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PERSISTENCE AND NEUTRALITY IN INTERACTING REPLICATOR DYNAMICS

November 2024

DOI:10.48550/arXiv.2310.02809

Authors:

Leonardo Videla

University of Santiago Chile

Cristóbal Quiñinao

Pontifical Catholic University of Chile

Pablo A Marquet

Pontifical Catholic University of Chile

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We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for a class of associated McKean-Vlasov dynamics. In particular, our results show that, unlike typical models of neutral ecology, fitness equivalence does not need to be assumed but emerges as a condition for the persistence of the system. Further, neutrality is associated with a unique Dirichlet invariant probability measure. We illustrate our findings with some simple case studies, provide numerical results, and discuss our conclusions in the light of Neutral Theory in ecology.

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PERSISTENCE AND NEUTRALITY IN INTERACTING REPLICATOR

DYNAMICS

LEONARDO VIDELA1, MAURICIO TEJO2, CRISTÓBAL QUIÑINAO3, PABLO A. MARQUET3,4,

AND ROLANDO REBOLLEDO5

Abstract. We study the large-time behavior of an ensemble of entities obeying replicator-

like stochastic dynamics with mean-ﬁeld interactions as a model for a primordial ecology. We

prove the propagation-of-chaos property and establish conditions for the strong persistence of

the N-replicator system and the existence of invariant distributions for a class of associated

McKean-Vlasov dynamics. In particular, our results show that, unlike typical models of

neutral ecology, ﬁtness equivalence does not need to be assumed but emerges as a condition

for the persistence of the system. Further, neutrality is associated with a unique Dirichlet

invariant probability measure. We illustrate our ﬁndings with some simple case studies,

provide numerical results, and discuss our conclusions in the light of Neutral Theory in

ecology.

Keywords: Stochastic replicator dynamics, Propagation of Chaos, Stochastic persis-

tence, McKean-Vlasov equation, Invariant distributions, Emergence of ecologies.

Mathematics Subject Classiﬁcation (2010): 60H10 and 92D25.

1. Introduction

The emergence of life corresponds to the emergence of an ecological system of interacting

self-replicating entities. Several characteristics of early life have endured through time and are

shared by all forms of life today, from the near universality of the genetic code to the inter-

mediate metabolism [32, 51]. It is unlikely that these biological universalities represent only

"frozen accidents" linked to a universal common ancestry, but may emerge as a consequence of

some fundamental principles associated with information and thermodynamics that aﬀect the

robustness and evolvability of biological systems [51, 60]. In a similar vein, it is expected that

early ecological systems faced important challenges to their persistence in ﬂuctuating environ-

ments, where the number of co-existing entities increases forming diverse communities, and that

some fundamental principles may also be invoked to understand their persistence. In this work,

we are primarily interested in the principle of neutrality, which is mathematically expressed

in a particular invariant probability distribution associated with the process of abundances of

diﬀerent types of entities (e.g. species) in an ecological system (e.g. community) where birth

and death rates are linear, and thus entities can be considered as weakly interacting (e.g., a

neutral ecology where species are equivalent in ﬁtness, [26]) due to the operation of ﬁtness

equalizing mechanisms associated to tradeoﬀs and incompatible optima. These mechanisms

render neutrality as a good approximation, even if entities weakly interact, as envisioned in

the Red Queen and in the Neutral Theory of Ecology [57, 26, 27, 28]. In particular, using the

neutrality approximation in a stochastic framework, we showed in [33] that a one-dimensional

diﬀusion approximation for the frequency or proportional abundance of species in a community

admits a Beta invariant distribution (the Proportional Species Abundance Distribution), the

Date: Received: date / Accepted: date.

arXiv:2310.02809v2 [math.PR] 13 Nov 2024

2 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

same as found, for instance, for gene frequencies in partially isolated populations [63, 31], which

generalizes to a Dirichlet in the multidimensional case.

To model an early ecological system where to shed light on the importance of neutrality in

early ecologies, we use a quintessential model for replicating entities, which has been applied to

the modeling of polynucleotides in a container, the dynamics of gene frequencies, selection (Price

equation), relative densities of interacting populations (Lotka-Volterra model), the frequency of

diﬀerent strategies in a population (game theoretical models), and adaptive dynamics [55, 49,

25, 43]. One of the most basic and general equations for all these dynamics is the deterministic

equation introduced by Taylor and Jonker [55] and dubbed replicator equation by Schuster and

Sigmund [55, 49].

Thus, in this article, any given entity in our primordial ecological system will be represented

by a replicator, and we will follow its proportional abundance through time. Now, since it can

be argued that in the primordial soup, many entities shared a common habitat in this article

we will extend the simple replicator dynamics to the setup of a large number of self-reproducing

entities (community), constituting in this way an interactive set. In the setting we have chosen,

the intraspeciﬁc dynamics of each singular replicator will be aﬀected, in the form of a mean-ﬁeld

eﬀect, by the dynamics of the other replicators in the system.

Is neutrality still plausible to emerge in this larger setting? and if so, What is its role? The

sections that follow are devoted to shed light on these questions.

1.1. Preliminaries. The main motivation of Taylor and Jonker’s replicator model ([55]) was

to unify many game-theoretic models that pursue the study of evolutionary stable strategies

(ESS) in the idealized dynamics of animal conﬂicts as introduced by Maynard Smith and Price

[52]. Recall that Taylor and Jonker’s replicator is a dynamical system (x(t) : t≥0) taking

values on the (d−1)-dimensional simplex:

Sd:= {x∈Rd:xi≥0,

i=1

xi= 1},

obeying the ordinary diﬀerential equation:

dt =x◦(Ax− ⟨x, Ax⟩1),(1)

for some initial condition x0∈Sd. Here, ◦stands for the Hadamard product (entry-wise), Ais

a suitable d×dreal matrix termed the payoﬀ matrix, and 1stands for the vector of dimensions

dwith all its entries equal to 1. Equation (1) is intended to model the dynamics of the

proportions (xi:i= 1, . . . , d)of individuals playing contesting strategies in (the continuous-

time approximation of) a multi-round, two-player game: if each player can choose among

strategies 1,2, . . . , d, the entry Aij is interpreted as the payoﬀ from using strategy iagainst

an individual using strategy j. The term − ⟨x, Ax⟩1modulates the ﬁtness variation in such

a way that the ﬁtness of a player of type iincreases if the payoﬀ is greater than the average

payoﬀ of the community (see [25, 20, 23] for a thorough study of these systems in the context

of evolutionary game theory).

Many generalizations of the original replicator dynamics can be considered. For example,

we can obtain the generalized replicator dynamics:

dt =x◦(F(x)− ⟨x, F (x)⟩1),

INTERACTING REPLICATOR DYNAMICS 3

for some suitable instantaneous ﬁtness function F:Sd7→ Rd. In this form, replicator dynam-

ics has become a general model for the evolution of populations under frequency-dependent

selection, “ﬂexible enough to cover a great deal of evolution models, suggesting a unifying view

of replicator selection from the primordial soup up to animal societies” ([49]). The mathemat-

ical analysis of general replicator equations has been extensive ([24], [25], [21], [61], [18], [9],

[41], [42], [10]), [8], testifying to its fundamental role in the development of evolutionary game

theory.

Stochastic versions of the replicator equation have a long tradition. From the initial eﬀorts

of Foster and Young ([14]) and Fudenberg and Harris ([17]) in exploring the replicator equation

in a stochastic diﬀerential equation form (see also [1], [46]), the analysis of stochastic replicator

equations have become increasingly complex, going beyond the case of only two pure strategies

([29]) to replicator dynamics with Stratonovich-type perturbations ([30]), to the analysis of

stochastic persistence ([3]), variations in game dynamics with a potential function ([2]) to the

inclusion of revision protocols such as imitation ([47], [37]).

It must be said that the interaction between the main system and the noise can be interpreted

in many diﬀerent ways. For example, Fudenberg and Harris’s version of the stochastic replicator

can be obtained from the following considerations regarding the instantaneous ﬁtness (see [17]).

Let Rd

++ be the cone of strictly positive real d-tuples, let Yt= (Y(1)

t, Y (2)

t, . . . , Y (d)

t)⊤∈Rd

be the vector of abundances at time t≥0of the dtypes in a focal population, and let:

Xt=Yt

i=1 Y(i)

∈Sd,

be the relative proportions at time tof these types. We prescribe a stochastic evolution of the

abundances through the equation.

dYt=Yt◦(AXtdt + ΣdWt),(2)

where Σ := diag(σ1, σ2, . . . , σd)is a diagonal non-singular matrix and W= (W1, . . . , Wd)is a

standard d-dimensional Wiener process deﬁned on a probability space (Ω,F,P)and adapted

to a right-continuous and augmented ﬁltration F:= (Ft:t≥0). The rationale behind this

evolution is easy to grasp: the instantaneous ﬁtness of type iis the payoﬀ associated with the

strategy of playing iat the density level X,plus a random perturbation. In (2) the stochastic

term is generally interpreted as environmental ﬂuctuations due, for example, to high-frequency

variations of the abiotic or biotic factors that underlie the ecological dynamics. Depending on

the time scale of the phenomenon we are interested in, these variations can come from diﬀerent

sources: temperature ﬂuctuations, changes in resource availability, among others.

From equation (2), it is easy to show that the dynamics of the proportion obeys the SDE:

dXt=Xt◦(˜

AXt−DXt,˜

AXtE1)dt +

i=1

Xt◦(Σei− ⟨Xt,Σei⟩1)dW (i)

t,(3)

where ˜

A=A−ΣΣ⊤and eiis the i-th vector of the canonical basis of Rd.

Hofbauer and Imhof [22] studied equation (3), with a focus on the time-averages of the

dynamics. Among other results, the authors show that under some conditions on the payoﬀ

matrix, the process is positive recurrent and its transition probability function admits an in-

variant Dirichlet distribution. Interestingly, this holds when the sum of the oﬀ-diagonal payoﬀs

and the diagonal obey a simple linear relation (this is their Corollary 3.12). This symmetry

can be regarded as equivalent to a type of neutrality, as described in the ﬁrst paragraphs of this

introduction, and in fact, induces the emergence of a neutral ecology in replicator dynamics,

4 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

which provides a remarkable connection to the existing neutral theory in community ecology

[26, 33]. Thus, in this contribution we will use the replicator equation in a stochastic framework

as a vehicle to understand the emergence of primordial ecologies and test for the emergence and

generality of the ﬁtness equivalence principle outlined in [22, 33], in the context of a community

of interacting replicators, as explained next.

1.2. Our model: interacting replicators. For measurable functions F:Sd7→ Rd, write:

ΠTF(x) := x◦(F(x)− ⟨x, F (x)⟩1),

and extend the deﬁnition to matrix-valued functions by tensorisation, namely: if F= (F1, . . . , Fm)

where Fi:Sd7→ Rdis bounded and measurable, then ΠTFis the d×mmatrix whose j-th

column is ΠTFj(x). With this, equation (3) can be written as:

dXt= ΠTΦ(Xt)dt + ΠTΨ(Xt)dWt,(4)

for a suitable vector ﬁeld Φand a suitable matrix-valued function Ψ; here Wis a standard

d-dimensional Brownian motion on a ﬁltered probability space. This is the basic replicator

equation we will be working with in this contribution.

Now, to capture the interaction of one replicator with other of the same type, we assume that

a smooth function Υ : Sd×Sd7→ Rdis given. The notation ΠTΥ(x,y)stands for the action

of ΠTon the ﬁrst variable. Let be given a natural number N, and consider the (Sd)N-valued

stochastic process deﬁned via

dX(N;i)

t= ΠTΦ(X(N;i)

t)dt

j=1

ΠTΥ(X(N;i)

t,X(N;j)

t)dt + ΠTΨ(X(N;i)

t)dW(i)

t,(5)

for i= 1,2, . . . , N . Here, (W(i), i = 1,2, . . . , N )is a family of independent F-adapted, d-

dimensional standard Wiener processes carrying the local noises (i.e., noise associated to each

species represented by a single replicator).

To ﬁx ideas, consider the case d= 2,Φlinear and Ψa constant matrix, and assume, for

instance, that Υhas the form:

Υ(x,y) = δy1Ex,E= 0 1

−1 0!

where δis a small parameter. We will return to this example in Section 5, and so it seems

just right to use it to illustrate the meaning of the equations above. Indeed, in this case, we

can provide an intuitive interpretation of equation (5): the payoﬀ obtained from using the ﬁrst

strategy against a player using the second one increases proportionally to the overall average

frequency of the usage of the ﬁrst strategy amongst the family of replicators. Thus, at least

intuitively, in this system of interacting replicators, the ﬁrst strategy becomes more resilient,

and thus one should expect higher average usage of the ﬁrst strategy as compared to the

situation with no interaction. Of course, the overall eﬀect must take account of the particular

form of the intrinsic payoﬀ matrix (the diagonal elements could restrain a net bias to the use of

the ﬁrst strategy) and the noise part (if it is large, randomness may dominate the dynamics),

but as we will see later, under appropriate assumptions, the intuition above is not far from the

actual long-term picture.

INTERACTING REPLICATOR DYNAMICS 5

Remark 1. In this article, we are considering the case of interacting replicators of just one

type, which could be akin to traditional species that are either phylogenetically and functionally

close and thus share a similar niche, trophic level, and interactions within communities, as

envisioned in the principle of niche conservatism [62]. As we will see below, this will translate

in that, for large N, the individual replicators behave as statistically identical entities. The

multi-type model (which would amount to mean-ﬁeld interactions of N1replicators of type 1,N2

replicators of type 2, etc.; see for instance models of this type associated to neurosciences [13],

social sciences [7] or statistical mechanics [6]) is technically more involved and represents a

reasonable second step in our research program.

Systems analogous to equation (5) are generically referred to as mean-ﬁeld interacting parti-

cle systems, and one of the main problems associated with them is the determination of condi-

tions under which, in the thermodynamic limit N→ ∞, the assembly of particles (in our case,

replicators) behave as if they were independent. This is the propagation of chaos property, whose

precise content will be given later. For the time being, assume that we are interested in the

study of the system (5) on a ﬁnite time horizon and for large values of N. Write µ(N)

tfor the em-

pirical probability measure induced on the simplex by 

X(N)

t:= (X(N;i)

t,X(N;2)

t,...,X(N;N)

t)⊤,

namely:

µ(N)

t(A) := 1

i=1

δX(N;i)

t(A).

With a similar notation, µ(N)

[0,t](A)stands for the random probability measure on C([0, t]; Sd)

given by:

µ(N)(A) = µ(N)

[0,t](A) := 1

i=1

δX(N;i)

[0,t]

(A).(6)

The following convention will be used in the sequel. For an arbitrary metric space Eand a

measurable function G:Sd×Sd7→ E, we will write:

G(x, µ) := ZSd

G(x,y)µ(dy),

and thus, we can regard Gas a map Sd× P(Sd)7→ E. With this convention, equation (5) can

be written as:

dX(N;i)

t= ΠTΦ(X(N;i)

t)dt + ΠTΥ(X(N;i)

t, µ(N)

t)dt(7)

+ ΠTΨ(X(N;i)

t)dW(i)

t, i = 1,2, . . . , N.

Assume that we know that for each tin a ﬁnite time horizon, the random probability measures

(µ(N)

t:N≥1) converge (almost surely in the weak topology of probability measures) to a

certain deterministic probability measure, µtsay. Then, at least intuitively, we should have

that for large values of N, the dynamics of a typical particle should be governed by:

dΞt= ΠTΦ(Ξt)dt + ΠTΥ(Ξt, µt)dt + ΠTΨ(Ξt)dWt,(8)

and, following a Glivenko-Cantelli-like reasoning, it should be apparent that µtis nothing but

the law of Ξt. Equation (8) is of the McKean-Vlasov type, and its solutions (whenever they

exist) induce a paradigmatic example of a non-linear Markov process. Here, the term non-linear

has nothing to do with the algebraic degree of the unknown Ξin (8) but rather with the fact

that the random dynamics depends on the very distribution of the Markov process; in turn, it

6 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

translates into a non-linearity at the level of the evolution equation associated to the dynamics.

There is a rich literature on McKean-Vlasov SDE, with the connection with nonlinear PDE

being an important research topic (see the classical [35] for the motivations underlying this

kind of process).

The aim of this article is to study the link between the mean-ﬁeld system (5) and the

McKean-Vlasov equation (8); to study the persistence (in a sense to be deﬁned below) for the

mean-ﬁeld system and to prove existence and uniqueness of non-trivial invariant measures for

the McKean-Vlasov equation in some simple cases. All these issues are treated with an eye

on the problem posed at the opening of the introduction, namely the emergence and role of

neutrality in large interacting sets of biological entities. The organization of the paper reﬂects

closely this sequence of problems. In Section 2 we present all the mathematical preliminaries

that allow us to put the models on ﬁrm grounds, and we describe our assumptions and main

results. Then, we prove the analytical properties of the processes and provide a result of

the propagation of chaos linking the N-replicator dynamics and the McKean-Vlasov system

(Section 3). Then, for the N-replicator systems we investigate conditions that guarantee the

persistence of the whole set of replicators (Section 4). For the non-linear replicator, we prove

the existence and uniqueness of invariant measures for a particular type of interaction functions

and noise (Section 5). Section 6 shows some simulations and numerical results related to the

model treated in Section 5. Finally, Section 7 discuss our results and points out open questions

regarding our research.

2. Main results

Here we present our main deﬁnitions and assumptions and summarise the main results.

2.1. Notation and deﬁnitions. Notation

•Rd

+and Rd

++ stand for the cones of real d-dimensional vectors with positive and strictly

positive entries, respectively. A generic element of Rd

+or Rd

++ will be denoted by

bold letters x,y,z, etc., and when we need to stress its components, we will write

x= (x1, x2, . . . , xd)⊤.

•For a metric space (E , ρ)the notation B(E)stands for both, its Borel σ-ﬁeld and the

algebra of real-valued Borel measurable functions. Also, we write P(E)for the set of

probability measures on (E, B(E)).

•If Ais a subset of a topological space we write int (A),A, and ∂A to denote its interior,

closure, and boundary, respectively. In particular, int(Sd) = {x∈Sd:Qd

i=1 xi= 0}

and ∂Sd={x∈Sd:Qd

i=1 xi= 0}.

•If A, B are subsets of a topological space, the notation A⋐Bmeans that Ais compactly

contained in B:A⊂B, and Ais compact.

•For a function f:E→E′, with (E , d)and (E′, d′)two metric spaces, the notation

Lip(f)stands for the best Lipschitz constant of f, namely: Lip(f)≤Kif and only if

d′(f(x), f (y)) ≤Kd(x, y)for every x, y ∈E.

•If Eis a (regular enough) subset of Rd, we denote by C(E)the space of continuous

real-valued functions on E. Analogously, Ck(E)stands for the k-times continuously

diﬀerentiable real-valued functions on E.

INTERACTING REPLICATOR DYNAMICS 7

•If Eis a compact metric space and A:E7→ Rd⊗Rmis a continuous map, we deﬁne

the (uniform) Frobenius norm through:

|||A||| := sup

x∈EqTr{A⊤(x)A(x)}.

•For a Banach space Ewe denote with ⟨·,·⟩ the duality product of E∗with E. In

particular, if E=Rd, the notation ⟨x,y⟩stands for the usual inner product of vector

of Rd.

•For f∈ Bb(E)and µaσ-ﬁnite measure on E, we write µf to denote the integral

REf(x)µ(dx). Regarding the point above, we also write ⟨µ, f ⟩with the same meaning.

•For probability measures µ, ν on a measurable space (E , E), the notation dTV (µ, ν)

stands for total variation distance between µand ν, namely:

dT V (µ, ν) = sup

A∈E

|µ(A)−ν(A)|.

If E=Rd, then Wasp(µ, ν )stands for the Wasserstein distance with respect to the

underlying p-th vector norm, namely:

Wasp(µ, ν ) = inf

π∈Couplings(µ,ν)ZRd×Rd

∥x−y∥pπ(dx, dy)1/p

whenever the above quantity exists. Here, Couplings(µ, ν )is the family of probability

measure on the product space Rd×Rdthat have ﬁrst and second marginals µand ν,

respectively.

2.2. The stochastic replicator. The structure of the SDEs (and later, McKean-Vlasov type

equations) that we are going to consider take their general form from a geometric consideration.

Recall that the Shahshahani metric on Rd

++ is given by the metric tensor:

(9) si,j (x) = ∥x∥1

δi,j ,(x∈E, i, j = 1, . . . , d).

(see [50] for further details). The gradient of a C1-function U:Rd\ {0} → Ris transformed by

the Shahshahani metric into S(x)−1∇U(x), where S−1(x)is the inverse of the diagonal matrix

S(x)whose entries are si,j, that is

S(x)−1∇U(x) = 





∥x∥1

∂

∂x1U(x)

∥x∥1

∂

∂xdU(x)







∥x∥1

x◦ ∇U(x),

where ◦denotes Hadamard product. In particular, for x∈Sdone gets S(x)−1∇U(x) =

x◦ ∇U(x). This gradient projected onto the tangent bundle at x(hereafter, Tx=T) gives:

(10) x◦ ∇U(x)− ⟨x◦ ∇U(x),1⟩x=x◦(∇U(x)− ⟨x,∇U(x)⟩1).

Thus, at least if Ain equation (1) is a symmetric payoﬀ matrix, the original deterministic

replicator has the form of a gradient-type dynamics on the simplex.

Notice, however, that the right-hand side of equation (10) makes sense if we replace ∇Uby

any Rd-valued function F. From now on, write:

ΠTF(x) := x◦(F(x)− ⟨F(x),x⟩1),

and extend in an obvious way the deﬁnition for matrix-valued functions G:Rd7→ Rd⊗Rm.

8 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

Let Φ : Sd7→ Rdand Ψ : Sd7→ Rd⊗Rdbe given. We assume that, at least, both

Φand Ψare C1(Sd)functions. This allows us to aﬃrm the existence of a uniquely deﬁned

matrix-valued function AΨsuch that ΠTΨ(x) = x◦AΨ(x)for every x∈Sd. Consider the

d-dimensional SDE given in (4),with suitable initial conditions on Sd. We note ﬁrst that this

equation deﬁnes a Sd-valued process that can be obtained via the projection on the simplex of

a simple model of population dynamics (see the Appendix for the details of this derivation),

and thus the paths remain in Sdby deﬁnition. Note also that by our assumptions, both drift

and diﬀusion components are Lipschitz on Sd, and thus it is well known that there exists

a unique (pathwise) solution to the SDE (4) which deﬁnes a well-posed Sd-valued Markov

process. Let (P(Φ,Ψ)

t:t≥0) be the semigroup associated to the Markov process (Xt:t≥0)

associated to the characteristics (Φ,Ψ). As usual, we consider the semigroup acting from the

left on the space of bounded Borel function and acting from the right on Borel measures on

Sd. Let Bbe a Borel set of Sd. It is a routine fact that (P(Φ,Ψ)

t:t≥0) is a Feller semigroup.

By compactness, it admits invariant measures [11, Theorem 12.39]. Recall that a probability

measure µ∈ P(B)is invariant for the semigroup (equivalently, invariant for the process) if

µP (Φ,Ψ)

t=µfor every t≥0, and in this case we write µ∈ Pinv(Φ,Ψ; B). Let Perg(Φ,Ψ; B)be

the subset of Pinv(Φ,Ψ; B)of extremal invariant probability measures supported on B(that

is, ergodic probability measures that put no mass on Sd\B).

It is well known that for well-behaved Markov processes (for example, for processes whose

semigroups enjoy the strong Feller property), diﬀerent ergodic measures have disjoint supports.

Thus, in order to obtain the uniqueness of an ergodic probability measure supported on the

interior of the simplex, it seems reasonable to impose conditions on the noise in such a way

that no submanifold of dimension less than d−1is kept invariant by the dynamics. So, we will

impose the following non-degeneracy of the noise.

Assumptions 1.

span(ΠTΨ(x)·,j, j = 1, . . . , m) = Tx,(11)

for all x∈int (Sd).

Following [3], to assess the stochastic persistence of the process X(and later, the stochastic

persistence of interacting replicators), we are going to impose conditions on the natural deter-

ministic process associated with it (namely, the process with no diﬀusion term). Deﬁne the

invasion growth rate:

hi(x) = Φ(x)i− ⟨x,Φ(x)⟩,(12)

and set h:= (h1, h2, . . . , hd)⊤. Observe that the i-th invasion rate is just the exponent asso-

ciated to the i-th population in the deterministic dynamics. Now, if µ∈ Pinv(Φ,0; Sd)is an

invariant probability measure for the deterministic replicator under which type iis absent, then

µcan be regarded as an equilibrium distribution for a given strict population with strategies

or types indexed by I⊂ {1,2, . . . , d}with i /∈I. Consider the following condition.

For every µ∈ Perg (Φ,0; ∂Sd):

max

i=1,...,d µhi>0.

The biological interpretation of this condition is just the requirement that any strict popula-

tion of types in ecological equilibrium (for the deterministic dynamics) allows for the invasion

INTERACTING REPLICATOR DYNAMICS 9

of at least one of the absent types. It turns out that the above condition is equivalent to the

following assumption (see e.g. [48]).

Assumptions 2. There exists p= (p1, p2, . . . , pd)⊤a vector of strictly positive components

such that:

inf

µ∈Perg (Φ,0;∂Sd)µ⟨p,h⟩=: ρ > 0.

The next theorem is just Theorem 1 in [3], which we state here for future comparison.

Theorem 2. Assume that Φis C2(Sd), and Assumptions 1 and 2 hold. Then, for every r > 0,

there exists ε=ε(r)>0such that for every Ψof class C3(Sd)such that |||AΨ||| < ε the

following holds:

(1) Pinv(int(Sd)) is a singleton, i.e., there exists a unique invariant probability measure µ

supported on the interior of the simplex for the system (4). Moreover µis absolutely-

continuous w.r.t. Lebesgue measure, and the function x7→ dist(x, ∂ Sd)−ris µ-integrable.

(2) There exist positive constants C, α such that:

dT V (δxPt, µ)≤Ce−αt

dist (x, ∂Sd)r,x∈int(Sd).

In this article, we extend the previous result to the setup of a family of replicators interacting

in a mean-ﬁeld regime. The next subsection provides the details.

2.3. Mean-ﬁeld interacting replicators and McKean-Vlasov evolutionary games. As

a ﬁrst step, we extend the deﬁnition of our parameters to account for the interaction among

replicators. Let be given a bounded measurable map Υ : Sd×Sd7→ Rd. Observe that:

ΠTΥ(x, µ) = µΠTΥ(x,·) = ΠTµΥ(x,·).

We assume the following regularity.

Assumptions 3. Υis a C1map.

In what follows, sometimes we will need to refer to the joint eﬀect of both the endogenous

and the interaction functions. To that aim, we deﬁne:

b:= ΠTΦ+ΠTΥ,

which is continuous as a map from Sd× P(Sd), where the relevant topology in P(Sd)is the

weak topology of probability measures. With this notation, equation (7) becomes

(13) dX(N;i)

t=b(X(N;i)

t, µ(N)

t)dt + ΠTΨ(X(N;i)

t)dW(i)

t, i = 1, . . . N,

and McKean-Vlasov equation (8) becomes

dΞt=b(Ξt, µt)dt + ΠTΨ(Ξt)dWt,(14)

where µt:= Law(Ξt).

Remark 3. Our hypotheses entail that b, when seen as a map from Sd× P(Sd), is Lipschitz

continuous for any Wasserstein distance in the measure variable, in the sense that for every

p≥1there exists a constant C(p, d)such that for every x,y∈Sdand µ, ν ∈ P(Sd):

∥b(x, µ)−b(y, ν)∥p≤C(p, d)∥x−y∥p+Wasp(µ, ν).

The following result is proved in the next section.

10 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

Proposition 4. Under the above assumptions, Equation (14) has a unique pathwise (hence

strong) solution.

Now we are in a position to settle the main results of this paper. We start with a proposition

that links the interacting particle system and the McKean-Vlasov replicator.

Theorem 5. Assume Φis C2(Sd), and Assumptions 1 and 3 hold. Let ˜µbe a probability

measure on the simplex, and consider the N-particle system (13) started from ˜µ⊗N. Consider

also the McKean-Vlasov replicator (14) started from ˜µ. Then the propagation-of-chaos property

holds in the following sense: for any positive time t, the sequence of random probability mea-

sures (µ(N)

[0,t])N≥1, converges almost surely in the topology of the weak convergence of probability

measures, to the deterministic probability measure µ[0,t]=Law(Ξ[0,t]).

It is well known that the thesis of the above theorem is equivalent to the fact that, in the

limit N→ ∞, a sort of asymptotic independence of the particles holds (see Proposition 2.2,

(i), at [53]). More precisely, under the above conditions, for any T > 0and any ﬁnite set

of functions gk∈ Cb(C([0, T ]; Sd)), with E(N)the expectation operator associated to the N-

replicator system, and E(MV )the corresponding expectation associated to the McKean-Vlasov

replicator, we have:

lim

N→∞ E(N)

˜µ⊗N m

k=1

gk(X(N;k)

[0,t])!=

k=1

E(MV )

˜µ(gk(Ξ[0,t])).

Let P(N)be the semigroup associated with the N-particle system. Let PN

inv (resp. PN

erg )be

the set of invariant (resp. ergodic) probability measures for P(N). The next theorem, which

extends Theorem 2, is proved in Section 4. It can be paraphrased as: if the typical replicator

of the assemblage satisﬁes Assumption 2, then the whole community is persistent.

Theorem 6. Assume that Φis C2(Sd), and Assumptions 1, 2, 3 hold. Fix N≥1,r > 0. Then

there exists ε:= ε(r)>0such that for any Ψof class C2(Sd)such that ∥Υ∥+|||AΨ||| < ε, the

following holds:

(1) PN

inv(int (Sd)N)is a singleton, i.e., the N-replicator systems admits a unique invariant

probability measure µsupported on the interior of (Sd)N. Moreover µis absolutely-

continuous w.r.t. Lebesgue measure, and the function x7→ dist (x, ∂(Sd)N)−r/N is µ-

integrable.

(2) There exist positive constants C, α such that:

dT V (δxP(N)

t, µ)≤Ce−αt

dist (x, ∂(Sd)N)r/N ,x∈int((Sd)N).

Finally, let us consider the case of Φ(x) = Axand Ψ(x) = σI, and assume that for the

constant σand the payoﬀ matrix Asatisfy the conditions:

[C1] For every i=j:

aij +aji −aii −ajj =σ2.

[C2] For the matrix A′with entries a′

ij =aij −1

2σ2, there exists a vector α= (α1, . . . , αd)

of strictly positive entries summing up to 1such that A′αis a multiple of 1.

In the case of small σ, condition [C1] implies the existence of ﬁtness equivalence (the diago-

nal and oﬀ-diagonal ﬁtness payoﬀs are similar to each other), in the sense that any perturbation

of the ﬁtness payoﬀs that results in that intraspeciﬁc and interspeciﬁc interactions compensate

INTERACTING REPLICATOR DYNAMICS 11

each other, will preserve the condition of neutrality. Notice that the condition of neutrality is

assumed in the neutral theory of ecology ([26]) but here it is derived as a condition for system

persistence. Further, a small σalso ensures the coexistence of strategies, such that the pro-

cess spends little time near the pure strategy state, and, as shown below, the process admits

a unique invariant distribution of Dirichlet type. Condition [C2] implies the existence of an

interior Nash equilibrium.

Consider the particular case where Υis given by:

Υ(x, µ) = δF (µ)x,

where F:P(Sd)7→ Rd⊗Rdis a Lipschitz function with respect to the Was1distance that

ranges in the space of skew-symmetric real matrices and δ∈Ris a parameter that represents

the interaction strength of a replicator with the rest of the alike replicators in the community.

Observe that, in this case, the condition [C1] is preserved, in the sense that if Asatisﬁes this

condition, then for every positive time the payoﬀ matrix associated with the dynamics of the

McKean-Vlasov replicator with characteristics (Φ,Υ,Ψ) also satisﬁes the condition. Thus, if

we regard [C1] as a proxy for neutrality, the prescription for Υcan be seen as a deﬁnition that

preserves neutrality in the extended set up of mean-ﬁeld replicator entities. In Section 5 we

prove the existence of invariant distribution for a class of McKean-Vlasov replicators associated

with these parameters.

Theorem 7. Assume Asatisﬁes [C1] and [C2]. Then there exists an interaction strength

δ0>0such that for every δ < δ0, the stochastic McKean-Vlasov replicator with Φ(x) = ˜

and Ψ(x) = σId×dconstant admits a unique invariant probability measure putting no mass on

the boundary of the simplex. Moreover, this invariant measure is of the Dirichlet family.

Although the above result states the existence of unique invariant measures for the above

class of McKean-Vlasov replicator, by now we have no rigorous arguments for the ergodicity of

the system. So, in Section 6 we provide statistical evidence to support this fact (at least in the

simple setup of the S2-Mckean-Vlasov replicator). The evidence is then confronted to a test

that allows us to accept the hypothesis that, for δ > 0small enough, every initial condition is

attracted to the Beta(s, 1−s)distribution, with s=a12 −a22

σ2−δ.

3. Propagation of chaos for the mean-field replicators

In this section, we settle some auxiliary results regarding the well-posedness of the equa-

tions we are dealing with, and we prove the propagation of chaos for the N-replicator system

(Theorem 5).

3.1. Elementary results. We start with a proposition that regards the existence and unique-

ness of solutions to the nonlinear McKean-Vlasov equation. Its proof is relatively standard,

and consequently, it is deferred to the Apprendix A.2.

Proposition 8. Under Assumptions 1 and 3, equation (14) has a unique pathwise (hence

strong) solution.

Observe that the process Ξis not a Markov process. Indeed, the family of maps (Ps:s≥0)

acting on Bb(Sd)deﬁned via

Psf(x) := Ex(f(Ξs)),

12 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

do not determine a semigroup of linear operators, in neat contrast to the Markovian setup.

However, the semigroup property still holds in the dual sense. Deﬁne the one-parameter family

of operators:

Qt:P(Sd)7→ P(Sd)(15)

given by the prescription: Qtµ=Lawµ(Ξt), the law of Ξtwhen started from an initial condition

distributed as µ.

Proposition 9. The family of operators (Qt:t≥0) enjoys the semigroup property:

Qt+s=QtQs,(16)

for all s, t ≥0.

Remark 10. Still, the operators Qsare non-linear: for µ, ν ∈ P(Sd),λ∈[0,1], in general:

(λµ + (1 −λ)ν)Qt=λµQt+ (1 −λ)νQt.

Proof. Let µ∈ P(Sd), and t, s ≥0. Put µt:= Qtµ. Consider random variables Ξ0and Ξ1

distributed as µand µtrespectively, and the processes (Ξ(Ξ0)

t+s:s≥0) and (Ξ(Ξ1)

s:s≥0). By

(14), for s≥0, on the one hand we have:

Ξ(µ)

t+s

Law

= Ξ(Ξ0)

t+s= Ξ0+Zt

E(b(Ξ(Ξ0)

r,˜

Ξ(Ξ0)

r))dr +Zt

ΠTΨ(Ξ(Ξ0)

r)dWr

+Zs

E(b(Ξ(Ξ0)

r+t,˜

Ξ(Ξ0)

r+t))dr +Zs

ΠTΨ(Ξ(Ξ0)

t+r)dWt+r

Law

= Ξ0+Zt

E(b(Ξ(Ξ0)

r,˜

Ξ(Ξ0)

r))dr +Zt

ΠTΨ(Ξ(Ξ0)

r)dWr

+Zs

E(b(Ξ(Ξ0)

r+t,˜

Ξ(Ξ0)

r+t))dr +Zs

ΠTΨ(Ξ(Ξ0)

t+r)d˜

Wr.

where ˜

Wis a Wiener process independent of (Wr: 0 ≤r≤t)and ˜

Estands for expectation

with respect to the law of an independent copy of the process Ξ(Ξ0). Observe that the sum of

the ﬁrst three terms in the last equation is distributed as Ξ1and is independent of ˜

W. Writing:

Y·:= Ξ(Ξ0)

t+·, the last line becomes:

Law

= Ξ1+Zs

E(b(Yr,˜

Yr))dr +Zs

ΠTΨ(Yr)dWr

=Ys

Law

= Ξ(µt)

where the ﬁrst-to-last line follows from pathwise uniqueness of the solutions of (14) (Proposition

8), and the last line is obvious. Our claim follows.

For future reference, we establish a simple result. For the interested reader, we present the

proof in Appendix A.3.

Lemma 11. For every Ξ0∈int(Sd):

P(MV )

Ξ0min

i=1,...,d Ξ(i)

t>0for all t≥0= 1.

INTERACTING REPLICATOR DYNAMICS 13

An analogous result holds for the N- particle system, namely: for every 

x0∈int ((Sd)N),

P(N)



x0min

j=1,...,N ∥X(N;j)

t∥>0for all t≥0= 1.

In words: with probability 1, in ﬁnite time there is no absorption on the boundary of Sd(for

the McKean-Vlasov replicator) nor on the boundary of (Sd)N(for the N-particle system).

As a direct result of Lemma 11, we have that for any x∈int (Sd)and t > 0

Qtδx(∂Sd)=0,

i.e., the trajectories of the solution of (14) do not hit the boundary of the simplex in ﬁnite time.

The following result regards the existence of a density of the law of the nonlinear process.

Proposition 12. For any x∈int(Sd)and t > 0,Qtδx≪Lebd−1.

Proof. For t≥0and y∈Sd, deﬁne:

b(y, t) := b(y,Qtδx),

and consider the following diﬀusion with time-dependent coeﬃcients:

Ξ(x,δx)

t=x+Zt

b(Ξ(x,δx)

s, s)ds +Zt

ΠTΨ(Ξ(x,δx

s))dWs.

Observe that the coeﬃcients are Lipschitz-continuous and bounded, and moreover, on account

of Assumption 1, for every compact D⋐int (Sd):

z,ΠTΨ(y)ΠTΨ(y)⊤z> ε∥z∥2,

for a certain constant ε > 0, for every y∈ D and z∈ T . Now, for any x∈int(Sd)consider:

τD:= inf{t≥0:Ξ(x,δx)

t/∈ D,x∈ D},

and deﬁne the ﬁnite measure on Sd:

t(A) := P(Ξ(x,δx)

t∈A, t ≤τD).

By a classical piece of the theory of parabolic PDEs with uniformly elliptic coeﬃcients (see

for example Chapter 6 of [16] together with the representation given by Theorem 3.46, pag.

207, of [44]), we deduce that the measure mD

t(·)is absolutely-continuous with respect to the

Lebesgue measure.

Consider next a sequence of compact subsets {Dϵ}ϵin int (Sd), such that Dϵ′⊂ Dϵif ϵ<ϵ′,

and limϵ→0Dϵ=int (Sd). Analogously as we have done with Dabove, deﬁne τDϵand mDϵ

t(·).

Let Abe a measurable subset of int (Sd)with Lebd−1(A) = 0. Then mDϵ

t(A) = 0 for every

ϵ > 0. Since the trajectories of the solution of (14) do not hit the boundary of Sdin ﬁnite time,

we have that mDϵ

t(A)↑δxQt(A)as ϵ↓0. This proves the claim.

Remark 13. The result still holds for the N-replicator systems, i.e., for every initial condition

in int ((Sd)N)the law of 

Xtis absolutely continuous w.r.t. the N-fold product of the Lebesgue

measure.

14 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

3.2. Convergence of the empirical laws. Fix t > 0. Let µ(N)be the random probability

measure on C([0, t]; Sd)given by (6). Now we will prove that (µ(N))Nalmost surely converges,

in the weak sense, to the deterministic limit µ:= Law(Ξ[0,t]). Recall that this means: if

mN∈ P(P(C([0, t]; Sd))) is the law of the random measure µ(N), we aim to prove that for every

bounded continuous function F∈ Cb(P(C([0, t]; Sd))), we have:

ZP(C([0,t];Sd))

mN(dν)F(ν)→F(µ),

as Ngoes to inﬁnity. If we write E(N)for the expectation operator associated with the N-

replicator system, the above convergence is equivalent to asking that:

E(N)((µ(N)f−µf)2)→0,

as N→ ∞, for every f∈ C(C([0, t]; Sd). In turn, it can be proved that this will follow if we are

able to prove that the family of probability measures E(N)(µ(N)) := Law(X(N;1)

[0,t])converges in

P(C([0, t]; Sd)(see [54], pages 177 et seq.; see also [36], pages 66 et seq.). So, this is what we

will do now.

Assume that the N-particle system (X(N;1)

·, ..., X(N;N)

·)in (13) is issued from independent

and identically distributed conditions (X(N;1)

0, ..., X(N;N)

0), and are driven by independent d-

dimensional standard Brownian motions (W(1)

·, ..., W(N)

·). Notice that this implies that the

law of (X(N;1)

·, ..., X(N;N)

·)is symmetric.

Theorem 14. (Propagation of Chaos). Let (Ξ(i):i= 1, . . . , N )be a family of independent

copies of the non-linear process solution of (14), such that Ξ(i)

0=X(N;i)

0for every i= 1, . . . , N

almost surely, and both X(N;i)

·and Ξ(i)

·are driven by indistinguishable Brownian motions.

Then, under the above assumptions, for any ﬁnite time horizon [0, S]there exists a positive

constant K=K(S, d)such that for every i= 1, . . . , N:

E sup

t∈[0,S]



X(N;i)

t−Ξ(i)

t



2!≤K

Proof. Let X(N;i)

·and Ξ(i)

·be the processes speciﬁed above, and let S > 0. Then, for any

t∈[0, S]we have:



X(N;i)

t−Ξ(i)

t



2≤2SZt

0



b(X(N;i)

s, µ(N)

s)−b(Ξ(i)

s, µs)



2ds

+ 2 



Zt

[ΠTΨ(X(N;i)

s)−ΠTΨ(Ξ(i)

s)]dW(i)

s



Now, for the ﬁrst term of the right-hand side, by the Lipschitz condition of b, there exists a

K1=K1(S, d)>0such that:

0



b(X(N;i)

s, µ(N)

s)−b(Ξ(i)

s, µs)



2ds ≤K1Zt

0



X(N;i)

s−Ξ(i)

s



2ds

+K1Zt

0



j=1

X(N;j)

s−ZSd

xµs(dx)



ds.

INTERACTING REPLICATOR DYNAMICS 15

On the other hand, notice that the term R·

0[ΠTΨ(X(N;i)

s)−ΠTΨ(Ξ(i)

s)]dW(i)

sis an (Ft:t≥0)

square-integrable martingale, and thus by Doob’s Maximal inequality we have that

E sup

t∈[0,S]



Zt

[ΠTΨ(X(N;i)

s)−ΠTΨ(Ξ(i)

s)]dW(i)

s



≤4E





ZS

[ΠTΨ(X(N;i)

s)−ΠTΨ(Ξ(i)

s)]dW(i)

s



2

.

By Itô’s isometry and the Lipschitz assumption on ΠTΨ(x), there exists a constant K2=

K2(S, d)>0such that:

E





ZS

[ΠTΨ(X(N;i)

s)−ΠTΨ(Ξ(i)

s)]dW(i)

s



2



≤K2E ZS

0



X(N;i)

s−Ξ(i)

s



2ds!.

Therefore, we have that:

E sup

t∈[0,S]



X(N;i)

t−Ξ(i)

t



2!≤K3ZS

Esup

u≤s



∥X(N;i)

u−Ξ(i)

u



2ds

+K3Zt

E



sup

u≤s



j=1

X(N;j)

u−ZSd

xµu(dx)



2



ds,

for some a constant K3=K3(S, K1, K2).

Recall that we have assumed that (Ξ(1)

·,...,Ξ(N)

·)are Ni.i.d. copies obeying (14) such

that for all j= 1, ..., N ,Ξ(j)

0=X(N;j)

0almost surely, and Ξ(j)

·and X(N;j)

·are driven by

indistinguishable Brownian motions. Then:

E



sup

u≤s



j=1

X(N;j)

u−ZSd

xµu(dx)



2



ds

≤2

N2ZS

E



sup

u≤s



j=1

(X(N;j)

u−Ξ(j)

u)



2



ds

N2ZS

E



sup

u≤s



j=1

(Ξ(j)

u−ZSd

xµu(dx))



2



ds.

The last term above is O(N−1)by the Law of Large Numbers and, on the other hand, we have

that:

N2ZS

E



sup

u≤s



j=1

(X(N;j)

u−Ξ(j)

u)



2



ds ≤ZS

Esup

u≤s



X(N;i)

u−Ξ(i)

u



2ds,

16 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

since, by symmetry, the laws of the (X(N;j)

·−Ξ(j)

·)’s are exchangeable. Finally, by applying

Grönwall’s inequality, we obtain:

E sup

t∈[0,S]



X(N;i)

t−Ξ(i)

t



2!≤O(1/N) exp{2K3S},

and therefore the theorem follows.

We deduce from the previous result that the sequence of empirical measures (µ(N))Ncon-

verges to the deterministic limit µ:= Law(Ξ[0,t]), in the weak sense. This fact plus the property

that the law of the N-replicator (X(N;1)

·, ..., X(N;N)

·)is symmetric (or exchangeable), implies

that, for any g1,g2, ...., gkcontinuous and bounded functions on Sd, with k≤N, we have:

lim

N→∞ E(N)

µ⊗N k

i=1

gk(X(N;σ(i))

·)!=

i=1

⟨µ, gi⟩,

where (X(N;σ(1))

·, ..., X(N;σ(N))

·)corresponds to any permutation of the vector (X(N;1)

·, ..., X(N;N)

·)

(see Proposition 2.2.i) in [54] or Proposition 4.2 in [36]). That is, the N-particle system in the

limit N→ ∞ becomes a sequence of independent and identically distributed solutions of the

McKean-Vlasov equation.

4. Strong stochastic persistence for the N-replicator system

In this section we address the problem of persistence of the Markov processes (

X(N)

t:t≥0).

For N≥1, the notation (Sd)Nstands for the N-fold product of Sd. The space (Sd)Nis endowed

with the topology of the topological sum. Since Sdis Polish, the same holds for its N-fold sum,

for the metric:

dist N(

x,

y) := max

dist (x(i),y(i)),

where the notation dist in the right-hand side stands for the L1distance on the simplex. We

further set:

dist N(

x, ∂(Sd)N) := min

dist (x(j), ∂Sd).

Let P(N)be the semigroup associated with the N-replicator system (7). The action of

the generator associated with the semigroup on smooth functions can be expressed as follows.

Given x(1),...,x(N)arbitrary vectors in Rd, we will write 

xfor the vector in RdN resulting

from the concatenation, in lexicographic order, of the components of the x(i)’s. For a given

smooth function φ: (Sd)N7→ R, for i= 1, . . . , N and 

x∈(Sd)Nwe denote by φ(i)



xthe map

Sd∋y7→ φ(x(1),x(2) ,...,y,...,x(N))where yis in the place of the i-th block of order d.

A straightforward computation shows that whenever φis a twice-continuously diﬀerentiable

real-valued function, the action of the generator is given by:

INTERACTING REPLICATOR DYNAMICS 17

L(N)φ(

x) =

i=1 D∇φ(i)



x(x(i)),ΠTΦ(x(i))E

i=1 D∇φ(i)



x(x(i)),ΠTΥ(x(i), µ

x)E

i=1

Tr ΠTΨ(x(i))⊤∇∇⊤φ(i)



x(x(i))ΠTΨ(x(i)),

where: µ

x=1

NPN

j=1 δx(i), and ∇∇⊤φ(i)



xis the Hessian matrix associated to φ(i)



Deﬁnition 15. (see [19],[59]) We say that a (Sd)N-valued, right-continuous Markov process

(Yt:t≥0) is strongly stochastically persistent (SSP) if there exists a unique invariant

probability measure πsuch that π(∂((Sd)N)) = 0 and such that for every x∈int ((Sd)N):

dT V (Px(Yt∈ ·), π(·)) →0,

as t→ ∞.

Here we will prove that under our running assumptions, (

X(N)

t:t≥0) is a SSP processes

taking values on (Sd)N. In fact, we will prove an extension of Theorem 2 and, in particular,

we will obtain geometric rates of convergence for both processes.

To this end, we introduce some notation. We will say that x∈int(Sd)is ε-close to extinction

if there exists i∈Isuch that xi< ε. For given ε, deﬁne:

Ext(ε) := {xis ε−close to extinction}.

We write ExtN(ε) := {

x∈(Sd)N:x(i)∈Ext(ε)for some i}.

To prove convergence to an invariant measure for (

X(N)

t:t≥0), we ﬁrst observe that it is

enough to prove that for every t > 0there is convergence for the chain (

X(N)

n:= 

X(N)

tn :n≥0),

i.e. the chain sampled from 

X(N)every ttime units. This claim follows from the Feller

property of the semigroup (P(N)

t:t≥0) (which can be derived in a rather standard way from

the Lipschitz continuity of the parameters) and from the following general fact about Feller

processes.

Lemma 16. Let (Pt:t≥0) be the Feller semigroup associated with a Markov process (Yt:t≥

0) taking values on a compact metric space (E , ρ). Denote by (P∗

t:t≥0) the dual semigroup

acting on ﬁnite measures. Fix F∈ B(E). Assume that for every s∈(0,1] there exists a unique

probability measure πson (E , B(E)) with πs(FC) = 0 and such that P∗

sπs=πs. Then the

process Yhas a unique invariant probability measure πwith π(FC)=0.

Proof. For every t∈(0,1] ∩Q, the unique invariant measure for P∗

tis also invariant for P∗

Hence, πt=π1for every rational t. Then, for general tand any n∈N, we have:

π1=P∗

⌊nt⌋

π1,

Since Pis Feller and Eis compact, P∗is a weakly-continuous semigroup. Hence, sending

nto ∞in the last equation, we obtain π1f=P∗

tπ1ffor every f∈ C(E). Since the class

of continuous functions is measure-determining on E, we get that π1=P∗

tπ1, and by the

uniqueness assumption, πt=π1.

18 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

Fix t > 0, and let P:= P(N)

t. Consider the following statement.

[H] There exists a continuous function H:int((Sd)N)7→ R+and constants α∈(0,1)

and C > 0such that:

(1) limε→0inf

x∈ExtN(ε)H(x) = ∞.

(2) PH(

x)≤αH(

x) + Cfor every 

x∈int ((Sd)N).

Observe that [H] entails that for any x∈int ((Sd)N), any weak limit point, say ν, of the

sequence of probability (δ

xPn)nsatisﬁes ν(∂((Sd)N)) = 0. For suppose this is not the case,

and given 

x∈int ((Sd)N), assume that a limit point νof (δ

xPn)nsatisﬁes ν(∂((Sd)N)) >2ε

for a given ε > 0. Apply ntimes the second condition of [H] to get:

PnH(

x)≤αnH(

x) + D,

where D:= C

1−α. Deﬁne η:= H(

x) + D

εand consider the set Kε:= H−1([0, η]). Since His

continuous, Kεis a closed (indeed, compact) set contained in int ((Sd)N)(by the ﬁrst point at

[H]). Then, for every n, Markov inequality yields:

Pn1KC

ε(

x)≤ε.(17)

Consider a continuous function g: (Sd)N7→ [0,1] with the property g≡1in ∂((Sd)N),g≡0on

Kε. Since g≤1KC

ε, by (17) we obtain δ

xPng≤εfor every n. But νg > 2εwhich contradicts

the fact that νis a weak-limit point.

Remark 17. Consider for a while the dynamics given by the McKean-Vlasov replicator Ξ

discussed in the previous sections, and for every t≥1consider the operator ˜

Ptacting on

bounded functions as ˜

Ptf(x) := Ex(f(Ξt)). As remarked above, in general, we do not have

Pnt =˜

t. So, the previous argument breaks down from the very start in the case of McKean-

Vlasov replicators, and thus new tools are needed to face the problem of persistence in this

setup.

Thus, [H] guarantees that the limit points of the sequence (Law(

X(N)

tn ) : n≥0) support the

whole ensemble of replicators. Also, observe that if νis an invariant probability measure for

the process 

X(N)that supports the whole ensemble of replicators, then by deﬁnition νP =ν.

Let A⊆int ((Sd)N)be a measurable set with Leb(A)=0. Then:

ν(A) = νP (A) = Z(Sd)N

ν(d

x)δ

xP1A=Zint((Sd)N)

ν(d

x)δ

xP1A,

since νdoes not charge the boundary of (Sd)N. By Proposition 12, for every 

xthere exists

a density, say p(

x,·), of the probability measure δ

xPwith respect to the Lebesgue measure.

Consequently:

ν(A) = Zint((Sd)N)

ν(d

x)δ

xP1A

=Zint(Sd)

ν(d

x)ZA

p(

x,

y)d

y=ZA

d

y Zint((Sd))N

ν(d

x)p(

x,

y)!,

INTERACTING REPLICATOR DYNAMICS 19

by Fubini’s theorem. Since Leb(A) = 0, the last integral vanishes, and we deduce ν(A) = 0.

In other words, if νis an invariant measure for the process 

X(N)under which the full system

persists, then it is continuous with respect to the Lebesgue measure.

Finally, assume that we are able to prove that there exists a unique invariant measure, say

µ, for the semigroup Pwhich is supported at the interior of (Sd)N. We claim that H, as in

[H], is µ-integrable. Indeed, for n≥0, deﬁne Hn:= H ∧ n. Then, for every n≥1:

µHn= lim

k→∞ µP kHn≤µ(lim sup

k→∞

PkHn)≤D,

where the equality follows since µis invariant for P, and the ﬁrst inequality is (reverse) Fatou’s

Lemma. Apply (direct) Fatou’s Lemma to obtain:

µH=Zint((Sd)N)

lim

n→∞ Hn(

x)µ(dx)≤lim inf

n→∞ Zint((Sd)N)

Hn(

x)µ(d

x)≤D,

and we deduce that His µ-integrable.

Actually, the existence of a function Has in [H] implies stronger results (see [3], page 186,

and the references therein, for the following lemma).

Lemma 18. Assume that Hexists as in [H]. Assume also that Ppossesses a density with

respect to the Lebesgue measure. Then, there exists a unique invariant probability measure π

for the kernel P, and moreover:

(1) π≪Leb.

(2) His π-integrable.

(3) There exists ρ∈(0,1) and C′>0such that:

dT V (δ

xPn, π)≤C′(1 + H(

x))ρn,

x∈int ((Sd)N).(18)

Remark 19. Observe that (18) implies that πattracts every initial measure νwith ν(∂(Sd)N) =

0. Indeed, for any compact set K⊂int (Sd)N, by the deﬁnition of total variation norm and

Fubini’s theorem:

∥νP n−π∥T V = sup

fBorel ,|f|≤1 Z(Sd)N

ν(d

x)Pnf(

x)−Z(Sd)N

π(d

y)f(

y)!

= sup

fBorel ,|f|≤1Z(Sd)N

ν(d

x)Z(Sd)N

(Pn(

x, d

y)f(

y)−π(d

y)f(

y))

≤Z(Sd)N

ν(d

x) sup

fBorel ,|f|≤1Z(Sd)N

(Pn(

x, d

y)f(

y)−π(d

y)f(

y))

≤ZK

ν(d

x)∥δ

xPn−π∥T V + 2ν(int ((Sd)N)\K),

since the total variation norm is bounded by 2and νdoes not charge the boundary. Since (Sd)N

is Polish, νis regular, and then given ε > 0we can choose Kεsuch that the last term is smaller

than ε. The result follows since the term inside the integral converges uniformly to 0on Kεas

n→ ∞.

Theorem 20. Assume that Φis C2(Sd), and Assumptions 1-2-3 hold. Then for every r > 0

there exists δ > 0such that for any Υ,Ψof class C2such that ∥ΠTΥ∥+|||AΨ||| < δ, the

following hold:

20 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

(1) P(N)

inv (int ((Sd)N)) = {ν}; i.e., there exists a unique invariant probability measure sup-

ported on the interior of the N-fold topological sum of Sdfor the system (13). Moreover

νis absolutely continuous w.r.t. Lebesgue measure, and the function int ((Sd)N)∋

x7→

dist N(

x, ∂((Sd)N))−r/N is ν-integrable.

(2) There exist positive constants C′, α′such that:

∥δxP(N)

t−ν∥T V ≤C′e−α′t

dist N(x, ∂((Sd)N))r/N ,x∈int((Sd)N).

Proof. (of Theorem 20). We follow closely the steps in the proof Theorem 3.1 of [3]. Recall

the deﬁnition of the invasion rates (12). There exists α > 0, a neighbourhood Uof ∂Sdand a

function W:Sd7→ Rof class C2(Sd)such that:

⟨p,h(x)⟩+⟨∇W(x),ΠTΦ(x)⟩> α,(19)

for every x∈ U (see [21], Theorems 3.4 and 4,4, and Remark 3.5), and where pis taken as in

Assumption 2. Deﬁne the map VN: (U \ ∂Sd))N7→ Rgiven by

VN(

x) := 1

i=1

V(x(i)),

where V(x) := Pd

j=1 pjln(xj) + W(x). Since h(x) = diag(x)−1ΠTΦ(x), we have:

i=1 D(∇VN(

x))(i),ΠTΦ(x(i))E=1

i=1 D∇V(x(i)),ΠTΦ(x(i))E> α,

on (U \ ∂Sd)N. Apply the generator L(N)to VNto obtain:

L(N)VN(

x) =

i=1 D∇V(i)

N,

x(x(i)),ΠTΦ(x(i))E

i=1 D∇V(i)

N,

x(x(i)),ΠTΥ(x(i), µ

x)E

i=1

Tr ΠTΨ(x(i))⊤∇∇⊤V(i)

N,

x(x(i))ΠTΨ(x(i)),

where the map V(i)

N,

xis given by Sd∋y7→ VN(x(1),x(2),...,y,...,x(N)), where yis in the

place of the i-th block of order d.

Since Wis of class C2(Sd), the above expression can be made greater than α/2, say, for



x∈(U \ ∂Sd))Nby choosing ˜ε1>0small enough and imposing ∥ΠTΥ∥+|||AΨ||| <˜ε1.

Deﬁne λ:= r

inf pi

, and consider the function H: (int (Sd))N7→ R++ given by H:=

exp(−λVN), which is of class C2((int (Sd))N). Furthermore

lim

η→0inf

{

z:z(i)∈Ext(η)\∂Sd,∀i=1,...,N}H(

z) = ∞.

INTERACTING REPLICATOR DYNAMICS 21

Since Wis bounded on Sd, with K:= exp(−λinfz∈SdW(z)), we have:

H(

x)≥Kexp

−λ(1/N)

i=1

j=1

pjlnx(i)

j



≥K

i=1

exp

−(r/N)

j=1

lnx(i)

j



i=1

(x(i)

1x(i)

2. . . x(i)

d)r/N

≥K

i=1

dist (x(i), ∂(Sd))r/N

≥K

dist N(

x, ∂(Sd)N)r/N .(20)

Let Γ(N)be the carré du champ operator associated to the operator L(N), whose action on

smooth functions φ: (Sd)N7→ Ris given by:

Γ(N)φ(

x) = 1

i=1 X

j,k

[ΠTΨ(x(i))ΠTΨ(x(i))⊤]j,k∂jφ

x(x(i))∂kφ

x(x(i)).

For smooth functions f:R7→ R, φ : (Sd)N7→ R, we have:

L(N)f(φ(

x)) = f′(φ(

x))L(N)φ(

x) + f′′(φ(

x))Γ(N)φ(

x).

Hence:

L(N)H(

x) = −λH(

x)L(N)VN(

x)−λΓ(N)VN(

x),

and then we can choose ˜ε < ˜ε1and impose ∥ΠTΥ∥+|||AΨ||| <˜εsuch that the term inside the

parentheses is not smaller than α/3, say, on (U \ ∂Sd)N. And then, for this choice:

L(N)H(

x)≤ −λα

3H(

x),

on (U \ ∂Sd)N. Observe that L(N)His bounded on (int(Sd)\ U )Nby some absolute constant

C. Thus, with β:= λα

3, we obtain:

L(N)H(

x)≤ −βH(

x) + C,(21)

for all 

x∈(int (Sd))N.

Let ηn:= inf{t≥0 : H(

X(N)

t)> n}, and observe that since His bounded on compact sets

of (int (Sd))N, by Lemma 11, we have ηn→ ∞ a.s. For given t≥0, set tn:= t∧ηn. For every

n≥1, the process:

t=eβtnH(X(N)

tn)− H(

X(N)

t0)−Ztn

eβs(βH(

X(N)

s) + L(N)H(

X(N)

s))ds, t ≥0,

is a martingale, and thus for every n≥1, by (21):

E

x(eβtnH(

X(N)

tn)) = H(

x) + EZtn

Ceβ s≤ H(

x) + C

β(1 −eβt).

22 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

Hence:

E

x(eβtH(

X(N)

t)1t≤tn)≤ H(

x) + C

β,

and then,

E

x(H(

X(N)

t)1t≤tn)≤e−βtH(

x) + C

β.

Since tn→ta.s., by Monotone Convergence, we deduce that for every t≥0and 

x∈(int (Sd))N:

E

x(H(

X(N)

t)) ≤e−βtH(

x) + D,

where Dis a constant. In particular, there exists γ∈(0,1) such that, at time 1:

P(N)

1H ≤ γH+D,

on (int (Sd))N. By Proposition 12, P(N)

1possesses a density with respect to the Lebesgue

measure. Hence, Lemma 18 applies to P(N)

1and we deduce the existence of a unique invariant

probability measure µfor P(N)

1such that µ(∂(Sd)N)=0, and moreover µattracts every initial

condition on (int (Sd))Nat geometric rate. Of course, the same line of reasoning applies to

P(N)

tfor every t∈(0,1], and we deduce that for every t∈(0,1] there exists a unique invariant

measure, say πt, for the chain (

X(N)

nt :n≥0). The uniqueness follows from Lemma 16.

The remaining parts of the result are exactly those in the reference [3].

5. Invariant distributions for the McKean-Vlasov stochastic replicator

We go back to the initial model of Fudenberg and Harris, as treated in [22], and consider

a non-linear, McKean-Vlasov-type extension of the model. Assume ﬁrst that d= 2, that Ais

given, and that Σ = σI. Recall that in this case, each replicator solves (3). From Theorem 3.6

of [22] we deduce the following result.

Proposition 21. Assume that a12 −a22 >0and a21 −a11 >0, and furthermore:

a12 +a21 −a11 −a22 =σ2.

Then, the system (3) has a unique invariant distribution that does not charge the boundary,

namely the absolutely continuous measure with density Beta with parameters (a12 −a22)σ−2

and (a21 −a11)σ−2.

Observe that for the deterministic system associated to (3) the unique ergodic probability

measures that support strict subsets of the community are delta distribution, namely: δ1with

all the mass at the ﬁrst population, and δ2with all the mass at the second population. It is

direct to show that the average positivity of the invasion rates (Assumption 2) holds in this

case.

Assume that we choose 0< δ < min{σ2, a12 −a22, a21 −a11 }. For an arbitrary probability

measure µon the simplex, deﬁne Φ(x) = ˜

Ax,Ψ(x)=Σand

Υ(x, µ) = δZy1µ(dy1, dy2)0 1

−1 0!x,(22)

and the associated non-linear replicator:

dΞt= ΠTΦ(Ξt)dt + ΠTΥ(Ξt, µt)dt + ΠTΨ(Ξt)dWt.(23)

INTERACTING REPLICATOR DYNAMICS 23

Theorem 22. Equation (23) admits a unique invariant distribution, namely the Beta(s, 1−s)

distribution, where s=a12 −a22

σ2−δ.

The proof of the above theorem will be transparent as soon as we prove its extension to

higher dimensions (and the computation of the invariant measure will follow after some simple

algebraic manipulation).

Before starting the next Lemma, observe that if a payoﬀ matrix Asatisﬁes [C1], for every

x= (x1, . . . , xd)⊤∈ T we have:

x⊤Ax=1

i,j

(aij +aji )xixj=1

i,j

(aij +aji −aii −aj j )xixj

i=j

(aij +aji −aii −aj j )xixj

2σ2

i=1

xi(x1+x2+. . . +xi−1+xi+1 +. . . +xd) = −1

2σ2

i=1

In particular, if x∈ T is not the null vector, Axcannot be a constant vector. We deduce that

the equation A′α=constant vector has at most one solution summing up to 1.

Lemma 23. Assume that Asatisﬁes [C1] and [C2]. Then there exists δ > 0such that for

any skew-symmetric matrix Ewith ∥E∥< δ the matrix ˆ

A:= A+Esatisﬁes [C2].

Proof. First, observe that for Aand Eas in the claim, the matrix A+Esatisﬁes [C1] as

well. Next, observe that we can assume that Ehas the block-form:

E=δE202×(d−2)

0(d−2)×20,with E2=0 1

−1 0,

since, if the result is true for this kind of perturbation, applying repeatedly the lemma we arrive

(via permutations of rows and columns) at the desired result.

We aim to ﬁnd a strictly positive vector ˆα= (ˆα1,..., ˆαd)⊤such that ˆ

A′ˆαis a constant vector.

Let α:= (α1, . . . , αd)⊤∈Rdand c∈Rbe the vector and the real number ensured by [C2]

associated to the matrix A. For i= 1,2, . . . , d −1, write ˆαi:= αi+εi, and ˆαd=αd−Pd−1

j=1 εj.

Then for a certain constant ˆc, for i= 1, . . . , d:

ˆc=

j=1

(ˆaij −σ2/2) ˆαj

d−1

j=1

(aij +δeij −σ2/2)(αj+εj)+(aid +δeid −σ2/2)(αd−

d−1

k=1

εk),

24 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

must hold. Then:

ˆc=c+δ(α2+ε2) +

d−1

j=1

(a1j−a1d)εj

=c−δ(α1+ε1) +

d−1

j=1

(a2j−a2d)εj

=c+

d−1

j=1

(aij −aid)εj, i = 3,4, . . . , d,

and consequently:

δ(α2+ε2) +

d−1

j=1

(a1j−a1d)εj=−δ(α1+ε1) +

d−1

j=1

(a2j−a2d)εj

d−1

j=1

(aij −aid)εj,

for i= 3,4, . . . , d. This reduces to a linear system of d−1equations for the d−1unknowns

ε1, . . . , εd−1. Let H(δ)be the matrix representing the above system. Then ε= (ε1, . . . , εd−1)⊤

satisﬁes the system (Sδ):

H(δ)ε(δ)=δ(α1+α2)δα2δα2· · · δα2⊤.(24)

Since the equation A′α=constant is uniquely solvable for αstrictly positive, summing up to

1and satisfying [C1], the equation H(0)ε′= 0 has a unique solution, and hence H(0) is non

singular. On the other side, it is clear that (H(δ):δ∈[0,1]) is a Lipschitz continuous family

of matrices (with respect to a ﬁxed reference norm in Rd⊗Rd, say the L2subordinate matrix

norm), and thus there exists 1≥δ > 0such that for δ < δ there exists a unique solution of

(Sδ). Let (ε(δ):δ∈[0, δ ]) be the family of solutions, and let ℓδbe the right hand side of (Sδ).

Then for δ1, δ2∈[0, δ], and some constant C > 0that can change from line to line:

∥ε(δ1)−ε(δ2)∥≤∥(H(δ1))−1(ℓδ1−ℓδ2)∥+∥((H(δ1))−1−(H(δ2))−1)ℓδ2∥

≤C∥(H(δ1))−1∥∥ℓδ1−ℓδ2∥+∥(H(δ1))−1−(H(δ2))−1∥∥ℓδ2∥

≤C∥(H(δ1))−1∥|δ1−δ2|

+C∥(H(δ1))−1∥∥(H(δ2))−1∥∥H(δ1)−H(δ2)∥.

As mentioned earlier, (H(δ):δ∈[0, δ]) is a Lipschitz family; in particular, it has bounded norm

and, consequently, the family of inverses is bounded as well (this is so since the coeﬃcients of

H(δ)are continuous in δ, and consequently they are bounded on [0, δ ]; the determinant across

the family, being the determinant a continuous functions of the coeﬃcients, is a continuous

function of a parameter ranging over a compact interval, and thus is bounded; again, since [0, δ]

is compact, the determinants are bounded away from zero; Cramer’s rule gives the conclusion).

Again, by Lipschitz continuity, the second term is bounded by C′|δ1−δ2|. We deduce that

(ε(δ):δ∈[0, δ]) is a Lipschitz family. So, we can ﬁnd ˆ

δ < δ such that for δ < ˆ

δthe vector ˆαis

still strictly positive. Since, by deﬁnition, Pd

j=1 ˆαi= 1, this proves our claim.

INTERACTING REPLICATOR DYNAMICS 25

As a last ingredient in the proof of the main Theorem of this section, we need an estimate of

Wasserstein’s distance between Dirichlet distributions in terms of the diﬀerences between their

parameters. Hereafter, for a positive vector a= (a1, . . . , ad)with Pd

i=1 ai= 1, the notation

Dastands for the absolutely continuous measure with Dirichlet(a)density.

Lemma 24. For Daand Dbas above:

Was1(Da, Db)≤

d−1

i=1

2(d−i)|ai−bi|.

We could not ﬁnd this result in the available literature, so we provide a full proof in the

Appendix.

Theorem 25. Let Abe a payoﬀ matrix satisfying [C1] and [C2], and F:P(Sd)7→ Rd⊗Rd

be a Was1-Lipschitz function with range included in the set of skew-symmetric matrices. Let

Φ(x) = ˜

Ax,Ψ(x) = Σ, and Υ(x, µ) = δF (µ)x. Then there exists δ0>0such that for every

δ < δ0the stochastic McKean-Vlasov replicator with the previous parameter admits a unique

invariant probability measure putting no mass on the boundary of the simplex.

Proof. For Aas above, let δbe the tolerance guaranteed by Lemma 23, and ﬁx δ < δ. For

ﬁxed µ∈ P(int (Sd)), let (X(µ,δ)

t:t≥0) be the unique solution of the “frozen” replicator:

dX(µ,δ)

t= ΠTΦ(X(µ,δ)

t)dt + ΠTΥ(X(µ,δ)

t, µ)dt + ΠTΨ(X(µ,δ)

t)dWt.(25)

By Theorem 3.6 of [22], there exists a vector α=α(µ)of strictly positive entries summing

up to 1such that Dα(µ)is invariant for X(µ,δ). Moreover, by Corollary 3.8 of [22], this is the

unique invariant distribution that does not charge the boundary. Let Tµ=Dα(µ). Moreover,

since Fis Lipschitz, from the proof of Lemma 23, it transpires that there exists a constant

C > 0such that:

∥αµ−αν∥1≤δCWas1(µ, ν ),

for every µ, ν ∈ P(Sd). But then, by Lemma 24:

Was1(Tµ, Tν)≤2(d−1)δC Was1(µ, ν).

We deduce that for δ < min{δ, (2(d−1)C)−1}:= δ0, the map µ7→ Tµis a contraction. By the

Contraction Mapping Theorem (see e.g. Theorem 5.7, page 138 of [4]), there exists a unique

µ∈ P(Sd)such that µ=Tµ. Clearly, µis invariant for the McKean-Vlasov replicator. Since

the range of Tis a subset of the Dirichlet family, the last claim follows.

6. Simulations

In Section 5, for the case of the McKean-Vlasov replicator with linear interaction function

and (small) Lipschitz dependence on the instantaneous measure (see Theorem 25), we have

stated the existence of a unique invariant probability measure putting no mass on the boundary.

In this section, via numerical simulations, we investigate whether this invariant probability

attracts the solutions of the system at large times. To that aim, we use the well-known Euler-

Maruyama scheme to simulate the associated N-replicator system, with step size h= 0.01.

26 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

Our objective is to study the behaviour of the solutions to (23) as t→ ∞, i.e. the long term

solutions to

dX(N;i)

t= ΠTΦ(X(N;i)

t)dt +1

j=1

ΠTΥ(X(N;i)

t, X(N;j)

t)dt + ΠTΨ(X(N;i)

t)dWt,

where X(N;i)

t= (x(N;i)

1(t), x(N;i)

2(t)) for Nlarge. Here Φ,Ψand Υ, correspond to those deﬁned

in (22). In particular, we simulated the following sets of parameters

(PS1) A=0.5 1

1 0.5, σ = 1, δ = 0.05, s = 0.5263,

and

(PS2) A=0.6 0.9

1 0.4, σ = 0.9487, δ = 0.04, s = 0.5814,

For each of the two sets of parameters, to test the eﬀect of the initial distribution, we perform

the simulations for initial conditions uniformly scattered in the unit interval, and also for

initial conditions uniformly scattered in the subset [0.2,0.4]. We called those distributions

uniform and lump initial conditions (UIC and LIC for short). For each of the four experiments

described, we approximate the probability density function from a normalized histogram of

the ﬁrst coordinate of each of the N-replicator {x(N;i)

1}N

i=1. This gives us some qualitative

understanding of the time convergence of the probability density.

The contour plot shown in Figure 1 (top panel) corresponds to the time evolution of such

histograms for PS1 and UIC. Results for LIC and PS2 can be found in the appendix Figure 3.

We notice that the histograms stabilize after approximately 3000-time steps (t= 30), with small

ﬂuctuations due to ﬁnite-size eﬀects. At the bottom panel of Figure 1 we see that the ﬁnal

empirical stationary pattern is similar to the Beta distribution stated at that Theorem 22, and

it seems to not depend on the initial distributions UIC or LIC (see appendix Figure 3). If such

convergence holds true, then as the time tgoes by, the position of the ﬁrst coordinate of each

of the Ninteracting replicators would constitute a sample from the unique Beta distribution.

In that case, the empirical mean 1

NPN

i=1 x(N;i)

1would be converging towards the mean of the

limit distribution which is, in this case, s/(s+ (1 −s)) = s. To test this conjecture we calculate

- for each parameter set and initial distribution - at each time step the empirical mean and

calculate the relative percentage error with respect to s. For any of the numerical experiments

performed, the results showed a relatively fast convergence of the empirical mean towards s,

with small ﬂuctuations around ssmaller than 2%.

To test that the convergence is not only restricted to the empirical mean but also to the

empirical distribution given by the simulated N-replicator system, we calculate the empirical

cumulative distribution function (ECDF) for each set of parameters and initial distributions.

In Figure 2 we plot the ECDF for diﬀerent times and each numerical experiment. In all cases,

the shape of the empirical CDF and the theoretical CDF given by Theorem 22, are relatively

similar. To further study this hypothesis, we performed at each time step tnlarger than

t= 30 an Anderson-Darling test to see whether the sample vector x(N;i)

1(tn)is drawn from

the theoretical Beta(s,1-s) distribution. For a 1% of signiﬁcance level, we see that there is no

statistical evidence to reject the null hypothesis for 64.91%, 72.47%, 76.32%, and 80.31% of

the times for the four PS1-UIC, PS1-LIC, PS2-UIC and PS2-LIC respectively.

As a consequence, it seems reasonable to conjecture that ergodicity must hold true. We

perform an additional statistical analysis, but now consider true independent trajectories of

INTERACTING REPLICATOR DYNAMICS 27

PS1-UIC

0 10 20 30 40 50 60 70 80 90 100

time

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

relative abundance probability

0.05

0.1

0.15

0.2

0.25

0.3

Figure 1. Simulations for the (S2)10000 interacting particle system. For the

N-replicator system, at each time step, we calculate the empirical histogram

from the ﬁrst coordinate of each replicator under probability normalization.

The top panel shows the time evolution of those empirical relative abundance

probabilities for a set of parameters and initial conditions PS1-UIC. The bot-

tom panel shows the histogram for N-replicator at t= 100 (bins) and for a

sample of 107random numbers drawn for the theoretical Beta(s,1-s) (red line).

N-replicators at a signiﬁcantly lower Nthan the previously used ones. To that end, we now

simulate 103times the 500-replicator dynamics with UIC for both sets parameters, saving the

trajectory of only one randomly selected replicator each time. We report a goodness-of-ﬁt test

regarding the theoretical stationary Beta distribution speciﬁed in Theorem 22 at times t= 40,

and t= 50. In all cases, we will see now that the laws of the trajectories are quite close to

the corresponding theoretical stationary Beta distribution (for more details see also appendix

Figure 4).

The implemented test is based on an L2-type test statistic proposed in [12, Theorem 1 -

Corollary 1]: based on a random sample Y1, ..., Yn, their test statistic is given by:

Tn(ˆan,ˆ

bn) = nZ1

0

j=1 (ˆan+ˆ

bn)Yj−ˆan1{Yj≥t}−tˆan(1 −t)ˆ

B(ˆan,ˆ

bn)

dt,

where B(·,·)is the Beta function, and ˆanand ˆ

bnare consistent estimators of the Beta pa-

rameters, such as moment estimators or maximum likelihood estimators (MLEs). However,

28 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

0 0.2 0.4 0.6 0.8 1

Relative abundance

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Empirical CDF

PS1-UIC

Empirical samples (t=10)

Empirical samples (t=25)

Empirical samples (t=50)

Empirical samples (t=75)

Empirical samples (t=100)

Theoretical CDF

0 0.2 0.4 0.6 0.8 1

Relative abundance

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Empirical CDF

PS1-LIC

Empirical samples (t=10)

Empirical samples (t=25)

Empirical samples (t=50)

Empirical samples (t=75)

Empirical samples (t=100)

Theoretical CDF

0 0.2 0.4 0.6 0.8 1

Relative abundance

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Empirical CDF

PS2-UIC

Empirical samples (t=10)

Empirical samples (t=25)

Empirical samples (t=50)

Empirical samples (t=75)

Empirical samples (t=100)

Theoretical CDF

0 0.2 0.4 0.6 0.8 1

Relative abundance

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Empirical CDF

PS2-LIC

Empirical samples (t=10)

Empirical samples (t=25)

Empirical samples (t=50)

Empirical samples (t=75)

Empirical samples (t=100)

Theoretical CDF

Figure 2. Empirical cumulative distribution function of the ﬁrst coordinate of

the N-replicator system. At the top (left) panels, we show the ﬁrst parameter

set (uniform initial distribution), and in the bottom (right) panels we show

the second parameter set (lump initial distribution). In all plots, we show the

ECDF for times t= 10,25,50,75 and t= 100, and the dashed black line is

the CDF associated with the theoretical Beta(s,1-s) distribution. Again, for

times larger than 25, the shape of the empirical and theoretical distribution

are qualitatively similar supporting the conjecture of ergodicity.

since in our simulations, the stationary distribution is completely speciﬁed by the resulting

parameterization given in Theorem 22, we can introduce such values, aand b, instead of its

estimates. Thus, for our null hypothesis, H0: “the law of the ﬁrst coordinate trajectories of

the one-dimensional McKean-Vlasov replicator (23) follows a Beta(a, b)for time t”, our test

statistic will become:

(26) Tn=Tn(a, b) = nZ1

0

j=1

((a+b)Yj−a)1{Yj≥t}−ta(1 −t)b

B(a, b)

dt,

where in our case a=sand b= 1 −a. The critical region to reject the H0is delimited by

1−αquantiles of the distribution of the above test statistic under H0. Since Tnis a distance,

the larger the observed values of Tnare, the further we will be from the null hypothesis. To

obtain an approximation of the distribution of Tnunder H0, we construct the density of B

values of Tn, each one obtained by an independent simulated random sample from a Beta(a, b).

In our case, we used B= 5000. We found that the 90% quantiles are 0.1056 (PS1) and 0.1119

INTERACTING REPLICATOR DYNAMICS 29

(PS2). For t= 40 the computed tests statistics are Tn= 0.0175 (PS1) and Tn= 0.0046

(PS2). For t= 50 the computed tests statistics are Tn= 0.0817 (PS1) and Tn= 0.0492 (PS2).

In all cases, the null hypothesis is not rejected. The corresponding R-code can be found at

https://github.com/leonardo-videla/beta_test.git.

Given these results, it seems reasonable to conjecture that ergodicity must hold, at least in

this case. We have no proof yet, and thus this remains an open question.

7. Discussion and outlook

This paper advances the modeling of biological phenomena, which we have proposed in

previous papers using the framework of Open System Dynamics ([33], [45], [15], [34], [56]). We

regard our contribution as a step forward in understanding how the living develops in continuous

interaction with the environment, including this time the interaction within communities of

living entities in what we envisioned as an ecology of primordial or early life represented by

a community of replicators. In particular, we were interested in assessing if neutrality and

associated ﬁtness equivalence could emerge in this system and under which conditions. To this

end, we generalized the stochastic replicator equation by adding a new component that aﬀects

the instantaneous ﬁtness, the idea being that this component accounts for the interaction of

one particular replicator population with others, alike ones, within a community in a mean-ﬁeld

regime.

We show that under suitable hypotheses, the asymptotic behavior of interacting replicator

dynamics may be approached by independent non-markovian, McKean-Vlasov particles (Sec-

tion 3). For the mean-ﬁeld replicator aggregates, we have shown that the conditions used in

[3], with some simple modiﬁcations, still apply to guarantee the coexistence (in the sense of

persistence) of large (but ﬁnite) systems of interacting communities (Section 4). In simple

cases (Section 5), we show the existence of a unique non-trivial invariant probability measure

for the McKean-Vlasov replicator. Indeed, in simple setups, we have been able to prove that

the Dirichlet family is stable (in the obvious sense that a system with Dirichlet invariant still

has a Dirichlet invariant when we introduce the dependence on the measure). This result,

which partially extends the work [22] and [29], admits further extensions, and it is a part of our

current research eﬀorts. Likewise, regarding the ergodicity of the McKean-Vlasov replicator,

we still have no concluding results (although we have shown in Section 6 that numerical exper-

iments give strong empirical support in this direction). This is an open question we continue

investigating.

We point out some salient features of this article. The ﬁrst one refers to the fact that, to the

best of our knowledge, this is the ﬁrst attempt to import the McKean-Vlasov machinery into

the evolutionary-game-theoretic realm. Also, the study of persistence for interacting, ﬁnite

aggregates of complex communities seems to be a novel feature. The use of the ﬁxed-point

argument to prove existence/uniqueness of invariant measures for McKean-Vlasov systems (see

Theorem 7) also seems to be a technique that has not been previously informed in this literature.

From the point of view of theoretical ecology, we note that the condition [C1] can be regarded

as the expression of neutrality and ﬁtness equivalence among types, as assumed in the neutral

ecological theory ([26]); in turn, in previous work ([33]) we have shown that this condition is

associated with Beta invariant probability measure. In light of the results of Section 5, this

observation provides theoretical support for the idea that neutrality, or ﬁtness equivalence,

is associated with the emergence of ecologies from simple, replicating molecules to complex

ecosystems, and is an emergent condition that is related to the persistence of the ecological

30 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

system. Thus, unlike traditional models in ecology, we did not assume neutrality but neutrality

naturally emerged as a condition for persistence.

There are many open directions to continue this work. The ﬁrst one was alluded to above,

namely: a proof for ergodicity of the McKean-Vlasov replicator is still missing. Secondly, we

were able to prove persistence for the ﬁnite-size particle system under small mean-ﬁeld and

noise terms. The problem of proving the analogous property for the McKean-Vlasov replicator

was not faced in this article, but we notice in advance that the techniques used in Section 4 (a

modiﬁcation of the well-known Foster-Lyapunov techniques as treated for example in the classic

articles of Meyn and Tweedie [38, 39, 40]) are unlikely to work in the non-linear case since they

are strongly grounded on the Markovian nature of the objects they are meant to be applied

to. Finally, another aspect that must be investigated is the extension of the model proposed

here to the setup of many types of interacting replicators. More speciﬁcally, we can think of a

myriad of entities of diﬀerent types, say Ni≥1replicators of type ifor i= 1, . . . , M , where M

is the ﬁxed number of types. A preliminary assessment of this extended setup shows that the

propagation-of-chaos (under suitable conditions) must still hold. The problem of persistence in

this multi-type interacting replicator seems to be a challenging issue, and we think it deserves

further investigation.

Acknowledgements This work was funded by project FONDECYT 1200925 The emergence of ecolo-

gies through metabolic cooperation and recursive organization, Centro de Modelamiento Matemático

(CMM), Grant FB210005, BASAL funds for centers of excellence from ANID-Chile, Exploration-ANID

13220168, Biological and quantum Open System Dynamics: evolution, innovation and mathematical

foundations, FONDECYT Iniciación project number 11240158-2024 Adaptive behavior in stochastic

population dynamics and non-linear Markov processes in ecoevolutionary modeling, and FONDECYT

Iniciación 11200436, Excitation and inhibition balance as a dynamical process, and “Programa de In-

serción Académica 2024 Vicerrectoría Académica y Prorrectoría de la Pontiﬁcia Universidad Católica

de Chile”. We thank Professor Rodrigo Plaza for his valuable help in the implementation of statistical

tests. We also thank the editor and an anonymous referee who provided us with valuable comments

that greatly improved our manuscript.

Declarations and conﬂict of interest

The authors declare that have no ﬁnancial or personal relationship with other people or

organizations that could inappropriately inﬂuence or bias the content of this paper. Also, the

authors declare that they have no conﬂicts of interest.

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INTERACTING REPLICATOR DYNAMICS 33

Appendix A. Appendix

A.1. Derivation of the replicator equation from a model of population dynamics.

Deﬁne G:Rd\ {0} 7→ Sdvia:

G(y) = ∥y∥−1

1y,

and consider the SDE:

dYt=Yt◦(˜

Φ(G(Yt))dt + Ψ(G(Yt))dWt).

Here, Yis to be understood as a Rd

+-valued process of abundances. The characteristic ˜

and Ψare suitable vector- and matrix-valued functions, respectively. Observe that in this

formulation, the instantaneous ﬁtness rates of the populations depend on Ytonly through the

relative proportions of the populations. Consider the case where Ψis a diagonal matrix (the

general case is analogous). Write Xt= (X(i)

t)i=1,...,d =G(Yt)=(gi(Yt))i=1,...,d, and let ˜

ϕi, ψi

the components of ˜

Φand the diagonal elements of Ψ, respectively. We easily compute:

∂kgi(y) = δik

∥y∥1

−yi

∥y∥2.

∂kkgi(y) = 2 −δik

∥y∥2+yi

∥y∥3.

Thus, Itô’s formula gives:

dX(i)

t= X

Y(k)

t˜

ϕk(Xt) δik

∥Yt∥−Y(i)

∥Yt∥2!

(ψk(Xt)Y(k)

t)2 Y(i)

∥Yt∥3−δik

∥Yt∥2!!dt

Y(k)

t˜

ψk(Xt) δik

∥Yt∥−Y(i)

∥Yt∥2!dW (i)

= X

X(k)

t˜

ϕk(Xt)δik −X(i)

t

(ψk(Xt)X(k)

t)2X(i)

t−δik!dt

X(k)

t˜

ψk(Xt)δik −X(i)

tdW (i)

Hence, we see that Xsatisﬁes the equation (4) with:

Φ(x) = ˜

Φ(x)−ΨΨ⊤(x)x,

A.2. Proof of Proposition 8. Proposition 2: Under Assumptions 1 and 3, equation (14)

has a unique pathwise (hence strong) solution.

Proof. Since b= ΠTΦ + ΠTΥis Lipschitz, in particular, it is Dini-continuous. On the

other hand, Sdis compact, and consequently the drift and diﬀusion coeﬃcients are bounded.

34 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

Finally, Assumption 4.2 implies that for every x∈Sd, we have ∥ΠTΨ(x)⊤y∥ = 0 whenever

y∈ T ,y= 0. But then:

inf

x∈Sd,y∈T ,∥y∥=1 y⊤ΠTΨ(x)ΠTΨ(x)⊤y>0,

since the expression inside the inﬁmum is a continuous function of x,yand it is taken on a

compact set. Theorem 1 of [58] yields the claim.

A.3. Proof of Lemma 11. Lemma 1: For every Ξ0∈int (Sd):

P(MV )

Ξ0min

i=1,...,d Ξ(i)

t>0for all t≥0= 1.

An analogous result holds for the N- particle system, namely: for every 

x0∈int ((Sd)N),

P(N)



x0min

j=1,...,N ∥X(N;j)

t∥>0for all t≥0= 1.

In words: with probability 1, in ﬁnite time there is no absorption on the boundary of Sd(for

the McKean-Vlasov replicator) nor on the boundary of (Sd)N(for the N-particle system).

Proof. We prove the result for the McKean-Vlasov system, since the proof for the N-replicator

is completely analogous. For any k∈N, set:

Vk:= {x∈Sd: min

ixi< e−k},

and consider the function Uk:Sd\Vk7→ R+given by:

Uk(x) := −X

ln(xi).

Then Ukis uniformly continuous for each k, and indeed it is a C∞function in its domain. For

µ∈ P(Sd)consider the operator Lµacting on a smooth function φ∈ C2(Sd)as:

Lµφ(x) = ⟨b(x, µ),∇φ(x)⟩+1

2Trace(ΠTΨ(x)⊤∇∇⊤φ(x)ΠTΨ(x)),(27)

where µtsolves the non-linear Fokker-Planck equation:

dt ⟨µt, φ⟩=⟨µt, Lµtφ⟩,(28)

for every φ∈ C2(Sd)([36], [5], [54]).

So, for x∈Sd\Vkand µ∈ P(Sd), we can compute:

LµUk(x) = −X

iΦi(x)− ⟨x,Φ(x)⟩−X

iΥi(x, µ)− ⟨x,Υ(x, µ)⟩+1

Tr(AΨA⊤

Ψ)(x),

and we observe that LµUkis, for each k, a uniformly continuous function on Sd. Moreover,

there exists an absolute constant, say M, such that LµUk< M for every k. Let:

τk:= inf{t≥0:Ξt∈Vk}.

INTERACTING REPLICATOR DYNAMICS 35

Now, ﬁx Ξ0∈int (Sd). There exists k0such that Ξ0∈Sd\Vkfor every k≥k0. It is known that

for k≥k0, the process (Uk(Ξt∧τk)−Uk(Ξ0)−Rt

0LLaw(Ξs∧τk)(Ξs∧τk)ds :t≥0) is a martingale.

Thus, for every N∈N:

EΞ0(Uk(Ξτk∧N)) = Uk(Ξ0) + EΞ0 Zτk∧N

LLaw(Ξs∧τk)Uk(Ξs∧τk)ds!

≤Uk(Ξ0) + MN.(29)

Let:

ΩN:= \

k≥k0

{ω∈Ω : τk< N},

and deﬁne ϵN:= PΞ0(ΩN). We have:

EΞ0(Uk(Ξτk∧N)) ≥EΞ0(Uk(Ξτk)·1ΩN)

≥kϵN,

by path-continuity. Using (29), we deduce that for ﬁxed Nand every k≥k0:

kϵN≤Uk(Ξ0) + M N ≤C+M N.

for some constant Cthat depends on Ξ0but not on k. Letting kgo to inﬁnity, we obtain

ϵN= 0 for every N, and consequently:

PΞ0(lim

kτk<∞) = PΞ0 [

N∈N

ΩN!= 0,

and this proves our claim.

A.4. Proof of Lemma 24. Lemma 5: Let Daand Dbbe the Dirichlet distributions with

paramters aand b, both summing up to 1. Then:

Was1(Da, Db)≤

d−1

i=1

2(d−i)|ai−bi|.

Proof. By dominated convergence, it is direct that the map a7→ Dais weakly continuous, i.e.

continuous when we endow the set of probability measures on the simplex with the topology of

weak convergence of probability measures. Thus, it suﬃces to consider the case where a,bhave

rational entries. So, for a certain natural number Nthere exists natural numbers m1, . . . , md

and n1, . . . , ndsuch that for ai=mi/N , bi=ni/N. Put Mi=Pj≤imj, and analogously

Ni=Pj≤inj. Let (Gj:j= 1, . . . , N )be an i.i.d. family of Gamma(1/N, 1) random variables.

Put:

Vi:=

j=Mi−1+1

Gj;Wi:=

j=Ni−1+1

GjZ:=

j=1

Gj.

Then X= (Xi)d

i=1 := (Vi/Z)d

i=1 and Y= (Yi)d

i=1 := (Wi/Z)d

i=1 have law Daand Dbrespec-

tively. We say that the ivariates overlap if [Mi−1+ 1, Mi]∩[Ni−1+ 1, Ni]=∅. Observe that

if the ivariates overlap, then |Vi−Wi|is bounded by the sum of two independent Gamma-

distributed random variables of parameters (|Ni−1−Mi−1|

N,1) and (|Ni−Mi|

N,1). If the ivariates

36 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

do not overlap, then |Vi−Wi|is bounded by the sum of two (not independent) Gamma-

distributed random variables of parameters (|Ni−1−Mi−1|

N,1) and (|Ni−Mi|

N,1). Denote by Ri

and Sithese random variables. We have:

E(|Xi−Yi|)≤E(Ri/Z) + E(Si/Z)

=|Ni−1−Mi−1|

N+|Ni−Mi|

N≤

i−1

j=1

|aj−bj|+

j=1

|aj−bj|.

Finally, by considering the ﬁrst and last variates separately, we obtain:

Was1(Da, Db)≤

i=1

E(|Xi−Yi|)

≤2(d−1)|a1−b1|+ 2(d−2)|a2−b2|+. . .

+. . . + 2 ×2|ad−2−bd−2|+ 2|ad−1−bd−1|,

and this is the claim. □

A.5. Supplementary ﬁgures. In this section, we present some supplementary results for the

model studied in Section 6. Figure 3, shows the time evolution of the N-replicator for as second

set of initial conditions (LIC) and also for the second set of parameters (PS2-UIC and PS2-

LIC) along with the comparative between the ﬁnal histogram for the N-replicator dynamics

and the Beta theoretical distribution of Theorem 22. These numerical results support the claim

that the long-term convergence and the shape of the steady state do not depend on the initial

conditions chosen or the used parameter set.

The second supplementary ﬁgure shows the result for the goodness-of-ﬁt for diﬀerent param-

eter sets at times t= 40 and t= 50. This is a graphical representation of the results explained

in the main text, along with the 90%, 95%, and 99% quantiles. We can see that in none of the

cases we can reject the null hypothesis, supporting the conjecture that as the system evolves

in time, the solutions converge to the stationary distribution given by Theorem 22

1Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencia, Universidad

de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile

Email address:leonardo.videla@usach.cl

2Instituto de Estadística, Universidad de Valparaíso, Valparaíso, Chile

Email address:mauricio.tejo@uv.cl

3Facultad de Ciencias Biológicas, Pontificia Universidad Católica de Chile, Santiago, Chile

Email address:cquininao@uc.cl

Email address:pmarquet@puc.bio.cl

4Centro de Modelamiento Matemático (CMM), Universidad de Chile-IRL 2807 CNRS Beauchef

851, Santiago, Chile

4The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe NM 87501, USA

4Instituto de Sistemas Complejos de Valparaíso, Subida Artillería 470, Valparaíso, Chile

5Instituto de Ingeniería Matemática, Universidad de Valparaíso

Email address:rolando.rebolledo@uv.cl

INTERACTING REPLICATOR DYNAMICS 37

PS1-LIC

0 10 20 30 40 50 60 70 80 90 100

time

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

relative abundance probability

0.05

0.1

0.15

0.2

0.25

0.3

PS2-UIC

0 10 20 30 40 50 60 70 80 90 100

time

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

relative abundance probability

0.05

0.1

0.15

0.2

0.25

0.3

PS2-LIC

0 10 20 30 40 50 60 70 80 90 100

time

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

relative abundance probability

0.05

0.1

0.15

0.2

0.25

0.3

Figure 3. Simulations for the (S2)10000 interacting particle system. For the

N-replicator system, at each time step, we calculate the empirical histogram

from the ﬁrst coordinate of each replicator under probability normalization.

From top to bottom, the plot shows the time evolution of the empirical relative

abundance probability for PS1-LIC, PS2-UIC, and PS2-LIC.

38 VIDELA, TEJO, QUIÑINAO, MARQUET, AND REBOLLEDO

Tn obs = 0.01756

quantile 99% = 0.24979

quantile 95% = 0.15579

quantile 90% = 0.10562

0.0 0.1 0.2 0.3

Probability density

Tn obs = 0.08174

quantile 99% = 0.24979

quantile 95% = 0.15579

quantile 90% = 0.10562

0.0 0.1 0.2 0.3

Probability density

Tn obs = 0.00465

quantile 99% = 0.23385

quantile 95% = 0.14588

quantile 90% = 0.11191

0.0 0.1 0.2 0.3

Probability density

Tn obs = 0.04922

quantile 99% = 0.23385

quantile 95% = 0.14588

quantile 90% = 0.11191

0.0 0.1 0.2 0.3

Probability density

Figure 4. Graphical hypothesis testing for contrasting H0: “the law of the

ﬁrst coordinate trajectories of (23) follows a Beta(a, b)at time t”. The sta-

tistical test is based on the work of [12]. The upper panels show the results

for PS1 and the lower panels are the corresponding results for PS2, with their

corresponding theoretical sgiven in the main text. The left column corre-

sponds to time t= 40 and the right one to time t= 50. The test is based

on the statistic Tn, constructed according to a L2distance given in Equation

(26), whose observed values are shown by the red dashed lines. We can notice

that, in all the cases, they are much lower than the quantiles that delimit the

rejecting region at each indicated (1 −α)×100 percentile, with α= 0.1(yel-

low dashed lines), α= 0.05 (purple dashed lines) and α= 0.01 (green dashed

lines), regarding the theoretical probability density of the statistic Tnunder

H0, approximated by simulations.

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