competitive Lotka–Volterra system can depend on the initial state. Moreover, the flows of three dimensional Lotka–Volterra systems do not necessarily converge to a stable equilibrium but can exhibit cyclical behaviours. We will see that the time of appearance of the second mutation is crucial to determine the final population state. For example, the dynamics described in Proposition 7 can only be obtained if the second mutation occurs when the fraction of the wild type-individuals in the population is small. In Sections 1 and 2 we present the model and the main results of the paper. Sections 3–5 are devoted to the study of the dynamics of stochastic and deterministic phases. Section 6 is dedicated to the proofs of the main results. Finally in the Appendices we present technical results (Appendix A) and provide a complete description of the possible population dynamics (Appendix B). 1. Model We consider an asexual haploid population and focus on one locus. The individuals may carry three distinct alleles 0, 1 or 2, and we denote by E := {0, 1, 2} the type space. The population process N K = (N K (t), t ≥ 0) = ((N0K (t), N1K (t), N2K (t)), t ≥ 0), where NiK (t) denotes the number of i-individuals at time t when the carrying capacity is K , is a multitype birth and death process. The birth rate of i-individuals is bi (n) = βi n i ,

(1.1)

where βi > 0 is the individual birth rate of i-individuals and n = (n 0 , n 1 , n 2 ) ∈ Z3+ denotes the current state of the population. An i-individual can die either from a natural death (rate δi > 0),

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or from type-dependent competition: the parameter Ci, j > 0 models the impact an individual of type j has on an individual of type i, where (i, j) ∈ E 2 . The strength of the competition also depends on the carrying capacity K ∈ N, which is a measure of the total quantity of food or space available. This results in the total death rate of individuals carrying the allele i ∈ E:   Ci,1 Ci,2 Ci,0 K (1.2) n0 + n1 + n2 ni . di (n) = δi + K K K Let us now introduce the notions of invasion, final state and time of invasion that we will use in the sequel. We say that a type i ∈ E invades if there exist n¯ i ∈ R+ , β, V > 0 and K ∈ N such that the following event holds: I (i, K , n¯ i , β, V ) := {∀ β log K < t < e V K , NiK (t) > n¯ i K }.

(1.3)

When K is large, it means that i-individuals represent a non negligible fraction of the population during a very long time. It is necessary to bound this duration (by e V K ) as a birth and death process with density dependent competition almost surely gets extinct in finite time. To characterize the long time behaviour of the 0-, 1-, and 2-population sizes, we define the final state set for ε, β, V > 0 and K ∈ N by:   F S(ε, K , β, V ) := n¯ ∈ R3+ : sup ∥N K (t)/K − n∥ ¯ ≤ε , (1.4) β log K
where ∥ · ∥ denotes the L 1 -Norm on RE . Finally, we introduce the time needed for the rescaled population process N K /K to reach a vicinity of n¯ ∈ R3+ and stay in the latter during an exponential time. For ε, V > 0 and K ∈ N:   (ε,K ) TF S(V ) (n) ¯ := inf s > 0, ∀s ∈ (t, e V K ), ∥N K (t)/K − n∥ ¯ ≤ε . (1.5) As a quantity summarizing the advantage or disadvantage a mutant with allele type i has in a j-population at equilibrium, we introduce the so-called invasion fitness Si j through Si j := βi − δi − Ci, j n¯ j ,

(1.6)

where the equilibrium density n¯ i is defined by n¯ i :=

βi − δi . Ci,i

(1.7)

The role of the invasion fitness Si j and the definition of the equilibrium density n¯ i follow from the properties of the two-dimensional competitive Lotka–Volterra system:  (z) (z) (z) (z) (z) n˙ i = (βi − δi − Ci,i n i − Ci, j n j )n i , n i (0) = z i , (1.8) (z) (z) (z) (z) (z) n˙ j = (β j − δ j − C j,i n i − C j, j n j )n j , n j (0) = z j , for z = (z i , z j ) ∈ R2+ . If we assume n¯ i > 0,

n¯ j > 0,

and

S ji < 0 < Si j ,

(1.9)

then n¯ i is the equilibrium size of a monomorphic i-population and the system (1.8) has a unique stable equilibrium (n¯ i , 0) and two unstable steady states (0, n¯ j ) and (0, 0). Thanks to Theorem 2.1 p. 456 in [19] we can prove that if NiK (0) and N jK (0) are of order K , K is large and

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NkK (0) = 0 for k = E \{i, j}, the process (NiK /K , N jK /K ) is close to the solution of (1.8) during any finite time interval (see (A.4) for a precise statement). The invasion fitness Si j corresponds to the per capita initial growth rate of the mutant i when it appears in a monomorphic population of individuals j at their equilibrium size ⌊n¯ j K ⌋. Hence the dynamics of the type i is very dependent on the properties of the system (1.8). It is proven in [9] that under condition (1.9) one mutant i appearing in a monomorphic j-population at its equilibrium size ⌊n¯ j K ⌋ has a positive probability to fix. More precisely, for ε > 0, there exist two finite constants β(ε) and V (ε) such that    Si j  P n¯ i ei ∈ F S(ε, K , β(ε), V (ε))N K (0) = ei + ⌊n¯ j K ⌋e j = + O K (ε), βi

(1.10)

where (ei , i ∈ E) is the canonical basis of RE and O K (ε) is a function of K and ε satisfying lim sup |O K (ε)| ≤ cε,

(1.11)

K →∞

for a finite c. Similarly,    Si j  P n¯ j e j ∈ F S(ε, K , β(ε), V (ε))N K (0) = ei + ⌊n¯ j K ⌋e j = 1 − + O K (ε). βi

(1.12)

Moreover, the invasion time of the mutant i, as defined in (1.5), satisfies for a finite c   log K log K  K (ε,K ) P (1 − cε) < TF S(V (ε)) (n¯ i ei ) < (1 + cε) N (0) = ei + ⌊n¯ j K ⌋e j Si j Si j =

Si j + O K (ε). βi

(1.13)

If we assume on the contrary n¯ i > 0,

n¯ j > 0,

Si j > 0,

S ji > 0

and Ci,i C j, j ̸= Ci, j C j,i ,

(1.14)

then the system (1.8) has three unstable steady states (n¯ i , 0), (0, n¯ j ) and (0, 0) and a unique (i) ( j) stable equilibrium (n¯ i j , n¯ i j ), where (i)

n¯ i j = ( j)

n¯ i j =

C j, j (βi − δi ) − Ci, j (β j − δ j ) Ci,i C j, j − Ci, j C j,i Ci,i (β j − δ j ) − C j,i (βi − δi ) . Ci,i C j, j − Ci, j C j,i

and (1.15)

It is proven in [11] that under condition (1.14) one mutant i has a positive probability to invade then coexist with a j-population during a long time. More precisely, for ε > 0, there exist two finite constants β(ε) and V (ε) such that    Si j  ( j) (i) P n¯ i j ei + n¯ i j e j ∈ F S(ε, K , β(ε), V (ε))N K (0) = ei + ⌊n¯ j K ⌋e j = + O K (ε), βi    Si j  P n¯ j e j ∈ F S(ε, K , β(ε), V (ε))N K (0) = ei + ⌊n¯ j K ⌋e j = 1 − + O K (ε). βi

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Moreover, the invasion time of the mutant i, as defined in (1.5), satisfies for a finite c   log K log K  K ( j) (i) (ε,K ) P (1 − cε) < TF S(V (ε)) (n¯ i j ei + n¯ i j e j ) < (1 + cε) N (0) = ei + ⌊n¯ j K ⌋e j Si j Si j Si j = + O K (ε). βi Analogously, the fate of a mutation of type k occurring when the types i and j coexist in a population of large carrying capacity K , or the result of the competition between the three types of populations when they all have a size of order K is very dependent on the properties of the three-dimensional competitive Lotka–Volterra system:  (z) (z) (z) (z) (z) (z) n i (0) = z i ,  n˙ i = (βi − δi − Ci,i n i − Ci, j n j − Ci,k n k )n i , (z) (z) (z) (z) (z) (z) (1.16) n˙ j = (β j − δ j − C j,i n i − C j, j n j − C j,k n k )n j , n j (0) = z j ,   (z) (z) (z) (z) (z) (z) n˙ k = (βk − δk − Ck,i n i − Ck, j n j − Ck,k n k )n k , n k (0) = z k , where z = (z i , z j , z k ) ∈ R3+ . Similarly as in (1.6) we can define the invasion fitness of the type k in a two-type i/j-population, (i)

( j)

Ski j = Sk ji := βk − δk − Ck,i n¯ i j − Ck, j n¯ i j ,

(1.17)

where we recall definition (1.15). It corresponds to the initial per capita growth rate of an individual of type k appearing in a large population in which the individuals of type i and j ( j) (i) are at their coexisting equilibrium, (⌊n¯ i j K ⌋, ⌊n¯ i j K ⌋). The class of three-dimensional competitive Lotka–Volterra systems has been studied in detail by Zeeman and coauthors [48–50]. They exhibit much more variety than the two-dimensional ones, and we will recall the long time behaviour of their flows in Section 4.2. The complexity of the three-dimensional competitive Lotka–Volterra systems entails a broad variety of dynamics in the case of two mutations occurring successively in a population. In particular, depending on the population state when the second mutation occurs, the new type can be advantageous or not when rare: for example, we can have S20 > 0 and S210 < 0, which implies that if the mutant 2 occurs in a 0-population at equilibrium the 2-population size has a positive probability to hit a non-negligible fraction of the population, whereas if it appears in a two-type 0/1-population at its coexisting equilibrium it gets extinct immediately. In the sequel, we assume that a mutant of type 1 occurs in a population of type 0 (also called wild type) at equilibrium at time 0, and that the mutants 1 are beneficial when rare. In other words we assume Assumption 1. N K (0) = ⌊n¯ 0 K ⌋e0 + e1

and

S10 > 0.

Let us describe in few words what happens when there is only the first mutation (we refer to [9] for the proofs). We can distinguish three phases in the mutant invasion (see Fig. 1): an initial phase in which the fraction of mutant-individuals does not exceed a fixed value ε > 0 and where the dynamics of the wild-type population is nearly undisturbed by the invading type. A second phase where both types account for a non-negligible percentage of the population and where the dynamics of the population can be well approximated by the system (1.8). And finally a third phase where either the roles of the types are interchanged and the wild-type population

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500 1000 1500 2000 2500

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Fig. 1. The three phases of a mutant invasion. The y-axis corresponds to the two type population sizes (0 in black, 1 in red), and the x-axis to the time. In this simulation, K = 1000, (β0 , β1 ) = (2, 3), δi = 0.5, Ci, j = 1, (i, j) ∈ {0, 1}2 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

is near extinction (when (1.9) holds), or the two types stay close to their coexisting equilibrium (when (1.14) holds). The duration of the invasion process is of order log K with a deterministic phase which only lasts an amount of time of order 1. More precisely, during the first phase, when the population 1 has a size smaller than ⌊εK ⌋, this latter can be approximated by a supercritical birth and death process, with respective individual birth and death rates β1

and

δ1 +

C1,0 n¯ 0 K = β1 − S10 < β1 . K

Hence from well known properties of supercritical birth and death processes (see [1] for example) we know that with a probability close to S10 /β1 the 1-population size hits the value ⌊εK ⌋ in a time of order log K /S10 for large K . The second phase can be approximated by the dynamical system (1.8) with (i, j) = (0, 1) and the system takes a time of order 1 to get (0) (1) close to (N0K , N1K ) = (⌊n¯ 01 K ⌋, ⌊n¯ 01 K ⌋) (if S01 > 0) or to (N0K , N1K ) = (⌊ε 2 K ⌋, ⌊n¯ 1 K ⌋) (if S01 < 0). Finally, if S01 > 0 the populations 0 and 1 stay close to their coexisting equilibrium (0) (1) (⌊n¯ 01 K ⌋, ⌊n¯ 01 K ⌋) during a time of order e V K for a positive V and if S01 < 0 the 0-population size is comparable to a subcritical birth and death process, with respective individual birth and death rates β0

and

δ0 +

C0,1 n¯ 1 K = β0 + |S01 | > β0 . K

Hence it gets extinct almost surely, in a time close to log K /|S01 |. To summarize, for large K , the first and third phases have a duration of order at least log K and the second phase has a duration of order 1. Hence if a second mutation appears during the sweep, this occurs with high probability during the first or the third phase. We denote by α log K the time of occurrence of the second mutation and we distinguish three cases: • Either the mutation 2 occurs during the first phase of the mutant 1 invasion and the 1-population size hits ⌊εK ⌋ before the 2-population, which corresponds to Assumption 1 and: Assumption 2. S20 > 0

and

0∨

 1 1  1 − <α< . S10 S20 S10

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• Or it occurs during the first phase but the invasion fitness S20 > 0 is large enough for the 2-population to hit size ⌊εK ⌋ before the 1-population, which corresponds to Assumption 1 and: Assumption 3. S20 > 0

and

0 < α < 1 S20 >S10

 1 1  − . S10 S20

• Or it occurs during the third phase, which corresponds to Assumption 1 and: Assumption 4. One of the following condition holds (i) S01 > 0 and α > 1/S10 ; (ii) S01 < 0 and 1/S10 < α < 1/S10 + 1/|S01 |. In Appendix B we give a complete description of the possible population dynamics. Assumptions 2 and 3 are similar in terms of final states of the system: it is enough to interchange the types 1 and 2 and to modify the duration before hitting the final state (see Appendix B.2 in the Appendices). As a consequence, we will only highlight the outcomes under Assumption 2 in the main results of the paper. We will nevertheless study the properties of the different phases of the process under both assumptions (see Sections 3–5) to get the needed tools to study all the cases. Remark 1 (About Our Assumptions on Mutation). Two remarks can be done. First, the time of occurrence of the second mutation is a parameter in our model, α log K , while it is a random variable in natural and experimental conditions, depending on the mutation rate. We expect that the dynamics observed in our model and described hereafter are most probable only under the condition that the mutation rate is neither too high, otherwise much more than three mutants would co-occur in the population, nor too low, otherwise the dynamics essentially consist in the competition between two mutants. Since the total time taken by a beneficial mutant 1 to invade a resident population is of order logK and that the total number of individuals is of order K , we expect the dynamics that we describe when the probability of beneficial mutation per reproductive event is of order 1/(K log K ) (beneficial in the sense that its probability of invasion is positive). Of course, this remark holds true only if the mutants 1 do not stably coexist with the resident individuals of type 0, otherwise it is expected that a third type of individuals will eventually occur, as soon as the mutation probability per reproductive event is larger than 1/(K e V K ). Second, we assume in our model that the ecological parameters of mutation 2 are independent of its occurrence time. However, in natural or experimental conditions, the second mutation occurs in a given background: either mutant 0 or 1 gives birth to the new mutant 2. The probability that mutation 2 occurs in the 0 or 1 genetic background only depends on their relative frequencies in the population. Mutant 2 has a higher probability to be derived from wild type 0 if it occurs during the first stochastic phase, or from mutant 1 if it occurs during the second stochastic phase, or depending on their expected relative frequencies if both 0 and 1 stably coexist. The ecological parameters of mutation 2 can thus be very different depending on its ancestor. We chose to remain as general as possible in the present work and decided not to take into account such a phenomenon. However we study it, and especially its biological implications, in a companion paper [3]. We now highlight the biologically relevant outcomes.

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2. Results 2.1. Transitive versus non-transitive fitnesses Let us say a word about classical models of clonal interference (see [2,23,17,8,16] for instance). The population size is constant (or infinite). The fitness si of an i-individual corresponds to its exponential growth rate (Malthusian parameter) and only depends on its type i. Suppose that in a population of type 0 a beneficial mutation 1 is followed by a second beneficial mutation 2 before the fixation of the type 1 individuals. In population genetics, “beneficial” means that s1 > 0 and s2 > 0, which implies that the mutant populations have a positive probability to escape genetic drift and reach a positive fraction of the population. Then there are two possibilities: 1. Either s1 > s2 : then the 1-population outcompetes the 0- and 2-populations and the mutation 1 becomes fixed. 2. Or s2 > s1 : then the 2-population outcompetes the 0- and 1-populations and the mutation 2 becomes fixed. If a third mutation (individuals of type 3) occurs with fitness satisfying s3 > s2 , then not only the type 3 individuals outcompete the type 2 individuals, but by transitivity of the total order > on R, they also outcompete the type 1 and type 0 individuals, and so on. In other words, in population genetics of haploid asexuals, the fitnesses are transitive in the sense that if 1 outcompetes 0 and 2 outcompetes 1, then necessarily 2 outcompetes 0. Such a model is natural when competitive interactions between individuals are simple: in an experiment with only one limiting resource, beneficial mutations often correspond to an increase of resource consumption efficiency. But let us imagine an environment with two resources, A and B. A mutant which prefers resource A (resp. B) will be favoured in a population of individuals which consume preferentially resource B (resp. A). In this example there is no transitivity. Moreover, experiments on cancer and viral cells have shown the long term coexistence of several mutant strains [27], which cannot be explained by a model with transitive fitnesses. More generally, it is known [39] that ecological interactions cause non-transitive phenotypic interactions. Hence to model complex interactions, we need another definition of “fitnesses”, to incorporate the dependency on the population state. This is achieved by the notion of invasion fitnesses (defined in (1.6) and (1.17)), which naturally follows from the model that we have presented. The main novelty of our approach is to consider type dependent competitive interactions. Indeed, if all the competitive interactions (Ci, j , (i, j) ∈ E 2 ) are the same, then for three individual’s types i, j and k, the invasion fitnesses satisfy: Si j > 0 ⇐⇒ S ji = β j − δ j − C n¯ i = β j − δ j − C(βi − δi )/C = −Si j < 0, and Si j > 0

and

S jk > 0 =⇒ Sik = Si j + S jk > 0.

In other words, it boils down to the case of transitive fitnesses. We can have a more precise result on the form of competitions allowing non transitive relations between several mutants. To state this latter we introduce the following order for (i, j) ∈ E 2 : i ≺ j ⇐⇒ Si j < 0 < S ji ,

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i = j ⇐⇒ Si j .S ji ≥ 0, and the notations C˜ i, j =

Ci, j C j, j

and ρi = βi − δi .

(2.1)

Then we can state the following result which will be proven in Section 6.1 Proposition 1. Let C1 ≤ C2 be two positive real numbers and i, j, k ∈ E. Assume that for every m, n ∈ {i, j, k}, m ̸= n we have C1 ≤ C˜ m,n ≤ C2 . 1. If  1 2 > C2 C1 ∨ C2

 1 2 ∧ C2 < C1 C1

and

(2.2)

then i≺ j

and

j ≺ k =⇒ i ≺ k.

2. If  1 2 C1 ∨ > C2 C2

 1 2 ∧ C2 < C1 C1

or

(2.3)

then i≺ j

and

j ≺ k =⇒ i ≼ k.

3. If  1 2 C1 ∨ < C2 C2

 1 2 ∧ C2 > C1 C1 then there exist some ecological parameters (ρm , C˜ m,n , m ̸= n ∈ {i, j, k}) such that i ≺ j,

j ≺k

and

and

(2.4)

k ≺ i.

Remark 2. By looking at all the possible subcases we can show the following equivalencies: (2.2) ⇐⇒ C22 < C1 < C2 < 1

or

1 < C1 < C2 < C12 ,

(2.3) ⇐⇒ C1 < C22 < C2 < 1

or

1 < C1 < C12 < C2 ,

(2.4) ⇐⇒ C1 < 1 < C2 . This shows that transitive relations are more likely when the rescaled competitions (C˜ i, j , (i, j) ∈ E 2 ) are close to each other and both smaller or greater than 1. By allowing dependency of competitive interactions on the individual’s types, we are able to model new ecological patterns found in nature and describe them mathematically. In particular, we will show that it allows us to model coexistence of several mutant populations, and complex dynamics as Rock–Paper–Scissors cycles. To study the effect of interactions between the two mutations we need to compare the case where the two mutations occur with the case where only one mutation occurs. To this aim, we introduce the following notation P(k) (.) = P(.|the mutation(s) k occur(s)),

k ∈ {{1}, {2}, {1, 2}},

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where we recall that the mutation 1 occurs at time 0 and the mutation 2 at time α log K . In other words, P({2}) (.) means that the initial population state is ⌊n¯ 0 K ⌋e0 and not ⌊n¯ 0 K ⌋e0 + e1 , P({1}) (.) means that the initial population state is ⌊n¯ 0 K ⌋e0 +e1 (Assumption 1 holds) and that the mutation 2 never occurs, and finally P({1,2}) (.) means that Assumption 1 holds and mutation 2 occurs at time α log K . For the sake of readability we omit the curly bracket in the sequel and write P(1) (.), P(2) (.) and P(1,2) (.). 2.2. Does clonal interference speed up or slow down invasion? In population genetic models of clonal interference, the authors concluded that the presence of other mutants slowed down the invasion of the fittest mutant as it had to outcompete some individuals fitter than the wild type individuals [35,36,23,17]. In our individual based model, the result of the competition between two mutant subpopulations depends on the state of the total population, and the interplay between different mutations can take various forms. For some values of the parameters, clonal interference does indeed slow down the invasion of a beneficial mutant. Recall Eqs. (1.10)–(1.13) about the invasion of one mutant. Then the presence of the mutation 1 may not modify the invasion probability of the mutation 2 and the final state set of the population. However the presence of the first mutant may slow down the invasion of the second one. Proposition 2. Let Assumption 2 be satisfied and suppose: S01 , S02 < 0,

S12 < 0 < S21 ,

S21 < S20 ,

and

(2.5)

or S01 > 0, S02 < 0,

S12 < 0 < S201

and

S201 < S20 .

Then for ε > 0, there exist two finite constants β(ε) and V (ε) such that P(k) (n¯ 2 e2 ∈ F S(ε, K , β(ε), V (ε))) =

S20 + O K (ε), β2

for k = {1, 2} or {2}.

However there exists a positive constant c such that for every ε > 0:  log K log K  (ε,K ) P(1,2) (1 − cε) < TF S(V (ε)) (n¯ 2 e2 ) − α log K < (1 + cε) S˜20 S˜20 S20 S10 = + O K (ε), β2 β1 where S˜20 > S20 is defined by  1 S20  1 1  1 = + −α 1− > , ˜S20 S S10 S S20 where S = S21 in case (2.5) and S = S201 in case (2.6). Finally,  log K log K  (ε,K ) P(1,2) (1 − cε) < TF S(V (ε)) (n¯ 2 e2 ) − α log K < (1 + cε) S20 S20 S20  S10  = 1− + O K (ε). β2 β1

(2.6)

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Fig. 2. Invasion times of the mutation 2 without and with the first mutation. The y-axis corresponds to the three type population sizes (0 in black, 1 in red and 2 in green), and the x-axis to the time. In these simulations, K = 1000, (βi , δi ) = (2, 0), i ∈ E, C0,0 = 1.8, C0,1 = 4, C0,2 = 3, C1,0 = 1, C1,1 = 2.3, C1,2 = 3, C2,0 = 1.5, C2,1 = 1, C2,2 = 2.1, and α = 0.5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

But due to the non-transitive phenotypic interactions that our model is able to take into account, the presence of the first mutant can also speed up the invasion of the second one, without modifying its invasion probability and the final state of the population. More precisely we have the following result, which is illustrated in Fig. 2 and will be proven in Section 6.2. Proposition 3. Let Assumption 2 be satisfied and suppose: S01 , S02 < 0,

S12 < 0 < S21 ,

S21 > S20 ,

and

(2.7)

or S01 > 0, S02 < 0,

S12 < 0 < S201

and

S201 > S20 .

Then for ε > 0, there exist two finite constants β(ε) and V (ε) such that P(k) (n¯ 2 e2 ∈ F S(ε, K , β(ε), V (ε))) =

S20 + O K (ε), β2

for k = {1, 2} or {2}.

However there exists a positive constant c such that for every ε > 0:  log K log K  (ε,K ) P(1,2) (1 − cε) < TF S(V (ε)) (n¯ 2 e2 ) − α log K < (1 + cε) S˜20 S˜20 S20 S10 = + O K (ε), β2 β1 where S˜20 > S20 is defined by  1 1  1 S20  1 −α 1− < , = + ˜S20 S S10 S S20 where S = S21 in case (2.7) and S = S201 in case (2.8). Finally  log K log K  (ε,K ) P(1,2) (1 − cε) < TF S(V (ε)) (n¯ 2 e2 ) − α log K < (1 + cε) S20 S20 S20  S10  = 1− + O K (ε). β2 β1

(2.8)

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2.3. How does clonal interference modify the invasion probability? Previous models predicted that clonal interference could only decrease the invasion probability of a mutant [17]. It was a direct consequence of the fact that the fate of a competition between two mutants was only dependent on the relative values of their fitnesses. Moreover, this implied that individuals with different phenotypes could not coexist for a long time. In our model, both decrease and increase of the invasion probability due to clonal interference may occur, depending on the competitive interactions between individuals and the time of appearance of the second mutation. Moreover, long term coexistence of several beneficial mutations is possible when we do not assume that the mutant fitnesses are totally ordered. Recall the definition of invasion in (1.3). Then we have the following result: Proposition 4. Let one of the following conditions hold:  S201 < 0 and Assumption 2 holds,  S01 > 0, S01 < 0, S21 < 0 and Assumption 2 holds  S01 < 0, S21 < 0, S20 > 0 and Assumption 4 holds Then for every n ∈ R3+ \ {0} and β, V > 0: lim P(1,2) (I (2, K , n, β, V )) = 0,

K →∞

whereas there exist n ∈ R3+ \ {0} and β, V > 0 such that: lim P(2) (I (2, K , n, β, V )) =

K →∞

S20 . β2

Notice that even if we recover here a classical result, saying that a mutant can get extinct because of the competition with another mutant, we do not require S10 > S20 > 0, which would be the equivalent of assumptions done in population genetic models. Proposition 5. Let Assumption 4 hold, and suppose that S20 < 0. Then under one of the following additional conditions (i) S01 < 0, S21 > 0, S12 > 0, S012 < 0 and (ii) S01 < (iii) S01 < (iv) S01 >

1 1 1 1 S10 < α < S10 + |S01 | − S21 , 0, S21 > 0, S12 < 0, S02 < 0 and S110 < α < S110 + |S101 | − S121 , 0, S21 > 0 and α > S110 + |S101 | − S121 , 0 and S201 > 0,

there exist n ∈ R3+ \ {0} and β, V > 0 such that: lim P(1,2) (I (2, K , n, β, V )) =

K →∞

S S10 , β2 β1

where S = S21 in case (i) to (iii) and S = S201 in case (iv), whereas for every n ∈ R3+ \ {0} and β, V > 0: lim P(2) (I (2, K , n, β, V )) = 0.

K →∞

This result will be proven in Section 6.3.

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Fig. 3. Illustration of Proposition 6. The y-axis corresponds to the three type population sizes (0 in black, 1 in red and 2 in green), and the x-axis to the time. Such a trajectory occurs with a probability close to S10 S21 /(β1 β2 ), and with a probability 1 − S10 /β1 the first mutant does not invade. In these simulations, K = 1000, (βi , δi ) = (2, 0), i ∈ {0, 1, 2}, C0,0 = 1.8, C1,0 = C2,1 = 1, C1,1 = 2.3, C1,2 = 2C0,1 = 5, C2,0 = 2C0,2 = 3, C2,2 = 2.1, and α = 1.9. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Concerning the interplay of invasion and clonal interference, let us mention recent works which have taken into account the case where many beneficial mutations occur before any can fix [29,4,16,31,32]. The authors still assume transitivity of mutant fitnesses, but consider a regime of frequent mutations (high mutation rate or very large population). New mutations constantly occur in individuals already carrying other mutations, in their way of invasion, and the fate of a mutation depends on the genetic background of the individual where it occurs more than on its intrinsic advantage. They argue that this dynamical equilibrium is a way to preserve genetic diversity despite clonal interference, where the amount of variation results in a subtle balance between selection, which reduces it, and new mutations, which increase it. This approach is interesting and relates on experimental data which confirm that populations with so frequent mutations do exist [32]. However, due to the transitivity assumption, the authors need to assume that a large number of mutants co-occur in order to explain the possible coexistence of several types in the population, which is not necessary in our model. 2.4. When do beneficial mutations annihilate adaptation? Another interesting phenomenon can happen in our model: the occurrence of the second mutation can annihilate the effects of the first one and lead to the final fixation of the wild type population 0 which would have been outcompeted by the first mutation alone. Such a phenomenon is also impossible in case of transitive fitnesses, as in this setting S10 > 0 and S21 > 0 necessarily imply S20 > 0. Proposition 6 is illustrated in Fig. 3. Proposition 6. Let Assumptions 1 and 4 hold and suppose that S01 < 0, S12 < 0 < S21 , S20 < 0 < S02 1 1 1 S02 <α− + < . |S12 ||S01 | S10 S21 |S01 |

and

Then for every ε > 0, there exist two finite constants β(ε) and V (ε) such that   S10 S21 (1,2) P (n¯ 0 e0 ∈ F S(ε, K , β(ε), V (ε))) = 1 + − 1 + O K (ε). β1 β2

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2.5. At the origin of the Rock–Paper–Scissors cycles Rock–Paper–Scissors (RPS) is a children’s game where rock beats scissors, which beat paper, which in turn beats rock. Such competitive interactions between morphs or species in nature can lead to cyclical dynamics, and have been documented in various ecological systems [6,46,44,28, 30,7,37]. Let us describe two examples of such cycles. The first one [44] is concerned with pattern of sexual selection on some male lizards. Males are associated to their throat colours, which have three morphs. Type 0 individuals (orange throat) are polygamous and very aggressive. They control a large territory. Type 1 individuals (dark-blue throat) are monogamous. They control a smaller territory. Finally type 2 individuals (prominent yellow stripes on the throat, similar to receptive females) do not engage in female-guarding behaviour but roam around in search of sneaky matings. As a consequence of these different strategies, the type 0 outcompetes the type 1 (because males are more aggressive), which outcompetes the type 2 (as males of type 1 are able to control their small territory and guard their female), which in turn outcompetes the type 0 (as males of type 0 are not very efficient in defending their territory, having to split their efforts on several females). The second example [30] is concerned with the interactions between three strains of Escherichia coli bacteria. Type 0 individuals release toxic colicin and produce an immunity protein. Type 1 individuals produce the immunity protein only. Type 2 individuals produce neither toxin nor immunity. Then type 0 is defeated by type 1 (because of the cost of toxic colicin production), which is defeated by type 2, (because of the cost of immunity protein production), which in turn is defeated by type 0 (not protected against toxic colicin). Neumann and Schuster [38] modelled such interactions by a three dimensional competitive Lotka–Volterra system. In particular they proved that migrations or recurrent mutations were not necessary ingredients to obtain limit cycles, as it was assumed in previous models (see [15,42] for example). They studied the long time behaviour of the system but payed little attention to initial conditions, only assuming “the presence of all three strains in one homogeneous medium”. But the question of initial conditions is crucial. Indeed, how to explain the appearance of such cycles whereas when only two strains are present one of them is outcompeted by the other one and disappears? Our simple model provides a framework explaining how a cyclical RPS dynamics emerges in an ecological system thanks to the interplay of two successive mutations. Proposition 7, stated after the following definitions, is illustrated in Fig. 4 and proven in Section 6.4. We call cycle an interval between two successive global maxima of order K of the type 1 population size on a time interval of order log K , between which type 2 and type 0 population sizes also hit one global maxima of order K on a time interval of order log K (see Fig. 4 where we delimited the first cycle with two vertical lines). We also introduce N K := number of cycles,

(l)

D K := duration of cycle l,

l ∈ N,

and  1 1  |S01 | |S01 ||S12 |  |S01 ||S12 ||S20 | l−1 + 1+ + log K . T (α, l, K ) := α − S10 S21 S02 S02 S10 S02 S21 S10 Proposition 7. Let Assumptions 1 and 4 and the following inequalities hold: S01 < 0 < S10 , and

S12 < 0 < S21 ,

S20 < 0 < S02 ,

(2.9)

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Fig. 4. RPS cycles. The y-axis corresponds to the three type population sizes (0 in black, 1 in red and 2 in green), and the x-axis to the time. In this simulation, K = 1000, (βi , δi ) = (2, 0), i ∈ {0, 1, 2}, C0,0 = C1,1 = C2,2 = 2, C0,1 = 2.5, C0,2 = C1,0 = C2,1 = 1, C1,2 = C2,0 = 3, and α = 1.1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0<α−

 1  S02 S02 S10 1 1 < min . , , − S10 |S01 | |S12 ||S01 | |S12 ||S01 ||S20 | S21

(2.10)

Then for every l ∈ N, lim P(1,2) (N K ≥ l) =

K →∞

S10 S21 . β1 β2

(l)

Moreover, for l ∈ N, D K satisfies for a finite c   S10 S21 (l) + O K (ε). P(1,2) N K ≥ l, (1 − cε)T (α, l, K ) < D K < (1 + cε)T (α, l, K ) = β1 β2 Remark 3. We say that System (1.16) is permanent if there exists a compact attractor B in the interior of R3+ of its solutions. Under condition (2.9), Theorem 2 in [38] states that (1.16) is permanent if and only if |S01 ||S12 ||S20 | < S02 S21 S10 . This condition is satisfied under the assumptions of Proposition 7. Hence if one of the types gets extinct, this is due to the demographic stochasticity and not to the behaviour of the approximating dynamical system. Moreover, we are not able to know in general if the interior fixed point is globally attracting or if there exist stable periodic orbits for the flows of the three dimensional deterministic Lotka–Volterra system. In Appendix A we give two examples of systems satisfying the conditions of Proposition 7 but with distinct long time behaviours. 3. First phase In Sections 3–5 we describe the dynamics of the successive phases. For the sake of readability, we do not indicate anymore the superscript (k) of the probabilities. The context will make clear the mutations which occur. The first phase is rigorously defined as follows, for ε > 0 and K ∈ N: Phase 1(K , ε) := {t ≥ 0, N1K (t) + N2K (t) ≥ 1, sup NiK (t) < ⌊εK ⌋}. i∈{1,2}

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Under Assumption 4 the second mutation occurs during the third phase. This corresponds to the case already studied in [9] and we will recall the outcomes in this case in Section 3.4. Under Assumption 2 or 3 the second mutation occurs during the first phase and we have to study the resulting dynamics during the first phase. 3.1. Couplings with auxiliary birth and death processes under Assumption 2 The number of type 1 and type 2 individuals is small during the first phase. As a consequence, they have a small impact on the dynamics of type 0 individuals and their number stay almost constant. This allows us to compare the dynamics of type 1 and type 2 populations with dynamics of birth and death processes without competition. Let i, j and k be distinct in {0, 1, 2}, ε > 0 and K ∈ N. We introduce a finite subset of N containing the equilibrium size of a monomorphic i-population,   Ci, j + Ci,k  Ci, j + Ci,k   , K n¯ i + 2ε ∩ N, Iε(K ,i) := K n¯ i − 2ε Ci,i Ci,i

(3.1)

(K ,i j) (K ,i) (K ,i) and T˜ε and the stopping times Ta , Ta , which denote respectively the hitting time of size ⌊a⌋ for a ∈ R+ by the population of type i and by the total population of types i and j, and (K ,i) the exit time of Iε by the population of type i,   Ta(K ,i) := inf t ≥ 0, NiK (t) = ⌊a⌋ and ∃s < t, NiK (t) ̸= ⌊a⌋ , (3.2)   (K ,i j) Ta := inf t ≥ 0, NiK (t) + N jK (t) = ⌊a⌋ , (3.3)   T˜ε(K ,i) := inf t ≥ 0, NiK (t) ̸∈ Iε(K ,i) . (3.4)

In the vein of Fournier and M´el´eard [22] we represent the population process in terms of Poisson measures. Let Q 0 (ds, dθ ), Q 1 (ds, dθ ) and Q 2 (ds, dθ ) be three independent Poisson random measures on R2+ with intensity dsdθ , and recall that (ei , i ∈ {0, 1, 2}) is the canonical basis of R3 . Then the process N K can be written as follows: N K (t) = N K (0) +

1  t  0

i=0

  + e2 1t≥α log K 1 +

t α log K

R+

  ei 1θ≤βi N K (s − ) − 10<θ−βi N K (s − )≤d K (N K (s − )) Q i (ds, dθ) i

i

i

   1θ≤β2 N K (s − ) − 10<θ−β2 N K (s − )≤d K (N K (s − )) Q 2 (ds, dθ) , 2

2

2

(3.5)

where (diK , i ∈ {0, 1, 2}) have been defined in (1.2). The idea is to couple the population process with birth and death processes to get bounds on the different hitting times. To this aim, let us introduce approximations of the so called rescaled invasion fitnesses Si0 /βi , for i ∈ {1, 2} and ε > 0 small enough: (ε,−)

si0

(ε,+)

si0

  C0,1 + C0,2  1 βi − δi − Ci,0 n¯ 0 + 2ε − (Ci,1 + Ci,2 )ε βi C0,0   1 C0,1 + C0,2  := βi − δi − Ci,0 n¯ 0 − 2ε . βi C0,0 :=

(3.6) (3.7)

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These real numbers satisfy for ε small enough   Si0  (ε,+) (ε,+)  (ε,+) ≤ si0 < 1, and si0 − si0  βi   1 C0,1 + C0,2 ≤ 4 + Ci,1 + Ci,2 ε. βi C0,0 (ε,−)

0 < si0

≤

(3.8)

Thanks to these definitions we can introduce, for ∗ ∈ {−, +} the processes  t  (ε,∗) N0 (t) = ⌊n¯ 0 K ⌋ + 1θ ≤β N (ε,∗) (s − ) 0

0

R+

0

− 10<θ −β

(ε,∗) (ε,∗) (ε,∗) − (s )≤(δ0 +C0,0 N0 (s − )+1{∗=−} ε(C0,1 +C0,2 ))N0 (s − ) 0 N0



Q 0 (ds, dθ ),

and the supercritical birth and death processes (ε,∗)

N1

(t)  t

=1+ 0

R+

 1θ ≤β

(ε,∗)

1 N1

(s − )

− 10<θ −β

(ε,∗)

1 N1

(ε,∗)

(ε,∗)

(s − )≤(1−s10 )β1 N1

 (s − )

Q 1 (ds, dθ), (3.9)

and (ε,∗)

N2

t

  (t) = 1t≥α log K 1 +



α log K

R+



1θ ≤β

(ε,∗) − (s ) 2 N2

− 10<θ −β

(ε,∗) − (ε,∗) (ε,∗) (s )≤(1−s20 )β2 N2 (s − ) 2 N2



 Q 2 (ds, dθ ) .

(3.10)

Then recalling definition (3.2) we get (ε,−)

N0

(ε,+)

(t) a.s. ∀ t < TεK

(ε,+)

(t)

(t) ≤ N0K (t) ≤ N0

(K ,1)

(K ,2)

(3.11)

∧ TεK

and for i ∈ {1, 2}, (ε,−)

Ni

(t) ≤ NiK (t) ≤ Ni

(K ,1)

a.s. ∀ t < TεK

(K ,2)

∧ TεK

∧ T˜ε(K ,0) . (ε,∗ )

(3.12) (ε,∗ )

Moreover by construction for every (∗1 , ∗2 ) ∈ {−, +}2 the processes N1 1 and N2 2 are independent. We can show that with a probability close to one, Couplings (3.11) and (3.12) hold during the whole first phase. More precisely, if we introduce the event (K ,1)

AεK := {TεK

(K ,2)

∧ TεK

(K ,12)

∧ T0

≤ T˜ε(K ,0) },

(3.13)

its probability satisfies Lemma 4. Let Assumptions 1 and 2 hold. lim inf P(AεK ) ≥ 1 − cε K →∞

(3.14)

for a finite c and ε small enough. Proof. First applying (A.5) we get the existence of a positive V such that for ∗ ∈ {−, +} lim P(T˜ε(K ,∗) > e V K ) = 1,

K →∞

(3.15)

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where   (ε,∗) T˜ε(K ,∗) := inf t ≥ 0, N0 (t) ̸∈ Iε(K ,0) .

(3.16)

(K ,0) We decompose the probability in (3.14) into two parts according to the position of T˜ε with respect to e V K : (K ,1)

1 − P(AεK ) = P(T˜ε(K ,0) < TεK

(K ,2)

∧ TεK

= P(T˜ε(K ,0) ≤ e V K , T˜ε(K ,0) < + P(e V K < T˜ε(K ,0) <

(K ,12)

∧ T0

(K ,1) TεK

(K ,1) TεK

∧

) (K ,2)

∧ TεK

(K ,2) TεK

∧

(K ,12)

∧ T0

(K ,12) T0 ).

) (3.17)

Thanks to (3.11) and (3.15) we get (K ,1)

lim P(T˜ε(K ,0) ≤ e V K , T˜ε(K ,0) < TεK

K →∞

(K ,2)

∧ TεK

(K ,12)

∧ T0

) = 0.

(K ,0) Consider now the second probability in the right hand side of (3.17). The event {e V K < T˜ε } (K ,1) (K ,2) means that Couplings (3.12) hold at least until time e V K ∧ TεK ∧ TεK . Hence   (K ,1) (K ,2) (K ,12) e V K < T˜ε(K ,0) < TεK ∧ TεK ∧ T0   (1,+) (2,+) (1,−) (2,−) ⊂ e V K < (T0 ∨ T0 ) ∧ TεK ∧ TεK     (1,+) (1,−) (2,+) (2,−) ⊂ e V K < T0 ∧ TεK ∪ e V K < T0 ∧ TεK .

But thanks to Eqs. (A.2) and (A.3) we know that for i ∈ {1, 2},     (i,+) (i,+) lim P e V K < T0 △ T0 =∞ K →∞     (i,−) (i,−) = lim P e V K < TεK △ T0 < ∞ = 0, K →∞

where △ denotes the symmetric difference: for two sets B and C, B △ C = (B ∩ C c ) ∪ (C ∩ B c ). This implies that   (K ,2) (K ,12) (K ,1) P e V K < T˜ε(K ,0) < TεK ∧ TεK ∧ T0 ≤

2    (i,+) (i,−) P T0 = ∞, T0 < ∞ + O K (ε) = O K (ε), i=1

where the last equality is a consequence of definitions (3.9) and (3.10), and Eq. (A.1). This ends the proof of (3.14).  3.2. First phase under Assumptions 1 and 2 Let us introduce a finite subset of N which may contain the type 2 population size at the end of the first phase.   S ( 1 −α−ε) S ( 1 −α+ε) Jε(K ,2) := K 20 S10 , K 20 S10 ∩ N, (3.18) where we recall that α log K is the time of occurrence of the second mutation, and the invasion fitnesses have been defined in (1.6). In the following lemma we (approximately) compute the probabilities of possible states of the process at the end of the first phase.

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Lemma 5. Under Assumptions 1 and 2,    S10  S20  (K ,i) (K ,i) < TεK < T˜ε(K ,0) , ∀i ∈ {1, 2} = 1 − P T0 1− + O K (ε), β1 β2   (K , j) (K , j) (K ,i) (K ,i) < TεK < T˜ε(K ,0) < T˜ε(K ,0) , T0 P TεK < T0 =

S j0  Si0  1− + O K (ε), βi βj

(3.19)

i ̸= j ∈ {1, 2}.

(3.20)

Furthermore, there exists a positive constant M1 such that   S10 S20 (K ,1) (K ,1) (K ,2) + O K (ε). P TεK < T˜ε(K ,0) , N2K (TεK ) ∈ J M1 ε = β1 β2

(3.21)

Proof. We assume that AεK holds, which is true with a probability close to one according to (K ,1) (K ,2) (K ,12) (3.14), and implies that Coupling (3.12) holds on the time interval [0, TεK ∧ TεK ∧ T0 ]. (3.19): Let us recall definitions (3.6) and (3.7). Thanks to the independence of the processes (ε,+) (ε,+) N1 and N2 we get by applying (A.1) (1,+)

P(T0

(1,+)

< TεK



(2,+)

, T0

(ε,+)

s10

= 1−

(2,+)

< TεK ) 

(ε,+)

1 − (1 − s10 )⌊εK ⌋  S10  S20  ≥ 1− 1− − cε β1 β2

(ε,+)

1−

s20



(ε,+) ⌊εK ⌋ )

1 − (1 − s20

(3.22)

for a finite c, K large enough and ε small enough. Moreover we have the following inclusion:     (1,+) (1,+) (2,+) (2,+) (K ,1) (K ,1) (K ,2) (K ,2) AεK , T0 < TεK , T0 < TεK ⊂ AεK , T0 < TεK , T0 < TεK . We then get (K ,1)

P(AεK , T0

(1,+) P(T0

(K ,1)

< TεK

(K ,2)

, T0

(2,+) (1,+) TεK , T0

< <   S10 S20 1− − cε, ≥ 1− β1 β2

≥

(K ,2)

< TεK

(1,+)

) ≥ P(AεK , T0

(2,+) TεK ) − P((AεK )c )

(1,+)

< TεK

(2,+)

, T0

(2,+)

< TεK



)

(3.23)

for a finite c, K large enough and ε small enough, where we used (3.14) and (3.22). (ε,+)

(3.20): The independence of the processes N1 (1,−) P(TεK

<

(1,−) (2,+) T0 , T0

<

(2,+) TεK )

(ε,+)

and N2

(ε,−)

=

s10

(ε,−)

again yields 

1 − (1 − s10 )⌊εK ⌋ S10  S20  ≥ 1− − cε β1 β2

1−

(ε,+)

s20

(ε,+) ⌊εK ⌋ )

1 − (1 − s20

for a finite c, K large enough and ε small enough. Moreover, thanks to Coupling (3.12) we get   (1,−) (1,−) (2,+) (2,+) (2,+) (1,+) AεK , TεK < T0 , T0 < TεK , T0 < TεK



S. Billiard, C. Smadi / Stochastic Processes and their Applications ( (K ,1)



(K ,1)

⊂ AεK , TεK

< T0

(K ,2)

, T0

(K ,2)

< TεK

(K ,2)

, T0

(1,−)

(K ,1)

< TεK

(1,−)



)

21

–

.

(2,+)

(2,+)

But according to Lemma A.1, on the event {TεK < T0 , T0 < TεK } we can find (2,+) a finite constant M such that for K large enough with a probability close to one T0 ≤ (ε,−) (1,−) α log K + M, and TεK is close to log K /(β1 s10 ). We finally get: (K ,1)

(K ,1)

P(AεK , TεK

< T0

(K ,2)

(K ,2)

, T0

< TεK

(K ,2)

, T0

(K ,1)

< TεK

)

≥

(1,−) (1,−) (2,+) (2,+) (2,+) (1,+) P(TεK < T0 , T0 < TεK , T0 < TεK ) − P((AεK )c ) (1,−) (1,−) (2,+) (2,+) (2,+) (1,−) P(TεK < T0 , T0 < TεK , T0 < TεK ) (2,+) (1,+) (2,+) (1,−) − P({T0 < TεK } △ {T0 < TεK }) − P((AεK )c )

≥

S10  S20  1− − cε β1 β2

≥

(3.24)

for a finite c, K large enough and ε small enough, where we used (3.14) and (A.1). By interchanging the roles of types 1 and 2 we derive a lower bound for (3.20). (3.21): Let us now focus on the last inequality of Lemma 5. First using again independence (ε,−) (ε,−) between N1 and N2 , and (A.1) we get (1,−)

P(TεK

(1,−)

< T0

(2,−)

, TεK

(2,−)

< T0

)≥

S10 S20 − cε, β1 β2 (K ,1)

(K ,2)

for a finite c and ε small enough. But as Coupling (3.12) only holds before time TεK ∧ TεK we have to determine which process, N2K or N1K hits ⌊εK ⌋ first. For i ∈ {1, 2}, from (A.2) and (A.3), (i,−)

P(TεK

(i,−)

< T0

(i,−)

, T0

< ∞) ≤ cε.

Using again (A.3) we get the existence of a finite constant c such that:   S10 S20 (i,−) = ∞, i ∈ {1, 2} = + O K (ε) P T0 β1 β2 and  1 − cε  (i,+) (i,−) P 1{i=2} α + log K ≤ TεK ≤ TεK Si0   1 + cε  (i,−) ≤ 1{i=2} α + log K , T0 = ∞, i ∈ {1, 2} Si0 S10 S20 ≥ + O K (ε). β1 β2

(3.25) (ε,∗)

Hence, as under Assumption 2, 1/S10 < 1/S20 + α, processes (N1 , ∗ ∈ {+, −}) hit ⌊εK ⌋ (ε,∗) (i,−) before processes (N2 , ∗ ∈ {+, −}) on the event {T0 = ∞, i ∈ {1, 2}} with a probability (ε,∗) close to one. The last step consists in determining the values of (N2 , ∗ ∈ {+, −}) on the time interval [(1 − cε) log K /S10 , (1 + cε) log K /S10 ]. First we notice that (A.3) implies for

22

S. Billiard, C. Smadi / Stochastic Processes and their Applications (

)

–

∗ ∈ {+, −}: T

(2,∗) K

S20 ( S1 −α∗2cε) 10

log K

→α+

 1  − α ∗ 2cε , (ε,∗) S 10 β2 s S20

=

= ∞}.

20

Moreover, we get from (A.1)  (2,−) (2,−) P T S ( 1 −α−3cε) < T S ( K

(2,∗)

a.s. on {T0

20 S10

K

(ε,−) K )

(1 − s20

1 20 S10 −α−cε)

S20 ( S1 −α−cε) 10 1

(ε,−) K S20 ( S10 −α−cε) )

(1 − s20

  S ( 1 −α−2cε)  (2,−) (0) = K 20 S10 N (ε,−) K )

− (1 − s20

S20 ( S1 −α−2cε) 10 1

(ε,−) K S20 ( S10 −α−3cε) )

→ 0,

K → ∞.

− (1 − s20

This completes the proof of the lower bound for (3.21). Adding (3.23) and (3.24) ends the proof of Lemma 5.  We end the study of the first phase dynamics under Assumptions 1 and 2 by an approximation of the duration of this phase in the case we are interested in: Lemma 6. Recall the definition of M1 in Lemma 5. Under Assumptions 1 and 2,   log K log K  (K ,1) (K ,1) (K ,1) (K ,2) < TεK < (1 + cε) P (1 − cε) TεK < T˜ε(K ,0) , N2K (TεK ) ∈ J M1 ε S10 S10 = 1 + O K (ε). We do not detail the proof of this Lemma. It is a direct consequence of Couplings (3.12) and Eq. (3.25). 3.3. Assumption 3 Under Assumptions 1 and 3, the type 2 population size has a positive probability to become larger than the type 1 population size during the first phase. Let us introduce a finite subset of N which may contain the type 1 population size at the end of the first phase.   S ( 1 +α−ε) S ( 1 +α+ε) Jε(K ,1) := K 10 S20 ∩ N. (3.26) , K 10 S20 Then we have the following possible states at the end of the first phase. Lemma 7. Under Assumptions 1 and 3,    S10  S20  (K ,i) (K ,i) P T0 < TεK < T˜ε(K ,0) , ∀i ∈ {1, 2} = 1 − 1− + O K (ε), β1 β2   (K , j) (K , j) (K ,i) (K ,i) P TεK < T0 < T˜ε(K ,0) , T0 < TεK < T˜ε(K ,0) S j0  Si0  = 1− + O K (ε), i ̸= j ∈ {1, 2}. βi βj Furthermore there exists a positive constant M2 such that   S10 S20 (K ,2) (K ,1) (K ,2) P TεK < T˜ε(K ,0) , N1K (TεK ) ∈ J M2 ε = + O K (ε). β1 β2

S. Billiard, C. Smadi / Stochastic Processes and their Applications (

)

–

23

We do not prove this result as it is very similar to Lemma 5. The idea is that when the 2-population survives the first phase, it takes a time of order log K /S20 to hit the value ⌊εK ⌋, whereas the 1-population size needs a time of order log K /S10 to hit such a value. As under Assumption 3, α + 1/S20 < 1/S10 , the 2-population size is the first to represent a positive fraction of the total population size. The value of the 1-population size at the end of the first phase is obtained thanks to Coupling (3.12) and Eq. (A.3). We have also an equivalent of Lemma 6: Lemma 8. Recall the definition of M2 in Lemma 7. Under Assumptions 1 and 3,    log K  (K ,2) log K  (K ,2) < TεK < (1 + cε) α + P (1 − cε) α + TεK S20 S20  (K ,2) (K ,1) < T˜ε(K ,0) , N1K (TεK ) ∈ J M2 ε = 1 + O K (ε). 3.4. Assumption 4 This case has already been studied in [9] and we said a few words about it just after the formulation of Assumption 1. In the following lemma we reformulate the corresponding results from [9] using our notations: Lemma 9. Under Assumptions 1 and 4,    S10  (K ,1) + O K (ε), P T0 < T˜ε(K ,0) = 1 − β1

  S10 (K ,1) + O K (ε), P TεK < T˜ε(K ,0) = β1

and   log K log K  (K ,1) (K ,1) P (1 − cε) < TεK < (1 + cε) TεK < T˜ε(K ,0) = 1 + O K (ε). S10 S10 4. Phases 2 and 2 p In this section, we describe the dynamics of “deterministic phases”, when some of the population sizes are well approximated by the solution of a two- or three-dimensional competitive Lotka–Volterra system. 4.1. Two-dimensional case Let us denote by φ1K the end of the first phase, when at least one of the mutant populations survives: (K ,1)

φ1K := TεK

(K ,2)

∧ TεK

,

by i(φ1K ) the label of the first mutant population which hits the size ⌊εK ⌋, and by j (φ1K ) the label of the other mutant population. We will focus on the case when the population j (φ1K ) does not get extinct, as the other ones have already been studied in [9,11]. We introduce the event:  (K ,i(φ K ))  (K ,i(φ K ))   (K , j (φ K )) ∈ J Mi ε 1 . (4.1) BεK := TεK 1 < T˜ε(K ,0) , N jK(φ K ) TεK 1 1

We will now consider the second phase of the sweep. It corresponds to the interval between (K ,i(φ1K ))

the time TεK

when the mutant population i(φ1K ) hits the value ⌊εK ⌋ and the time when

24

S. Billiard, C. Smadi / Stochastic Processes and their Applications (

)

–

K the rescaled population process (N0K , Ni(φ K ) )/K is close enough to the stable equilibrium of 1

the dynamical system (1.8) with labels 0 and i(φ1K ). To define rigorously the duration of the (z) second phase we need to introduce a deterministic time tε (i, j) (see (4.3)) after which the (z) (z) solution n (z) with n (z) (t) = (n i (t), n j (t)) of the dynamical system (1.8) with initial condition z = (z i , z j ) ∈ R2+ is close to the stable equilibrium. To do that in a simple way we introduce a notation for the stable equilibrium of (1.8) independent of conditions (1.9) and (1.14):  n¯ j e j , if (1.9) holds, eq eq (4.2) n i j = n ji := ( j) (i) n¯ i j ei + n¯ i j e j , if (1.14) holds, (z)

where we recall definition (1.15). Hence for ε > 0, tε (i, j) can be defined by   eq (z) (z) tε(z) (i, j) := inf s ≥ 0, ∀t ≥ s, ∥n i (t)ei + n j (t)e j − n i j ∥ ≤ ε 2 ,

(4.3)

where ∥ · ∥ denotes the L 1 -norm on R2 . Notice that unlike the end of the first phase which corresponds to the hitting time of ⌊εK ⌋ for a mutant population, the beginning of the third phase is linked to the distance ⌊ε 2 K ⌋ of the population sizes to the equilibrium sizes. Doing so allows us to use the fact that the ‘small populations’ will not hit the size ⌊εK ⌋ during the third phase and to use similar bounds as in the first phase (in particular definitions (3.2)–(3.4)). As we do not K know precisely the value of the rescaled process (N0K , Ni(φ K ) )/K at the beginning of the second (z)

1

phase, we consider the supremum over the possible tε (0, i(φ1K )): tε (0, i(φ1K )) := sup{tε(z) (0, i(φ1K )), z ∈ BεK },

(4.4)

where BεK := {(z 0 , z i(φ K ) ), |z 0 − n¯ 0 | ≤ 3ε(C0,1 + C0,2 )/C0,0 , ε/2 ≤ z i (φ1K ) ≤ ε}. 1

(z)

Notice that tε (0, i(φ1K )) is finite. We are now able to define rigorously the second phase: for ε > 0 and K ∈ K ,   (K ,i(φ K )) (K ,i(φ K )) Phase 2(K , ε) := t, TεK 1 ≤ t ≤ TεK 1 + tε (0, i(φ1K )) . During the second phase two populations have a size of order K , the wild type population and the mutant population of type i(φ1K ). We will prove that the dynamics of these two population sizes are well approximated by the deterministic two-dimensional competitive Lotka–Volterra system (1.8), which stays in a neighbourhood of its stable equilibrium after the time tε (0, i(φ1K )). The duration of the second phase, tε (0, i(φ1K )), does not tend to infinity with the carrying capacity K , unlike the durations of the first and third phases. As a consequence, the size of the j (φ1K )population stays negligible with respect to K during the second phase. Recall definition (4.1). Then we have the following result: Lemma 10. Recall the definitions of the constants M1 and M2 in Lemmas 5 and 7, respectively. Then    K K  K P N jK(φ K ) (φ1K + s) ∈ J2εM , ∀s ≤ t (0, i(φ )) (4.5) Bε = 1 + O K (ε) ε 1 K 1

j (φ1 )

S. Billiard, C. Smadi / Stochastic Processes and their Applications (

)

25

–

and   1   eq K K K P  (N0K e0 + Ni(φ  K ) ei )(φ1 + tε (0, i(φ1 ))) − n K 0i(φ1 ) K 1 (z) n (tε (0, i(φ1K )))  K  ≤ ε 2 ∧ inf 0 Bε 2 z∈BεK = 1 + O K (ε). Remark 11. Here we cannot apply directly (A.5) because it requires positive lower bounds for the jump rates of each population divided by K . Indeed in our case, the jump rate of the population 2 is of order K

S j (φ K )0 (1/Si(φ K )0 +α(1{ j (φ K )=1} −1{ j (φ K )=2} )) 1

1

1

1

which is negligible with respect to K . Hence we will couple the process N jK(φ K ) and the process K (N0K , Ni(φ K ) ) with well known processes to get bounds on their dynamics.

1

1

Proof of Lemma 10. For the sake of simplicity we will write (i, j) instead of (i(φ1K ), j (φ1K )) along the proof. Let us first prove that during the second phase the process N jK does not evolve a lot. First, notice that the processes N0K and NiK are bigger than if they were evolving alone. Hence if we introduce the event   CεK := sup {N0K (s) + NiK (s)} > 2(n¯ 0 + n¯ i )K , (4.6) s≤tε (0,i)

we deduce from (A.4) that, lim

K →∞

sup (K ,0) ,Ni ∈[εK −1,εK ] N0 ∈Iε

P(N0 ,Ni ) (CεK ) = 0,

(4.7)

where we used the convention P(Ni ,N j ) (.) := P(.|(NiK , N jK )(0) = (Ni , N j )),

(i, j) ∈ E 2 .

To control the number of j-individuals during the second phase, we introduce two birth and death (K ,2,+) (K ,2,−) constructed with the same Poisson random measure Q j as N jK and N j processes, N j (see representation (3.5)) (K ,2,−)

Nj

(t) := K

S j0 ( S1 +α(1{ j=1} −1{ j=2} )−M j ε) i0

 t − 0 (K ,2,+) Nj (t)

:= K

R+

1θ ≤(δ

(2,−) − (s ) j +2(n¯ 0 +n¯ i )(C j,0 +C j,i ))N j

S j0 ( S1 +α(1{ j=1} −1{ j=2} )+M j ε) i0

Q j (ds, dθ ),

 t + 0

R+

1θ ≤β

(2,+) − (s ) j Nj

Q j (ds, dθ ).

Recall definitions (4.1) and (4.6). We see that on the event BεK ∩ CεK , (K ,2,−)

Nj

(K ,2,+)

(t) ≤ N jK (φ1K + t) ≤ N j

(K ,2,−)

Let us first focus on the process N j

(t),

a.s. ∀t ≤ tε (0, i).

. It is a pure death process with individual death rate:

δ := δ j + 2(n¯ 0 + n¯ i )(C j,0 + C j,i ),

26

S. Billiard, C. Smadi / Stochastic Processes and their Applications (

)

–

and we can construct the following martingale associated with this process: R Kj (t)

(K ,2,−) Nj (t)eδt

:=

−

(K ,2,−) Nj (0)

 t =− 0

R+

1θ≤δ N (K ,2,−) (s − ) eδs Q˜ j (ds, dθ ), j

where Q˜ j is the compensated Poisson measure Q˜ j (ds, dθ ) := Q j (ds, dθ ) − dsdθ. Moreover, its quadratic variation can be expressed as  t

⟨R Kj ⟩t

= 0



t

= 0

R+

1θ≤δ N (K ,2,−) (s) e

2δs

 dsdθ =

j

(K ,2,−)

δ(R Kj (s) + N j

0

t

(K ,2,−)

δN j

(s)e2δs ds

(0))eδs ds.

Then Markov Inequality leads to   (K ,2,−) (K ,2,−) P Nj (tε (0, i)) < e−2δtε (0,i) N j (0)     (K ,2,−) = P R Kj (tε (0, i)) < e−δtε (0,i) − 1 N j (0)  2  2  2  (K ,2,−) ≤ P R Kj (tε (0, i)) > 1 − e−δtε (0,i) Nj (0)   E ⟨R Kj ⟩tε (0,i) ≤ 2  2 (K ,2,−) 1 − e−δtε (0,i) Nj (0)   (K ,2,−) eδtε (0,i) − 1 N j (0) = 2  2 , (K ,2,−) 1 − e−δtε (0,i) Nj (0) which implies that   (K ,2,−) (K ,2,−) P inf {N j (s)} < e−2δtε (0,i) N j (0) = O K (ε), s≤tε (0,i)

as a pure death process is non increasing. In the same way we prove that   (K ,2,+) (K ,2,+) (0) = O K (ε). P sup {N j (s)} > e2β j tε (0,i) N j s≤tε (0,i)

(4.8)

From (4.7) to (4.8) we deduce (4.5). Now we want to control the dynamics of populations 0 and i during the second phase. (K ,2,−) (K ,2,+) (K ,2,+) (K ,2,−) We introduce two pairs of processes, (N0 , Ni ) and (N0 , Ni ) whose dynamics are well known and such that with a probability close to one, (K ,2,−)

Nk

(K ,2,+)

≤ NkK ≤ Nk

,

k ∈ {0, i},

during the second phase. These processes are defined as follows for t ≥ 0 and (a, b) = (i, 0) or (0, i):

S. Billiard, C. Smadi / Stochastic Processes and their Applications (

  Na(K ,2,−) (t) := NaK (φ1K ) +         −1 (K ,2,−)

 t 0

R+

)

–

 Q a (ds, dθ ) 1θ≤β

(K ,2,−) − (s ) a Na



S j0 ( S1 +α(1{ j=1} −1{ j=2} )+2M j ε) (K ,2,+) − (K ,2,−) − (K ,2,−) − i0 (s )+Ca,b Nb (s )+Ca, j K (s − )≤(δa +Ca,a Na (s ) )Na

0<θ−βa Na  t    (K ,2,+) K K  ) + Q b (ds, dθ ) 1θ≤β N (K ,2,+) (s − ) N (t) := N (φ  b 1 b  b b  0 R+     − 10<θ −β N (K ,2,+) (s − )≤(δ +C N (K ,2,−) (s − )+C N (K ,2,+) (s − ))N (K ,2,+) (s − ) . b

b

b

b,a

27

b,b

a

b

,

b

Notice that conditionally on the initial condition of the second phase, N K (φ1K ), the processes (K ,2,∗) (Nk , k ∈ {0, i}, ∗ ∈ {−, +}) are independent of the process N jK , and that on the event K {φ1K < ∞} ∩ {N jK (φ1K + s) ∈ J2M , ∀s ≤ tε (0, i)}, jε

we have (K ,2,−)

Nk

(K ,2,+)

(t) ≤ NkK (φ1K + t) ≤ Nk

(t),

a.s. ∀t ≤ tε (0, i) and k ∈ {0, i}.

Moreover, a direct application of (A.4) leads to  (K ,2,∗) (K ,2,¯∗) lim sup P sup ∥(N0 , Ni )(t)/K K →∞

(K ,0)

(z 0 ,z i )∈Iε (z)

/K ×[ε/2,ε]

(z)

0≤t≤tε (0,i)



− (n 0 (t), n i (t))∥ > δ = 0, (K ,0) for ε and δ > 0, where ∗¯ denotes the complement of ∗ in {−, +}, Iε has been defined in (3.1) and tε (0, i) in (4.4). This completes the proof of Lemma 10. 

The dynamics of the population process is a succession of phases where at least two types of populations have a size of order K (2 pth phases, p ∈ N) and of phases were at most one population type has a size of order K ((2 p + 1)th phases, p ∈ Z+ ). The following lemma completes the description of the dynamics of phases 2 p, in the case where two populations have a size of order K . It is a generalization of Lemma 10 and we do not give the proof. Recall definition (4.4). Then Lemma 12. Let {i, j, k} = {0, 1, 2}, ε > 0, n, ci > 0, γ ∈ (0, 1) and assume that (NiK (0), N jK (0), NkK (0)) = (⌊n K ⌋, ⌊εK ⌋, ⌊K γ ⌋),

with |n − n¯ i | ≤ ci ε.

Then  P NkK (s) ∈ [K γ −ε , K γ +ε ], ∀s ≤ tε (i, j), (z) 1  n (tε (i, j))   (eq)  = 1 + O K (ε).  (NiK ei + N jK e j )(tε (i, j)) − n i j  ≤ ε 2 ∧ inf i K 2 z∈BεK

4.2. Three-dimensional case From a probabilistic point of view, the case where the three types of populations have a size of order K is simpler, as we can approximate the rescaled population process by the three-dimensional deterministic Lotka–Volterra system (1.16) according to Eq. (A.4). But the behaviour of the solutions of three-dimensional Lotka–Volterra systems is much more various than those of two-dimensional systems. They have been studied in detail by Zeeman and coauthors [48–50] and we will now present some of their findings.

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Fig. 5. The phase portraits on Σ . A fixed point is represented by a close dot • if it attracts on Σ ; by an open dot ◦ if it repels on Σ , and by the intersection of its hyperbolic manifolds if it is a saddle on Σ . (Figures found in [48] and modified to include results of [49] about classes 32 and 33).

As in the case of two dimensional systems, the invasion fitnesses (Si j , Si jk , i, j, and k distinct in {0, 1, 2}) will determine the overall behaviour of the flows. In the two dimensional case for interacting populations of types i and j, there are three possibilities: • Either Si j < 0 < S ji ; then the only stable fixed point is n¯ j e j (see (1.7)) (i)

( j)

• Or Si j , S ji > 0; then the only stable fixed point is n¯ i j ei + n¯ i j e j (see (1.15)) • Or Si j , S ji < 0; then there are two stable fixed points, n¯ i ei and n¯ j e j . In the three-dimensional case, Zeeman [48] numbered 33 equivalence classes. An “equivalence class” is a given combination of invasion fitness signs modulo permutation of the indices. Flows of systems belonging to the same class have the same type of long time behaviour, which is well determined in the 25 first classes and more complex in the 8 remaining classes. These behaviours are represented in Fig. 5 and we will now explain their meaning. By an application of Hirsch’s Theorem [25] Zeeman proved that there exists an invariant hypersurface of R3+ , denoted Σ , such that every non zero trajectory of a three-dimensional competitive Lotka–Volterra system is asymptotic to one in Σ for large times. Σ is called the carrying simplex, being a balance between the growth of small populations and the competition of large populations. A locally attracting fixed point is represented by a close dot •, a locally repelling one by an open dot ◦, and a saddle by the intersection of its hyperbolic manifolds. The classes 1 to 25 have no interior fixed points, and their dynamics are well known: the system converges to one of the stable fixed points with at most two positive coordinates. The solutions of systems in class 32 converge to one of the monomorphic equilibrium density n¯ i for i ∈ {0, 1, 2}. The solutions of systems in class 33 converge to the unique interior fixed points (see [49] for the two last assertions). In the classes 26 to 31 the system can have periodic orbits

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Fig. 6. Phase portraits in classes 26 to 31 in the absence of periodic orbits. (Figures found in [48]).

depending on the value of the parameters (see Fig. 6). More precisely Zeeman proved that there exist systems with and without periodic orbits in each class. We have no general criteria to discriminate between cyclical and converging (or diverging) behaviours of the flows in these classes. Note however that Hofbauer and Sigmund (Theorem 15.3.1 in [26]) and Zeeman and Zeeman (Theorem 6.7 in [50]) provided sufficient conditions to have a global attractor (or repeller). 5. Phases 3 and 2 p + 1 At the beginning of the third phase there are two possible cases: either there are already individuals of type 2 in the population; this corresponds to Assumption 2 or 3. Then two types (eq) of individuals have sizes close to n¯ i j K (defined in (4.2)), and the last type has a size of order K γ for some γ ∈ (0, 1). Or the second mutation has not occurred yet (Assumption 4) and there are only individuals of types 0 and 1. Let us first focus on Assumptions 2 and 3. Then at the beginning of the third phase we have, with a probability close to S10 S20 /(β1 β2 ), two kinds of initial conditions: K • Either S0i(φ K ) < 0 < Si(φ K )0 ; then Ni(φ ¯ i(φ K ) K , N0K equals ⌊n K ⌋ for K ) is close to n 1

(z)

1

1

1

(z)

some n ∈ [infz∈BεK n 0 (tε (0, i(φ1K ))), supz∈BεK n 0 (tε (0, i(φ1K )))] and N j (φ K ) belongs to 1 J2M j (φ K ) ε , 1

(i(φ K ))

(0)

K • Or S0i(φ K ) , Si(φ K )0 > 0; then (N0K , Ni(φ ¯ 0i(φ K ) K , n¯ 0i(φ1K ) K ) and N j (φ K ) K ) ) is close to (n 1

1

belongs to J2M j (φ K ) ε .

1

1

1

1

1

In fact such initial conditions will also be found in phases 2 p + 1, with p ≥ 2. Indeed we will see that in some cases there will still be two populations with sizes of order K and one population with a size of a smaller order after the two first alternations of stochastic and deterministic phases (see Fig. 4 for instance). Lemma 13 describes the dynamics of a stochastic phase with such initial conditions. Before stating this lemma, we introduce a finite subset of N2 and a stopping time: 1     ( j) (i) Iε(K ,i j) := (Ni , N j ),  (Ni ei + N j e j ) − (n¯ i j ei + n¯ i j e j ) K |Ci j C jk − Cik C j j | + |C ji Cik − C jk Cii |  ≤ 2ε , |Cii C j j − Ci j C ji |   T˜ε(K ,i j) := inf t ≥ 0, (NiK (t), N jK (t)) ̸∈ Iε(K ,i j) . Lemma 13. Let us take i, j and k distinct in {0, 1, 2}. Assume that Si j > 0 and NkK (0) = ⌊K γ ⌋ for some 0 < γ < 1.

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1. If S ji < 0, Ski > 0, NiK (0) ∈ [(n¯ i − ε 2 )K , (n¯ i + ε 2 )K ] and N jK (0) = ⌊δ K ⌋ for some 0 < δ < ε2 ,   1 1−γ (K , j) (K ,k) < TεK < T˜ε(K ,i) = 1 + O K (ε), for P T0 < , |S ji | Ski (1−γ +ε)|S ji | (1−γ −ε)|S ji |    1− 1− (K , j) (K ,k) (K ,k) Ski Ski , N jK (TεK ) ∈ K P TεK < T˜ε(K ,i) ∧ T0 ,K = 1 + O K (ε),

for

1−γ 1 > . |S ji | Ski (l)

(l)

2. If S ji > 0, Ski j > 0, NlK (0) ∈ [(n¯ i j − ε 2 )K , (n¯ i j + ε 2 )K ] for l ∈ {i, j},   (K ,k) P TεK < T˜ε(K ,i j) = 1 + O K (ε), 3. If S ji < 0, Ski < 0, NiK (0) ∈ [(n¯ i − ε 2 )K , (n¯ i + ε 2 )K ] and N jK (0) = ⌊δ K ⌋ for some 0 < δ < ε2 ,   (K , j) (K ,k) P T0 ∨ T0 < T˜ε(K ,i) = 1 + O K (ε), (l)

(l)

4. If S ji > 0, Ski j < 0, NlK (0) ∈ [(n¯ i j − ε 2 )K , (n¯ i j + ε2 )K ] for l ∈ {i, j},   (K ,k) P T0 < T˜ε(K ,i j) = 1 + O K (ε). The proof of this Lemma is very similar to the proof of Lemma 5. It consists in comparing the population process with birth and death processes, which can be subcritical in this case, ( j) (i) and to show that the equilibrium sizes ⌊n¯ i K ⌋ or (⌊n¯ i j K ⌋, ⌊n¯ i j K ⌋) are not modified much by population(s) with size(s) smaller than ⌊εK ⌋. Hence we do not detail the proof. We are also able to get approximation of the stochastic phase duration: Lemma 14. Let us take i, j and k distinct in {0, 1, 2}. Assume that Si j > 0 and NkK (0) = ⌊K γ ⌋ for some 0 < γ < 1, and that the conditions of point 1 ≤ l ≤ 5 are the same as in Lemma 13. Then there exists a finite constant c such that,    1−γ 1−γ  (K ,k) (K ,k) log K < TεK < (1 + cε) log K TεK < T˜ε(K ,i) 1. P (1 − cε) Ski Ski = 1 + O K (ε).    1−γ 1−γ  (K ,k) (K ,k) 2. P (1 − cε) log K < TεK < (1 + cε) log K TεK < T˜ε(K ,i j) Ski j Ski j = 1 + O K (ε).   1  1 γ  γ  (K , j) (K ,k) 3. P (1 − cε) ∨ log K < T0 ∨ T0 < (1 + cε) ∨ log K |S ji | |Ski | |S ji | |Ski |    (K , j) (K ,k) ∨ T0 < T˜ε(K ,i) = 1 + O K (ε). T0    1 1  (K ,k) (K ,k) 4. P (1 − cε) < T˜ε(K ,i j) log K < T0 < (1 + cε) log K T0 |Ski j | |Ski j | = 1 + O K (ε).

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Let us now describe the dynamics of the third phase when Assumption 4 holds. There are five possibilities, depending on the signs of the invasion fitnesses and on the value of α. To take into account the state of the population at the end of the first phase, we introduce the following notation under Assumption 4: Pφ1 (N0 ,N1 ) (.) := P(.|(N0K , N1K )(0) = (N0 , N1 ) and the second mutant appears at time (α − 1/S10 ) log K ). Recall that on the event {φ1K < ∞}, Lemma 9 ensures that φ1K is of order log K /S10 with a probability close to 1. Then we can state the following properties for the population dynamics during the third phase: Lemma 15. Under Assumptions 1 and 4, • If S01 < 0, S21 > 0 and 1/|S01 | < α − 1/S10 + 1/S21 ,   S21 (K ,0) (K ,2) inf Pφ1 (N0 ,N1 ) T0 < TεK < T˜ε(K ,1) = + O K (ε), (K ,1) β2 N0 ≤ε2 K ,N1 ∈Iε • If S01 < 0, S21 > 0 and 1/|S01 | > α − 1/S10 + 1/S21 ,  (K ,2) (K ,1) (K ,0) inf Pφ1 (N0 ,N1 ) TεK < T˜εK ∧ T0 , N0K (Tε(K ,2) ) (z)

(K ,1)

inf n 0 (tε (0,1))/2≤N0 /K ≤ε2 ,N1 ∈Iε

z∈BεK

  S21 1−|S01 |(α− S1 + S1 −ε) 1−|S01 |(α− S1 + S1 +ε) 10 21 10 21 + O K (ε), ,K = ∈ K β2 • If S01 < 0, S21 > 0, inf

(K ,1)

N0 ≤ε 2 K ,N1 ∈Iε

=1−

  (K ,0) (K ,2) Pφ1 (N0 ,N1 ) T0 ∨ T0 < T˜ε(K ,1)

S21 + O K (ε), β2

∀α < ∞,

• If S01 > 0, S201 > 0,   S201 (K ,2) Pφ1 (N0 ,N1 ) TεK < T˜ε(K ,01) = + O K (ε), (0) (1) β2 ∥(N0 ,N1 )/K −(n¯ ,n¯ )∥≤ε inf

01

01

• If S01 > 0, S201 > 0,   S201 (K ,2) Pφ1 (N0 ,N1 ) T0 < T˜ε(K ,01) = 1 − + O K (ε), (0) (1) β2 ∥(N0 ,N1 )/K −(n¯ ,n¯ )∥≤ε inf

01

01

∀α < ∞. Once again the proof is very similar to the proof of Lemma 5, and we can also derive approximations for the total duration of the three third phases: Lemma 16. Under Assumptions 1 and 4, there exists a finite constant c such that, • If S01 < 0, S21 > 0 and 1/|S01 | < α − 1/S10 + 1/S21 ,   1 1  (K ,0) + log K < T0 P (1 − cε) < (1 + cε) S10 |S01 |

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   1 1   (K ,0) + < T˜ε(K ,1) ∧ Tε(K ,2) = 1 + O K (ε). × log K T0 S10 |S01 | • If S01 < 0, S21 > 0 and 1/|S01 | > α − 1/S10 + 1/S21 ,   1  (K ,2) P (1 − cε) α + log K < TεK < (1 + cε) S21    1   (K ,2) (K ,0) × α+ log K TεK < T˜ε(K ,1) ∧ T0 = 1 + O K (ε). S21 • If S01 > 0, S201 > 0,   1  (K ,2) P (1 − cε) α + log K < TεK S201    1   (K ,2) < (1 + cε) α + log K TεK < T˜ε(K ,01) = 1 + O K (ε). S201 6. Proofs of Propositions 1–7 We now prove the main results of this paper. We begin with Proposition 1 which precises the conditions needed to have transitive interactions between several types of individuals: 6.1. Proof of Proposition 1 Let i, j, k be in E and recall notations (2.1). The relations i≺ j

and

j ≺k

are equivalent to ρj ρi − C˜ i j < 0 < − C˜ ji ρj ρi or in other terms ρi 1 < C˜ i j ∧ ρj C˜ ji

and

and

ρj 1 < C˜ jk ∧ . ρk C˜ k j

1. If (2.2) holds, then ρ  ρ i i − C˜ ik = ρk Sik = ρk ρk ρj  1  ˜ < ρk C˜ i j ∧ C jk ∧ C˜ ji and

ρj ρk − C˜ jk < 0 < − C˜ k j , ρk ρj

 ρj − C˜ ik ρk   1  ˜  1 2 − Cik ≤ ρk C2 ∧ − C1 < 0, C1 C˜ k j

 ρ ρ  k j − C˜ ki = ρi − C˜ ki ρi ρ j ρi     1 1  ˜  1 2 > ρi C˜ k j ∨ C˜ ji ∨ − Cki ≥ ρk C1 ∨ − C2 > 0, C2 C˜ jk C˜ i j

Ski = ρi

ρ

k

which implies that i ≺ k. 2. If (2.3) holds, then one of the two previous inequalities is not satisfied. Hence either i ≺ k or i = k.

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3. Let us now assume that (2.4) holds. We can choose η > 0 such that 2  −2  1  1  − η > C1 and C2 ∧ −η < C2 . C2 ∧ C1 C1 Assume now that  ρj 1  ρi = = C2 ∧ − η. ρj ρk C1 Then

ρi − C2 < 0 < ρj ρi − C1 > 0 > ρk

ρj − C1 , ρi ρk − C2 . ρi

ρj ρk − C2 < 0 < − C1 ρk ρj

and

This ends the proof of Proposition 1. The proofs of Propositions 2–7 are direct consequences of Lemmas stated in Sections 3–5. As the proofs of Propositions 2 and 3 are very similar, we only prove the second one. 6.2. Proof of Proposition 3 Let us first consider that Assumptions 1 and 2, and condition (2.7) are satisfied. (1 − S10 /β1 )S20 /β2 : According to Lemma 5, with a probability close to (1 − S10 /β1 )S20 /β2 only the 2-population survives the first phase and hits the size ⌊εK ⌋. Then it ‘deterministically’ competes with the 0-population and outcompetes it as S02 < 0. In this case the duration of the invasion, log K /S20 , is not modified by the presence of the mutant 1. S10 S20 /(β1 β2 ): According to Lemmas 5 and 6, with a probability close to S10 S20 /(β1 β2 ) both 1- and 2-populations survive the first phase, which has a duration close to log K . S10 (K ,2)

The 1-population size is the first to hit ⌊εK ⌋, whereas the 2-population size belongs to J M1 ε (defined in (3.18)) at the end of the first phase. Lemma 10 and Markov Property imply that with a probability close to one, the 2-population size stays of the same order during the second phase, and the 0- and 1-population sizes become close to (a K , n¯ 1 K ), with 0 < a < ε 2 depending on ecological parameters but not on K , at the end of the second phase. As S21 > 0, the 2-population size grows during the third phase and takes a time close to  1  1  1 − S20 − α log K S21 S10 to hit the size ⌊εK ⌋ (Lemma 14). Furthermore at the end of the third phase the 0-population has a size negligible with respect to K (which can be 0, see Lemma 13). During the fourth phase the 2-population ‘deterministically’ outcompetes the 1-population, as S12 < 0 < S21 (Lemma 12). Then the 0- and 1-populations get extinct, as S02 and S12 < 0 (Lemma 13). Such a trajectory is illustrated in the second simulation of Fig. 2. In this case the total duration of the mutant invasion is  1  1   1  + 1 − S20 − α − α log K , S10 S21 S10

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(the −α log K is due to the time of the second mutation occurrence) which has to be compared with the duration of the invasion in the absence of the first mutation, log K /S20 . The resolution of a quadratic equation leads to the condition S20 ̸∈ [S10 /(1 − αS10 ), S21 ]. More precisely, the inequality    1  1 1  log K + − α − α log K < 1 − S20 S10 S21 S10 S20 is equivalent to  1   1  1 2 1 S20 − α − S20 + − α + 1 > 0, S21 S10 S10 S21 whose solutions are S20 = S21 and S20 = S10 /(1 − αS10 ). As under Assumptions 1 and 2, 1 1 S10 > − α ⇐⇒ S20 < , S20 S10 1 − αS10 this ends the proof of this case. 1 − S20 /β2 : Finally with a probability close to 1 − S20 /β2 the 2-population gets extinct during the first phase (Lemma 5). We now consider that Assumptions 1 and 2, and condition (2.8) are satisfied. (1 − S10 /β1 )S20 /β2 : See the first point in the proof of Proposition 2 under the first condition. S10 S20 /(β1 β2 ): According to Lemmas 5 and 6, with a probability close to S10 S20 /(β1 β2 ) both 1- and 2-populations survive the first phase, which has a duration close to log K . S10 (K ,2)

The 1-population size is the first to hit ⌊εK ⌋, whereas the 2-population size belongs to J M1 ε (defined in (3.18)) at the end of the first phase. Lemma 10 and Markov Property imply that with a probability close to one, the 2-population size stays of the same order during the second phase, (0) (1) and the 0- and 1-population sizes become close to (n¯ 01 K , n¯ 01 K ) at the end of the second phase. As S201 > 0, the 2-population size grows during the third phase and takes a time close to  1  1  1 − S20 − α log K S201 S10 to hit the size ⌊εK ⌋ (Lemma 14). Then the population process is comparable to a three dimensional Lotka–Volterra system which belongs to class 7 described by Zeeman (see Fig. 5). The flows of the dynamical system converge to the stable equilibrium (0, 0, n¯ 2 ). In this case the total duration of the mutant invasion is  1  1   1  + 1 − S20 − α − α log K , S10 S201 S10 (the −α log K is due to the time of the second mutation occurrence) which has to be compared with the duration of the invasion in the absence of the first mutation, log K /S20 . The resolution of a quadratic equation leads to the condition S20 ̸∈ [S10 /(1 − αS10 ), S201 ]. More precisely, the inequality  1  1   1  log K + 1 − S20 − α − α log K < S10 S201 S10 S20

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is equivalent to  1   1  1 2 1 − α − S20 + − α + 1 > 0, S20 S201 S10 S10 S201 whose solutions are S20 = S201 and S20 = S10 /(1 − αS10 ). Under Assumptions 1 and 2, 1 1 S10 > − α ⇐⇒ S20 < , S20 S10 1 − αS10 and under the second condition of Proposition 3, S20 < S201 . Hence S20 ̸∈ [S10 /(1 − αS10 ), S201 ]. This ends the proof of this case. 1 − S20 /β2 : See the last point in the proof of Proposition 2. Again the proofs of Propositions 4 and 5 are very similar, and we only prove the second one. 6.3. Proof of Proposition 5 Let us first suppose that Assumptions 1 and 4 hold, and that S20 < 0, S01 > 0, S201 > 0. 1 − S10 /β1 : According to Lemma 9, with a probability close to 1 − S10 /β1 the mutants 1 do not survive the first phase. Then following (A.5) we get that the 0-population size stays close to its equilibrium n¯ 0 K during a time of order e V K where V is a positive constant independent of K . Hence it is still close to this value when the second mutant appears (time α log K ). The latter one has a negligible probability to survive as the invasion fitness S20 is negative. After the 2-population extinction, using again (A.5), we get that the 0-population size stays close to its equilibrium value n¯ 0 K during a time larger than e V K where V is a positive constant independent of K . S10 /β1 (1 − S201 /β2 ): According to Lemma 9, with a probability close to S10 /β1 the mutants 1 survive the first phase, and the 0- and 1-population sizes get close to their coexisting equilibrium (0) (1) (⌊n¯ 01 K ⌋, ⌊n¯ 01 K ⌋) during the second phase. Then we get from (A.5) that the 0- and 1-population sizes stay close to this coexisting equilibrium during a time larger than e V K for large K where V is a positive constant independent of K . Hence they are still close to this state when the second mutant appears (time α log K ). The latter one has a probability close to 1 − S201 /β2 to get extinct before hitting ⌊εK ⌋ (Lemma 15). After the 2-population extinction, the 0- and 1-population sizes (0) (1) stay close to their equilibrium value ⌊n¯ 01 K ⌋ and ⌊n¯ 01 K ⌋ during a time larger than e V K where V is a positive constant independent of K (Eq. (A.5)). S10 S201 /(β1 β2 ): Applying again Lemma 9 and Eq. (A.5), we get that with a probability close to (0) (1) S10 /β1 the 0- and 1-population sizes get close to (n¯ 01 K , n¯ 01 K ) and are still in this configuration when the second mutant occurs. The latter one has a probability close to S201 /β2 to hit a size of (K ,01) order K (Lemma 15) whereas (N0 , N1 ) still belongs to Iε . Then the final state depends on the signs of the other invasion fitnesses. The approximating deterministic Lotka–Volterra system after the third phase ✄ can belong to 7–12, 29, 31 or 33rd class described by Zeeman (see Fig. 5 and Table 2 case ✂B ✁). In all the cases the density of 2-type individuals does not tend to 0 during a time larger than e V K where V is a positive constant independent of K . The proof of the second case follows the same ideas. We just have to take into account the value of α log K compared to some invasion time to know whether the 0-population gets extinct

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before the hitting of ⌊εK ⌋ by the 2-population. We detail this kind of comparison in the proof of Proposition 7 and end here the proof of Proposition 4. Let us conclude this section with the proof of Proposition 7. The proof of Proposition 6 follows the same outline and is simpler; hence we leave it to the reader. 6.4. Proof of Proposition 7 Suppose that Assumptions 1 and 4, and conditions (2.9) and (2.10) hold. 1 − S10 /β1 : See the beginning of the proof of Proposition 4. (S10 /β1 )(1 − S21 /β2 ): Lemma 9 and Eq. (A.4) imply that with a probability close to S10 /β1 , the 1-population survives the first phase and the population state at the end of the second phase satisfies (N0 , N1 )(φ1K ) = (⌊n 0 K ⌋, ⌊n 1 K ⌋) with (z)

n 0 (tε (0, 1)) < n 0 < ε2 K 2 z∈Bε inf

and

|n 1 − n¯ 1 | ≤ ε,

where tε (0, 1) has been defined in (4.4). Then with a probability close to 1 − S21 /β2 the 2-population does not survive, and the 0-population size, which can be compared to a subcritical birth and death process as S01 < 0, also hits 0 while the 1-population size is still close to ⌊n¯ 1 K ⌋ (Lemma 15). In this case, the 1-population size stays close to its equilibrium value ⌊n¯ 1 K ⌋ during a time larger than e V K where V is a positive constant independent of K (Eq. (A.5)). S10 S21 /(β1 β2 ): First applying again Lemma 9 and Eq. (A.4) we get that with a probability close to S10 /β1 , the 1-population survives the first phase and the population state at the end of the second phase satisfies (N0 , N1 )(φ1K ) = (⌊n 0 K ⌋, ⌊n 1 K ⌋) with (z)

n 0 (tε (0, 1)) < n 0 < ε2 K 2 z∈Bε inf

and

|n 1 − n¯ 1 | ≤ ε.

Then Lemma 15 implies that with a probability close to S21 /β2 , the 2-population size hits the value ⌊εK ⌋ before the extinction of the 0-population, because  1 1  + log K , S10 |S01 | which is the approximate extinction time of the 0-population (Lemma 16), is bigger than  1  α+ log K , S21 which is the approximate hitting time of ⌊εK ⌋ by the 2-population size (Lemma 16). Combining Lemmas 9, 15 and 16 we even get that at the end of the third phase, the 0-population size is of order K

  1−|S01 | α− S1 + S1 10

21

,

and that the duration of the third phase is 

α−

1 1  + log K . S10 S21

(6.1)

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During the fourth phase we can approximate the dynamics of the 1- and 2-populations, which have a size of order K , by the system (1.8) with (i, j) = (1, 2) (Lemma 12). As S12 < 0 < S21 , the 1- and 2-population states at the end of the fourth phase are (⌊n 1 K ⌋, ⌊n 2 K ⌋) with (z)

n 1 (tε (1, 2)) < n 1 < ε2 2 z∈BεK inf

and

|n 2 − n¯ 2 | ≤ ε.

During the fifth phase, the 0-population size has an evolution comparable to this of a supercritical birth and death process (S02 > 0) and takes a time 1  1 |S01 |  + α− log K S02 S10 S21

(6.2)

to hit ⌊εK ⌋ (Lemmas 13 and 14). The 1-population size has an evolution comparable to this of a subcritical birth and death process (S12 < 0) and takes a time log K |S12 |

(6.3)

to get extinct. As according to condition (2.10) 1  |S01 |  1 1 + , α− < S02 S10 S21 |S12 | Eqs. (6.2) and (6.3) imply that the 0-population size hits ⌊εK ⌋ before the extinction of the 1-population. The fifth phase has a duration of order |S01 |  1 1  α− + log K , S02 S10 S21

(6.4)

and at the end of the fifth phase, the 1-population size is of order K

1−

|S12 ||S01 | S02

  α− S1 + S1 10

21

.

We then again apply the same reasoning: during the sixth phase, the 0-population outcompetes the 2-population. During the seventh phase, which has a duration of order 1 1  |S12 ||S01 |  α− + log K , S10 S02 S10 S21

(6.5)

the 1-population size hits the value ⌊εK ⌋ and the 2-population sizes end at a value of order K

1−

|S20 ||S12 ||S01 | S10 S02

  α− S1 + S1 10

21

,

where we used condition (2.10) which implies that |S12 ||S01 |  1 1  1 α− + < . S10 S02 S10 S21 |S20 | Adding the durations of the third, fifth and seventh phases in (6.1), (6.4) and (6.5), we get the duration of the first cycle,  1 1  |S01 | |S01 ||S12 |  α− + 1+ + log K . S10 S21 S02 S02 S10

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Then by induction we prove that at the end of the phase 3 + l, l ∈ N, the 0-population size is of order K

   |S12 ||S01 ||S20 | l 1−|S01 | α− S1 + S1 S S S 10

21

10 02 21

,

at the end of the phase 5 + l, l ∈ N, the 1-population size is of order K

1−

|S12 ||S01 | S02



α− S1 + S1 10



21

 |S12 ||S01 ||S20 | l S10 S02 S21

,

and at the end of the phase 7 + l, l ∈ N, the 2-population size is of order K

1−

|S20 ||S12 ||S01 | S10 S02



α− S1 + S1 10



21

 |S12 ||S01 ||S20 | l S10 S02 S21

.

We also prove that the 3 + lth phase, l ∈ N, has a duration close to  |S ||S ||S | l  1 1  12 01 20 α− + log K , S10 S02 S21 S10 S21 the 5 + lth phase, l ∈ N, has a duration close to  |S ||S ||S | l |S |  1  1 12 01 20 01 + α− log K , S10 S02 S21 S02 S10 S21 and the 7 + lth phase, l ∈ N, has a duration close to  |S ||S ||S | l |S ||S |  1  1 12 01 12 01 20 + α− log K . S10 S02 S21 S10 S02 S10 S21 This completes the proof of Proposition 7. Acknowledgements The authors would like to thank Olivier Tenaillon who suggested this research subject several years ago, and Sylvie M´el´eard for her comments. They also want to thank two anonymous reviewers for their careful reading of the paper, and several suggestions and improvements. This work was partially funded by project MANEGE “Mod`eles Al´eatoires en Ecologie, G´en´etique et Evolution” of the French national research agency ANR-09-BLAN-0215 and Chair “Mod´elisation Math´emathique et Biodiversit´e” of Veolia Environnement — Ecole Polytechnique - Mus´eum National d’Histoire Naturelle — Fondation X and the French national research agency ANR-11-BSV7-013-03. Appendix A. Technical results This section is dedicated to technical results needed in the proofs. We first recall some facts about birth and death processes. They are used in Sections 3–5 and can be found in Lemma 3.1 in [43] and in [1, p. 109 and 112]: Lemma A.1. Let Z = (Z t )t≥0 be a birth and death process with individual birth and death rates b and d. For a ∈ R+ , Ta = inf{t ≥ 0, Z t = ⌊a⌋} and Pi (resp. Ei ) is the law (resp. expectation) of Z when Z 0 = i ∈ N. Then

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• For (i, j, k) ∈ Z3+ such that i < j < k, P j (Tk < Ti ) =

1 − (d/b) j−i . 1 − (d/b)k−i

(A.1)

• If 0 < d ̸= b, for every i ∈ Z+ and t ≥ 0, Pi (T0 ≤ t) =

 d(1 − e(d−b)t ) i b − de(d−b)t

.

(A.2)

• If 0 < d < b, on the non-extinction event of Z , which has a probability 1 − (d/b) Z 0 , the following convergence holds: TN / log N → (b − d)−1 , N →∞

a.s.

(A.3)

We also need large deviation results to quantify the difference between the rescaled population process N K /K and the approximating deterministic Lotka–Volterra processes (1.8) and (1.16). The following statements can be found in [9] Theorem 3 (b) and (c) and in [11, Proposition A.2]. They follow from Dupuis and Ellis [18, Theorem 10.2.6 in Chapter 10]: Lemma A.2. Let C be a compact subset of (R+ r {0})2 or (R+ r {0})3 , and T a finite positive constant. Then for every positive δ,     lim sup P sup ∥N K (t)/K − n (z) (t)∥ > δ N0K (0) = ⌊z K ⌋ = 0, (A.4) K →∞ z∈C

t≤T

where ⌊z K ⌋ = (⌊z 0 K ⌋, ⌊z 1 K ⌋, ⌊z 2 K ⌋), and n (z) is the solution of (1.16) with initial condition z. Let n¯ denote a stable equilibrium of a competitive Lotka–Volterra system in dimension one, two or three, with all coordinates positive. Let ε > 0 and N K denote the population process with the same ecological parameters as the considered Lotka–Volterra system and carrying capacity K . Then there exists a positive constant V such that    N K (0)   N K (t)       lim P sup  − n¯  ≤ 2ε   − n¯  ≤ ε = 1. K →∞ K K t≤e V K

(A.5)

We now present two examples of three dimensional Lotka–Volterra systems satisfying conditions of Proposition 7 and exhibiting different long time behaviours. In the first one, the solutions converge to a limit cycle with constant period whereas in the second one they converge to the unique globally attracting fixed point of the system. Lemma A.3. (i) There exist three dimensional Lotka–Volterra systems satisfying conditions of Proposition 7 and whose solutions exhibit limit cycles. (ii) There exist three dimensional Lotka–Volterra systems satisfying conditions of Proposition 7 and whose solutions converge to a stable fixed point. Before proving this Lemma, we need to give a definition and recall a result of Hofbauer and Sigmund.

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Definition A.1. A matrix A = (ai j )0≤i, j≤2 ∈ R3×3 is called Volterra–Lyapunov stable if there exist positive numbers d0 , d1 , d2 such that  di ai j n i n j < 0, ∀n ∈ R3+ r {0}. 0≤i, j≤2

Let us introduce the matrix   C0,0 C0,1 C0,2 A = C1,0 C1,1 C1,2  . C2,0 C2,1 C2,2 Then we have: Theorem A.1 (Theorem 15.3.1 in [26]). If −A is Volterra–Lyapunov stable then the Lotka–Volterra system (1.16) has one globally stable fixed point. We can now prove Lemma A.3. Proof of Lemma A.3. (i) In [38], the authors model the RPS interactions of Escherichia coli strains by the following system:  n˙ 0 = (δ − κ1 n 0 − µn 1 − µn 2 )n 0 , n˙ 1 = (η − µn 0 − κ3 n 1 − µn 2 )n 1 ,  n˙ 2 = (β − (µ + γ )n 0 − µn 1 − µn 2 )n 2 , where all the parameters belong to R+ r {0}. They give an example for which the solutions converge to a limit cycle with constant period. It corresponds to the following values of the parameters: η = µ = 1,

κ1 = β = 2,

κ3 = 0, 844 . . .

and

κ2 = 1, 75,

γ = 2, 84,

δ = 1, 156 . . .

For these values of the parameters we have: S10 = 0, 422 . . . ,

S01 = −0, 0288 . . .

S21 = 0, 816 . . . ,

S12 = −0, 0143

S02 = 0, 0131 . . . ,

S20 = −0, 220 . . . .

These invasion fitnesses satisfy (2.9) and S21 > |S01 |,

S02 S21 > |S01 ||S12 |

and

S02 S10 S21 > |S01 ||S12 ||S20 |.

Hence we can choose α such that (2.10) holds, which proves point (i). (ii) Recalling the definition of invasion fitnesses in (1.6) we get that −A is Volterra–Lyapunov stable if there exist positive constants d0 , d1 , d2 such that for every n ̸= 0,  (βi − δi )di Ci,i n i C j, j n j (βi − δi − Si j ) > 0, C β i,i i − δi β j − δ j 0≤i, j≤2 which amounts to the existence of positive constants d0 , d1 , d2 such that for every n ̸= 0,  di (βi − δi − Si j )n i n j > 0. 0≤i, j≤2

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Fig. 7. Different dynamics under Assumption 2. See Appendix B.1 for the caption.

Expanding the sum and using the notation ρi = βi − δi for i ∈ E we get the condition   d0 ρ0 n 20 + d1 ρ1 n 21 + d2 ρ2 n 22 + d0 (ρ0 + |S01 |) + d1 (ρ1 − S10 ) n 0 n 1     + d0 (ρ0 − S02 ) + d2 (ρ2 + |S20 |) n 0 n 2 + d1 (ρ1 + |S12 |) + d2 (ρ2 − S21 ) n 1 n 2 > 0.

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Table 1 Possible dynamics under Assumption 2. Encircled letters correspond to those in Fig. 7. See Appendix B.1 for the caption. Case

Duration of the sweep: log K .

>0

<0

Class

(0)

(1)

(n¯ 01 , n¯ 01 , 0)

1 S10

S21 S12 , S21





1 1 1 S10 + S201 1 − S20 S10 − α



S12 , S02

7

(0, 0, n¯ 2 )

S12 , S02 , S21

8

(0, 0, n¯ 2 )

S02 , S012

9

(0, n¯ 12 , n¯ 12 )

1 1 S10 + S21



1 S10 − α



|S01 | 1 1 S10 + S21 1 + S012

S02 , S12 , S21

S102

12

S012

12

S12 , S102

9

S02 , S12

S21 , S102

10

S02

S21 , S12

11

S12

S21 , S02

29

012

S21 , S12 , S012

S02

31

012

S02 , S12 , S102

S21

31

012

S102 , S21 , S02

S12

31

012

33

012 (0, n¯ 12 , n¯ 12 )



1 S10 − α



S102



1 S10 − α

(2)

(0)

10

(n¯ 02 , 0, n¯ 02 )

31

012 (0, 0, n¯ 2 )

10



(2)

(1)



S102

|S01 | 1 1 S10 + S21 1 + S02

(2)

S02 , S12 , S21

S02 , S21

   1 1−S 1 1 20 S − α S + S 21

(1)

(0) (2) (n¯ 02 , 0, n¯ 02 ) (0) (2) (n¯ 02 , 0, n¯ 02 ) (0) (2) (n¯ 02 , 0, n¯ 02 ) (0) (2) (n¯ 02 , 0, n¯ 02 ) (1) (2) (0, n¯ 12 , n¯ 12 )

S21 , S12 , S02 , S012 , S102

10

FS

(2)

(0)



(n¯ 02 , 0, n¯ 02 )



|S01 | 1 1 S10 + S21 1 + S02 +   |S01 |S|12 | 1 S02 S102 S10 − α

29

1 S10

012

(0, n¯ 1 , 0)

Let us now make the following choice: ρi di = 1,

i ∈ E,

ρ0 − S02 ρ1 − S10 ρ2 − S21 = = = η, ρ0 ρ1 ρ1

|S01 | |S12 | |S20 | = = = η, ρ0 ρ1 ρ2 where 0 < η < 1/2. The condition to satisfy becomes n 20 + n 21 + n 22 +

1 + 2η (2n 0 n 1 + 2n 0 n 2 + 2n 1 n 2 ) 2

(A.6)

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Table 2 Possible dynamics under Assumption 4. Encircled letters correspond to those in Fig. 9. See Appendix B.3 for the caption. Case

Duration of the sweep: log K .

>0

<0

Class

FS (0)

1 S10

(1)

(n¯ 01 , n¯ 01 , 0)

α + S1

210

S21 , S20 S20 S21

S12 , S02 S12 , S02 , S21 S12 , S02 , S20

7 8 8

(0, 0, n¯ 2 ) (0, 0, n¯ 2 ) (0, 0, n¯ 2 )

S20 , S12 , S21

S02

9

(0, n¯ 12 , n¯ 12 )

S20 , S02 , S21

S12

9

S20 , S02 , S12

S21 , S102

10

S21 , S12 , S02

S20 , S012

10

(1)

S20 , S02

S21 , S12

11

S21 , S12

S20 , S02

11

S20 , S02 , S12 , S21

S102

12

S20 , S02 , S12 , S21 S20 , S12 S21 , S02

S012 S21 , S02 S20 , S12

12 29 29

012 012

S20 , S21 , S12 , S012 S20 , S21 , S02 , S102 S20 , S02 , S12 , S102 S21 , S02 , S12 , S012 Si j , Si jk , i, j, k ∈ {0, 1, 2}

S02 S12 S21 S20

31 31 31 31 33

012 012 012 012 012 (1)

α + S1

(2)

(0) (2) (n¯ 02 , 0, n¯ 02 ) (2) (0) (n¯ 02 , 0, n¯ 02 ) (1) (2) (0, n¯ 12 , n¯ 12 ) (0) (2) (n¯ 02 , 0, n¯ 02 ) (1) (2) (0, n¯ 12 , n¯ 12 ) (0) (2) (n¯ 02 , 0, n¯ 02 ) (1) (2) (0, n¯ 12 , n¯ 12 )

(2)

(0, n¯ 12 , n¯ 12 )

201

S20 

|S | α + S1 + S 01 α − S1 + S1 21 012 10 21

 S20 , S102

(0)

(2)

S102

10

(n¯ 02 , 0, n¯ 02 )

S20

29

012

31

012

α + S1

(0, 0, n¯ 2 )

  |S | α + S1 + S01 α − S1 + S1 21 02 10 21

(n¯ 02 , 0, n¯ 02 )

  |S | α + S1 + S01 α − S1 + S1 21 02 10 21

(n¯ 0 , 0, 0)

21

 |S | α + S1 + S01 1 + 21  02 1 1 S10 + S21  |S | α + S1 + S01 1 + 21  02 1 1 S + S 10

1 S10

(0)

(2)

|S12 | S102

 α−

29

012

|S12 | S10

 α−

27

cycles RPS

21

(0, n¯ 1 , 0)

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Fig. 8. From Assumption 2 to Assumption 3 (see Appendix B.2).

=

1 − 2η 2 1 + 2η (n 0 + n 21 + n 22 ) + (n 0 + n 1 + n 2 )2 > 0, 2 2

which holds for every non null n ∈ R3 . We still have to prove that we can indeed choose the parameters as in (A.6). Assume for example that the ρi ’s and the Ci,i ’s are given. Then it is enough to take ρi+1 ρi Ci+1,i = η Ci+1,i+1 , where i is modulo 2. Ci,i , Ci,i+1 = (1 + η) ρi ρi+1 Finally it is easy to check that the conditions of Proposition 7 are satisfied. Applying Theorem A.1 we get that the Lotka–Volterra system (1.16) has one globally stable fixed point. This ends the proof of point (ii).  Appendix B. Complete description of possible dynamics We now give a complete description of the possible population dynamics. Appendices B.1 and B.3 are dedicated to Assumptions 2 and 4, respectively. In Appendix B.2 we explain how the dynamics under Assumptions 1 and 3 can be deduced from the dynamics under Assumptions 1 and 2. B.1. Assumption 2 In Fig. 7 and Table 1, we assume that the 1- and 2-populations survive the first phase. This happens with a probability close to S10 S20 /(β1 β2 ) (Lemma 6) and is the case we are interested in, as otherwise there is no clonal interference and we are brought back to the invasion of one mutant already studied in [9,11]. The behaviour of the population process after the first phase depends on the relations between the ecological parameters of the different individual types. We describe the different conditions which discriminate between different scenarios in Fig. 7, and list them in their chronological order of appearance. We also indicate the phase during which the conditions have an impact on the population process behaviour.

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Fig. 9. Different dynamics under Assumption 4. Same legend as in Fig. 7.

✄ In Table 1 we describe the “final state” of the population processes under the conditions ✂A ✁ ✄ 3 to ✂K ✁. For the sake of simplicity we call “final state” in this setting the element of R+  F S := {n ∈ R3+ , ∃β(ε), V (ε) > 0, lim inf P(1,2) (n ∈ F S(ε, K , β(ε), V (ε))) > 0}, (B.1) ε>0

K →∞

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when this intersection is non empty, otherwise we call “final state” the long time behaviour of the three dimensional Lotka–Volterra system close to the rescaled population process once the three population types have a size of order K . In this case we indicate the corresponding class of the dynamical system in the Zeeman representation (Fig. 5) and we write “012” when the invasion fitness signs do not allow to discriminate between a cyclical or stable coexistence of the three types of populations. We have also indicated in Table 1 the durations of the sweep. They have to be understood in the sense of Lemma 6 for instance, when the final state F S described in (B.1) is non empty. They are good approximations of the durations with a probability one up to a constant times ε. When F S is empty, these durations correspond to the time needed for the three populations to hit a size of order K . B.2. Assumption 3 Let us now explain how we deduce the possible dynamics under Assumption 3 from the possible dynamics under Assumption 2. Applying Lemma 5 to 8 we get the dynamics represented in Fig. 8 with a probability close to S10 S20 /(β1 β2 ) under Assumptions (1 and) 2 and 3, respectively. We deduce that to obtain an equivalent of Fig. 1 it is enough to: 1. Interchange the roles of the 1- and 2-populations, 2. Replace α by −α in the conditions of Fig. 7 and in the duration of the sweep 3. Add α log K to the duration of the sweep. We see that we do not get new long time behaviours by comparison with Assumptions 1 and 2. This is why we did not detail this case. B.3. Assumption 4 Fig. 9 and Table 2 are the analogues of Fig. 7 and Table 1 for Assumption 4. Here we only assume that the 1-population survives the first phase (probability close to S10 /β1 according to Lemma 5), as the second mutation occurs during the third phase. The captions are the same, except that we add in Table 2 a final state “cycles Rock–Paper–Scissors” which corresponds to the class 27 in Zeeman’s classification (Proposition 7). References [1] K.B. Athreya, P.E. Ney, Branching Processes, Dover Publications Inc., Mineola, NY, 2004, Reprint of the 1972 original [Springer, New York; MR0373040]. [2] N.H. Barton, Linkage and the limits to natural selection, Genetics 140 (2) (1995) 821–841. [3] S. Billiard, C. Smadi, Beyond clonal interference: Scrutinizing the complexity of the dynamics of three competing clones, 2016 (submitted for publication). [4] J.P. Bollback, J.P. Huelsenbeck, Clonal interference is alleviated by high mutation rates in large populations, Mol. Biol. Evol. 24 (6) (2007) 1397–1406. [5] R. Brink-Spalink, C. Smadi, Genealogies of two linked neutral loci after a selective sweep in a large population of stochastically varying size, 2015. arXiv preprint arXiv:1502.03837. [6] L. Buss, J. Jackson, Competitive networks: nontransitive competitive relationships in cryptic coral reef environments, Amer. Nat. (1979) 223–234. [7] D.D. Cameron, A. White, J. Antonovics, Parasite-grass-forb interactions and rock–paper–scissor dynamics: predicting the effects of the parasitic plant rhinanthus minor on host plant communities, J. Ecol. 97 (6) (2009) 1311–1319. [8] P.R. Campos, C. Adami, C.O. Wilke, Modelling stochastic clonal interference, in: Modelling in Molecular Biology, Springer, 2004, pp. 21–38.

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