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Scaling limits of population and evolution processes in random environment

We propose a general method for investigating scaling limits of ﬁnite dimensional Markov chains to diffusions with jumps. The results of tightness, identiﬁcation and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively deﬁned as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.


Introduction
This work is a contribution to the study of scaling limits of discrete population models.
The parameter N ∈ N scales the population sizes. The population processes (Z N n : n ∈ N) are N d -valued Markov chains inductively defined by where F N is a function giving the number of individual events. For each z, e, N , (L N i,n (z, e) : i, n ≥ 1) is a family of independent identically distributed random variables and E N n is a R d -random variable describing the environment at generation n. This class of processes includes well known processes in population dynamics and population genetics. In particular, Galton-Watson processes correspond to the case when F N (z) = z and L N i,n = L N , i.e it does not depend on (z, e), while Wright-Fisher processes are obtained when F N (z) = N and L N i,n (z, e) are Bernoulli random variables with parameter z/N . More generally, these population models can also take into account the effect of random environment and include many additional ecological forces such as competition, cooperation and sexual reproduction.
We are interested in the convergence of the sequence of processes (Z N [v N t] /N : t ≥ 0), as N tends to infinity, v N being a time scale tending to infinity with N . We provide a unified framework adapted to population models and characterize this convergence through the asymptotic properties relying on v N , F N and L N . Many works have been devoted to the approximation of Markov processes. They are essentially based on tightness arguments and identification of the martingale problem, see for example [12,19]. Unfortunately, this general method does not satisfactorily apply to our framework since the required assumptions are difficult to check. Applying for instance this method to the classical Galton-Watson framework seems to lead to moment assumptions. However, it is well known from the works of Lamperti [24,25] and Grimvall [16] that the finite dimensional convergence of the renormalized processes (Z N [v N t] /N : t ≥ 0) with a time scale v N → ∞ is equivalent to the convergence of a characteristic triplet associated with (v N , L N ) when N tends to infinity. In this case, the sequence of processes (Z N [v N t] /N : t ≥ 0) converges as N → ∞ to a Continuous State Branching Process (CSBP) defined as the unique strong solution of a Stochastic Differential Equation (SDE). The parameters of this SDE are given by the limiting characteristic triplet of (v N , L N ). Note that the proof is based on the branching property, using either the Laplace exponent [16], or the relation with the convergence of the associated random walk to a spectrally positive Lévy process via a Lamperti time change (cf. [25] [8]). Lamperti also introduced a powerful transform in the stable framework, see e.g. [25] and [29] and [5]. Other time changes have been successfully used to obtain scaling limits of discrete processes, in particular for some diffusion approximations, see for instance [21] for branching processes in random environment, [9] for branching processes with immigration and [33] for controlled branching processes, amongst others. Such time change techniques seem essentially restricted to branching processes or stable processes or diffusion approximations. In our work, we are interested in the convergence in law of discrete Markov processes (Z N ) N which do not enjoy the branching property and may jump in the limit. The limiting processes may even be explosive and are not necessarily stable.
It is well known that the law of the process (Z N [v N .] /N ) is determined by its initial law and the family of functions for H continuous and bounded on R d . Moreover, the asymptotic behavior of G N x (H) as N → ∞ for a large enough class of functions H captures the convergence of the processes. In such discrete cases, Jacod and Shiryaev in [19,II.3,IX] prove that the tightness and the identification are deduced from the convergence as N → ∞ of the characteristics of the semimartingales Scaling limits of population and evolution processes processes. This is why we prove that the functions H can be chosen differently, belonging to some (rich enough) functional space H, dense in the set of regular functions vanishing at zero for a norm equivalent to The choice of the space H depends on the assumptions on the model. In our applications, we exploit the independence property of the variables (L N i,. (., .) : i ≥ 1). The characterization of the law by the Laplace exponent is then used at the level of conditional increments, which is well adapted to the sum of independent non-negative random variables.
Our main motivations were the famous frameworks of population genetics and population dynamics. The efficiency of our method can be seen in the generalizations obtained for the approximation of Wright-Fisher and Galton-Watson chains. We first study a Wright-Fisher model with selection in a random environment impacting the selective advantage. The environments are assumed to be independent and identically distributed and the associated random walk converges to a Lévy process. We obtain the convergence of the joint law of the processes and random walks, by using the functional space We thus derive a diffusion with jumps in random environment, which generalizes the Wright-Fisher diffusion with selection and takes into account small random fluctuations and punctual dramatic advantages in the selective effects.
The second application focuses on generalized Galton-Watson processes with reproduction law that is both density and environment dependent. We obtain a result of convergence in law to the so called continuous state branching process with interaction in Lévy environment henceforth called BPILE (introduced in [31,17]). These processes have unbounded characteristics and the result is deduced from the convergence of the compactified processes exp(−Z N k /N ).
To deal with the joint laws of the latter and the environment random walk, we use the space of functions from [−1, 1] × (−1, ∞) to R defined by Our results extend the criterion for the convergence of a sequence of Galton-Watson processes as well as the results we know in random environment [21,4] or with interactions [11,32]. They are further applied to Galton-Watson processes with cooperation and to branching processes with logistic growth in random environment.
The paper is organized as follows. In Section 2, we give general results for the tightness, the identification and the convergence in law of a scaled Markov process to a diffusion with jumps in R d . The functional space H is introduced in Section 2.1.
Tightness and identification results are stated in Section 2.2 by assuming the uniform convergence and boundedness of characteristics G N . (H) for any H ∈ H. Convergence requires an additional uniqueness assumption, obtained from pathwise uniqueness in the applications, using standard techniques for non-negative SDE [18,14]. Proofs of these general statements are given in Section 2.3. In Section 3, we apply our method to a Wright-Fisher model with selection in a random environment. We obtain in a suitable scaling limit a Wright-Fisher diffusion in random environment for which we prove uniqueness of solution. In Section 4 (Sections 4.1, 4.2, 4.3), we apply our method to Galton-Watson processes with reproduction law both density dependent and environment dependent. Section 4.4 is devoted to explosive CSBP with interaction in random environment. In particular, we consider Galton-Watson processes with cooperative effects. Section 4.5 is dedicated to the conservative case and an application to Galton-Watson processes with logistic competition and small environmental fluctuations is studied. Finally, we expect the method to be applied in various contexts, in particular for structured populations models with sexual reproduction, competition or cooperation, see Section 5.
Notation. For x ∈ R d , we denote by |x| the euclidian norm of x.
The functional norms are denoted by . . In particular the sup norm of a bounded function f on a set U is denoted by f U ,∞ . The sets C b (U, R) and C c (U, R) denote the spaces of continuous real functions defined on U respectively bounded and with compact support.
For any U subset of R d containing a neighborhood of 0, we define U * as U \ {0}.

A criterion for tightness and convergence in law
Let X be a Borel subset of R d and U be a closed subset of R d containing a neighborhood of 0.
Let us introduce a scaling parameter N ≥ 1. For any N , we consider a discrete time X -valued Markov chain (X N k : k ∈ N) satisfying for any k ≥ 0, where for any N ∈ N, (F N x , x ∈ X ) denotes a measurable family of X -valued random variables such that for any x ∈ X , the random variable F N x − x takes values in U. The natural filtration of the process X N is denoted by (F N k ) k . Note that the increments X N k+1 − X N k take values in U.
Our aim is the characterization of the convergence in law of the sequence of processes is a given sequence of positive real numbers going to infinity when N tends to infinity. It is based on the criteria for tightness and identification of semimartingales by use of characteristics given in [19,IX], which consists in studying the asymptotic behavior of for real valued bounded measurable functions H defined on U.

Hypothesis (H0)
We first assume that the family of random variables This hypothesis avoids to get infinite jumps in the limit. We will see in the examples that this condition affects both the population and the environment dynamics.
Under (H0), we will prove that the study of (2.1) can be reduced to a rich enough and tractable subclass H of functions H. The choice of H depends on the particular models and is illustrated in the examples.

Specific and truncation functions
We consider a closed subset U of R d containing a neighborhood of 0 and introduce the functional space The functions of C 2 b,0 can be decomposed in a similar way with respect to any smooth function which behaves like the identity at 0, as stated in the next lemma. The proof uses the uniqueness of the second order Taylor expansion in a neighborhood of 0.
is a continuous and bounded function and α f i (H), β f i,j (H), i, j = 1 · · · d are real coefficients and β f is a symmetric matrix.
We introduce • the specific function h which satisfies • the truncation function h 0 , as defined in [19] : Obviously, h i 0 h j 0 ∈ C 2 b,0 for any i, j = 1, . . . , d.
Note that in general a specific function is not a truncation function since it may not coincide with the identity function in a neighborhood of 0. Its choice will be driven by the processes we are considering. We will give different choices of functions h in the next sections, for instance h(x) = 1 − exp(−x) on [−1, ∞) when d = 1. These specific functions will play a crucial role in the whole paper.

General statements
We introduce a functional space H containing the coordinates of the specific function h and their square products and which "generates" the continuous functions with compact support in U in the sense described below. The space H will be a convergence determining class.

Hypotheses (H1)
There exists a functional space H such that For any g ∈ C c (U, R) with g(0) = 0, there exists a sequence (g n ) n ∈ C 2 b,0 such that lim n→∞ g − g n ∞,U = 0 and |h| 2 g n ∈ V ect(H).
3. There exists a family of real numbers (G x (H); x ∈ X , H ∈ H) such that for any H ∈ H,

Remark 2.2.
In the examples of the next sections, (H1.2) is proved with the use of the locally compact version of the Stone-Weierstrass Theorem. We refer to the Appendix for a precise statement.
Contrary to the "convergence determining class" of [19], the functions of H will not be vanishing (or o(u 2 )) in a neighborhood of 0.
Hypothesis (H1.3) implies that the map x ∈ X → G x (H) is measurable and bounded for any H ∈ H.
We first obtain a tightness result based on the space H of test functions. Theorem 2.3. Assume that the sequence (X N 0 ) N is tight in X and that (H0) and (H1) hold. Then the sequence of processes (X N The next hypothesis (H2) in addition to (H1) is sufficient to get the identification of the limiting values by their semimartingale characteristics, and then their representation as solutions of a stochastic differential equation.

Hypotheses (H2)
1. For any H ∈ H, the map x ∈ X → G x (H) is continuous and extendable by continuity to X .

2.
For any x ∈ X and any H ∈ H, The elements (b, σ, V, µ, K) will be specified in the applications.
Theorem 2.4. If the sequence (X N 0 ) N is tight in X and (H0), (H1), (H2) hold then any limiting value of (X N [v N .] , N ∈ N) is a semimartingale solution of the stochastic differential system where X 0 ∈ X and B is a d-dimensional Brownian motion and N is a Poisson point measure on R + × V with intensity dsµ(dv). Moreover X 0 , B , N are independent andÑ is the compensated martingale measure of N .
To obtain the convergence in law of the sequence of processes (X N Hypothesis (H3) The law of the initial condition X 0 ∈ X being given, the uniqueness in law of the solution of (2.5) holds in D([0, ∞), X ).
We are now in position to state the convergence result.

Proofs
From now on, we assume that hypotheses (H0) and (H1) hold. We recall that In the proofs, we use the space R b of continuous and bounded functions which are small enough close to 0 : Using Lemma 2.1 and (H1.1), we have We work with the norm defined for H ∈ R b such that sup u∈U * |H(u)|/|h(u)| 2 < +∞. In that case, the positivity and linearity of G N x for all x ∈ X and N ≥ 1 imply that

Proof of Theorem 2.3
We first extend the assumptions (H1.3) to C 2 b,0 in order to prove the tightness. We note that (H1.3i) and (H1.3ii) extend immediately to H ∈ V ect(H) by linearity of H → G N x (H) for any x ∈ X and N ≥ 1.  Proof. Using (2.6) and linearity, we only have to prove the extension to R b . Let us first prove the result for the compactly supported functions of R b . We consider H ∈ R b with compact support and show that the sequence (G N x (H)) N converges when N tends to infinity. The function H/|h| 2 defined on U * can be extended to a continuous function g on U with compact support and g(0) = 0. Then by (H1.2), there exists a sequence (g n ) n of functions of C 2 b,0 uniformly converging to g and such that H n = |h| 2 g n ∈ V ect(H). Since H = |h| 2 g, H n − H h → 0 when n → ∞. Moreover the sequence G N . (H n ) N converges to G . (H n ) when N tends to infinity for any fixed n and uniformly on X . Let us now consider two integers m and n. Equation (2.7) tells us that and letting N go to infinity, we obtain that (G x (H n )) n is a Cauchy sequence. Then it converges to a limit denoted by G x (H), which satisfies sup X |G . (H)| < ∞. Moreover an appropriate choice of n and then of N allows us to upper bound the left hand side by any > 0 and this ensures Let us now consider H ∈ R b . We introduce a non-decreasing sequence (ϕ n ) n ∈ For x ∈ X and N ≥ 1, where C n → 0 as n → ∞ by (H0). Letting N tend to infinity, we obtain that for any x ∈ X , the sequence (G x (Hϕ n )) n is Cauchy and converges to some real number G x (H). We now prove that a σ-finite measure can be associated to G x for each x ∈ X . It describes the jumps of the limiting process.  For any x ∈ X , G x is then extended by (2.9) to any measurable and bounded function H on (R d ) * such that H(u) = o(|u| 2 ). Moreover  Proof. For any x ∈ R d and H ∈ C c (U * , R), the map H → G x (H) is a positive linear operator. Adding that U * is locally compact, Riesz Theorem leads to the existence of a σ-finite measure µ x on U * such that for any H ∈ C c (U * , R), G x (H) = U * H(u)µ x (du). The extension of this identity to any H ∈ R b follows again from an approximation procedure, using ϕ n defined in the proof of Lemma 2.6. Indeed, on the one hand monotone convergence ensures that U * Hϕ n µ x goes to U * Hµ x . On the other hand, |G x (Hϕ n )−G x (H)| ≤ C n H ∞ goes to 0. Finally (2.10) comes from (H0) with a monotone approximation of 1 B(0,b) c by elements of R b and the convergence of G N to G.
We now prove the convergence of conditional increments functionals, defined for any function H ∈ C 0 b,2 and t > 0 by where the last identity follows from the Markov property.
Proposition 2.8. For any function H ∈ C 0 b,2 and t > 0, Proof. Using (2.8), we have and the conclusion follows from (H1.3i), which holds for H thanks to Lemma 2.6.
We define on the canonical space D([0, ∞), X ) a triplet which characterizes the limiting values of the sequence (X N [v N .] , N ∈ N). Using the measurability and boundedness of x → G x (f ) for x ∈ X and f ∈ C 2 b,0 and the truncation function h 0 introduced in (2.3), we define for any ω = (ω s , for any H ∈ C b (R d , R) such that H(u) = o(|u| 2 ). The last identity comes from (2.9). As in Chapters II. 2 & 3 in [19] adapted to the state space X (instead of R d ), the characteristic triplet associated with the semimartingale X N is given for i, j ∈ {1, . . . , d} by where U N k = X N k − X N k−1 and H is a continuous bounded function on R d vanishing in a neighborhood of 0. Proposition 2.8 implies the convergence of the characteristics, as stated in the next proposition. Proposition 2.9. For any T > 0 and any i, j = 1, · · · , d and any H ∈ C b (U, R) equal to 0 in some neighborhood of 0, we have the following almost-sure convergences Proof. From Proposition 2.8, we immediately obtain the first and last convergences and Hence the second term in C N,ij t tends to 0 as N → ∞, which yields the result.
We are now in position to provide a proof of Theorem 2.3. In order to apply Theorem 3.9 IX p543 in [19] and get the tightness, we need to check the strong majoration hypothesis and the condition on big jumps required in its statement.
Second, to control the big jumps, we use the fact that ν t (., which tends to 0 as b tends to infinity from (2.10). We thus obtain The tightness of (X N [v N .] , N ∈ N) follows from (2.14)-(2.18) and from the tightness of the initial condition, by an application of the forementioned theorem in [19].

Proofs of Theorems 2.4 and 2.5
Let us now assume the additional Hypothesis (H2). We wish to identify the limiting values of (X N [v N .] , N ∈ N) as solutions of the stochastic differential system (2.5). We first need to extend continuously the limiting characteristic triplet to the boundary. Lemma 2.10. (i) For any H ∈ R b , the map x ∈ X → G x (H) is continuous and extendable by continuity to X . Moreover and where k, µ are defined in (H2) and Proof. Let H ∈ R b . Using the sequences ϕ n and (H n ) n defined in the proof of Lemma 2.6 and approximating ϕ n H for . h by H n ∈ V ect(H) ∩ R b as in the proof of Lemma 2.7, we obtain sup x∈X ,N ≥1 with compact support. We note that H = |h| 2 g with g ∈ C c (U, R). By (H1.2), the function g is uniformly approximated by a sequence g n such that |h| 2 g n ∈ V ect(H) ∩ R b . The identity (2.4) implies that (2.20) holds for any |h| 2 g n and We let n tend to infinity in both terms using (2.8) and the assumption The extension to R b follows again from a monotone approximation by the compactly supported functions Hϕ n , which ends the proof.
This lemma allows us to extend the definitions of the characteristics and the iden-

The discrete model
Let us consider the framework of the Wright-Fisher model: at each generation, the alleles of a fixed size population are sampled from the previous generation. We consider a population of N individuals characterized by some allele. The number of individuals carrying this allele is a process (Z N k , k ∈ N) whose dynamics depends on the environment. When N ≥ 1 is fixed, we consider the coupled process describing the discrete time dynamics of the population process and the environment process. It is is a finite random variable, (E N k ) k are independent and identically distributed with values in (−1, +∞) and the family of random variables and extends the classical Wright Fisher model with rare selection to random environments.
Following [19] [chap.VII Corollary 3.6,p.415], we state an assumption for the random walk S N [N.] to converge in law to a Lévy process with characteristics (α E , β E , ν E ). Let us consider a truncation function h E defined on (−1, +∞), i.e. continuous and bounded and satisfying h E (w) = w in a neighborhood of 0. For convenience, we also assume that h E (w) = 0 for any w = 0. Assumption A. There exist α E ∈ R, σ E ≥ 0 and a measure ν E on (−1, +∞) satisfying for any f vanishing in a neighborhood of 0, continuous and bounded.
The small fluctuations of the environment are given by σ E , while the dramatic events are given by the jump measure ν E . Negative jumps will correspond to dramatic disadvantages of allele A and an usual set of selection coefficient is (−1, ∞), as illustrated in Section 3.4.
The limiting environment process Y can thus be defined by where B E is a Brownian motion and N E is a Poisson point measure on R + × (−1, +∞) independent of B E with intensity measure ν E . By construction, this Lévy process has jumps larger than −1.
Let us first prove a consequence of Assumption A which will be needed in the proof of the next theorem.
The first two terms converge uniformly as N → ∞ by a direct application of Assumption A. Moreover the last part of Assumption A can be extended to any continuous function f (w) = o(w 2 ) using a monotone approximation of f by functions vanishing in a neighborhood of 0. Then the last term converges for fixed z and it remains to prove that the convergence is uniform on [0, 1]. First, let us consider a compact subset and its first derivative with respect to z are well defined and extendable by continuity to [0, 1] × (−1, ∞). Thus the derivative of g(z, w)/h E (w) 2 with respect to z is bounded on K. As (N E(h E (E N ) 2 )) N is bounded by the second part of Assumption A, there exists C > 0 such that for any N ≥ 1, Moreover, since all functions involved in the definition of g are bounded, there exists C > 0 such that and by the last part of Assumption A, Combining the last two inequalities, we obtain that the family of functions (N E( g(., E N ))) N is uniformly equicontinuous on [0, 1] and the convergence is uniform by Ascoli Theorem.
We can now generalize the classical convergence in law to the Wright-Fisher diffusion with selection to i.i.d. environments.

Tightness and identification
We are interested in the asymptotic behavior of the Markov chain For the statement, we introduce the drift coefficient inherited from the fluctuations of the environment: and any limiting value of this sequence is solution of the following stochastic differential equation Proof. We apply our results to the Markov chain The state space of the random variables F N x − x is U = [−1, 1] × (−1, +∞). We first prove that (H0), (H1) and (H2) are satisfied with v N = N .
(i) Let us first check (H0). We take b > 0 and consider is satisfied.
Scaling limits of population and evolution processes The space H is the subset of real functions on U defined as We can apply the local Stone-Weierstrass Theorem to the algebra V ect(H) ∩ C 0 (U * ), U * = U \ {0, 0} being a locally compact Hausdorff space (see Appendix 6.4). This algebra in dense in C 0 (U * ) and then any function in C c (U) vanishing at zero is the uniform limit of elements of V ect(H). Moreover V ect(H) is stable by multiplication by |h| 2 . We deduce that (H1.2) is satisfied, while (H1.1) is obvious.
Let us now prove that (H1.3) is satisfied. We need to study the limit of The following Taylor expansion gives By expansion, we deduce that uniformly for x = (z, y) ∈ [0, 1] × R. Then (H1.3i) is satisfied and for any x ∈ X , 0). The assumptions on the function p allow us to conclude that (H1.3ii) is also satisfied.
(iii) We now check (H2). The continuity of G . on [0, 1] × R comes from the regularity of p, from the integrability assumption on ν E and from Lebesgue's Theorem (by Assumption A).

Now, let us introduce the truncation function defined on
With the notation of Section 2, recall that h 1 0 (u, w) = u and h 2 0 (u, w) = h E (w).
With the notation of Lemma 2.1 we have Then (3.6) can be written as Thus (H2) holds for any H = H k, ∈ H.
We can now apply Theorems 2.3 and 2.4 for tightness and identification and conclude.

Pathwise uniqueness and convergence in law
To get the uniqueness for (3.3), we will use the pathwise uniqueness result from Li-Pu [28]. The monotonicity assumption on p is natural regarding the model since the more individuals carry an allele in a generation, the more this allele should be carried in the next generation.
Proof. In order to apply Theorem 2.5, let us first show that (H3) holds. The pathwise uniqueness of the process Y is well known. Let us focus on the first equation of (3.3) and prove the pathwise uniqueness of the process Z. First, we rewrite the SDE for Z as We are in the conditions of application of Theorem 3.2 in [28]. Indeed, we observe first that b 1 is Lipschitz since p ∈ C 3 ([0, 1], (−1, ∞)) and We remark also that the Brownian part of (3.3) writes with W Brownian motion since B D and B E are two independent Brownian motions. We easily prove that for any z 1 , for any z 1 , z 2 ∈ [0, 1]. Then all the required assumptions for [28] Theorem 3.2 are satisfied and we get the pathwise uniqueness of the solution of (3.3).

Example
We consider the following main example where the environment w acts as the selection factor. By construction, this selection coefficient w is larger than −1. The particular case when the environment is non-random, i.e. E N k = s/N a.s. for some real number s ∈ (−1, +∞), yields the classical Wright-Fisher process with weak selection. It is well known that in this case, the processes (Z N [N.] ) N converge in law to the Wright-Fisher diffusion with selection coefficient s whose equation is given by Here we generalize this result for random independent identically distributed environments.
First, we observe that Under Assumption A, we can apply Corollary 3.3 to obtain the proposition stated below.
In particular if σ E = 0 and ν E = 0, we recover the classical Wright-Fisher diffusion with deterministic selection α E . This extension allows us to consider small random fluctuations (asymptotically Brownian) and punctual dramatic advantage of the selective effects.

Continuous state branching process with interaction in Lévy environment
In this section, we are interested in approximations of large population dynamics with random environment and interaction. We generalize in different directions the classical convergence of Galton-Watson processes to Continous State Branching processes (CSBP), see for example [16,24,8]. We focus on models where the environment and the interaction mainly affect the mean of the reproduction law and thus modify the drift term of the CSBP by addition of stochastic and nonlinear terms. Our method based on Section 2 allows us to obtain new statements both for convergence of discrete population models and for existence of solutions of SDE with jumps, as can be seen in the following theorems. In particular we obtain a discrete population model approximating the so-called CSBP with interaction in Lévy environment (BPILE) for large populations.
The CSBPs in random environment or with interaction have recently been subject of great attention. We refer to [31] for existence of the solution of the associated SDE under general assumptions, [4,3,29] for approximations and study of some classes of CSBP in random environment (without interaction), [2,11,27] for CSBP with interaction in continuous time (without random environment) and [33] for diffusion approximations.

The discrete model
Let us now describe our framework. The population size is scaled by the integer N ≥ 1. As in Section 3, we introduce for any N a sequence of independent identically distributed real-valued random variables (E N k ) k≥0 with same law as E N . The asymptotic behavior of (E N k ) k≥0 is similar to the one in Section 3 (Assumption A). Nevertheless, the scaling parameter is no longer N but can be any sequence (v N ) N tending to infinity with N . As in the previous section, h E denotes a truncation function defined on (−1, +∞).
for any f vanishing in a neighborhood of zero.
We also consider the associated random walk defined by We recall (as in Section 3) that A1 is equivalent to the convergence of the random walk S N [v N .] to the Lévy process Y with characteristics (α E , β E , ν E ) defined in (3.2). We reduce the set of jumps to (−1, ∞) to avoid degenerated cases when a catastrophe below −1 could kill all the population in one generation.
Let us fix N . We assume that given a population size n and an environment w, each individual reproduces independently at generation k with the same reproduction law L N (n, w).We thus introduce random variables Z 0 ≥ 0 and L N i,k (n, w) such that the family of random variables (Z 0 , (L N i,k (n, w), n ∈ N, w ∈ (−1, +∞)), E N j ; i, k ∈ N * , j ∈ N) is independent and for each n ∈ N, w ∈ (−1, +∞), the random variables L N i,k (n, w) are all distributed as L N (n, w) for i, k ≥ 1. We also assume that the function L N i,k defined on Ω × N × (−1, +∞) endowed by the product σ-field is measurable.
The population size Z N k at generation k is recursively defined as follows, To investigate the convergence in law of the process   (4.4) and observe that for any z ∈ N/N , conditionally on X N k = (exp(−z), y), the random variable X N k+1 is distributed as F N x .
To apply the theoretical framework developed in Section 2, we define χ = (0, 1] × R, U = [−1, 1] × (−1, ∞) and for (u, w) ∈ U, The presence of the term −1 in (4.6) may look strange at first glance, but it ensures that P N k → 0 as N → ∞. Using the binomial expansion and by independence of the reproduction random variables conditionally on E N , we obtain that We also obtain that for ≥ 1, The convergence of A N j, characterizes the effect of the reproduction law on the population dynamics, including density dependence and random environment. The uniform convergence and boundedness of exp(−kz)A N j, (z) will ensure the tightness of X N [v N .] by Theorem 2.3. The continuity of the limiting functions will be checked for the identification of the characteristic triplet. Finally, the representation of the limiting semimartingales as solutions of a stochastic differential equation and its associated uniqueness will give the convergence (Theorem 2.5).

Remark 4.1. In the case of Galton-Watson processes, E
It can easily be proved that the uniform convergence of e −jz A N j (z) is equivalent to the convergence of v N N E(g((L N − 1)/N )) for any g in a set containing a truncation function, its square and regular functions null in a neighborhood of zero. Thus this uniform convergence is equivalent to the classical necessary and sufficient condition for convergence in law of Galton-Watson processes [16,4].
In the next section, we generalize this criterium to reproduction random variables depending on the population size and the environment.

Tightness
We first prove the tightness of 1 uniformly for z ≥ 0.
Then we state a tightness criterion for the original scaled process in the state space Proof. Let us prove the tightness of (X for any x ∈ X = (0, 1] × R. Let us now define G . for H k, ∈ H and k ≥ 1, ≥ 0. We set

Identification
Now, we identify the limiting values of (X N [v N .] ) N as diffusions with jumps. We are interested in models where the environment and the interaction affect the mean reproduction law.
We introduce a truncation function h D on the state space (0, +∞), parameters α D ∈ R and σ D ≥ 0 and a σ-finite measure ν D on (0, +∞) such that ∞ 0 (1 ∧ z 2 )ν D (dz) < +∞. (4.12) We also consider a locally Lipschitz function g defined on R + such that e −z z g(z) z→∞ −→ 0. The function g models the interaction between individuals. In the applications to population dynamics, the most relevant functions will be polynomials.
We provide now the scaling assumption on the reproduction random variable L N so that the limiting values of Z N /N can be identified to a BPILE. This assumption will become more explicit and natural through the identification and examples of the next sections.
Assumption A2. Setting for z ≥ 0, we assume that for any 1 ≤ j ≤ k and ≥ 0, (4.14) where A N j, has been defined in (4.7). (ii) We believe that the pointwise convergence induced by A2 is actually necessary for the convergence of the process Z N /N to a BPILE. It does not seem sufficient in general since some integration argument is involved. Uniformity in A2 provides a sufficient condition. It can be proved for many classes of reproduction laws via uniform continuity, using monotonicity or convexity arguments or boundedness of derivative on compact sets, see the examples below. (iii) Finally, let us remark that we only need to prove the previous convergence for z ∈ N/N in A2, using the definition of A N j, (z) and the uniform continuity of the limit. It will be more convenient for examples.
We observe that under Assumption A2, Assumption A1' is satisfied with Indeed, this expression is bounded using (4.13) and the boundedness of exp(−kz)γ E jz+ , since |γ E jz+ | ≤ C ,j (z + z 2 + e jz/2 z 2 β E + e jz ν E (−1, −1/2)) for j ≤ k and k ≥ 1. Let us then observe from the proof of Theorem 4.2 that (X N [v N .] ) N is tight in D(R + , [0, 1] × R). Moreover we can simplify the expression (4.11) of the limiting characteristic G x , which writes For k ≥ 3, it follows from (4.18) and (4.19) and straightforward computation that
Our choice of parameters in (4.23)-(4.29) ensures that (H2.2) is satisfied for any H and H k, . Applying Theorem 2.4 to X N allows us to conclude.
Let us now write explicitly the stochastic differential equation (4.30) for X t = (X 1 t , Y t ): Using Itô's formula (see [18]), a straightforward computation leads to the equation satisfied by Z t = − log X 1 t . More precisely, we define the explosion time T exp by We obtain on the time interval [0, T exp ) and Z t = +∞ for t ≥ T exp . When T exp = +∞ almost surely, the process is said to be conservative (or nonexplosive). Grey's condition gives a criteria for CSBP, which has been recently extended to CSBP in random Lévy environment in [17].
We have thus proved the tightness of the process and identified the limiting values of (X N [v N .] ) N as weak solutions of a SDE. Uniqueness of the SDE (4.31) (Hypothesis H3) has to be proven to conclude convergence. From the pioneering works of Yamada and Watanabe, several results have been obtained for pathwise uniqueness relaxing the Lipschitz conditions on coefficients. In particular, general results for positive processes with jumps have been obtained in [14,28] and used in random environment, see in particular [31]. This technique allows us to conclude strong uniqueness before explosion.
Here, the process may explode in finite time, which is already the case for classical CSBP and in our framework, explosion can also be due to cooperation or random environment. This leads us to consider two cases. In the first case, we obtain a convergence in law on the state space [0, ∞] under an additional regularity assumption on the drift term close to infinity. This result extends the classical criterion for convergence of Galton-Watson processes, adding both random environment and interaction. In the second case, we ] in [0, ∞) when the limiting values of the sequence of processes are non-explosive. We observe that it also extends results of [4] to Lévy environment with infinite variation and of [11] by relaxing moment assumptions for interaction. The pathwise uniqueness of the SDE allows us to capture limiting processes where infinity is either absorbing or non-accessible. Other situations are interesting, where infinity is regular and uniqueness in law could be invoked. In particular, we refer to [13] for a criterion for reflection at infinity of CSBP with quadratic competition and [22] and [5] for similar issues.

Explosive CSBP with interaction and random environment
In this section, the process may be non-conservative, i.e. T exp may be finite. In order to obtain the strong uniqueness and following [14,29], we consider the following assumption concerning the regularity of the drift term.
Assumption A3. There exist continuous functions r, b r and b d such that for any z ∈ [0, ∞), Proof of Theorem 4.5. We first remark that the convergence in law of (X N We recall from the previous section that X N satisfies (H1) and (H2). To apply Theorem 2.5, it remains to check that X defined in (4.30) is unique in law.
Recently, Pardoux and Dramé [11] have proven the convergence of some continuous time and discrete space processes to CSBP with interaction. Here we relax their conservative assumption and extend to random environments and to general classes of reproduction laws, in a discrete time setting.
Application to Galton-Watson processes with cooperative effects. Note that Theorem 4.5 allows us to recover the convergence in law of the Galton-Watson processes ] ) N defined as in (4.2) with the reproduction laws L N ∈ N satisfying: The limiting process is the (possibly explosive) CSBP with characteristics (α D , β D , ν D ) solution of the stochastic differential equation where N D is a Poisson measure with intensity dtdθν D (dr).
As a new application of Theorem 4.5, we extend the convergence above by taking into account a cooperative effect. In this case, the interactions prevent the use of the classical generating function tool. The reproduction random variable L N (n) depends on the total population size n and we set L N (n) = L N + E N (n), (4.37) where for each n ≥ 0, E N (n) ∈ {0, 1} is a Bernoulli random variable independent of L N and P E N (n) for some function g ∈ C 1 ([0, ∞), [0, ∞)). The process Z N is defined as in (4.2) with this reproduction random variable L N (n).
We obtain the following convergence result.
Proposition 4.6. We assume that v N → ∞ and that (4.35), (4.37) and (4.38) hold. We also assume that z → exp(−z)zg(z) is non-increasing for z large enough and goes to 0 as z → ∞.
for t < T exp and Z t = +∞ for t ≥ T exp .
The monotonicity assumption on z → exp(−z)zg(z) is chosen for sake of simplicity to obtain the pathwise uniqueness. It captures in particular simple cooperative functions We observe that the limiting process Z may be explosive, due to the heavy tails of the reproduction random variable L N (i.e. the CSBP part is explosive) or due to cooperative effects (note for instance that y t = y t g(y t ) is explosive if g(z) = z α , α > 0).
Finally, we add that extensions of the last convergence to random environments are possible in several ways, in particular catastrophes can be added and A3 still holds. But if σ E > 0, the function g has to compensate the quadratic term so that A3 can be fulfilled.
Otherwise, other arguments have to be invoked and one may expect to get uniqueness in law using quenched Laplace exponent (without interaction) or duality arguments.

Proof. Let us introduce
By a Taylor expansion (developed in Appendix 6.2), one can prove that We conclude using Theorem 4.5.

Conservative CSBP with interaction and random environment
We focus on the conservative case. Now +∞ is not accessible and the pathwise uniqueness is obtained without Assumption A3.
Theorem 4.7 allows us to obtain various scaling limits to diffusions with jumps due either to the environment or to demographic stochasticity. The conditions for tightness and identification are very general. The conservativeness can be obtained by different methods as moment estimates or comparison with a conservative CSBP or conservative CSBP in random environment when the process is competitive or with bounded cooperation.
Proof. Using that T exp = +∞ a.s., one can check that pathwise uniqueness holds for (4.31). It can be achieved by using the pathwise uniqueness for Z obtained in [31] before T exp or by adapting the proof of Theorem 4.5. We recall from Theorem 4.4 that weak existence also holds for (4.30) under A1 and A2, so that both strong existence and weak uniqueness hold.
Then (H3) is fulfilled and we can apply Theorem 2.5 to X N and get the weak , [0, ∞) × R) and the pathwise uniqueness of (Z, Y ) follow, which ends up the proof.
Application to logistic Feller diffusion in a Brownian environment. The next example illustrates the result. We consider a reproduction law which takes into account logistic competition and small fluctuations of the environment.
Let us now prove that A2 holds. First, from (4.42), we get E e − j N (L N (n,e)−1) where o(1/N 2 ) is uniform with respect to z and e. Then, for any z ∈ N/N , by considering the cases z ≤ √ N and z ≥ √ N . We obtain that for any 1 ≤ j ≤ k and ≥ 0, Writing g(z) = cz and using that γ We recall from Remark 4.2(iii) that this uniform convergence then holds for z ≥ 0 and A2 is satisfied.
Finally, a coupling with the Feller diffusion in Brownian environment (c = 0, studied in [6]) allows us to prove that the process Z is conservative. The result is then an application of Theorem 4.7.

Perspectives and multidimensional population models
The general results of Section 2 have been applied in the two previous sections to the Wright-Fisher processes in a Lévy environment and to the Galton-Watson processes with interaction in a Lévy environment with jumps larger than −1. These generalizations of historical population models were our initial motivation for this work. The results of Section 2 can be applied in other interesting contexts. We mention here some suggestions in these directions and ongoing works.
First, we could consider environments which are not independent and identically distributed or not restricted to (−1, ∞). This restriction to (−1, ∞) allowed us to consider a functional space generated by the functions exp(−k.) (k ≥ 0) which are bounded on (−1, ∞). To extend the results to random walks converging to Lévy processes with a jump measure ν on R such that R (1 ∧ w 2 )ν E (dw) < ∞, one could consider the functional space of compactly supported for studying Wright Fisher in a Lévy environment and for studying branching processes with interaction in random environment. Indeed these spaces satisfy (H1.1,2). This would require to check that (H1.3) holds.
Such functional spaces could also help to study cases when the environment E N k depends on S N k and S N converges to a diffusion with jumps. Second, as explained in the introduction, we are more generally interested in k-type population models, where the population at generation n is described by a vector Such processes allow to model competition, prey-predators interactions, sexual reproduction, mutations .... Some examples have been well studied, as multitype branching processes, controlled branching processes or bisexual Galton-Watson processes, see e.g. respectively [30], [15] and [1]. One way to obtain the scaling limits is to consider the compactified proces and to use the functional space Indeed H satisfies Assumption (H1.1,2) and the exponential transformation combined with this functional space may allow to exploit the independence structure of the model as for extended branching processes in Section 4. Some work will then be required to check that Assumption (H1.3) holds. Moreover uniqueness can be delicate. In an ongoing work, we consider bisexual Galton-Watson processes and their scaling limits to bisexual CSBPs under general conditions. It is also worth noticing that in the scaling limits, the nonlinearity or the environment can impact the diffusion or jump terms, and not only the drift as for BPILE considered in Section 4. One could also prove limits to CSBP with Lévy environment, where the jump measure associated with the demographical stochasticity (large jumps coming from the offsprings of one single individual, at a rate proportional to the number of individuals) is impacted by the environment, see [4], [29] for an example. Note also that one may want to go beyond the boundedness assumptions on the characteristics G N . This seems to be a challenging question but our approach may be extendable. Indeed, we obtain the boundedness assumptions in Section 4 by a compactification of the state space using the function z → exp(−z), which allows us to consider explosive processes.
The last point to mention is that our criteria concern semimartingales in general. The Markov setting allows us to simplify the form of the characteristics G N and to reduce the problem to analytical approximations, nevertheless we could try to work with non Markovian processes with similar techniques.

General construction of a discrete random variable satisfying A2
We first consider the case σ D = 0 and assume E N ∈ (−1 + 1/ √ N , ∞) for simplicity. We also introduce g N which converges to g and such that where N ψ N (z, e) is continuous bounded and N |ψ N (z, e)| ≤ c(1/N 4/3 + |φ N (z, e)|) and c is a constant which may change from line to line. Thus to infinity (see (4.9) for details). To conclude and prove (4.14), we prove and combine the asymptotic results stated below.
Lemma 6.1. For any j ≥ 1, Proof. (i) First, by Taylor expansion and using that g N (z)/N 1/3 is bounded, there exists c > 0 such that for any z ≤ N 2/3 , Moreover, for any z ≥ 0, N E f jz+ (E N ) → γ E jz+ by Assumption A1 (see again (4.9) for details) and the convergence is uniform on [0, A] by convexity of z → N E f jz+ (E N ) and by continuity of z → γ E jz+ (third Dini's theorem). It proves (ii) on compacts sets. Let us now prove that sup z≥A,N ≥1 exp(−jz) N A N 2 (z)| → 0 as A → ∞. Let us fix ε > 0 and Recalling that E N ≥ −1 + 1/ First, we recall that sup N P(E N < −1 + ε) → 0 as ε → 0 and N E h E (E N )1 E N ≥−1+ε is bounded (actually convergent by Assumption A1). Second, we prove that sup z≥A,N ≥1 exp(−jz) N A N 2 (z)| → 0 as A → ∞ and ends the proof of (ii) by recalling that exp(−jz)γ E jz+ → 0 as z → ∞. We finally prove (iii). where N 0 is chosen such that 1 − 2/ √ N 0 > 1/2.

Taylor expansion for a Galton-Watson process with cooperation
Recalling (4.40) and (4.14), To conclude, we observe that min{z : z + g(z)z ≥ v N } → ∞ as N → ∞. Then sup z+g(z)z≥v N e −jz zγ N,D j + jzg(z) → ∞. Let us now prove that Indeed, g ≥ 0 and either z ≥ v N /2 and e −jz v N C N j (z)| ≤ e −jz v N e zc/v N ≤ 2ze −zj/2 for N such that j − c/v N ≥ j/2 or z ≤ v N /2 and v N ≤ 2zg(z) and there exists c > 0 such that e −jz v N C N j (z)| ≤ e −jz 2zg(z) e c . Recalling that min{z; z + g(z)z ≥ v N } → ∞ as N → ∞ and that zg(z) exp(−z) → 0 as z → ∞, we obtain the desired result.
If y ∈ V 0 and |xy| ≥ 1, we have which ends the proof.
Let us now prove the forthcoming inequality (6.4). Proof. Let us first assume that min(x, x ) ≥ |x − x |. In this case, it is immediate that |x log x − x log x | ≤ |x − x |(1 + log(|x − x |)) by the mean value theorem. We now assume that 0 ≤ x ≤ |x − x | ≤ x , which implies that x ≤ 2|x − x |. We have using that x/x ∈ [0, 1] and that the function α ∈ [0, 1] → α log α is bounded by some constant C. We obtain that |x log x − x log x | ≤ 2C|x − x | + | log(|x − x |)| |x − x |, which ends the proof. Proof. For the first inequality, one can use that −x log x is bounded for the first term in (4.33) and the mean value theorem for the second one.

Stone-Weierstrass theorem on locally compact space
We recall here the local version of Stone-Weierstrass Theorem and assume that the space X is a locally compact Hausdorff space.
Let C 0 (X, R) the space of real-valued continuous functions on X which vanish at infinity, i.e. given ε > 0, there is a compact subset K such that f (x) < ε whenever the point x lies outside K. In other words, the set {x, f (x) ≥ ε} is compact.
Let us consider a subalgebra A of C 0 (X, R). Then A is dense in C 0 (X, R) for the topology of uniform convergence if and only if it separates points and vanishes nowhere.