The Polymorphic Evolution Sequence for Populations with Phenotypic Plasticity
TTHE POLYMORPHIC EVOLUTION SEQUENCE FOR POPULATIONS WITHPHENOTYPIC PLASTICITY
MARTINA BAAR AND ANTON BOVIERA bstract . In this paper we study a class of stochastic individual-based models that describe theevolution of haploid populations where each individual is characterised by a phenotype and agenotype. The phenotype of an individual determines its natural birth- and death rates as wellas the competition kernel, c ( x , y ) which describes the induced death rate that an individual oftype x experiences due to the presence of an individual or type y . When a new individual isborn, with a small probability a mutation occurs, i.e. the o ff spring has di ff erent genotype as theparent. The novel aspect of the models we study is that an individual with a given genotypemay express a certain set of di ff erent phenotypes, and during its lifetime it may switch betweendi ff erent phenotypes, with rates that are much larger then the mutation rates and that, moreover,may depend on the state of the entire population. The evolution of the population is describedby a continuous-time, measure-valued Markov process. In [4], such a model was proposed todescribe tumor evolution under immunotherapy. In the present paper we consider a large classof models which comprises the example studied in [4] and analyse their scaling limits as thepopulation size tends to infinity and the mutation rate tends to zero. Under suitable assumptions,we prove convergence to a Markov jump process that is a generalisation of the polymorphicevolution sequence (PES) as analysed in [8, 10].
1. I ntroduction
Over the last decade there has been increasing interest in the mathematical analysis of so-called stochastic individual based models of adaptive dynamics . These models were introducedin a series of papers by Bolker, Pacala, Dieckmann, and Law [6, 7, 12]. They describe the evo-lution of a population of individuals characterised by their phenotypes under the influence of theevolutionary mechanisms of birth, death, mutation, and ecological competition in an inhomoge-neous "fitness landscape" as a measure valued Markov process. In these models there appear twonatural scaling parameters. The carrying capacity , K , which regulates the size of the populationand that can reasonably considered as a large parameter, and the mutation rate (of advantageousmutations), u , that in many biological situations can be taken as a small parameter. In a seriesof remarkable papers, Champagnat and Méléard [8, 10] (and others) have analysed the limitingprocesses that arise in the limit when K is taken to infinity while at the same time u = u K tendsto zero. Under conditions that ensure the separation of the ecological and evolutionary time Key words and phrases. adaptive dynamics, canonical equation, large population limit, mutation-selectionindividual-based model.M. B. is supported by the German Research Foundation through the Priority Programme 1590 “ProbabilisticStructures in Evolution”. A.B. is partially supported by the German Research Foundation in the Collaborative Re-search Center 1060 "The Mathematics of Emergent E ff ects", the Priority Programme 1590 “Probabilistic Structuresin Evolution”, the Hausdor ff Center for Mathematics (HCM), and the Cluster of Excellence “ImmunoSensation” atBonn University. a r X i v : . [ m a t h . P R ] A ug ES WITH PHENOTYPIC PLASTICITY 2 scales. This means that the mutation rates are so small that the system has time to equilibrate(ecological time scale) between two mutational events. On the time scale where mutations occur(evolutionary time scale), the evolution of the population can then be described as a Markovjump process along a sequence of equilibria of, in general, polymorphic populations. An impor-tant (and in some sense generic) special case occurs when the mutant population fixates whilethe resident population dies out in each step. The corresponding jump process is called the
TraitSubstitution Sequence (TSS) in adaptive dynamics. Champagnat [8] derived criteria in the con-text of individual-based models under which convergence to the TSS can be proven. The generalprocess is called the
Polymorphic Evolution Sequence (PES) [10]. Here the limit is describes asa jump process between possibly polymorphic equilibria of systems of Lotka-Volterra equationsof increasing dimension.In the present paper we extend this analysis to models where an additional biological phenom-enon is present, the so-called phenotypic plasticity . By this we mean the following. Individualsare no longer described by their phenotype, but by both their genotype and their phenotype.Moreover, an individual of a given phenotype can express several phenotypes and it can changeits phenotype during the course of its lifetime.Our original motivation for this comes from applications to cancer therapy, where it is well-known that phenotypic switches ( “phenotypic plasticity" ) is of utmost importance and in facta major obstacle to successful therapies (see, e.g. [17] and references therein). For a first at-tempt at modelling specific scenarios in the framework of individual based stochastic models,see [4]. However, phenotypic switches without mutations are certainly relevant in many if notmost biological systems.Here we take a broader look at a large class of models. By expanding the techniques of [10] weprove that the microscopic process converges on the evolutionary time scale to a generalisationof the Polymorphic Evolution Sequences (PES) (cf. Thm. 3.3). The main di ff erence in the proofis that we have to couple the process with multi-type branching processes instead of normalbranching processes, which leads also to a di ff erent definition of invasion fitness in this setting.Note that we gave in [4] already heuristic arguments why the process should converge to aMarkov jump process. The aim of this paper is to give the rigorous statement and its proof.The remainder of this paper is organised as follows. In Section 2 we define the model, give apathwise description of the Markov process we are studying and state the convergence towardsa quadratic system of ODEs in the large population limit. In Sections 3 we consider the caseof rare mutations and fast switches. More precisely, we state the convergence to the Polymor-phic Evolution Sequences with phenotypic Plasticity (PESP) in Subsection 3.2 and prove it inSubsection 3.3. 2. T he microscopic model In this section we introduce the stochastic individual-based model we analyse (cf. [4, 15, 8,10, 5]). The evolutionary process changes populations on a macroscopic level, but the basicmechanisms of evolution, heredity, variation (in our context caused by mutation and phenotyp-ical switching), and selection, act on the microscopic level of the individuals. We describe theevolving population as a stochastic system of interacting individuals, where each individual ischaracterised by its phenotype and its genotype.
ES WITH PHENOTYPIC PLASTICITY 3
Let l ≥ X a finite set of the form X = G × P , where G is the set of genotypes and P is the set of phenotypes. We call X the trait space of the population. As usual, we introduce aparameter K ∈ N , called the carrying capacity . This parameter allows to scale the populationsize and can be interpreted as the size of available space or the amount of available resources.Let M ( X ) be the set of finite, non-negative measures on X , equipped with the topology of weakconvergence, and let M K ( X ) ⊂ M ( X ) be the set of finite point measures on X rescaled by K, i.e. M K ( X ) ≡ K n (cid:88) i = δ x i : n ∈ N , x , . . . x n ∈ X , (2.1)where δ x denotes the Dirac mass at x ∈ X . We model the time evolution of a population asan M K ( X )-valued, continuous time Markov process ( ν Kt ) t ≥ . To account for the process basicmechanisms of evolution and the phenotypic plasticity, we introduce the following parameters:(i) b ( p ) ∈ R + is the rate of birth of an individual with phenotype p ∈ P .(ii) d ( p ) ∈ R + is the rate of natural death of an individual with with phenotype p ∈ P .(iii) c ( p , ˜ p ) K − ∈ R + is the competition kernel which models the competitive pressure an in-dividual with phenotype p ∈ P feels from an individual with phenotype ˜ p ∈ P and isinversely proportional to the carrying capacity K .(iv) s g nat. ( p , ˜ p ) ∈ R + is the natural switch kernel which models the natural switching from phe-notype p to ˜ p of individuals with genotype g .(v) s g ind. ( p , ˜ p )( ˆ p ) K − ∈ R + is the induced switch kernel which models the switching from phe-notype p to ˜ p of individuals with genotype g induced by an individual with phenotypeˆ p . (Compare with the cytokine-induced switch of [4], especially the one of TNF- α (TumourNecrosis Factor).)(vi) u K m ( g ) with u K , m ( g ) ∈ [0 ,
1] is the probability that a mutation occurs at birth from anindividual with genotype g ∈ G , where u K is a scaling parameter.(vii) M (( g , p ) , ( ˜ g , ˜ p )) is the mutation law , i.e. if a mutant is born from an individual with trait( g , p ), then the mutant’s trait is ( ˜ g , ˜ p ) with probability M (( g , p ) , ( ˜ g , ˜ p )).Note that most of the parameters depend on the phenotype only and that we explicitly allowthat individuals with di ff erent genotypes can express the same phenotype and conversely thatindividuals with the same genotype can express di ff erent phenotypes. Assumption 1.
For simplicity we assume that s g ind. ( p , ˜ p )( ˆ p ) K − = p ∈ X whenever s g nat. ( p , ˜ p ) =
0, i.e. depending on the environment the total switching rate can be larger or smallerbut not zero or non-zero.At any time t ≥
0, we consider a finite population which consist of N t individuals and eachindividual is characterised its trait x i ( t ) ∈ X . The state of a population at time t is the measure ν Kt = K N t (cid:88) i = δ x i ( t ) . (2.2) ES WITH PHENOTYPIC PLASTICITY 4
The population process ν K is a M K ( X )-valued Markov process with infinitesimal generator L K , defined, for any bounded measurable function φ : M K ( X ) → R and for all µ K ∈ M K ( X ) by (cid:16) L K φ (cid:17) ( µ K ) (2.3) = (cid:88) ( g , p ) ∈G×P (cid:16) φ (cid:16) µ K + δ ( g , p ) K (cid:17) − φ ( µ K ) (cid:17) (1 − u K m ( g )) b ( p ) K µ K ( g , p ) + (cid:88) ( g , p ) ∈G×P (cid:88) (˜ g , ˜ p ) ∈G×P (cid:16) φ (cid:16) µ K + δ (˜ g , ˜ p ) K (cid:17) − φ ( µ K ) (cid:17) u K m ( g ) M (cid:0) ( g , p ) , ( ˜ g , ˜ p ) (cid:1) b ( p ) K µ K ( g , p ) + (cid:88) ( g , p ) ∈G×P (cid:16) φ (cid:16) µ K − δ ( g , p ) K (cid:17) − φ ( µ K ) (cid:17) (cid:18) d ( p ) + (cid:88) ˜ p ∈P c ( p , ˜ p ) µ K ( ˜ p ) (cid:19) K µ K ( g , p ) + (cid:88) ( g , p ) ∈G×P (cid:88) ˜ p ∈P (cid:16) φ (cid:16) µ K + δ ( g , ˜ p ) K − δ ( g , p ) K (cid:17) − φ ( µ K ) (cid:17) (cid:18) s g nat. ( p , ˜ p ) + (cid:88) ˆ p ∈P s g ind. ( p , ˜ p )( ˆ p ) µ K ( ˆ p ) (cid:19) K µ K ( g , p ) . The first and second terms describe the births (without and with mutation), the third termdescribes the deaths due to age or competition, and the last term describes the phenotypic plas-ticity. Observe that the first and second terms are linear (in µ K ), but the third and fourth termsare non-linear. The only di ff erence to the standard model is the presence of the fourth term thatcorresponds to the phenotypic switches. However, this term changes the dynamics substantially.In particular, the system of di ff erential equations which arises in the large population limit with-out mutation ( u K =
0) is not a generalised Lotka-Volterra system anymore, i.e. has not the form˙ n = n f ( n ), where f is linear in n (cf. Thm. 2.1 and Def. 3.1). Remark . (i) Since X is finite, we could also represent the population state as an |X| -dimen-sional vector. More precisely, let E be a subset of R |X| and E K ≡ E ∪ { n / K : n ∈ N } , thenfor fixed K ≥
1, the population process can be constructed as Markov process with statespace E K by using independent standard Poisson processes (cf. [14] Chap. 11).(ii) For an extension to a non-finite trait space, e.g. if G and P are compact subsets of R k forsome k ≥
1, the modeling of switching the phenotype has to be changed in the followingway: Each individual with trait ( g , p ) ∈ G × P has instead of the natural switch kernel s g nat. ( p , ˜ p ) a natural switch rate s nat. ( g , p ) combined with a probability measure S ( g , p )nat. ( d ˜ p )on P and instead of the induced switch kernel s g ind. ( p , ˜ p )( ˆ p ) K − a induced switch kernel s ind. (( g , p ) , ˆ p ) K − combined with a family of probability measure { S (( g , p ) , ˆ p )ind. ( d ˜ p ) } on P .2.1. Explicit construction of the population process with phenotypic plasticity.
It is usefulto give a pathwise description of ν K in terms of Poisson point measures (cf. [15]). Let us recallthis construction. Let ( Ω , F , P ) be an abstract probability space. On this space, we define thefollowing independent random elements:(i) a convergent sequence ( ν K ) K ≥ of M K ( X )-valued random measures (the random initial pop-ulation),(ii) |X| independent Poisson point measures ( N birth( g , p ) ( ds , di , d θ ) ) ( g , p ) ∈X on [0 , ∞ ) × N × R + withintensity measure ds (cid:80) n ≥ δ n ( di ) d θ ,(iii) |X| independent Poisson point measures ( N mut.( g , p ) ( ds , di , d θ, dx ) ) ( g , p ) ∈X on [0 , ∞ ) × N × R + × X with intensity measure ds (cid:80) n ≥ δ n ( di ) d θ (cid:80) ˜ x ∈X δ ˜ x ( dx ). ES WITH PHENOTYPIC PLASTICITY 5 (iv) |X| independent Poisson point measures ( N death( g , p ) ( ds , di , d θ ) ) ( g , p ) ∈X on [0 , ∞ ) × N × R + withintensity measure ds (cid:80) n ≥ δ n ( di ) d θ ,(v) |X| independent Poisson point measures ( N switch( g , p ) ( ds , di , d θ, d p ) ) ( g , p ) ∈X on [0 , ∞ ) × N × R + ×P with intensity measure ds (cid:80) n ≥ δ n ( di ) d θ (cid:80) ˜ p ∈P δ ˜ p ( d p ),Then, ν K is given by the following equation ν Kt = ν K + (cid:88) ( g , p ) ∈X (cid:90) t (cid:90) N (cid:90) R + { i ≤ K ν Ks − ( g , p ) , θ ≤ b ( p )(1 − u K m ( g )) } K δ ( g , p ) N birth( g , p ) ( ds , di , d θ ) (2.4) + (cid:88) ( g , p ) ∈X (cid:90) t (cid:90) N (cid:90) R + (cid:90) X { i ≤ K ν Ks − ( g , p ) , θ ≤ b ( p ) u K m ( g ) M (( g , p ) , x ) } K δ x N mut.( g , p ) ( ds , di , d θ, dx ) − (cid:88) ( g , p ) ∈X (cid:90) t (cid:90) N (cid:90) R + (cid:110) i ≤ K ν Ks − ( g , p ) , θ ≤ d ( p ) + (cid:80) ˜ p ∈P c ( p , ˜ p ) ν Ks − ( ˜ p ) (cid:111) K δ ( g , p ) N death( g , p ) ( ds , di , d θ ) + (cid:88) ( g , p ) ∈X (cid:90) t (cid:90) N (cid:90) R + (cid:90) P { i ≤ K ν Ks − ( g , p ) , θ ≤ s g nat. ( p , ˜ p ) + (cid:80) ˆ p ∈P s g ind. ( p , ˜ p )( ˆ p ) ν Ks − ( ˆ p ) }× K (cid:16) δ ( g , ˜ p ) − δ ( g , p ) (cid:17) N switch( g , p ) ( ds , di , d θ, d ˜ p ) . Remark . This construction uses that X is a discrete set and is in some sense closer to thedefinition given in [14] (p. 455). For non-discrete trait spaces the process can be constructed asin [15].2.2. The Law of Large Numbers.
If the mutation rate is independent of K and the initialconditions converge to a deterministic limit, then the sequence of rescaled processes, ( ν K ) K ≥ ,converges, almost surely, as K ↑ ∞ to the solution of system of ODEs. This follows directlyfrom the law of large numbers for density depending processes, see, e.g. Ethier and Kurtz [14],Chap. 11. The following theorem gives a precise statement. Theorem 2.1.
Let u K ≡ . Suppose that the initial conditions converge almost surely to adeterministic limit, i.e. lim K ↑∞ ν K = ν , where ν is a finite measure on X . Then, for every T > ,exists a deterministic function ξ ∈ C ([0 , T ] , M F ( X )) such that lim K ↑∞ sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Kt − ξ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = , a.s., (2.5) ES WITH PHENOTYPIC PLASTICITY 6 where || . || TV is the total variation norm. Moreover, let n be the unique solution to the dynamicalsystem ˙ n ( g , p ) ( t ) = n ( g , p ) ( t ) (cid:32)(cid:0) − m ( g ) (cid:1) b ( p ) − d ( p ) − (cid:88) (˜ g , ˜ p ) ∈G×P c ( p , ˜ p ) n (˜ g , ˜ p ) ( t ) (2.6) − (cid:88) ˜ p ∈P (cid:18) s gnat. ( p , ˜ p ) + (cid:88) (ˆ g , ˆ p ) ∈G×P s gind. ( p , ˜ p )( ˆ p ) n (ˆ g , ˆ p ) ( t ) (cid:19)(cid:33) + (cid:88) ˜ p ∈P n ( g , ˜ p ) ( t ) (cid:18) s gnat. ( ˜ p , p ) + (cid:88) (ˆ g , ˆ p ) ∈G×P s gind. ( ˜ p , p )( ˆ p ) n (ˆ g , ˆ p ) ( t ) (cid:19) + (cid:88) (˜ g , ˜ p ) ∈G×P n (˜ g , ˜ p ) ( t ) m ( ˜ g ) b ( ˜ p ) M (( ˜ g , ˜ p ) , ( g , p )) , ( g , p ) ∈ G × P , with initial condition n x (0) = ν ( x ) for all x ∈ X .Then, ξ is given as ξ t = (cid:80) x ∈X n x ( t ) δ x .Proof. This result follows from Theorem 2.1 in Chapter 11 of [14], since we can construct theprocess as described in Remark 1 (i). For more details see [21]. (cid:3)
Remark . If the trait spaces is not finite, one can obtain a similar result, cf. [15].3. T he interplay between rare mutations and fast switches .In this section we state our main results. As in previous work, we place ourselves under theassumptions ∀ V > , exp( − V K ) (cid:28) u K (cid:28) K ln K , as K ↑ ∞ , (3.1)which ensure that a population reaches equilibrium before a new mutant appears. Under theseassumptions we prove that the individual-based process with phenotypic plasticity convergencesto a generalisation of the PES. Let us start with describing the techniques used in [10].The key element in the proof of the convergence to the PES used by Champagnat and Méléard[10] is a precise analysis of how a mutant population fixates. A crucial assumption in [10] isthat the competitive Lotka-Volterra systems that describes the large population limit always havea unique stable fixed point ¯ n . Thus, the main task is to study the invasion of a mutant thathas just appeared in a population close to equilibrium. The invasion can be divided into threesteps: First, as long as the mutant population size is smaller than K (cid:15) , for a fixed small (cid:15) > K (cid:15) ,the whole system is close to the solution of the corresponding deterministic system and reachesan (cid:15) -neighbourhood of ¯ n in finite time. Third, the subpopulations which have a zero coordinatein ¯ n can be approximated by subcritical branching processes until they die out.The first and third steps require a time of order ln( K ), whereas the second step requires only atime of order one, independent of K . Since the expected time between two mutations is of order1 / ( u K K ), the the upper bound on u K in (3.1) guarantees that, with high probability, the threesteps of an invasion are completed before a new mutation occurs. ES WITH PHENOTYPIC PLASTICITY 7
In the first invasion step the invasion fitness of a mutant plays a crucial role. Given a populationin a stable equilibrium that populates a certain set of traits, say M ⊂ X , the invasion fitness f ( x , M ) is the growth rate of a population consisting of a single individual with trait x (cid:60) M in thepresence of the equilibrium population ¯ n on M . In the case of the standard model, it is given by f ( x , M ) = b ( x ) − d ( x ) − (cid:88) y ∈ M c ( x , y ) ¯ n y . (3.2)Positive f ( x , M ) implies that a mutant appearing with trait x from the equilibrium population on M has a positive probability (uniformly in K ) to grow to a population of size of order K ; negativeinvasion fitness implies that such a mutant population will die out with probability tending to one(as K ↑ ∞ ) before it can reach a size of order K . The reason for this is that the branching process(birth-death process) which approximates the mutant population is supercritical if f ( x , M ) ispositive and subcritical if f ( x , M ) is negative.In order to describe the dynamics of a phenotypically heterogeneous population on the evolu-tionary time scale, we have to adapt the notion of invasion fitness to the case where fast pheno-typic switches are present. Since switches between phenotypes associated to the same genotypehappen at times of order one, the growth rate of the initial mutant phenotype does not determinethe probability of fixation. See [11] for a similar issue in a model with sexual reproduction. Inthe proof of Theorem 3.3 we approximate the dynamics of the mutant population by a multi-type branching process until the reaches a size K (cid:15) (or dies out). A continuous-time multi-typebranching process is supercritical if and only if the largest eigenvalue of its infinitesimal gen-erator of is strictly positive (cf. [2, 23]). Therefore, this eigenvalue will provide an appropriategeneralisation of the invasion fitness.3.1. The competitive Lotka-Volterra system with phenotypic plasticity.
We first considerthe large population limit without mutation ( u K ≡ d traits, ( g , p ) = (( g , p ) , . . . , ( g d , p d )) ∈ ( G × P ) d , and that the sequence of the initialconditions converges almost surely to a deterministic limit, i.e.lim K ↑∞ ν K = d (cid:88) i = n i (0) δ ( g i , p i ) , a.s., where n i (0) > i ∈ { . . . d } . (3.3)By Theorem 2.1, for every T >
0, the sequences of processes ν K ∈ D ([0 , T ] , M K ( X )) generatedby L K with initial state ν K converges almost surely, as K ↑ ∞ , to a deterministic function ξ ∈ C ([0 , T ] , M ( X )). Since u K ≡
0, no new genotype can appear in the population process ν K .Moreover, not every genotype can express every phenotype. < let us describe the support of ν Kt more precisely.For all g ∈ G , let X g be a stationary discrete-time Markov chain with state space P andtransition probabilities P [ X gi = ˜ p | X gi − = p ] = s g ( p , ˜ p ) (cid:80) ˆ p ∈P s g ( p , ˆ p ) , if (cid:88) ˆ p ∈P s g ( p , ˆ p ) > P [ X gi = p | X gi − = p ] = , if (cid:88) ˆ p ∈P s g ( p , ˆ p ) = . (3.5) ES WITH PHENOTYPIC PLASTICITY 8
The Markov chains { X g , g ∈ G} contain only partial information on the switching behaviour ofthe process ν K , but we see that this is the key information needed later.In the sequel we work under the following simplifying assumption: Assumption 2.
For all g ∈ G , all communicating classes of X g are recurrent.We denote the communicating class associated with ( g , p ) ∈ G × P by [ p ] g . This is thecommunicating class of X g which contains p , i.e. p can be seen as a representative of the class,which has an equivalence relation depending on g . By Assumption 1, this ensures that if we start p p p p p p p p p p F igure Example of a Markov chain X g . Here, P = { p , . . . , p } and X g has fourcommunicating classes: { p , p , p , p } , { p , p , p } , { p } , { p , p } . The class { p } hasonly one element, i.e. (cid:80) i = s g ( p , p i ) = with a large enough population consisting only of individuals carrying the same trait ( g , p ), then,after a short time, all phenotypes in the class [ p ] g will be present in the population, but none ofthe other classes. Observe that these Markov chains do not describe the dynamics of the wholeprocess. If we allowed transient states this would not imply that the trait would get extinct, sinceits growth rate could be larger than the switching rate.Thus, [ p ] g is the set of phenotypes which are reachable in the Markov chain X g with X g = p and the set of traits which can appear in the population process ν K is given by X ( g , p ) ≡ d (cid:91) i = { g i } × [ p i ] g i . (3.6)With this notation, ξ is given by ξ ( t ) = (cid:80) x ∈X ( g , p ) n x ( t ) δ x , where n is the solution of the competi-tive Lotka-Volterra system with phenotypic plasticity defined below. Definition 3.1.
For any ( g , p ) ∈ ( G × P ) d , we denote by LVS ( d , ( g , p )) the competitive Lotka-Volterra system with phenotypic plasticity . This is an |X ( g , p ) | -dimensional system of ODEs givenby˙ n ( g , p ) = n ( g , p ) (cid:18) b ( p ) − d ( p ) − (cid:88) (˜ g , ˜ p ) ∈X ( g , p ) c ( p , ˜ p ) n (˜ g , ˜ p ) − (cid:88) ˜ p ∈ [ p ] g (cid:18) s g nat. ( p , ˜ p ) + (cid:88) (ˆ g , ˆ p ) ∈X ( g , p ) s g ind. ( p , ˜ p )( ˆ p ) n (ˆ g , ˆ p ) ( t ) (cid:19)(cid:19) + (cid:88) ˜ p ∈ [ p ] g n ( g , ˜ p ) (cid:18) s g nat. ( ˜ p , p ) + (cid:88) (ˆ g , ˆ p ) ∈X ( g , p ) s g ind. ( ˜ p , p )( ˆ p ) n (ˆ g , ˆ p ) ( t ) (cid:19) , ( g , p ) ∈ X ( g , p ) . (3.7)We choose the (possibly misleading) name competitive Lotka-Volterra system with phenotypicplasticity to emphasise that we add phenotypic plasticity (induced by switching rates) in theusual competitive Lotka-Volterra system. Note, however, that the system LVS is not a system ofLotka-Volterra equations. ES WITH PHENOTYPIC PLASTICITY 9
We now introduce the notation of coexisting traits in this context (cf. [10]).
Definition 3.2.
For any d ≥
2, we say that the distinct traits ( g , p ) , . . . , ( g d , p d ) coexist if thesystem LVS ( d , ( g , p )) has a unique non-trivial equilibrium ¯ n ( g , p ) ∈ (0 , ∞ ) |X ( g , p ) | which is locallystrictly stable, meaning that all eigenvalues of the Jacobian matrix of the system LVS ( d , ( g , p ))at ¯ n ( g , p ) have strictly negative real parts.Note that if ( g , p ) , . . . , ( g d , p d ) coexist, then all traits of X ( g , p ) coexist and the equilibrium¯ n ( g , p ) is asymptotically stable. We will prove later that if the traits ( g , p ) , . . . , ( g d , p d ) coexist,then the invasion probability of a mutant trait ( ˜ g , ˜ p ) which appears in the resident population X ( g , p ) close to ¯ n ( g , p ) is given by the function1 − q ( g , p ) ( ˜ g , ˜ p ) , (3.8)where q ( g , p ) ( ˜ g , ˜ p ) is given as follows: Let us denote the elements of [ ˜ p ] ˜ g by ˜ p , ˜ p , . . . , ˜ p | [ ˜ p ] ˜ g | andassume without lost of generality that ˜ p = ˜ p . Then, q ( g , p ) ( ˜ g , ˜ p ) is the first component of thesmallest solution of u ( y ) = , (3.9)where u is a map from R | [ ˜ p ] ˜ g | to R | [ ˜ p ] ˜ g | defined for all i ∈ { , . . . , | [ ˜ p ] ˜ g |} by u i ( y ) ≡ (3.10) b ( ˜ p i ) y i + | [ ˜ p ] ˜ g | (cid:88) j = (cid:18) s ˜ g nat. ( ˜ p i , ˜ p j ) + (cid:88) ( g , p ) ∈X ( g , p ) s ˜ g ind. ( ˜ p i , ˜ p j )( p ) ¯ n ( g , p ) (cid:19) y j + d ( ˜ p i ) + (cid:88) ( g , p ) ∈X ( g , p ) c ( ˜ p i , p ) ¯ n ( g , p ) ( g , p ) − (cid:32) b ( ˜ p i ) + | [ ˜ p ] ˜ g | (cid:88) j = (cid:18) s ˜ g nat. ( ˜ p i , ˜ p j ) + (cid:88) ( g , p ) ∈X ( g , p ) s ˜ g ind. ( ˜ p i , ˜ p j )( p ) ¯ n ( g , p ) (cid:19) + d ( ˜ p i ) + (cid:88) ( g , p ) ∈X ( g , p ) c ( ˜ p i , p ) ¯ n ( g , p ) ( g , p ) (cid:33) y i . In fact, (1 − q ( g , p ) ( ˜ g , ˜ p )) is the probability that a single mutant survives in a resident populationwith traits X ( g , p ) . We obtain this by approximating the mutant population with multi-type branch-ing processes (cf. proof of Thm. 3.6). The function (1 − q ( g , p ) ( ˜ g , ˜ p )) plays the same role as thefunction [ f ( y ; x )] + / b ( y ) in the standard case (cf. [10]).To obtain that the process jumps on the evolutionary time scale from one equilibrium to thenext, we need an assumption to prevent cycles, unstable equilibria or chaotic dynamics in thedeterministic system (cf. [10] Ass. B). Assumption 3.
For any given traits ( g , p ) , . . . , ( g d , p d ) ∈ G × P that coexist and for any mutanttrait ( ˜ g , ˜ p ) ∈ X \ X ( g , p ) such that 1 − q ( g , p ) ( ˜ g , ˜ p ) >
0, there exists a neighbourhood U ⊂ R |X ( g , p ) | + | [ ˜ p ] ˜ g | of ( ¯ n ( g , p ) , , . . . ,
0) such that all solutions of
LVS ( d + , (( g , p ) , ( ˜ g , ˜ p ))) with initial condition in U ∩ (0 , ∞ ) |X ( g , p ) | + | [ ˜ p ] ˜ g | converge as t ↑ ∞ to a unique locally strictly stable equilibrium in R |X ( g , p ) | + | [ ˜ p ] ˜ g | denoted by n ∗ (( g , p ) , ( ˜ g , ˜ p )).We write n ∗ and not ¯ n to emphasise that some components of n ∗ can be zero. We use theshorthand notation (( g , p ) , ( ˜ g , ˜ p )) for (( g , p ) , . . . , ( g d , p d ) , ( ˜ g , ˜ p )). Assumption 3 does not haveto hold for all traits in X \ X ( g , p ) , but only for those traits ( ˜ g , ˜ p ) which can appear in the residentpopulation by mutation, i.e. only if (cid:80) ( g , p ) ∈X ( g , p ) m ( g ) M (( g , p ) , ( ˜ g , ˜ p )) is positive. ES WITH PHENOTYPIC PLASTICITY 10
Remark . It is possible to extend the definitions and assumptions for the study of rare mutationsand fast switches in populations with non-discrete trait space if one assumes that an individualcan change its phenotype only to finitely many other phenotypes. This must be encoded in theswitching kernels. More precisely, for all ( g , p ) ∈ G × P the communicating class [ p ] g shouldcontain finitely many elements.3.2. Convergence to the generalised Polymorphic Evolution Sequence.
In this subsectionwe state the main theorem of this paper and give the general idea of the proof illustrated by anexample.
Theorem 3.3.
Suppose that the Assumptions 1, 2 and 3 hold. Fix ( g , p ) , . . . , ( g d , p d ) ∈ G × P coexisting traits and assume that the initial conditions have support X ( g , p ) and converge almostsurely to ¯ n ( g , p ) , i.e. lim K ↑∞ ν K = (cid:80) x ∈X ( g , p ) ¯ n x ( g , p ) δ x a.s.. Furthermore, assume that ∀ V > , exp( − V K ) (cid:28) u K (cid:28) K ln( K ) , as K ↑ ∞ . (3.11) Then, the sequence of the rescaled processes ( ν Kt / Ku K ) t ≥ , generated by L K with initial state ν K ,converges in the sense of finite dimensional distributions to the measure-valued pure jump pro-cess Λ defined as follows: Λ = (cid:80) ( g , p ) ∈X ( g , p ) ¯ n ( g , p ) ( g , p ) δ ( g , p ) and the process Λ jumps for all ( ˆ g , ˆ p ) ∈ X ( g , p ) from (cid:88) ( g , p ) ∈X ( g , p ) ¯ n ( g , p ) ( g , p ) δ ( g , p ) to (cid:88) ( g , p ) ∈X (( g , p ) , (˜ g , ˜ p )) n ∗ ( g , p ) (( g , p ) , ( ˜ g , ˜ p )) δ ( g , p ) (3.12) with infinitesimal rate m ( ˆ g ) b ( ˆ p ) ¯ n (ˆ g , ˆ p ) ( g , p )(1 − q ( g , p ) ( ˜ g , ˜ p )) M (( ˆ g , ˆ p ) , ( ˜ g , ˜ p )) . (3.13) Remark . (i) The convergence cannot hold in law for the Skorokhod topology (cf. [8]). Itholds only in the sense of finite dimensional distributions on M F ( X ), the set of finite posi-tive measures on X equipped with the topology of the total variation norm.(ii) The process Λ is a generalised version of the usual PES. Therefore, we call Λ PolymorphicEvolution Sequence with phenotypic Plasticity (PESP).(iii) Assumption 3 is essential for this statement. In the case when the dynamical system hasmultiple attractors and di ff erent points near the initial state lie in di ff erent basins of at-traction, it is not clear and may be random which attractor the system approaches. Thecharacterisation of the asymptotic behaviour of the dynamical system is needed to describethe final state of the stochastic process. This is in general a di ffi cult and complex prob-lem, which is not doable analytically and requires numerical analysis. Thus, we restrictourselves to the Assumption 3.We describe in the following the general idea of the proof, which is quite similar to the onegiven in [10]. The population is either in a stable phase or in an invasion phase. Until the firstmutant appears the population is in a stable phase, i.e. the population stays close to a givenequilibrium. From the first mutational event until the population reaches again a stable state,the population is in an invasion phase. In fact, the mutant either survives and the populationreaches fast a new stable state (where the mutant trait is present) or the mutant goes extinct and ES WITH PHENOTYPIC PLASTICITY 11 the population is again in the old stable state. After this the populations is again in a stable phaseuntil the next mutation, etc..Note that we prove in the following that the invasion phases are relatively short ( O (ln( K )))compared to the stable phase ( O (1 / u K K )). Since we study the process on the time scale 1 / Ku K ,the limit process proceeds as a pure jump process which jumps from one stable state to another. The stable phase:
Fix (cid:15) >
0. Let X ( g , p ) be the support of the initial conditions. For large K , thepopulation process ν K is, with high probability, still in a small neighbourhood of the equilibrium¯ n ( g , p ) when the first mutant appears. In fact, using large deviation results on the problem of exitfrom a domain (cf. [16]), we obtain that there exists a constant M > ν K leave the M (cid:15) -neighbourhood of ¯ n ( g , p ) is bigger than exp( V K ) for some V > x ∈ X ( g , p ) appear with a rate which is close to u K m ( x ) b ( x ) K ¯ n x ( g , p ) . The condition 1 / ( Ku K ) (cid:28) exp( V K ) for all V > The invasion phase:
We divide the invasion of a given mutant trait ( ˜ g , ˜ p ) into three steps , as in[8] and [10] (cf. Fig. 2). In the first step, from a mutational event until the mutant populationgoes extinct or the mutant density reaches the value (cid:15) , the number of mutant individuals is small(cf. Fig. 2, [0 , t ]). Thus, applying a perturbed version of the large deviation result we usedin the first phase, we obtain that the resident population stays close to its equilibrium density¯ n ( g , p ) during this step. Using similar arguments as Champagnat et al. [8, 10], we prove thatthe mutant population is well approximated by a | [ ˜ p ] ˜ g | -type branching process Z , as long as themutant population has less than (cid:15) K individuals. More precisely, let us denote the elements of[ ˜ p ] ˜ g by ˜ p , . . . , ˜ p | [ ˜ p ] ˜ g | , then, for each 1 ≤ i ≤ | [ ˜ p ] ˜ g | , each individual in Z (carrying trait ( ˜ g , ˜ p i ))undergoes(i) birth (without mutation) with rate b ( ˜ p i ),(ii) death with rate d ( ˜ p i ) + (cid:80) ( g , p ) ∈X ( g , p ) c ( ˜ p i , p ) ¯ n ( g , p ) ( g , p ) and(iii) switch to ˜ p j with rate s ˜ g nat. ( ˜ p i , ˜ p j ) + (cid:80) ( g , p ) ∈X ( g , p ) s ˜ g ind. ( ˜ p i , ˜ p j )( p ) ¯ n ( g , p ) for all 1 ≤ j ≤ | [ ˜ p ] ˜ g | .This continuous-time multi-type branching process is supercritical if and only if the largesteigenvalue of its infinitesimal generator, which we denote by λ max , is larger than zero. Hence, themutant invades with positive probability if and only if λ max >
0. Moreover, the probability thatthe density of the mutant’s genotype, ν K ( ˜ g ), reaches (cid:15) at some time t is close to the probabilitythat the multi-type branching process reaches the total mass (cid:15) K , which converges as K ↑ ∞ to(1 − q ( g , p ) ( ˜ g , ˜ p )).In the second step, we obtain as a consequence of Theorem 2.1 that once the mutant densityhas reached (cid:15) , for large K , the stochastic process ν K can be approximated on any finite timeinterval by the solution of LVS ( d + , (( g , p ) , . . . , ( g d , p d ) , ( ˜ g , ˜ p ))) with a given initial state. ByAssumption 3, this solution reaches the (cid:15) -neighbourhood of its new equilibrium n ∗ (( g , p ) , ( ˜ g , ˜ p ))in finite time. Therefore, for large K , the stochastic process ν K also reaches with high probabilitythe (cid:15) -neighbourhood of n ∗ (( g , p ) , ( ˜ g , ˜ p )) at some finite ( K independent) time t . ES WITH PHENOTYPIC PLASTICITY 12
In the third step, we use similar arguments as in the first atep. Since n ∗ (( g , p ) , ( ˜ g , ˜ p )) is astrongly locally stable equilibrium (Ass. 3), the stochastic process ν Kt stays close n ∗ (( g , p ) , ( ˜ g , ˜ p ))and we can approximate the densities of the traits ( g , p ) ∈ X (( g , p ) , (˜ g , ˜ p )) with n ∗ ( g , p ) (( g , p ) , ( ˜ g , ˜ p )) = | [ p ] g | - type branching processes which are subcritical and therefore become extinct, a.s..The duration of the first and third step are proportional to ln( K ), whereas the time of the secondstep is bounded. Thus, the second inequality in (3.11) guarantees that, with high probability,the three steps of invasion are completed before a new mutation occurs. After the last step theprocess is again back in a stable phase, but with a possibly di ff erent resident population, untilthe next mutation happens. population sizeO(ln(K)) O(1) O(ln(K)) υ ε t t t t t time F igure The three steps of one invasion phase.
An example:
Figure 2 shows the invasion phase of a single mutant with trait ( ˜ g , ˜ p ), which ap-peared (at time 0) in a population close to ¯ n ( g , p ) (indicated by the dashed lines). In this examplethe resident population consists of two coexisting traits ( g , p ) and ( g , p ) and the mutant individ-uals can switch to one other phenotype only, i.e. [ ˜ p ] ˜ g = { ˜ p , ˜ p } . The parameters of the simula-tion of Figure 2 are given in Table 1. The stable fixed point of the system LVS (2 , (( g , p ) , ( g , p ))) b ( p ) = d ( p ) = c ( p , p ) = c ( p , p ) = . c ( p , ˜ p ) = . c ( p , ˜ p ) = . s g nat. ( p , p ) = b ( p ) = d ( p ) = c ( p , p ) = . c ( p , p ) = c ( p , ˜ p ) = . c ( p , ˜ p ) = . s g nat. ( p , p ) = b ( ˜ p ) = d ( ˜ p ) = c ( ˜ p , p ) = . c ( ˜ p , p ) = . c ( ˜ p , ˜ p ) = c ( ˜ p , ˜ p ) = . s ˜ g nat. ( ˜ p , ˜ p ) = . b ( ˜ p ) = d ( ˜ p ) = c ( ˜ p , p ) = . c ( ˜ p , p ) = . c ( ˜ p , ˜ p ) = . c ( ˜ p , ˜ p ) = s ˜ g nat. ( ˜ p , ˜ p ) = K = u K = ν K ( g , p ) = . ν K ( g , p ) = . ν K (˜ g , ˜ p ) = / K ν K (˜ g , ˜ p ) = s . ind. ( . , . )( . ) ≡ T able Parameters of Figure 2 is ¯ n (( g , p ) , ( g , p )) ≈ (1 . , . (cid:32) .
879 1 . − . (cid:33) . (3.14) ES WITH PHENOTYPIC PLASTICITY 13
Since the largest eigenvalue of this matrix is positive ( ≈ . n ∗ ≈ (0 , , . , . LVS (4 , (( g , p ) , ( g , p ) , ( ˜ g , ˜ p ) , ( ˜ g , ˜ p ))). The dynamical system and hence also thestochastic process reach in finite time the (cid:15) -neighbourhood of this value. The infinitesimal gen-erator of the multi-type branching process that approximates the resident population in the thirdstep is approximately (cid:32) − .
951 21 − . (cid:33) . (3.15)The largest eigenvalue of this matrix is negative ( ≈ − . t such that all individuals which carrytrait ( g , p ) or ( g , p ) are a.s. dead at time t .3.3. Proof of Theorem 3.3.
In this paragraph we prove the convergence to the PESP. (Theproof uses the same arguments and techniques as [10], which were developed in [8]. However,some extensions are necessary, if fast phenotypic switches are included in the process, whichwe state and prove in this subsection.) We start with an analog of Theorem 3 of [8]. Part (i) ofthe following theorem strengthens Theorem 2.1, and part (ii) provides control of exit from anattractive domain in the polymorphic case with phenotypic plasticity.
Theorem 3.4. (i) Assume that the initial conditions have support { ( g , p ) , . . . ( g d , p d ) } and areuniformly bounded, i.e. , for all ≤ i ≤ d, ν K ( g i , p i ) ∈ A, where A is a compact subset of R > . Then, for all T > K ↑∞ sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Kt − (cid:88) x ∈X ( g , p ) n x ( t , ν K ) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = a.s., (3.16) where n ( t , ν K ) ∈ R |X ( g , p ) | denotes the value of the solution of LVS ( d , ( g , p )) at time t with ini-tial condition n x (0 , ν K ) = ν K ( x ) for all x ∈ X ( g , p ) . Note that the measure (cid:80) x ∈X ( g , p ) n x ( t , ν K ) δ x depends on K, since the initial condition and hence the solution of LVS ( d , ( g , p )) dependson K.(ii) Let ( g , p ) , . . . , ( g d , p d ) ∈ X coexist. Assume that, for any K ≥ , Supp ( ν K ) = X ( g , p ) .Let τ mut. be the first mutation time. Define the first exit time from the ξ -neighbourhood of ¯ n x ( g , p ) by θ K ,ξ exit ≡ inf (cid:110) t ≥ ∃ x ∈ X ( g , p ) : (cid:12)(cid:12)(cid:12) ν Kt ( x ) − ¯ n x ( g , p ) (cid:12)(cid:12)(cid:12) > ξ (cid:111) . (3.17) Then there exist (cid:15) > and M > such that, for all (cid:15) < (cid:15) , there exists V > such thatif the initial state of ν K lies in the (cid:15) -neighbourhood of ¯ n x ( g , p ) , the probability that θ K , M (cid:15) exit islarger than e KV ∧ τ mut. converges to one, i.e. lim K ↑∞ sup n K ∈ ( N / K ) |X ( g , p ) | ∩ B (cid:15) (¯ n ( g , p )) P (cid:20) θ K , M (cid:15) exit < e KV ∧ τ mut. (cid:12)(cid:12)(cid:12)(cid:12) ν K ( x ) = n Kx for all x ∈ X ( g , p ) (cid:21) = , (3.18) where n K ≡ ( n Kx ) x ∈X ( g , p ) and B (cid:15) ( ¯ n ( g , p )) denotes the (cid:15) -neighbourhood of ¯ n ( g , p ) .Moreover, (3.18) also holds if, for all ( g , p ) ∈ X ( g , p ) , the total death rate of an individualwith trait ( g , p ) , d ( p ) + (cid:88) (˜ g , ˜ p ) ∈X ( g , p ) c ( p , ˜ p ) ν Kt ( ˜ g , ˜ p ) , (3.19) ES WITH PHENOTYPIC PLASTICITY 14 and the total switch rates of an individual with trait ( g , p ) ,s gnat. ( p , p i ) + (cid:88) (˜ g , ˜ p ) ∈X ( g , p ) s gind. ( p , p i )( ˜ p ) ν Kt ( ˜ g , ˜ p ) for all p i ∈ [ p ] g , (3.20) are perturbed by additional random processes that are uniformly bounded by ¯ c (cid:15) respec-tively ¯ s ind . (cid:15) , where ¯ c and ¯ s ind . are upper bounds for the parameters of competition andinduced switch.Remark . (i) One consequence of the second part of (ii) is that, with high probability, theprocess stays in the M (cid:15) -neighbourhood of ¯ n x ( g , p ) until the first time that a mutant’s densityreaches the value (cid:15) . In other words, let θ K Invasion denote the first time that a mutant’s densityreaches the value (cid:15) , i.e θ K Invasion ≡ (cid:110) t ≥ ∃ ( g , p ) (cid:60) X ( g , p ) : (cid:80) ˜ p ∈ [ p ] g ν Kt ( g , ˜ p ) ≥ (cid:15) (cid:111) . (3.21)Then, the probability that θ K , M (cid:15) exit is larger than e KV ∧ θ K Invasion converges to one. We use thisresult also for the third invasion step.(ii) Since ¯ n ( g , p ) is a locally strictly stable fixed point of the system LVS ( d , ( g , p )), there existsa constant M > (cid:15) > n ( t ) with || n (0) − ¯ n ( g , p ) || < (cid:15) , it holds that sup t ≥ || n ( t ) − ¯ n ( g , p ) || < M (cid:15) . Proof.
The main task to prove (i) is to show that a large deviation principle on [0 , T ] holds for asightly modify process and that the ν K has the same law on the random time interval we need tocontrol it. In fact, Theorem 10.2.6 of [13] can be applied to obtain the large deviation principle.The main task to prove (ii) is to show that the classical estimates for exit times from a domain (cf.[16]) for the jump process ν K can be used. Note that Freidlin and Wentzell study in [16] mainlysmall white noise perturbations of dynamical systems. However, there also are some commentson the generalisation to dynamical systems with small jump-like perturbations (cf. [16], Sec.5.4). (cid:3) The following Lemma describes the asymptotic behaviour of τ mut. and can be seen as an ex-tension of Lemma 2 of [8] or Lemma A.3 of [10]. Lemma 3.5.
Let ( g , p ) , ..., ( g d , p d ) ∈ X coexist. Assume that, for any K ≥ , Supp ( ν K ) = X ( g , p ) .Let τ mut. denote the first mutation time. Then, there exists (cid:15) > such that if the initial states of ν K belong to the (cid:15) -neighbourhood of ¯ n x ( g , p ) , then, for all (cid:15) ∈ (0 , (cid:15) ) , lim K ↑∞ P τ mut. > ln( K ) , sup t ∈ [ln( K ) ,τ mut. ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Kt − (cid:80) x ∈X ( g , p ) ¯ n x ( g , p ) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV < (cid:15) = , (3.22) Moreover, ( τ mut. u K K ) K ≥ converges in law to an exponential distributed random variable withparameter (cid:80) ( g , p ) ∈X ( g , p ) m ( g ) b ( p ) ¯ n ( g , p ) ( g , p ) and the probability that the mutant, which appears attime τ mut. , is born from an individual with trait ( g , p ) ∈ X ( g , p ) converges tom ( g ) b ( p ) ¯ n ( g , p ) ( g , p ) (cid:80) (˜ g , ˜ p ) ∈X ( g , p ) m ( ˜ g ) b ( ˜ p ) ¯ n (˜ g , ˜ p ) ( g , p ) (3.23) as K ↑ ∞ . ES WITH PHENOTYPIC PLASTICITY 15
Proof.
There exist constants C > V >
0, such that on the time interval [0 , exp( KV )] thetotal mass of the population, ν Kt ( X ), is bounded from above by C . Therefore, we can constructan exponential random variable A with parameter C (cid:48) Ku K , where C (cid:48) = C max g ∈G , p ∈P m ( g ) b ( p ),such that A ≤ τ mut. on the event (cid:8) τ mut. < exp( KV ) (cid:9) . (3.24)Thus, P (cid:2) τ mut. > ln( K ) (cid:3) ≥ P [ A > ln( K )] = e − C (cid:48) ln( K ) Ku K . Since (3 .
11) implies that ln( K ) Ku K con-verges to zero as K ↑ ∞ , we get lim K ↑∞ P [ τ mut. > ln( K )] = n ( g , p ) is asymptotic stable. Thus, ∃ (cid:15) > ∀ ˜ (cid:15) ∈ (0 , (cid:15) ) ∃ T (˜ (cid:15) ): (cid:107) n ( g , p )(0) − ¯ n ( g , p ) (cid:107) < (cid:15) , implies sup t ≥ T (˜ (cid:15) ) | n ( g , p )( t ) − ¯ n ( g , p ) | < ˜ (cid:15)/ . (3.25)In words, there exists a finite time T (˜ (cid:15) ) such that all trajectories, which start in the (cid:15) neighbour-hood of the fixed point, stay after T (˜ (cid:15) ) in the ˜ (cid:15)/ (cid:15) ∈ (0 , (cid:15) ) ∃ T (˜ (cid:15) ) such that, for K large enough, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν KT (˜ (cid:15) ) − (cid:80) x ∈X ( g , p ) ¯ n x ( g , p ) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV < ˜ (cid:15) a . s .. (3.26)Then, by (ii), there exist (cid:15) > M >
0: for all ˜ (cid:15) ∈ (0 , (cid:15) ) there exists V > K ↑∞ P sup t ∈ [ T (˜ (cid:15) ) , e KV ∧ τ mut. ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Kt − (cid:80) x ∈X ( g , p ) ¯ n x ( g , p ) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV < M ˜ (cid:15) = . (3.27)Moreover, for all ˜ (cid:15) ∈ (0 , (cid:15) ) there exists K ∈ N such that T (˜ (cid:15) ) < ln( K ) for all K ≥ K . Thus,setting (cid:15) = M ˜ (cid:15) , ends the proof of (3.22), provided that lim K ↑∞ P [ τ mut. < e KV ] = (cid:15) > A , K ,(cid:15) and A , K ,(cid:15) withparameters a u K K ≡ (cid:88) ( g , p ) ∈X ( g , p ) u K m ( g ) b ( p )( ¯ n ( g , p ) ( g , p ) + (cid:15) ) K (3.28)and a u K K ≡ (cid:88) ( g , p ) ∈X ( g , p ) u K m ( g ) b ( p )( ¯ n ( g , p ) ( g , p ) − (cid:15) ) K (3.29)such that A , K ,(cid:15) ≤ τ mut. ≤ A , K ,(cid:15) on the event { T (˜ (cid:15) ) < τ mut. < e KV } , (3.30)where T (˜ (cid:15) ) is the time defined in equation (3.26) and ˜ (cid:15) = (cid:15)/ M . Moreover, we havelim K ↑∞ P [ τ mut. < ln( K )] = K ↑∞ P [ A , K ,(cid:15) > e KV ] = , (3.31)because u K K e KV ↑ ∞ as K ↑ ∞ . Therefore, for all (cid:15) >
0, the probability of the event { T (˜ (cid:15) ) <τ mut. < e KV } converges to one as K goes to infinity. Moreover, the random variables A , K ,(cid:15) u K K and A , K ,(cid:15) u K K converge both in law to the same exponential distributed random variable withparameter (cid:88) ( g , p ) ∈X ( g , p ) m ( g ) b ( p ) ¯ n ( g , p ) ( g , p ) (3.32)as first K ↑ ∞ and then (cid:15) →
0. The random variables A , A , K ,(cid:15) and A , K ,(cid:15) can easily be constructedby using the pathwise description of ν K (cf. [3] or [9]). (cid:3) ES WITH PHENOTYPIC PLASTICITY 16
Theorem 3.6 (The three steps of invasion) . Let ( g , p ) , . . . , ( g d , p d ) ∈ X coexist. Assume that,for any K ≥ , Supp ( ν K ) = X ( g , p ) ∪ { ( ˜ g , ˜ p ) } . Let τ mut. denote the next mutation time (after timezero) and define θ K ,ξ No Jump ≡ inf (cid:110) t ≥ ν Kt ( ˜ g ) = and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Kt − (cid:80) x ∈X ( g , p ) ¯ n x ( g , p ) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV < ξ (cid:111) (3.33) θ K ,ξ Jump ≡ inf (cid:110) t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Kt − (cid:80) x ∈X (( g , p ) , (˜ g , ˜ p )) n ∗ x (( g , p ) , ( ˜ g , ˜ p )) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV < ξ (3.34) and ∀ ˆ x (cid:60) { x ∈ X : n ∗ x (( g , p ) , ( ˜ g , ˜ p )) > } : ν Kt ( ˆ x ) = (cid:111) . Assume that we have a single initial mutant, i.e. ν K ( ˜ g , ˜ p ) = / K. Then, there exist (cid:15) > , C > , and M > such that for all (cid:15) ∈ (0 , (cid:15) ) if || ν K − (cid:80) x ∈X ( g , p ) ¯ n x ( g , p ) δ x || TV < (cid:15) , lim K ↑∞ P (cid:104) θ K , M (cid:15) No Jump < θ K , M (cid:15) Jump (cid:105) ≥ q ( g , p ) ( ˜ g , ˜ p ) − C (cid:15), (3.35)lim K ↑∞ P (cid:104) θ K , M (cid:15) Jump < θ K , M (cid:15) No Jump (cid:105) ≥ − q ( g , p ) ( ˜ g , ˜ p ) − C (cid:15), (3.36) where − q ( g , p ) ( ˜ g , ˜ p ) is the invasion probability defined in (3.8) and ∀ η > , lim K ↑∞ P (cid:34) θ K , M (cid:15) Jump ∧ θ K , M (cid:15) No Jump ≥ η u K K ∧ τ mut. (cid:35) ≤ C (cid:15). (3.37)The structure of the proof is similar to the one of Lemma 3 in [8] (cf. also Lem. A.4. of [10]).However, we have to extend the theory to multi-type branching processes. Thus, the proof is nota simple copy the arguments in [8]. Before proving the theorem, let us collect some propertiesabout multi-type continuous-time branching processes . Most of these can be found in [2] or [23].The limit theorems we need in the sequel were first obtained by Kesten and Stigum [19, 18, 20]in the discrete-time case and by Athreya [1] in the continuous-time case.Let Z ( t ) be a k -dimensional continuous-time branching process. Assume that Z ( t ) is non-singular and that the first moments exist. (Note that a process is singular if and only if eachindividual has exactly one o ff spring and that the existence of the first moments is su ffi cient forthe non-exposition hypothesis.) Then, the so-called mean matrix M ( t ) of Z ( t ) is the k × k matrixwith elements m i j ( t ) ≡ E [ Z j ( t ) | Z (0) = e i ] , ≤ i , j ≤ k , (3.38)and e i is the i -th unit vector in R k . It is well known (cf. [2] p. 202) that there exists a matrix A ,called the infinitesimal generator of the semigroup { M ( t ) , t ≥ } , such that M ( t ) ≡ exp( A t ) = ∞ (cid:88) n = t n ( A ) n n ! . (3.39)Furthermore, let r = ( r , . . . , r k ) be the vector of the branching rates, meaning that every indi-vidual of type i has an exponentially distributed lifetime of parameter r i and let M be the meanmatrix of the corresponding discrete-time process, i.e. M ≡ { m i j , i , j = , . . . , k } , where m i j isthe expected number of type j o ff spring of a single type- i -particle in one generation. Then, wecan identify the infinitesimal generator A as A = R ( M − I ) , (3.40)where R = diag ( r , . . . , r k ), i.e. r i j = r i δ i ( j ) and I is the identity matrix of size k . ES WITH PHENOTYPIC PLASTICITY 17
Under the basic assumption of positive regularity , i.e. that there exists a time t such that M ( t ) has strictly positive entries, the Perron-Frobenius theory asserts that(i) the largest eigenvalue of M ( t ) is real-valued and strictly positive,(ii) the algebraic and geometric multiplicities of this eigenvalue are both one, and(iii) the corresponding eigenvector has strictly positive cmponents.By (3.39), the eigenvalues of M ( t ) are given by exp( λ i t ), where { λ i ; i = , . . . , k } are the eigen-values of A , and both matrices have the same eigenvectors, which implies that the left and righteigenvectors u and v of λ max ( A ) can be chosen with strictly positive components and satisfying (cid:80) ki = v i u i = (cid:80) ki = u i = . (3.41)The process Z is called supercritical, critical, or subcritical according as λ max ( A ) is larger, equal,or smaller than zero.Observe that the following properties are equivalent (cf. [23] p. 95-99 and [22]): Z is irreducible ⇔ M is irreducible ⇔ A is irreducible ⇔ M ( t ) is irreducible for all t > ⇔ M ( t ) > t > Lemma 3.7.
Let ( Z ( t )) t ≥ be a non-singular and irreducible k-dimensional continuous-timeMarkov branching process and q the extinction vector of Z, i.e.q i ≡ P [ Z ( t ) = for some t ≥ | Z (0) = e i ] for ≤ i ≤ k . (3.42) Furthermore, let ( t K ) K ≥ be a sequence of positive numbers such that ln( K ) (cid:28) t K , define T ρ ≡ inf { t ≥ (cid:80) ki = Z i ( t ) = ρ } and assume that, for all i , j ∈ { , . . . , k } and t ∈ [0 , ∞ ) , E [ Z j ( t ) ln( Z j ( t )) | Z (0) = e i ] < ∞ . (3.43) (i) If Z is subcritical , i.e. λ max ( A ) < , then for any (cid:15) > K ↑∞ P (cid:104) T ≤ t K ∧ T (cid:100) (cid:15) K (cid:101) (cid:12)(cid:12)(cid:12) Z (0) = e i (cid:105) = for all i ∈ { , . . . , k } (3.44) and lim K ↑∞ inf x ∈ ∂ B (cid:15) K P (cid:2) T ≤ t K (cid:12)(cid:12)(cid:12) Z (0) = x (cid:3) = , where ∂ B (cid:15) K ≡ { x ∈ N k : (cid:80) ki = x i = (cid:100) (cid:15) K (cid:101)} . (3.45) Moreover, for ¯ u = max ≤ i ≤ k u i min ≤ j ≤ k u j and for any (cid:15) > , lim K ↑∞ sup x ∈ B (cid:15) K P (cid:104) T (cid:100) (cid:15) K (cid:101) ≤ T (cid:12)(cid:12)(cid:12) Z (0) = x (cid:105) ≤ ¯ u (cid:15), where B (cid:15) K ≡ { x ∈ N k : (cid:80) ki = x i ≤ (cid:100) (cid:15) K (cid:101)} . (3.46) (ii) If Z is supercritical , i.e. λ max ( A ) > , then for any (cid:15) > (small enough) lim K ↑∞ P (cid:104) T ≤ t K ∧ T (cid:100) (cid:15) K (cid:101) (cid:12)(cid:12)(cid:12) Z (0) = e i (cid:105) = q i for all i ∈ { , . . . , k } (3.47) and lim K ↑∞ P (cid:104) T (cid:100) (cid:15) K (cid:101) ≤ t K (cid:12)(cid:12)(cid:12) Z (0) = e i (cid:105) = − q i for all i ∈ { , . . . , k } . (3.48) ES WITH PHENOTYPIC PLASTICITY 18
Moreover, conditionally on survival, the proportions of the di ff erent types present in thepopulation converge almost surely, as t ↑ ∞ , to the corresponding ratios of the componentsof the eigenvector: for all i = , . . . , k , lim t ↑∞ Z i ( t ) (cid:80) kj = Z j ( t ) = v i (cid:80) kj = v j , a.s. on { T = ∞} . (3.49) Proof.
We start with the proof of (i). Since Z ( t ) is in this case a subcritical irreducible continuous-time branching process and E [ Z j ( t ) ln( Z j ( t )) | Z (0) = e i ] < ∞ , we obtain by applying Satz 6.2.7 of[23] the existence of a constant C > t ↑∞ − q i ( t )e λ max ( A ) t = Cu i , (3.50)where q i ( t ) ≡ P [ Z ( t ) = | Z (0) = e i ]. Moreover, we have a non-explosion condition. Thus, forall (cid:15) >
0, either T (cid:100) (cid:15) K (cid:101) equals infinity or it converges to infinity as K ↑ ∞ . Putting both together,there exists a sequence s K with lim K ↑∞ s K = + ∞ such thatlim K ↑∞ P (cid:104) T ≤ t K ∧ T (cid:100) (cid:15) K (cid:101) (cid:12)(cid:12)(cid:12) Z (0) = e i (cid:105) ≥ lim K ↑∞ P (cid:2) T ≤ s K (cid:12)(cid:12)(cid:12) Z (0) = e i (cid:3) = lim K ↑∞ q i ( s K ) = . (3.51)The branching property implies that for all x ∈ N k , P [ Z ( t ) = | Z (0) = x ] = (cid:81) ki = ( q i ( t )) x i (cf. [22]p. 25). So, we getinf x ∈ ∂ B (cid:15) K P (cid:2) T ≤ t K (cid:12)(cid:12)(cid:12) Z (0) = x (cid:3) = inf x ∈ ∂ B (cid:15) K P [ Z ( t K ) = | Z (0) = x ] = inf x ∈ ∂ B (cid:15) K k (cid:89) i = ( q i ( t K )) x i . (3.52)For all i ∈ { , . . . , k } , 1 ≥ ( q i ( t K )) x i ≥ ( q i ( t K )) (cid:100) (cid:15) K (cid:101) and by (3.50) we have 1 − q i ( t K ) = O (e λ max ( A ) t K ).Moreover, for any sequence ( w K ) K ≥ such that lim K ↑∞ w K = K ↑∞ (cid:18) + w K K (cid:19) K = . (3.53)This implies that, for all t K with t K (cid:29) ln( K ) and C >
0, since lim K ↑∞ C e λ max ( A ) t k (cid:100) (cid:15) K (cid:101) = K ↑∞ (1 − C e λ max ( A ) t k ) (cid:100) (cid:15) K (cid:101) = . (3.54)Thus, taking the limit K ↑ ∞ in (3.52), we obtain the desired equation (3.45). To prove theinequality (3.46) we use the fact that ( (cid:80) ki = u i Z i ( t ))e − λ max t is a martingale (cf. [1], Prop. 2). Byapplying Doob’s stopping theorem to the stopping time T (cid:100) (cid:15) K (cid:101) ∧ T we obtain, for all x ∈ B (cid:15) K ,that E (cid:104)(cid:16)(cid:80) ki = u i Z i ( T (cid:100) (cid:15) K (cid:101) ) (cid:17) e − λ max ( A ) T (cid:100) (cid:15) K (cid:101) { T (cid:100) (cid:15) K (cid:101) < T } (cid:12)(cid:12)(cid:12)(cid:12) Z (0) = x (cid:105) = (cid:80) ki = u i x i . (3.55)Therefore, since λ max ( A ) < E (cid:20) min ≤ i ≤ k u i (cid:100) (cid:15) K (cid:101) { T (cid:100) (cid:15) K (cid:101) < T } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z (0) = x (cid:21) ≤ max ≤ i ≤ k u i (cid:100) (cid:15) K (cid:101) , for all x ∈ B (cid:15) K , (3.56)which implies (3.46).Let us continue by proving (ii). Since Z ( t ) is supercritical in this case, applying Theorem 5.7.2of [2] yields that lim t ↑∞ Z ( t )( ω )e − λ max ( A ) t = W ( ω ) v , a.s. , (3.57) ES WITH PHENOTYPIC PLASTICITY 19 where W is a nonnegative random variable. Since we assume that, for all i ∈ { , . . . , k } , E [ Z j ( t ) ln( Z j ( t )) | Z (0) = e i ] < ∞ , we get that P [ W = | Z (0) = e i ] = q i , E [ W | Z (0) = e i ] = u i , (3.58)and W has an absolutely continuous distribution on (0 , ∞ ). All components of v are strictlypositive and W >
0, a.s., on the event { ω : T ( ω ) = ∞} . Hence, we have Z ( t ) = O (cid:16) e λ max ( A ) t (cid:17) a.s. on { T = ∞} . (3.59)This implies, for K large enough, P [ Z ( t K ) < (cid:100) (cid:15) K (cid:101) , T = ∞ ] = K ↑∞ P [ T = ∞ , T (cid:100) (cid:15) K (cid:101) ≥ t K ] = . (3.60)Note that we used that t K (cid:29) ln( K ). Since P [ T = ∞| Z (0) = e i ] = − q i , we deduce (3.48).On the other hand, there exist two sequences s K and s K , which converge to infinity as K ↑ ∞ ,such that, for K large enough, s K ≤ t K ∧ T (cid:100) (cid:15) K (cid:101) ≤ s K a.s.. This implies (3.47), because for all i ∈ { , . . . k } and l = ,
2, hold lim K ↑∞ P [ T < s lK | Z = e i ] = q i . Note that equation (3.49) is asimple consequence of (3.57). (cid:3) Using these properties about multi-type branching processes we can now prove the theoremabout the three steps of invasion.
Proof of Theorem 3.6. The first invasion step.
Let us introduce the following stopping times θ K , M (cid:15) exit = inf (cid:110) t ≥ || ν Kt − (cid:80) x ∈X ( g , p ) ¯ n x ( g , p ) δ x || TV > M (cid:15) (cid:111) (3.61)˜ θ K (cid:15) = inf (cid:110) t ≥ ν Kt ( ˜ g ) ≥ (cid:15) (cid:111) (3.62)˜ θ K = inf (cid:110) t ≥ ν Kt ( ˜ g ) = (cid:111) (3.63)Until ˜ θ K (cid:15) the mutant population ν Kt ( ˜ g ) influences only the death and switching rates of the residentpopulation and this perturbation is uniformly bounded by (¯ c + ¯ s ind. ) (cid:15) . Thus, by applying Theorem3.4 (ii), we obtain lim K ↑∞ P (cid:104) θ K , M (cid:15) exit < e KV ∧ τ mut. ∧ ˜ θ K (cid:15) (cid:105) = . (3.64)On the time interval [0 , θ K , M (cid:15) exit ∧ τ mut. ∧ ˜ θ K (cid:15) ], the resident population can be approximated by (cid:80) x ∈X ( g , p ) ¯ n x ( g , p ) δ x and no further mutant appears. This allows us to approximate ν Kt ( ˜ g ) by multi-type branching processes.Let k ≡ | [ ˜ p ] ˜ g | . We construct two ( N ) k - valued processes X ,(cid:15) ( t ) and X ,(cid:15) ( t ), using the pathwisedefinition in terms of Poisson point measures of ν Kt , which control the mutant population ν Kt ( ˜ g ).To this aim let us denote the elements of [ ˜ p ] ˜ g by ˜ p , . . . , ˜ p k (w.l.o.g. ˜ p ≡ ˜ p ). Then, we define ES WITH PHENOTYPIC PLASTICITY 20 X ,(cid:15) by X ,(cid:15) ( t ) ≡ X ,(cid:15) (0) + k (cid:88) j = (cid:90) t (cid:90) N (cid:90) R + (cid:110) i ≤ X ,(cid:15) j ( s − ) , θ ≤ b ( ˜ p i ) − (cid:15) (cid:111) e j N birth(˜ g , ˜ p j ) ( ds , di , d θ ) (3.65) − k (cid:88) j = (cid:90) t (cid:90) N (cid:90) R + (cid:26) i ≤ X ,(cid:15) j ( s − ) , θ ≤ d ( ˜ p j ) + (cid:80) ( g , p ) ∈X ( g , p ) c ( ˜ p j , p )¯ n ( g , p ) ( g , p ) + ¯ cM (cid:15) (cid:27) e j N death(˜ g , ˜ p j ) ( ds , di , d θ ) + k (cid:88) j = (cid:90) t (cid:90) N (cid:90) R + (cid:90) [ ˜ p ] ˜ g (cid:110) i ≤ X ,(cid:15) j ( s − ) , i (cid:44) j (cid:111) (cid:18) (cid:26) θ ≤ s ˜ g nat. ( ˜ p j , ˜ p l ) + (cid:80) ( g , p ) ∈X ( g , p ) s ˜ g ind. ( ˜ p j , ˜ p l )( p )¯ n ( g , p ) ( g , p ) − ¯ s ind. M (cid:15) (cid:27) e l − (cid:26) θ ≤ s ˜ g nat. ( ˜ p j , ˜ p l ) + (cid:80) ( g , p ) ∈X ( g , p ) s ˜ g ind. ( ˜ p j , ˜ p l )( p )¯ n ( g , p ) ( g , p ) + ¯ s ind. M (cid:15) (cid:27) e j (cid:19) N switch(˜ g , ˜ p j ) ( ds , di , d θ, d ˜ p l ) , and similar X ,(cid:15) by X ,(cid:15) ( t ) ≡ X ,(cid:15) (0) + k (cid:88) j = (cid:90) t (cid:90) N (cid:90) R + (cid:110) i ≤ X ,(cid:15) j ( s − ) , θ ≤ b ( ˜ p i ) + (cid:15) (cid:111) e j N birth(˜ g , ˜ p j ) ( ds , di , d θ ) (3.66) − k (cid:88) j = (cid:90) t (cid:90) N (cid:90) R + (cid:26) i ≤ X ,(cid:15) j ( s − ) , θ ≤ d ( ˜ p j ) + (cid:80) ( g , p ) ∈X ( g , p ) c ( ˜ p j , p )¯ n ( g , p ) ( g , p ) − ¯ cM (cid:15) (cid:27) e j N death(˜ g , ˜ p j ) ( ds , di , d θ ) + k (cid:88) j = (cid:90) t (cid:90) N (cid:90) R + (cid:90) [ ˜ p ] ˜ g (cid:110) i ≤ X ,(cid:15) j ( s − ) , i (cid:44) j (cid:111) (cid:18) (cid:26) θ ≤ s ˜ g nat. ( ˜ p j , ˜ p l ) + (cid:80) ( g , p ) ∈X ( g , p ) s ˜ g ind. ( ˜ p j , ˜ p l )( p )¯ n ( g , p ) ( g , p ) + ¯ s ind. M (cid:15) (cid:27) e l − (cid:26) θ ≤ s ˜ g nat. ( ˜ p j , ˜ p l ) + (cid:80) ( g , p ) ∈X ( g , p ) s ˜ g ind. ( ˜ p j , ˜ p l )( p )¯ n ( g , p ) ( g , p ) − ¯ s ind. M (cid:15) (cid:27) e j (cid:19) N switch(˜ g , ˜ p j ) ( ds , di , d θ, d ˜ p l ) , where e j is the j -th unit vector in R k and N birth , N death , and N switch are the collections of Poissonpoint measures defined in Subsection 2.1. Note that X ,(cid:15) ( t ) and X ,(cid:15) ( t ) are k -type branching pro-cesses with the following dynamics: For each 1 ≤ i ≤ k , each individual in X ,(cid:15) ( t ), respectively X ,(cid:15) ( t ), with trait ( ˜ g , ˜ p i ) undergoes(i) birth (without mutation) with rate b ( ˜ p i ) − (cid:15) , respectively b ( ˜ p i ) + (cid:15) + k −
1) ¯ s ind. M (cid:15) ,(ii) death with rate D ( g , p ) ( ˜ p i ) + ¯ cM (cid:15) + k −
1) ¯ s ind. M (cid:15) , respectively D ( g , p ) ( ˜ p i ) − ¯ cM (cid:15) ,where D ( g , p ) ( ˜ p i ) ≡ d ( ˜ p i ) + (cid:80) ( g , p ) ∈X ( g , p ) c ( ˜ p i , p ) ¯ n ( g , p ) ( g , p ),(iii) switch to ˜ p j with rate S ( g , p ) ( ˜ p i , ˜ p j ) − ¯ s ind. M (cid:15) for all j (cid:44) i (for both processes),where S ( g , p ) ( ˜ p i , ˜ p j ) ≡ s ˜ g nat. ( ˜ p i , ˜ p j ) + (cid:80) ( g , p ) ∈X ( g , p ) s ˜ g ind. ( ˜ p i , ˜ p j )( p ) ¯ n ( g , p ) .Moreover, the processes X ,(cid:15) ( t ) and X ,(cid:15) ( t ) have the following property: There exists a K > p i ∈ [ ˜ p ] ˜ g and for all K ≥ K ∀ ≤ t ≤ θ K ,(cid:15) exit ∧ τ mut. ∧ ˜ θ K (cid:15) : X ,(cid:15) i ( t ) ≤ ν Kt ( ˜ g , ˜ p i ) K ≤ X ,(cid:15) i ( t ) . (3.67)Hence, if ˜ θ K (cid:15) ≤ θ K ,(cid:15) exit ∧ τ mut. , theninf (cid:110) t ≥ X ,(cid:15) ( t ) = (cid:100) (cid:15) K (cid:101) (cid:111) ≤ ˜ θ K (cid:15) ≤ inf (cid:110) t ≥ X ,(cid:15) ( t ) = (cid:100) (cid:15) K (cid:101) (cid:111) . (3.68) ES WITH PHENOTYPIC PLASTICITY 21
On the other hand, if inf { t ≥ X ,(cid:15) ( t ) = } ≤ inf { t ≥ X ,(cid:15) ( t ) = (cid:100) (cid:15) K (cid:101)} ∧ θ K ,(cid:15) exit ∧ τ mut. , then˜ θ K ≤ inf { t ≥ X ,(cid:15) ( t ) = } . (3.69)Next, let us identify the infinitesimal generator of the control processes X ,(cid:15) and X ,(cid:15) . There-fore, define, for i = , . . . , k , f ( g , p ) ( ˜ g , ˜ p i ) ≡ b ( ˜ p i ) − D ( g , p ) ( ˜ p i ) − (cid:80) j (cid:44) i S ( g , p ) ( ˜ p i , ˜ p j ) . (3.70)( f ( g , p ) ( ˜ g , ˜ p i ) would be the invasion fitness of phenotype ˜ p i if there was no switch back from theother phenotypes to ˜ p i .) Then, by Equation (3.40), the infinitesimal generators are given by thefollowing matrixes A ( X l ,(cid:15) ) = f l ,(cid:15) ( g , p ) ( ˜ g , ˜ p ) S ( g , p ) ( ˜ p , ˜ p ) − ¯ s ind. M (cid:15) . . . S ( g , p ) ( ˜ p , ˜ p k ) − ¯ s ind. M (cid:15) S ( g , p ) ( ˜ p , ˜ p ) − ¯ s ind. M (cid:15) f l ,(cid:15) ( g , p ) ( ˜ g , ˜ p ) ... . . . ... S ( g , p ) ( ˜ p k , ˜ p ) − ¯ s ind. M (cid:15) . . . f l ,(cid:15) ( g , p ) ( ˜ g , ˜ p k ) (3.71)for l ∈ { , } , where f ,(cid:15) ( g , p ) ( ˜ g , ˜ p i ) ≡ f ( g , p ) ( ˜ g , ˜ p i ) − (cid:15) (1 + ¯ cM + ( k −
1) ¯ s ind. M ) and f ,(cid:15) ( g , p ) ( ˜ g , ˜ p i ) ≡ f ( g , p ) ( ˜ g , ˜ p i ) + (cid:15) (1 + ¯ cM + k −
1) ¯ s ind. M ).We prove in the following that the number of mutant individuals grow with positive probabilityto (cid:15) K before dying out if and only if λ max of A (˜ g , ˜ p ) ≡ lim (cid:15) → A ( X ,(cid:15) ) is strictly positive. Thus, λ max ( A (˜ g , ˜ p ) ) is an appropriate generalisation of the invasion fitness of the class [ ˜ p ] ˜ g : F [ ˜ p ] ˜ g ( g , p ) ≡ λ max ( A (˜ g , ˜ p ) ) . (3.72)Since the birth and death rates of X ,(cid:15) and X ,(cid:15) are positive and since Assumption 2 implies that M ( X ,(cid:15) ) and M ( X ,(cid:15) ) are irreducible, we obtain that the processes X ,(cid:15) and X ,(cid:15) are non-singularand irreducible. Thus, X ,(cid:15) and X ,(cid:15) satisfy the conditions of Lemma 3.7. For l ∈ { , } , let q ( X l ,(cid:15) )denote the extinction probability vector of X l ,(cid:15) , i.e. q ( X l ,(cid:15) ) ≡ ( q ( X l ,(cid:15) ) , . . . , q k ( X l ,(cid:15) )) , where q i ( X l ,(cid:15) )) ≡ P (cid:104) X l ,(cid:15) ( t ) = t (cid:12)(cid:12)(cid:12) X l ,(cid:15) (0) = e i (cid:105) . Observe that q ( X l ,(cid:15) ) = (1 , . . . ,
1) if X l ,(cid:15) is not supercritical. To characterise q ( X l ,(cid:15) ) in the super-critical case, let us introduce the following functions u l : [0 , k × ( − η, η ) → R k , where η is some small enough constant and l ∈ { , } , (3.73)defined, for all 1 ≤ i ≤ k , by u i ( y , (cid:15) ) ≡ (cid:16) b ( ˜ p i ) − (cid:15) (cid:17) y i + (cid:88) j (cid:44) i (cid:16) S ( g , p ) ( ˜ p i , ˜ p j ) − ¯ s ind. M (cid:15) (cid:17) y j + D ( g , p ) ( ˜ p i ) + ¯ cM (cid:15) + k −
1) ¯ s ind. M (cid:15) − (cid:16) b ( ˜ p i ) + (cid:88) j (cid:44) i S ( g , p ) ( ˜ p i , ˜ p j ) + D ( g , p ) ( ˜ p i ) + (1 − ¯ cM + ( k −
1) ¯ s ind. M ) (cid:15) (cid:17) y i . (3.74) ES WITH PHENOTYPIC PLASTICITY 22 and u i ( y , (cid:15) ) ≡ (cid:16) b ( ˜ p i ) + (cid:15) + k −
1) ¯ s ind. M (cid:15) (cid:17) y i + (cid:88) j (cid:44) i (cid:16) S ( g , p ) ( ˜ p i , ˜ p j ) − ¯ s ind. M (cid:15) (cid:17) y j + D ( g , p ) ( ˜ p i ) − ¯ cM (cid:15) − (cid:16) b ( ˜ p i ) + (cid:88) j (cid:44) i S ( g , p ) ( ˜ p i , ˜ p j ) + D ( g , p ) ( ˜ p i ) + (1 − ¯ cM + ( k −
1) ¯ s ind. M ) (cid:15) (cid:17) y i . (3.75)Observe that u ( y , (cid:15) ) and u ( y , (cid:15) ) are the infinitesimal generating functions of X ,(cid:15) and X ,(cid:15) andthat u ( y , = u ( y , q ( X ,(cid:15) ) is the unique solution of u ( y , (cid:15) ) = y ∈ [0 , k (3.76)and q ( X ,(cid:15) ) is the unique solution of u ( y , (cid:15) ) = y ∈ [0 , k . (3.77)These solutions are in general not analytic. Applying Lemma 3.7 to X ,(cid:15) and X ,(cid:15) we obtain thatthere exists C > η > (cid:15) > ffi ciently small and K large enough, P (cid:104) θ K , M (cid:15) No Jump < η Ku K ∧ θ K , M (cid:15) exit ∧ τ mut. ∧ ˜ θ K (cid:15) (cid:105) ≥ P (cid:104) inf { t ≥ X ,(cid:15) ( t ) = } < η Ku K (cid:105) ≥ q ( X ,(cid:15) ) − C (cid:15) (3.78)and P (cid:104) ˜ θ K (cid:15) < η Ku K ∧ θ K , M (cid:15) exit ∧ τ mut. ∧ ˜ θ K (cid:105) ≥ P (cid:104) inf { t ≥ X ,(cid:15) ( t ) = } < η Ku K (cid:105) ≥ − q ( X ,(cid:15) ) − C (cid:15). (3.79)If X ,(cid:15) is sub- or critical for (cid:15) small enough, then lim (cid:15) ↓ q ( X ,(cid:15) ) = lim (cid:15) ↓ q ( X ,(cid:15) ) =
1. In thesupercritical case, let q ∈ [0 , k be the solution of u ( y , = u ( y , =
0. Then, by applyingthe implicit function theorem, there exist open sets U ⊂ R and U ⊂ R containing 0, opensets V ⊂ R k and V ⊂ R k containing q , and two unique continuously di ff erentiable functions g : U → V and g : U → V such that { ( (cid:15), g ( (cid:15) )) | (cid:15) ∈ U } = { ( (cid:15), y ) ∈ U × V | u ( y , (cid:15) ) = } . (3.80)and { ( (cid:15), g ( (cid:15) )) | (cid:15) ∈ U } = { ( (cid:15), y ) ∈ U × V | u ( y , (cid:15) ) = } . (3.81)By definition, g (0) = g (0) = q and q = q ( g , p ) ( ˜ g , ˜ p ). We can linearise and obtain that thereexists a constant C > q ( X ,(cid:15) ) ≤ q ( g , p ) ( ˜ g , ˜ p ) + C (cid:15) and q ( X ,(cid:15) ) ≥ q ( g , p ) ( ˜ g , ˜ p ) − C (cid:15) (3.82)Therefore, lim K ↑∞ P (cid:104) θ K , M (cid:15) No Jump ∧ ˜ θ K (cid:15) < η Ku K ∧ θ K , M (cid:15) exit ∧ τ mut. (cid:105) ≥ − C + C ) (cid:15). (3.83)Conditionally on survival, the proportions of the di ff erent phenotypes in X ,(cid:15) converge almostsurely, as t ↑ ∞ , to the corresponding ratios of the components of the eigenvector, which are all ES WITH PHENOTYPIC PLASTICITY 23 strictly positive (cf. Lem. 3.7, Eq. (3.49)). Moreover, there exists a constant C > (cid:15) small enough,lim K ↑∞ P (cid:104)(cid:110) ˜ θ K (cid:15) < η Ku K ∧ θ K , M (cid:15) exit ∧ τ mut. (cid:111) ∩ (cid:110) inf { t ≥ X ,(cid:15) ( t ) = } < ∞ (cid:111)(cid:105) < C (cid:15) (3.84)and ˜ θ K (cid:15) converges to infinity as K ↑ ∞ . Thus, conditionally on { ˜ θ K (cid:15) < η Ku K ∧ θ K , M (cid:15) exit ∧ τ mut. } , thereexists a (small) constant C > { ˜ p , . . . , ˜ p k } , are all larger than C (cid:15) at time ˜ θ K (cid:15) convergences to one as first K ↑ ∞ and then (cid:15) →
0. More precisely, there exists constants C > C > (cid:15) smallenough, lim K ↑∞ P (cid:104) ˜ θ K (cid:15) < η Ku K ∧ θ K , M (cid:15) exit ∧ τ mut. , ∃ i ∈ { , . . . k } : ν K ˜ θ K (cid:15) ( ˜ p i ) ≤ C (cid:15) (cid:105) ≤ C (cid:15). (3.85) The second invasion step.
By Assumption 3, any solution of
LVS ( d + , (( g , p ) , ( ˜ g , ˜ p ))) withinitial state in the compact set A ≡ (cid:110) x ∈ R |X ( g , p ) | : | x − ¯ n ( g , p ) | ≤ M (cid:15) (cid:111) × [ C (cid:15), (cid:15) ] k (3.86)converge, as t ↑ ∞ , to the unique locally strictly stable equilibrium n ∗ (( g , p )) , ( ˜ g , ˜ p )). Therefore,for all (cid:15) > T ( (cid:15) ) ∈ R such that any of these trajectories do not leave the set (cid:110) x ∈ R |X ( g , p ) | + k : | x − n ∗ (( g , p )) , ( ˜ g , ˜ p )) | ≤ (cid:15) / (cid:111) (3.87)after time T ( (cid:15) ). Back to the stochastic system, let us introduce on the event { ˜ θ K (cid:15) < η Ku K ∧ θ K , M (cid:15) exit ∧ τ mut. } the following stopping time θ K ,(cid:15) near n ∗ = inf (cid:110) t ≥ ˜ θ K (cid:15) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Kt − (cid:80) x ∈X (( g , p ) , (˜ g , ˜ p )) n ∗ x (( g , p ) , ( ˜ g , ˜ p )) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV < (cid:15) (cid:111) . (3.88)Then, we conclude by using the strong Markov property at ˜ θ K (cid:15) and Theorem 3.4 (i) on [0 , T ( (cid:15) )]that there exists a constant C > (cid:15) small enough,lim K ↑∞ P (cid:20) ˜ θ K (cid:15) < τ mut. ∧ η Ku K and sup s ∈ [˜ θ K (cid:15) , ˜ θ K (cid:15) + T ( (cid:15) )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Ks − (cid:80) x ∈X ( g , p ) n x ( s , ν K ) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV ≤ (cid:15) (cid:21) (3.89) ≥ − q ( g , p ) ( ˜ g , ˜ p ) − C (cid:15), which implies lim K ↑∞ P (cid:104) ˜ θ K (cid:15) < θ K ,(cid:15) near n ∗ < τ mut. ∧ η Ku K (cid:105) ≥ − q ( g , p ) ( ˜ g , ˜ p ) − C (cid:15). (3.90)We used that, at time ˜ θ K (cid:15) , the stochastic process ν K (considered as element of R |X ( g , p ) | + k ) lies in thecompact set A , where A is defined in (3.86). The third invasion step.
After time θ K ,(cid:15) near n ∗ we use again comparisons with multi-type branchingprocesses to show that all individuals carrying a trait which is not present in the new equilibrium n ∗ die out. To this aim let us define X n ∗ extinct = { ( g , p ) ∈ X (( g , p ) , (˜ g , ˜ p )) : n ∗ ( g , p ) (( g , p ) , ( ˜ g , ˜ p )) = } (3.91)For proving that the populations with traits in X n ∗ extinct stay small after θ K ,(cid:15) near n ∗ and that the pop-ulations with traits not in X n ∗ extinct stay close to its equilibrium value after θ K ,(cid:15) near n ∗ , let us define θ K ,(cid:15) not small = inf (cid:110) t ≥ θ K ,(cid:15) near n ∗ : ∃ ( g , p ) ∈ X n ∗ extinct such that ν Kt ( g , p ) > (cid:15) (cid:111) (3.92) ES WITH PHENOTYPIC PLASTICITY 24 and θ K , M (cid:15) exit n ∗ ≡ inf (cid:26) t ≥ θ K ,(cid:15) near n ∗ : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν Kt − (cid:80) x ∈X (( g , p ) , (˜ g , ˜ p )) n ∗ x (( g , p ) , ( ˜ g , ˜ p )) δ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV > M (cid:15) (cid:27) . (3.93)By using first the strong Markov property at θ K ,(cid:15) near n ∗ , we can apply Theorem 3.4 (ii) and obtainthat there exist constants M > C > (cid:15) small enough,lim K ↑∞ P (cid:104) ˜ θ K (cid:15) < θ K ,(cid:15) near n ∗ < τ mut. ∧ η Ku K and θ K , M (cid:15) exit n ∗ < e KV ∧ τ mut. ∧ θ K ,(cid:15) not small (cid:105) < C (cid:15) (3.94)This is obtained in a similar way as Equation (3.64) in the first step. Note that ( g , p ) ∈ X n ∗ extinct implies that ( g , p i ) ∈ X n ∗ extinct for all p i ∈ [ p ] g , which is a consequence of Assumption 2.Using the same arguments as in the first step, we can construct, for all ( g , p ) ∈ X n ∗ extinct , a | [ p ] g | -type continuous-time branching process Y (cid:15), ( g , p ) ( s ) with initial condition Y (cid:15), ( g , p ) i (0) = ν K θ K ,(cid:15) near n ∗ ( g , p i ) K for all p i ∈ [ p ] g (3.95)such that, for all K large enough and, for all t ∈ [ θ K ,(cid:15) near n ∗ , θ K , M (cid:15) exit n ∗ ∧ θ K ,(cid:15) not small ∧ τ mut. ], ν Kt ( g , p i ) K ≤ Y (cid:15), ( g , p ) i ( t − θ K ,(cid:15) near n ∗ ) for all p i ∈ [ p ] g . (3.96)Moreover, Y (cid:15), ( g , p ) ( t ) is characterised as follows: For each p i ∈ [ p ] g , each individual in Y (cid:15), ( g , p ) ( t )with trait ( g , p i ) undergoes(i) birth (without mutation) with rate b ( p i ) + | [ p ] g | − s ind. ( M + |X n ∗ extinct | ) (cid:15) (ii) death with rate d ( p i ) + (cid:80) (ˆ g , ˆ p ) ∈X ( g , p ) , (˜ g , ˜ p ) c ( p i , ˆ p ) n ∗ (ˆ g , ˆ p ) (( g , p ) , ( ˜ g , ˜ p )) − ¯ c ( M + |X n ∗ extinct | ) (cid:15) (iii) for all j (cid:44) i , switch to p j with rate s g nat. ( p i , p j ) + (cid:80) (ˆ g , ˆ p ) ∈X ( g , p ) , (˜ g , ˜ p ) s g ind. ( p i , p j )( ˆ p ) n ∗ (ˆ g , ˆ p ) (( g , p ) , ( ˜ g , ˜ p )) − ¯ s ind. ( M + |X n ∗ extinct | ) (cid:15) .Let A ( Y (cid:15), ( g , p ) ) denote the infinitesimal generator of the process Y (cid:15), ( g , p ) . Since the equilibrium n ∗ (( g , p ) , ( ˜ g , ˜ p )) is locally strictly stable (cf. Ass. 3), the eigenvalues of the Jacobian matrix ofthe dynamical system at n ∗ (( g , p ) , ( ˜ g , ˜ p )) are all strictly negative. If (cid:15) is small enough, this impliesthat all eigenvalues of { A ( Y (cid:15), ( g , p ) ) , ( g , p ) ∈ X n ∗ extinct } are strictly negative. (There exists an order ofthe elements of X ( g , p ) , (˜ g , ˜ p ) such that the Jacobian matrix is an upper-block-triangular matrix and { A ( Y , ( g , p ) ) , ( g , p ) ∈ X n ∗ extinct } are on the diagonal.) Thus, for all (cid:15) small enough, the branchingprocesses { Y (cid:15), ( g , p ) , ( g , p ) ∈ X n ∗ extinct } are all subcritical. Moreover, we can apply Lemma 3.7 andget, for all (cid:15) small enough and ( g , p ) ∈ X n ∗ extinct lim K ↑∞ P (cid:34) inf { t ≥ Y (cid:15), ( g , p ) ( t ) = } ≤ η Ku K (cid:35) = , (3.97)and there exists a constant C such that, for all (cid:15) small enough and ( g , p ) ∈ X n ∗ extinct ,lim K ↑∞ P (cid:104) inf { t ≥ Y (cid:15), ( g , p ) ( t ) = (cid:100) (cid:15) K (cid:101)} ≤ inf { t ≥ Y (cid:15), ( g , p ) ( t ) = } (cid:105) ≤ C (cid:15). (3.98)Hence, there exists a constant M > C > η > (cid:15) small enough,lim K ↑∞ P (cid:34) ˜ θ K (cid:15) < θ K , M (cid:15) Jump < τ mut. ∧ η Ku K ∧ θ K ,(cid:15) not small (cid:35) ≥ − q ( g , p ) ( ˜ g , ˜ p ) − C (cid:15), (3.99)which finishes the proof of the theorem. (cid:3) ES WITH PHENOTYPIC PLASTICITY 25
Combining all the previous results, we can prove similar as in [8] that for, all (cid:15) > , t > Γ ⊂ X ,lim K ↑∞ P (cid:104) Supp( ν Kt / Ku K ) = Γ , all traits of Γ coexist in LVS ( | Γ | , Γ ), (3.100)and || ν Kt / Ku K − (cid:88) x ∈ Γ ¯ n x ( Γ ) δ x || TV < (cid:15) (cid:105) = P [Supp( Λ t ) = Γ ]where Λ is the PES with phenotypic plasticity defined in Theorem 3.3. Finally, generalising thisto any sequence of times 0 < t < . . . < t n , implies that ( ν Kt / Ku K ) t ≥ converges in the sense offinite dimensional distributions to ( Λ t ) t ≥ (cf. [8], Cor. 1 and Lem. 1), which ends the proof ofTheorem 3.3.3.4. Examples.
Figure 3 shows two examples where in a population consisting only of type( g , p ) and being close to n ( g , p ) a mutation to genotype ˜ g occurs. In these example, ˜ g is associatedwith two possible phenotypes ˜ p and ˜ p .A BF igure Simulations of the invasion phase with K = p has a negative initial growth rate but can switch to ˜ p which has a positive one. Thefitness of the genotype ˜ g is positive. (B) The fitness of the mutant genotype ˜ g is positive,although each phenotype has a negative initial growth rate. This is possible because anoutgoing switch is a loss of a cell for a phenotype, but not for the whole genotype. In example (A), we start with a single mutant carrying trait ( ˜ g , ˜ p ) and which can switch to˜ p but the back-switch is relative weak (cf. Tab. 2). According to definition (3.70) we have f ( g , p ) ( ˜ g , ˜ p ) < f ( g , p ) ( ˜ g , ˜ p ) >
0. However, the global fitness of the genotype ˜ g is positive.More precisely, it is given by the largest eigenvalue of (cid:0) − . (cid:1) , which equals approximatively1 . g , ˜ p )). However, theprobability of invasion depends this. In this example, the invasion probability is given by thesolution of 2 y + y + − y = , (3.101)4 y + . y + . − y = . (3.102) ES WITH PHENOTYPIC PLASTICITY 26
Thus, if we start with the trait ( ˜ g , ˜ p ), the invasion probability is approximately 0 . .
338 if the first one has trait ( ˜ g , ˜ p ). In Figure 3 (A), the mutant population with genotype ˜ g survives and the stochastic process is attracted to the new equilibrium n ∗ (( g , p ) , ( ˜ g , ˜ p ) , ( ˜ g , ˜ p )) ≈ (0 , . , . b ( p ) = d ( p ) = c ( p , p ) = c ( p , ˜ p ) = c ( p , ˜ p ) = . s . ind. ( . , . )( . ) ≡ ν K ( g , p ) = b ( ˜ p ) = d ( ˜ p ) = c ( ˜ p , p ) = c ( ˜ p , ˜ p ) = c ( ˜ p , ˜ p ) = . s ˜ g ( ˜ p , ˜ p ) = ν K ( ˜ g , ˜ p ) = K − b ( ˜ p ) = d ( ˜ p ) = c ( ˜ p , p ) = . c ( ˜ p , ˜ p ) = . c ( ˜ p , ˜ p ) = s ˜ g ( ˜ p , ˜ p ) = . ν K ( ˜ g , ˜ p ) = T able Parameters of Figure 3 (A)
In example (B), f ( g , p ) ( ˜ g , ˜ p ) and f ( g , p ) ( ˜ g , ˜ p ) are both negative. Nevertheless, the fitness ofthe genotype is positive and thus the mutant invades with positive probability. (It is given bythe largest eigenvalue of (cid:0) − − . (cid:1) , which equals approximatively 0 . .
127 if we start with the trait ( ˜ g , ˜ p )and 0 .
207 else. In Figure 3 (B), the mutant population survives and the process is attracted to thestable fixed point n ∗ (( g , p ) , ( ˜ g , ˜ p ) , ( ˜ g , ˜ p )) ≈ (0 , . , . b ( p ) = d ( p ) = c ( p , p ) = c ( p , ˜ p ) = c ( p , ˜ p ) = . s . ind. ( . , . )( . ) ≡ ν K ( g , p ) = b ( ˜ p ) = d ( ˜ p ) = c ( ˜ p , p ) = c ( ˜ p , ˜ p ) = c ( ˜ p , ˜ p ) = . s ˜ g ( ˜ p , ˜ p ) = ν K ( ˜ g , ˜ p ) = / Kb ( ˜ p ) = d ( ˜ p ) = c ( ˜ p , p ) = . c ( ˜ p , ˜ p ) = . c ( ˜ p , ˜ p ) = s ˜ g ( ˜ p , ˜ p ) = ν K ( ˜ g , ˜ p ) = T able Parameters of Figure 3 (B) R eferences [1] K. B. Athreya. Some results on multitype continuous time Markov branching processes. Ann. Math. Stat. ,39:347–357, 1968.[2] K. B. Athreya and P. E. Ney.
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60, 53115 B onn , G ermany
E-mail address : [email protected] A. B ovier , I nstitut f ¨ ur A ngewandte M athematik , R heinische F riedrich -W ilhelms -U niversit ¨ at , E ndenicher A llee
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