From the distributions of times of interactions to preys and predators dynamical systems

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From the distributions of times of interactions to preys and predators dynamical systems

Vincent Bansaye, Bertrand Cloez

To cite this version:

Vincent Bansaye, Bertrand Cloez. From the distributions of times of interactions to preys and preda-

tors dynamical systems. 2021. �hal-03183538v2�

From the distributions of times of interactions to preys and predators dynamical systems

Vincent Bansaye ^∗ and Bertrand Cloez ^† May 18, 2021

Abstract

We consider a stochastic individual based model where each predator searches and then manipulates its prey or rests during random times. The time distributions may be non-exponential. An age structure allows to describe these interactions and get a Markovian setting. The process is characterized by a measure-valued stochastic differential equation. We prove averaging results in this infinite dimensional setting and get the convergence of the slow-fast macroscopic prey predator process to a two dimensional dynamical system. We recover classical functional responses. We also get new forms arising in particular when births and deaths of predators are affected by the lack of food.

Contents

1 Introduction 2

2 The stochastic individual based model 5

2.1 Existence and trajectorial representation . . . . 6 2.2 First estimates and properties . . . . 8

3 Scaling and averaging 12

3.1 Approximation of the scaled process by a dynamical system . . . . 13 3.2 Preliminary estimate and tightness . . . . 14 3.3 Identification of limiting values and proof of convergence . . . . 17

4 Examples and comments 23

4.1 Classical setting and functional responses : memory less interactions . . . 23 4.2 A first generalization : non-exponential time of interaction . . . . 24 4.3 Influence of distribution of time interaction . . . . 25

∗

CMAP, Ecole polytechnique, Palaiseau

†

MISTEA, INRAE, Institut Agro, Univ. Montpellier

4.4 About the behavior of the limiting ODEs . . . . 25 4.5 Discussion, scaling and extensions . . . . 26

1 Introduction

Functional responses are widely used to quantify interactions between species in ecology. The way functional responses arise at the macroscopic level and describe population dynamics or evolution is a fundamental issue for species conservation or sta- tistical inference of parameters. Indeed, their form influences the stability properties of dynamics, their long time behavior or speed of convergence. The link between individual behavior and macroscopic dynamics has attracted lots of attention for chemical reactions and population dynamics from the works of Michael and Menten.

Macroscopic derivation from individual based model rely in general on a large popula- tion approximation of finite dimensional Markov processes describing the number of indi- viduals of each species, possibly structured in status (searching, handling...), space or size.

In this setting, Kurtz and Popovic [KKP14] obtain the classical Michaelis Menten and Holling functional responses in limiting dynamical systems and fluctuations of processes around these limits. In our context of prey-pradators interactions, let us mention [DS13]

which starts from a stochastic individual based model. They derive a finite dimensional Markov chain and convergence to ODEs involving the classical functional responses. In [CKBG14], a simple decision tree based on game-theoretical approach response is devel- oped. Similarly, random walks and Poisson type process are used in [AKF11] to describe functional responses. The reduced model counting only the total number of preys and the total number of predators, without distinguishing their status, is also classically de- rived directly from the macroscopic ODEs [JKT02, BDBS96, HDB97]. Again, it uses a slow-fast scaling and the associated quasi-steady-state approximation. These Markov settings allow for justification of macroscopic equations in a context of absence of mem- ory of interactions. Indeed, the time for associated interactions are then exponentially distributed, potentially up to the addition of the relevant successive state to describe the interaction.

Random times involved in ecological or biological interactions are in general non- exponentially distributed, see [DKPvG15, BBC18] and references therein. Indeed, handling or manipulation times may have small standard deviations compared to the mean, while exponential distribution forces the value of variance once the mean is fixed.

Besides, foraging suggests that the probability of finding a prey eventually increases

with searching time for a given density of preys. Finally, as far as we see, these random

times seem to be distributed with one mode. The aim of the paper is to consider

general distribution for the times describing interactions. We extend approximation

results relying on absence of memory and obtain a reduced model. We also obtain new

features due to the fact that mortality depends on prey consumption and times are not

exponentially distributed. This paper can be seen as a continuation of [BBC18] and

propose a mathematical framework for the derivation of macroscopic dynamical systems,

when the interactions are modeled by renewal processes. Each predator successively

searches during a random time and then manipulates during an other random time,

which may include rest or other interactions. We assume that these time distributions

admit a density with respect to Lebesgue measure and are dependent of the prey density.

Extensions of the current approach to more than two status for predators would be straightforward.

We consider an age structure which enables us to exploit a Markov framework for the analysis. It consists in a classical extension of the state space of non-Markovian dynamics. In this vein, we mention in particular the epidemiological models, where age may be the biological age or infection age, or chemical reactions, where time of reaction may be non exponential. For this latter, a recent related and stimulating work [KKKR20]

propose averaging results. But the interpretation of the age is different and not density dependent. Furthermore, this work does not make rigorous proofs. In particular it does not deal with hypothesis required on time distribution to make convergence of stochastic models hold.

Let us first describe informally the model. We write n 1 ∈ N the number of preys and n 2 ∈ N the number of predators. Predators search preys during a random time distributed as a random variable T _S (n ₁ ). Typically the more n ₁ is large, the smaller T _S (n ₁ ) should be. At the end of this time, one prey is caught and the population of preys becomes n 1 − 1. The predator changes its status and now manipulates during a time distributed as T _M (n ₁ − 1). Several predators follow simultaneously and independently this dynamics, but they live with a common number of preys and impact each other through this common resource. Besides, each predator gives birth and dies with respective individual rates γ r (u) and β _r (u), which depends on their status r ∈ {S, M } and the time u from which they are in this status. Typically, the fact that the predator does not find a prey make its death rate β S (u) increase with u. Preys also give birth and die, with fixed rates γ and β.

We assume that the size of the populations of preys is of order of magnitude K ₁ and the size of the populations of predators is of order of magnitude K 2 and that K 1 K 2 . That means that preys are much more numerous than predators. A slow-fast dynamic is considered : the time scale of prey-predator interactions is short compared to the time scale of birth and death of predators and preys. It means that each predator eats many preys during its life and, if a prey is not eaten by a predator then its life length is comparable to that of predators, up to some factor. After scaling, we show, that the couple of stochastic processes describing the quantities of preys of predators converge in law in D ([0, ∞), R ² + ) as K 1 , K 2 tend to infinity, to the unique solution (x, y) of an ordinary differential equation:

x ⁰ (t) = (γ − β)x(t) − y(t)φ(x(t)) y ⁰ (t) = y(t)ψ(x(t)),

where

φ(x) = 1

E [T _S (x) + T _M (x)] , (1)

and

ψ(x) = E

h R T

S

(x)

0 (γ _S (u) − β _S (u))du + R T

M

(x)

0 (γ _M (u) − β _M (u))du i

E[T S (x) + T M (x)] . (2)

This limit theorem will be illustrated both by classical and new functional responses in

Section 4. We observe that the response φ of preys due to predatory is only sensitive

to mean time of interactions. It thus extends the exponential case to more general

distribution. At this macroscopic scale, for the population of preys, the distribution of times involved in interactions plays a role only through its mean. In contrast, the growth rate ψ of the population of predators is in general sensitive to the other characteristics of the distribution. We add that the distribution of time of interactions should also impact the dynamics of the population of preys at a second order, i.e. for fluctuations. This is relevant in particular when population are not too large and left for a future work.

The fact that the time of interactions is both density dependent and non-exponentially distributed leads us to extend the state space and consider rates of change of status which are depend on the age in the current statuts and the number of preys. We then exploit the generator and martingale problem and get also the age distribution of preys. The problem arising is then an averaging in infinite dimension. The strat- egy of proof follows the techniques developed in [KKP14] in finite dimension using the occupation measure. The averaging phenomenon in finite dimension is classical [KKP14, BKPR06, Cos16, MT12]. In infinite dimension, much less work has been done up to our knowledge. Let us mention [MT12] which considers averaging with an age structure and has also inspired this work. Two main differences appear in our context : the age structure is due to interactions and the rates involved are not bounded, since tail distribution of times may for instance decrease faster than exponentially. In this paper, at the macroscopic scale the infinite setting reduces to a finite one describing the number of preys and predators.

We consider a punctual measure whose atoms give the status and the age of predators, which is here the length of time since they have been in this status. Other relevant ages could be added, in particular the time from the birth. However, it seems superfluous at our stage. In our slow-fast dynamics, there is an averaging phenomenon and the numbers of predators in each status are instantaneously at equilibrium. This enables to reduce the infinite-dimensional model to a two-dimensional system of equations.

Following [Kur92, KKP14], the occupation measure Γ ^K , given by Γ ^K ([0, t]) = R t 0 δ _Y

K

s

ds deals with the fast time component Y ^K , here the predations. In our setting, Y ^K is the distribution of ages and status and is thus defined as a punctual measure. Instead of considering a measure whose atoms are punctual measures, we consider the mean measure Γ ^K ([0, t]) = R t

0 Y _s ^K ds, which is enough for our purpose. Consequently, our measure Γ ^K will not degenerate to some measure of the form R t

0 δ _{f (X}

_s

₎ ds, for some function f , but tends to some specific distribution.

The paper is structured as follows. In Section 2, we define the integer valued model without any time or population size scaling. We characterize the process as the unique strong solution of a stochastic differential equation and give first properties about its semimartingale decomposition. In particular, the key technical point is the control of the age distribution in our setting, by exploiting the local time associated to the renewal procedure. In Section 3, we introduce the scaled process and state our main result, Theorem 3.4. The result is proved by tightness and identification of the limit using the occupation measure. Finally, we give several examples in Section 4 and a few comments.

Notation. We write a ∞ ∈ (0, +∞] the maximal interaction age and X = {S, M } × [0, a ∞ ),

the state space of predators endowed with the product σ-algebra.

We denote by M(S ) the set of finite measures on any topological space S endowed with its Borel algebra. We endow M(S ) with the narrow (or weak) topology: that is µ _n tends to µ if and only if for every continuous and bounded function f on S,

n→∞ lim Z

S

f dµ n = Z

S

f dµ.

For r ∈ {S, M }, we write r the complementary status of r, i.e. the unique element of {S, M } \ {r}.

We finally denote by C ^1,b (X ) (resp. C ^1,b (U ×X ) and C ^1,b ([0, a ∞ ))) the space of measurable and bounded functions from {S, M } × [0, a ∞ ) (resp. U × {S, M } × [0, a ∞ ) and [0, a ∞ )) to R such that f is continuously differentiable with respect to its second (resp. third, resp.

first) variable, with bounded derivative.

2 The stochastic individual based model

In this section, we define the discrete individual based model using stochastic differ- ential equations with jumps. The accelerated normalized process and its approximation will be studied in the next section using properties and estimates obtained here.

Each predator is characterized by a status r ∈ {S, M } and an age a ∈ [0, a ∞ ) . To formal- ize conveniently this modeling, we label each predator using classical Ulam-Harris-Neveu notation and describe the associated genealogical tree. The set of individuals is

U = N × ∪ _k≥0 {1, 2} ^k .

For short, we write u = u ₀ u ₁ . . . u _k ∈ U and u then corresponds to an individual living in generation |u| = k and whose ancestor in generation i is u ₀ . . . u _i for 0 ≤ i ≤ k. At each reproduction event, we assume for simplicity that every predator u only gives birth to one predator and we label the mother by u1 and its child by u2. The population of predators alive at time t is denoted by P (t) and is a random subset of U . For each predator i ∈ P(t), we write r i (t) ∈ {S, M } its status at time t, which indicates respectively that it searches or manipulates. We write a i (t) its age, namely the time from which it searches or manipulates. Finally, we write X(t) ∈ N the number of preys at time t.

The state of the population is then given by the process Z = (Z(t)) t≥0 = (X(t), Y (t)) t≥0 ,

where the measure Y describes the predators and is given for all t ≥ 0 by Y (t) = X

i∈P(t)

δ _(i,r

_i

_(t),a

_i

_(t)) .

For any t ≥ 0, Y (t) ∈ M(U × X ), where we recall that X = {S, M } × [0, a ∞ ). Besides, for any U ⊂ U , the projected measure

Y (t, U, {r}, ·) = X

i∈P(t), r

_i

(t)=r

δ _a

_i

_(t)

gives the collection of ages of predators in status r at time t, whose labels belong to U .

The total number of predators at time t is then Y (t, U , {S, M }, R + ).

Let us now describe the population dynamic. For status r ∈ {S, M } and in the presence of x ∈ N preys, we assume that the time for interaction T _r (x) is a random variable with support [0, a ∞ ). We also assume that it admits a positive density on [0, a ∞ ), with respect to the Lebesgue measure, f _r (x, ·). The associated jump rate α is defined for (a, x) ∈ X by

α r (a, x) = f _r (a, x) R

[a,a

∞

) f _r (u, x)du ,

It gives the rate at which a predator changes its status when its age in its current status is equal to a and the number of preys is x. If the predator was searching, it provokes a death of a prey. We do not assume that these interactions rates α _r are lower and upper bounded. Indeed, for instance in the case when the time of interaction has a finite support (a ∞ < ∞) or a subexponential tail, it is not upper bounded, even for a given number of preys. Let us consider some classical distributions that will be captured in our setting:

• Exponential law: f(a, x) = λ(x)e ^−λ(x)a and α(a, x) = λ(x), for some bounded function λ.

• Log-normal distribution : f(a, x) = ¹

aσ(x) √

2π exp

− (log(a)−µ(x))

²

) 2σ(x)

²

and α has no explicit form.

• Uniform law: f(a, x) = 1 _[0,1] (a) and α(a, x) = (1 − a) ⁻¹ 1 _[0,1) .

• Pareto law : f(a, x) = k(x)(z(x) ^k(x) /a) ^k(x) 1 _a≥z(x) and α(a, x) = k(x)/a1 _a≥z(x) , for some bounded functions k : N → (0, +∞) and z : N → (1, +∞).

Finally, predators may give birth or die with respective measurable rates a 7→ γ r (a) and a 7→ β _r (a) which depend on their status r ∈ {S, M } and their interaction age a. In particular, lack of nutrition affects survival and reproduction and γ S may be decreasing and β S may be increasing. For sake of simplicity and realism, we assume these rates are bounded. For the preys, birth and death rates are non-negative numbers denoted by γ and β.

2.1 Existence and trajectorial representation

Following for instance [FM04, Tra06, BT10], we construct and characterize (Z(t)) t≥0

as the unique strong solution of a stochastic differential equation. For every i ∈ U, we let N ⁱ be independent Poisson punctual point measures on R ² + with intensity the Lebesgue measure. These measures provide the random times when a predator changes its status between searching and manipulating. We introduce also independent Poisson point measures M ⁱ and Q on R ² + with intensity the Lebesgue measure. They are independent of (N ⁱ , i ∈ U ) and describe births and deaths of preys and predators. For convenience, we write

α _i (s) = α _r

_i

_(s) (a _i (s), X(s)), γ _i (s) = γ _r

_i

_(s) (a _i (s)), β _i (s) = β _r

_i

_(s) (a _i (s)).

We consider the following equation for the evolution of the number of preys for t ≥ 0, X(t) = X(0) −

Z t 0

X

i∈P(s−) r

i

(s−)=S

Z

R

+

1 {u≤α

_i

(s−)} N ⁱ (ds, du)

+ Z t

0

Z

R

+

1 {u≤γX(s−)} − 1 {0<u−γX(s−)≤βX(s−)}

Q(ds, du). (3) Indeed, the number of preys decreases when they are caught by a predator and also varies independently by births and deaths. For every function f ∈ C ^1,b (U × X), we consider

hY (t),fi = hY (0), f i + Z t

0

X

i∈P (s−)

∂ a f (i, r i (s−), a _i (s−)) ds +

Z t 0

X

i∈P (s−)

Z

R

+

1 _u≤α

_i

_(s−) Df (i, s−) N ⁱ (ds, du) +

Z t 0

X

i∈P (s−)

Z

R

+

1 _u≤γ

_i

_(s−) ∆f(i, r _i (s−), a _i (s−)))

− 1 _0<u−γ

_i

_(s−)≤β

_i

_(s−) f(i, r _i (s−), a _i (s−))

Q ⁱ (ds, du), (4) where ∂ _a f stands for the partial derivative of f with respect to the third variable and Df (i, s) = f (i, r i (s), 0)−f (i, r i (s), a i (s)); ∆f (i, r, a) = f (i1, r, a)+f (i2, M, 0)−f (i, r, a).

Recall that r is the complementary status of r, i.e. the unique element of {S, M } \ {r}.

In this equation, status of new born are supposed to be in the manipulation state M . This choice seems natural but may sound somewhat arbitrary. However, more complex choices would require additional notations and should have no macroscopic impact.

Let us state the existence result and characterize the process using the previous stochastic differential equation. For convenience, we make the following boundedness and regularity assumptions, which are relevant for our purpose.

Assumption 2.1. There exists a 0 ∈ (0, a ∞ ) such that Y (0, {S, M }, [a 0 , a ∞ )) = 0 a.s.

Besides, for any r ∈ {S, M } and K > 0,

x∈[0,K] inf E (T r (x)) > 0, sup

a∈[0,a

∞

)

(γ r (a) + β r (a)) < ∞ and a → α r (a, x) is continuous on [0, a ∞ ) for any x ∈ N.

Proposition 2.2. Let Z (0) = (X(0), Y (0)) with X(0) ∈ N and Y (0) being a punctual measure in M(U × X ) a.s. Under Assumption 2.1, the system of stochastic differential equations (3-4) admits a unique strong solution Z = (X, Y ) in D ([0, ∞), N × M(U × X )) with initial condition Z(0).

Proof. The construction of the process and its uniqueness can be achieved iteratively,

using the successive random times between each event, see for instance [BM15]. The proof

is classical and we just give its sketch. The only point to justify is that the successive

times where an event (change of status, birth or death) occur do not accumulate. For that purpose, we proceed by a classical localization procedure and introduce the hitting time T _K = inf{t : X(t) ≥ K}. Before time T _K , by Assumption 2.1, mean time of change of status are lower bounded and birth and death rates of preys and predators are upper bounded. So a.s. no accumulation of jumps occurs. We need now to justify that T _K tends a.s. to infinity as K → ∞. It is achieved by dominating the process X by a pure linear birth process (Yule process) with birth rate per capita γ. The fact that this latter does not explode is well known and can be derived for instance from the finiteness of its first moment. Pathwise uniqueness of the system of stochastic differential equations is also obtained by induction on the successive jumps, which are provided by the common Poisson point measures. The argument above ensure that uniqueness holds for any positive time. Let us finally note that the system (3-4) is closed, since P (t) and (i, r i (t), a i (t)) are determined (uniquely) by the measure Y (t), which is itself determined by its projections hY (t), f i for f ∈ C ^1,b (U × X).

2.2 First estimates and properties

We start by a sharp and useful bound on the first moment of the punctual measures Y evaluated on tests functions which may be non bounded. For convenience, we write

Y(t, ·) = Y (t, U , ·) = X

i∈P(t)

δ _(r

_i

_(t),a

_i

_(t))

the projection of the measure Y (t) on X . We also introduce the exit time of the number of preys of (1/K, K) for K > 0 :

τ K = inf{t ≥ 0 : X t 6∈ (1/K, K)}.

We consider the associated bounds on the rate of transitions for r ∈ {S, M }, α _r (a, K) = sup

x∈(1/K,K)

α _r (a, x), α _r (a, K) = inf

x∈(1/K,K) α _r (a, x), which are continuous by continuity of α r and

¯

γ = sup

r∈{S,M }, a∈[0,a

∞

)

γ r (a).

Lemma 2.3. Under Assumption 2.1, there exists C > 0 such that for any continuous and non-negative function f : [0, a ∞ ) → R + and r ∈ {S, M } and K > 0,

E

"

Z T ∧τ

K

0

Z

[0,a

∞

)

f (a)Y (s, {r}, da)ds

#

≤ C(1 + T )e ^{¯ γT} Z

[0,a

∞

)

f ^? (a)e ⁻

1a>t0 4

R

[t0,a)

α

_r

(u,K)du

da,

where t ₀ = min(1, a ∞ /2) and f ^? (a) = 1 _[0,t

₀

₎ (a) sup _[0,t

₀

_] f + 1 _[t

₀

_,a

_∞

₎ (a)f (a).

Proof. We consider first f continuous, non negative with compact support in [0, A), where

A ∈ [0, a ∞ ). The conclusion for general f will follow with a monotone approximation

of f by compactly supported functions. Fix T ≥ 0 and consider an increasing sequence

(a n ) 0≤n≤n

0

, where a n+1 = a n + t n and (t n ) 0≤n≤n

0

−1 is a decreasing sequence of positive numbers and a _n

₀

= A. For a predator i ∈ U , a status r ∈ {R, M} and a level n ∈ N , we set

u ^i,r _n = E

Z τ

K

∧T 0

1 {i∈P(s), r

_i

(s)=r, a

i

(s)∈[a

_n

,a

n+1

)} ds

.

It is equal to the cumulative time spent by predator i, in status r and between ages a n

and a _n+1 . Let also

N _n ^i,r = X

s≤τ

K

∧T

1 _{{i∈P (s), r}

_i

_{(s)=r, a}

_i

_(s)=a

_n

_}

be the number of times that predator i ∈ U reaches age a n while it is in status r. In other words, writing b(i) the birth time of individual i, for every j ∈ N , we can define iteratively

T _j+1,n ^i,r = inf{t > T _j,n ^i,r | i ∈ P(t), r _i (t) = r, a i (t) = a n }, for j ≥ 0, with T _0,n ^i,r = b(i). We get

N _n ^i,r = X

j≥1

1 _{T

i,r

j,n

≤τ

K

∧T } .

With this notation and writing T j = T _j,n+1 ^i,r for convenience, we have for n ≤ n 0 − 2,

u ^i,r _n+1 ≤ E







N

_n+1^i,r

X

j=1

Z T

j

+t

n+1

T

j

1 _{{i∈P(s), a}

_i

_(s)∈[a

_n+1

_,a

_n+2

_{), r}

_i

_(s)=r} 1 {∀u∈[0,s]:X(u)∈K} ds





 . (5) Adding that (t _n ) _n decreases, the status cannot change twice during time t _n+1 and reach level a _n+1 again. So, on the event the process does not jump at all during the time period [T j , T j + t n+1 ] on the event considered and for s ≤ t n+1 ,

E h

1 _{{ i∈P (T}

_j

_{+s), a}

_i

_(T

_j

_+s)∈[a

_n+1

_,a

_n+2

_{), r}

_i

_(T

_j

_+s)=r, ∀u∈[0,s]:X(T

_j

+u)∈K}

T _j , (X(t)) t≤T

_j

+s

i

= E h

e ⁻

R

s

0

α

r

(a

n+1

+u,X(T

j

+u))du 1 {∀u∈[0,s]:X(T

_j

+u)∈K}

T j , (X(T j + u)) u≤s

i

≤ e ⁻

R

s

0

α

_r

(a

n+1

+u,K)du , we get for n ≤ n 0 − 2,

u ^i,r _n+1 ≤ E







N

_n+1^i,r

X

j=1

Z t

n+1

0

e ^{− R}

^0s

^α

^r

^(a

ⁿ⁺¹

^+u,K)du ds





 ≤ t _n p ^r _n E h

N _n+1 ^i,r i

, (6)

where

p ^r _n = 1 − e ^−α

^rn

^t

ⁿ⁺¹

α ^r _n t _n , α ^r _n = inf{α _r (a, x) : a ∈ [a _n+1 , a _n+2 ], x ∈ (1/K, K)}.

Besides, as ages increase at rate 1, either predator i is born at an age between a n+1 and a _n+2 or it has exactly spent the time t _n at level between ages [a _n , a _n+1 ). In any case,

E h

N _n+1 ^i,r i

≤ P (A ⁱ _n ) + u ^i,r n

t _n ,

where

A ⁱ _n = {b(i) ≤ T ∧ τ _K , a _i (b(i)) ∈ [a _n+1 , a _n+2 )}.

Combining these inequalities, we obtain

u ^i,r _n+1 ≤ p ^r _n u ^i,r _n + t n p ^r _n P (A ⁱ _n ), which then gives, by induction,

u ^i,r _n ≤ u ^i,r ₀

n−1

Y

j=0

p ^r _j +

n−1

X

k=0

t _k P (A ⁱ _k )

n−1

Y

j=k

p ^r _j .

Using now p ^r _j ≤ ^t

^j+1

_t

j

1 − ^α

r j

t

j+1

4

, when t _j+1 is small enough, and setting S _k ⁿ =

n−1

X

j=k

α ^r _j t _j+1 /4,

we get Q n−1

j=k p ^r _j ≤ ^t _t

ⁿ

k

e ^−S

^kn

and then u ^i,r _n ≤ t n

t ₀ u ^i,r ₀ e ^−S

⁰ⁿ

+ t n n−1

X

k=0

P (A ⁱ _k )e ^−S

^kn

.

To conclude for f compactly supported, we set f n = sup _a∈[a

_n

_,a

_n+1

₎ f(a) and get E

Z t 0

f (a _i (s)) 1 _i∈P(s),r

_i

_(s)=r ds

≤

n

0

−1

X

n=0

f n u ^i,r _n

≤ P (b(i) ≤ T ∧ τ K ) t ₀ ∧ 1

n

0

−1

X

n=0

f n t n e ^−S

⁰ⁿ

T +

n−1

X

k=0

P (A ⁱ _k | b(i) ≤ T ∧ τ K )e ^S

^k0

! ,

since u ^i,r ₀ ≤ T P (b(i) ≤ T ∧ τ K ). Let now t ₀ = min(1, a ∞ /2) and A ∈ (t ₀ , a ∞ ). For any h > 0, we consider a positive strictly decreasing sequence (t ^h _n ) _n≤n

^h

0

−1 such that t ^h ₀ = t 0 , lim

h→0 sup

n≥1

t ^h _n = 0, a ^h _n

h 0

=

n

^h₀

−1

X

k=0

t ^h _k = A.

Using the following uniform convergence of the Darboux sum sup

k=0,...,n

^h₀

−1

S ₀ ^k − 1 4

Z

[0,a

^h_k

)

α _r (u)du

h→0 −→ 0,

which comes from the continuity of α and uniform continuity of f on [0, A], we get, by letting h → 0,

E

Z τ

K

∧T 0

f(a _i (s))1 _{{i∈P (s), r}

_i

_(s)=r} ds

≤ C P (b(i) ≤ T ∧ τ K ) Z

[0,a

∞

)

f ^? (a)e ⁻

R

[0,a)

α

_r

(u,K)du/4

da

×

"

T + E 1 _{a

_i

_(b(i))≤a} exp Z

[0,a

i

(b(i)))

α _r (u, K)du/4

!!#

,

for some constant C, where we recall that f ^? (a) = 1 _[0,t

₀

₎ (a) sup _[0,t

₀

_] f + 1 _[t

₀

_,a

_∞

₎ (a)f (a).

Recall also that a _i (b(i)) has a compact support in [0, a ∞ ) by Assumption 2.1 and the fact that newborns have age 0. So the last term is bounded by a constant. Summing over all predators i yields the result since

X

i∈U

P (b(i) ≤ T ∧ τ K ) ≤ E (#{i ∈ U : b(i) ≤ T ∧ τ K }) ≤ e ^{T γ} E [Y(0, {S, M }, [0, a ∞ ))]

since #{i ∈ U : b(i) ≤ T ∧ τ K } is dominated by a pure birth process at time T , with individual birth rate γ .

We define F g,f : R + × M(X ) → R by

F _g,f (x, µ) = g(x) + hµ, f i, (7)

where g : R + → R and f : X → R are measurable and bounded functions. We introduce LF _g,f (x, µ) = γx(g(x + 1) − g(x)) + βx(g(x − 1) − g(x))

+ Z

X

∂

∂a f (r, a) + γ r (a)f (M, 0) − β r (a)f (r, a)

µ(dr, da)

+ Z

X

α _r (a, x)(1 _r=S (g(x − 1) − g(x)) + f (r, 0) − f (r, a))µ(dr, da).

The operator L is the generator of the Markov process (X(t), Y(t)) t≥0 . More precisely, our SDE representation (3-4) ensures the following classical martingale problem.

Lemma 2.4. Assume that Assumption 2.1 holds and that for any K > 0 and r ∈ {S, M }, Z

[0,a

∞

)

α r (a, K)e ⁻

¹⁴

R

[0,a)

α

_r

(u,K)du

da < ∞. (8)

Let g : R + → R be measurable and bounded and f ∈ C ^1,b (X ). Then (M(t)) t≥0 defined for t ≥ 0 by

M (t) = F _g,f (X(t), Y(t)) − F _g,f (X(0), Y(0)) − Z t

0

LF _g,f (X(s), Y(s))ds

is a local martingale. Besides (M (t ∧ τ K )) _t≥0 is a square-integrable martingale and its bracket is given, for all t ≥ 0, by

hM i(t ∧ τ K )

= Z t∧τ

K

0

γX(s)(g(X(s) + 1) − g(X(s))) ² + βX(s)(g(X(s) − 1) − g(X(s))) ² ds

+ Z t∧τ

K

0

X

i∈P(s)

α i (s) 1 _r

_i

_(s)=S (g(X(s) − 1) − g(X(s))) + (f (r i (s), 0) − f (r i (s), a i (s)) 2

ds

+ Z t∧τ

K

0

X

i∈P(s)

γ _i (s)f (M, 0) ² + β _i (s)f(r _i (s), a _i (s)) ²

ds.

Proof. The fact that M is a local martingale and the computation of its square variation is derived from our SDE representation (3-4) . Indeed one can write the semi-martingale decomposition of F _g,f (X, Y) using the compensation of the Poisson point measures, see for instance [FM04, BM15]. We only give details for the first component X:

g(X(t)) = g(X(0)) + Z t

0

X

i∈P(s−) r

i

(s−)=S

α i (s−)(g(X(s−) − 1) − g(X(s−)) ds

+ Z t

0

X

i∈P(s−) r

i

(s−)=S

Z

R

+

1 _{u≤α

_i

_(s−)} (g(X(s−) − 1) − g(X(s−)) N e ⁱ (ds, du)

+ Z t

0

(γX(s−)((g(X(s−) + 1) − g(X(s−)) + βX(s−)(g(X(s−) − 1) − g(X(s−))) ds +

Z t 0

Z

R

⁺

1 _{{u≤γX(s−)}} ((g(X(s−) + 1) − g(X(s−))

+ 1 {0<u−γX(s−)≤βX(s−)} (g(X(s−) − 1) − g(X(s−))

Q(ds, du), e

where N e ⁱ and Q e are the compensated measures of N ⁱ and Q.

Finally, square integrability of (M(t ∧ τ K )) t≥0 is a consequence of Lemma 2.3 applied to f = α r and Doob’s inequality and our integrability assumption (8).

3 Scaling and averaging

We introduce our scaling parameters K = (K 1 , K 2 ) ∈ (0, +∞) ² respectively for the size of the populations of preys and the predators. These sizes are going to infinity.

Besides, in our scaling,

λ _K = K ₁ K ₂

tends to infinity. The intial number of preys and predators satisfy

X ^K (0) = bK ₁ x ₀ c, Y ^K (0, U , {S, M }, [0, a ∞ )) = bK ₂ y ₀ c,

for some constants x 0 , y 0 > 0. The rates for interactions are now density-dependent (rather that population-size-dependent) and we set

α ^K _r (a, x) = α _r (a, x/K ₁ )

for r ∈ {S, M }, where α r is measurable on [0, a ∞ ) × R + . Interactions occur at a fast time scale which arises here through an acceleration of time. Birth and death of preys and predators (but the deaths of preys due to predation) occur at a slower time scale and we set

β _r ^K (a) = λ ⁻¹ _K β _r (a), γ ^K _r (a) = λ ⁻¹ _K γ _r (a), β ^K = λ ⁻¹ _K β, γ ^K = λ ⁻¹ _K γ,

where β _r ^K and γ _r ^K are non-negative, measurable and bounded functions and β, γ are non-

negative numbers. See Section 4.5 for a discussion on this scaling. To define the process

for any time, we make the following assumption.

Assumption 3.1. There exists a 0 ∈ (0, a ∞ ) such that Y ^K (0, {S, M }, [a 0 , ∞)) = 0 a.s.

for all K ≥ 1. Besides, for any r ∈ {S, M } and K > 0,

x∈[0,K] inf E (T r (x)) > 0, sup

a∈[0,a

∞

)

γ r (a) + β r (a) < ∞ and a → α r (a, x) is continuous on [0, a ∞ ) for any x ∈ R + .

Under this assumption, for each K = (K 1 , K 2 ) ∈ (0, +∞) ² , Proposition 2.2 ensures ex- istence and strong uniqueness of the solution Z ^K = (X ^K , Y ^K ) of the system of stochastic differential equations (3-4) with parameters α ^K _r , γ _r ^K , β ^K _r , γ ^K , β ^K given above. We consider now the accelerated and scaled process defined, for all t ≥ 0, by

Z ^K (t) = (Ξ ^K (t), Y ^K (t, dr, da)) = 1

K 1

X ^K (λ _K t), 1 K 2

Y ^K (λ _K t, U , dr, da),

.

For every T > 0, (Z ^K (t), t ∈ [0, T ]) belongs to the space D ([0, T ], R + ) × M([0, T ] × X ). Space D ([0, T ], R + ) is the classical Skorokhod space with its usual topology; see for instance [Bil13]. Space M([0, T ] × X ) is the space of finite positive measures on [0, T ] × {S, M } × [0, a ∞ ) embedded with narrow convergence.

3.1 Approximation of the scaled process by a dynamical system

Our result relies on the following assumption on the interaction rates. It is slightly stronger than (8) and is involved in tightness proof to localize age in compact sets of [0, a ∞ ). Following the previous section, we set for r ∈ {S, M } and K > 0,

α _r (a, K) = inf

x∈(−1/K,K) α _r (a, x), α _r (a, K) = sup

x∈(−1/K,K)

α _r (a, x), and

¯

γ = sup

r∈{S,M }, a∈[0,a

∞

)

γ _r (a).

Assumption 3.2. For any K > 0, there exists a continuous function V : [0, a ∞ ) → [1, ∞) such that lim a→a

∞

V(a) = +∞ and for r ∈ {S, M },

Z

[0,a

∞

)

V(a) (1 + α r (a, K))e ⁻

¹⁴

R

[0,a)

α

_r

(s,K) ds

da < ∞.

For r ∈ {S, M }, we write for convience

p _r (x, a) = e ^{− R}

^0a

^α

^r

^(x,u)du (9) the cumulative distribution of the interaction times. We define

φ(x) = 1

R

[0,a

∞

) (p _S (x, a) + p _M (x, a))da (10) and

ψ(x) = φ(x) Z

[0,a

∞

)

((γ _S (a) − β _S (a))p _S (x, a) + (γ _M (a) − β _M (a))p _M (x, a)) da. (11) Let us refer to Equations (1) and (2) in introduction for an expression of φ and ψ in terms of the random variables T r and the demographic rates. Our last assumption concerns uniqueness of the limiting equation and the fact that the limit does not reach a boundary.

For simplicity, we also assume here existence, but the limiting procedure we prove would

ensure existence up to this time when the process get close to the boundary.

Assumption 3.3. The following system of ordinary differential equations, x ⁰ (t) = (γ − β)x(t) − y(t)φ(x(t)),

y ⁰ (t) = y(t)ψ(x(t)), (12)

admits a unique global solution (x, y) ∈ C ¹ ( R + , ( R ^∗ + ) ² ) such that (x(0), y(0)) = (x ₀ , y ₀ ).

The preceding assumption holds under classical regularity assumption and in partic- ular, if φ and ψ are globally Lipschitz. Locally Lipschitz conditions are also sufficient when the system does not explode in finite time. That is enough for our purpose. Our main result can be stated as follows.

Theorem 3.4. Under Assumptions 3.1, 3.2 and 3.3, for every T > 0, the two following assertions hold :

i) the process (Ξ ^K (t), Y ^K (t, {S, M }, [0, a ∞ )) t∈[0,T ] converges in law to (x(t), y(t)) t∈[0,T]

in D ([0, T ], R ² + ),

ii) for each r ∈ {S, M }, the measure Y ^K (t, {r}, da)dt converges in law to the measure y r (dt, da) = y(t)p r (x(t), a)φ(x(t)) dt da

in the space M([0, T ] × [0, a ∞ )).

The fact that convergence of Y ^K (t, dr, da) hold on the associated Skorokhod space is left open.

3.2 Preliminary estimate and tightness

The proof is based on standard tightness and uniqueness arguments involving the occupation measures and averaging [Kur92, KKP14] and localization. The main new difficulties lie in the infinite dimension in the averaging procedure due to the age struc- ture combined with unboundedness of the interactions rates α _r inherent in our framework.

First, Lemma 2.3 above directly implies the following counterpart for the scaled pro- cess. It allows us to localize the age distribution. We set

τ _K ^K = inf {t ≥ 0 : Ξ ^K _t 6∈ (1/K, K)}.

Lemma 3.5. Under Assumption 3.1, there exists C > 0 such that for any continuous non-negative function f on [0, a ∞ ) and r ∈ {S, M } and K > 0,

E

"

Z T ∧τ

_K^K

0

Z

[0,a

∞

)

f (a)Y ^K (s, {r}, da)ds

#

≤ C(1 + T )e ^{γ ¯}

^r

^T Z

[0,a

∞

)

f ^? (a)e ⁻

1a>t0 4

R

[t0,a)

α

_r

(u,K)du

da,

where t 0 = min(1, a ∞ /2) and f ^? (a) = 1 _[0,t

₀

₎ (a) sup _[0,t

₀

_] f + 1 _[t

₀

_,a

_∞

₎ (a)f (a).

Proof. We have E

"

Z T ∧τ

_K^K

0

Z

[0,a

∞

)

f (a)Y ^K (s, {r}, da)ds

#

= 1

λ _K K ₂ E

"

Z _λ

_K

_(T ∧τ

_K^K

) 0

Z

[0,a

∞

)

f (a)Y ^K (s, {r}, da)ds

# .

Adding that λ K τ _K ^K is the exit time of (K 1 /K, K ₁ K) for X ^K and Y ^K (0, U , {S, M }, [0, a ∞ )) = bK ₂ y ₀ c and γ _r ^K (a) = λ ⁻¹ _K γ _r (a), the conclusion comes from Lemma 2.3.

We now give the counterpart of the martingales of Lemma 2.4 for the scaled process.

Recalling that F _g,f (x, µ) = g(x) + hµ, f i where g : R + → R is a bounded measurable function and f ∈ C ^1,b (X ), we set

L ^K F f,g (x, µ) = K 1 x γ (g(x + 1/K 1 ) − g(x)) + β(g(x − 1/K 1 ) − g(x)) +

Z

X

(γ r (a)f (M, 0) − β r (a)f (r, a)) µ(dr, da) +λ _K

Z

X

∂

∂a f (r, a) + α _r (a, x) (1 _r=S (g(x − 1/K ₁ ) − g(x)) + f (r, 0) − f(r, a))

µ(dr, da).

Lemma 3.6. Suppose that Assumptions 3.1 and 3.2 hold. Let g : R + → R be a bounded measurable function and f ∈ C ^1,b (X ). Then the process M ^K defined for t ≥ 0 by

M ^K (t) = F _f,g (Z ^K (t)) − F _f,g (Z ^K (0)) − Z t

0

L ^K F _f,g (Z ^K (s))ds,

is a local martingale. Besides (M ^K (t ∧ τ _K ^K )) t≥0 is a square integrable martingale and hM ^K i(t ∧ τ _K ^K )

= Z t∧τ

_K^K

0

Ξ ^K (s) γ (g(Ξ ^K (s) + 1/K 1 ) − g(x)) ² + β(g(Ξ ^K (s) − 1/K 1 ) − g(x)) ² ds

+λ _K Z t∧τ

_K^K

0

X

i∈P(s)

α _r

_i

_(s) (a _i (s), Ξ ^K (s))

×

1 _r

_i

_(s)=S (g(Ξ ^K (s) − 1/K ₁ ) − g(Ξ ^K (s))) + 1

K ₂ (f (r _i (s), 0) − f (r _i (s), a _i (s)) 2

ds

+ Z t∧τ

_K^K

0

X

i∈P (s)

γ _r

_i

_(s) (a i (s)) f (M, 0) ²

K ₂ ² + β _r

_i

_(s) (a i (s)) f(r i (s), a i (s)) ² K ₂ ²

ds.

We introduce now the measures Γ ^K _K on R + × {S, M } × [0, a ∞ ) defined a.s. for every bounded measurable functions H by

Γ ^K _K (H) = Z

R

+

Z

X

H(s, r, a)Γ ^K _K (ds, dr, da) = Z τ

_K^K

0

Z

X

H(s, r, a)Y ^K (s, dr, da)ds We also set

Ξ ^K _K (t) = Ξ ^K (t ∧ τ _K ^K ), Y _K ^K (t) = Y ^K (t ∧ τ _K ^K , {S, M }, [0, a ∞ ))

for the localized version of the processes counting preys and predators. Considering such space-time measures for proving averaging results is inspired from [Kur92, KKP14]. How- ever, we do not consider here the occupation measure of the fast variables Y ^K (t, dr, da).

Lemma 3.7. For every K > 0 and T > 0, the sequence (Ξ ^K _K , Y _K ^K , Γ ^K _K ) _K is tight in D ([0, T ], R + ) ² × M([0, T ] × X ).

Proof. On the first hand, using a domination of the process Y ^K (·, {S, M }, [0, a ∞ )) by a linear birth process, we have

sup

K≥1 E sup

t≤T

Y ^K (s, {S, M }, [0, a ∞ ))

!

< ∞. (13)

Then the first moment of the sequence of random variables (Γ ^K _K ([0, T ] × {S, M } × [0, a ∞ ))) _K is bounded and it is a tight sequence in R .

On the second hand, we can combine Assumption 3.2 with Lemma 3.5 to obtain sup

K≥1 E

"

Z τ

_K^K

∧T 0

Z

[0,a

∞

)

V(a)Y ^K (s, {r}, da) ds

#

< +∞ (14)

for r ∈ {S, M }. Then we get sup

K≥1 P (Γ ^K _K ([0, T ] × {S, M } × [a, a ∞ )) ≥ ε) ≤ ε

for a close enough to a ∞ . Moreover, as H ≥ 1, this also implies tightness of (Γ ^K _K ([0, T ] × X )) K≥1 . Lemma 1.1 of [Kur92] then entails the relative compactness of the sequence (Γ ^K _K ) K≥1 in the space of finite measures embedded with the weak (narrow) topology.

Now, we show that (Ξ ^K _K ) K≥1 is tight by using the Aldous-Rebolledo criterion.

Lemma 3.6 gives the semi-martingale decomposition Ξ ^K _K = Ξ ^K _K (0) + A ^K + M ^K , where

A ^K (t) = Z t∧τ

_K^K

0

(γ − β)Ξ ^K _K (s)ds − Z t∧τ

_K^K

0

Z

[0,a

∞

)

α S (a, Ξ ^K _K (s))Y ^K (s, {S}, da)ds, hM ^K i(t) = 1

K 1

Z t∧τ

_K^K

0

(γ + β)Ξ ^K _K (s)ds + 1 K 1

Z t∧τ

_K^K

0

Z

[0,a

∞

)

α _S (a, Ξ ^K _K (v))Y ^K (v, {S}, da)ds.

Hence, writing T ^K the set of stopping times associated to Ξ ^K _K , for any σ ∈ T ^K and h > 0, E

|A ^K (σ) − A ^K (σ + h)|

≤ h K |γ − β| + E

"

Z (σ+h)∧τ

_K^K

σ∧τ

_K^K

Z

[0,a

∞

)

α _S (a, Ξ ^K (v))Y ^K (v, S, da)dv

# .

Using again Assumption 3.2 and Lemma 3.5 with now f(a) = α S (a, K), we get

b→a lim

∞

sup

K≥1 E

"

Z τ

_K^K

∧T 0

Z

[b,a

∞

)

α _S (a, Ξ ^K (v)) Y ^K (s, {S}, da)ds

#

= 0.

Using (13) and that α S is bounded on compacts sets of [0, a ∞ ) × (0, ∞) by continuity, we obtain for any b ∈ [0, a ∞ ),

h→0 lim sup

K≥1, σ∈T

^K

, h≤δ

E

"

Z (σ+h)∧τ

_K^K

σ∧τ

_K^K

Z

[0,b]

α S (a, Ξ ^K (v))Y ^K (v, S, da)dv

#

= 0.

Combining these estimates yields

δ→0 lim sup

K≥1, σ∈T

^K

, h≤δ

E

|A ^K (σ) − A ^K (σ + h)|

= 0. (15)

Besides, for each t ≥ 0, the family of random variables (A ^K (t)) K is tight since the same integrability argument ensures that ( E (|A ^K (t)|)) _K is bounded. Proceeding analogously for the quadratic variation of M ^K and using [JM86, Theorem 2.3.2] ends the proof of tightness of (Ξ ^K _K ) K≥1 . The proof of tightness of (Y _K ^K ) K≥1 is similar since birth and death rates are bounded.

3.3 Identification of limiting values and proof of convergence

We proceed now with identification of limiting points. Recall that the survival function of interaction times is denoted by p r in (9) and response for prey is φ, see (10).

Lemma 3.8. Let T > 0, K ₀ > 0 and consider a limiting point (Ξ K

0

, Γ K

0

) of (Ξ ^K _K

₀

, Γ ^K _K

₀

) in D ([0, T ], R + ) × M([0, T ] × X ). For all but countably many K < K ₀ , it satisfies for any r ∈ {S, M }, and f continuous bounded on R + × [0, a ∞ ),

Z τ

K

0

Z

[0,a

∞

)

f (s, a)Γ K

0

(ds, {r}, da)

= Z τ

K

0

Z

[0,a

∞

)

f (s, a)p _r (Ξ K

₀

(s), a)φ (Ξ K

₀

(s)) Γ K

₀

(ds, {S, M }, [0, a ∞ ))da a.s., where

τ K = inf {t ≥ 0 | Ξ K

₀

(t) ∈ / (1/K, K)} .

Proof. To avoid the use of a sub-sequence, we assume that the sequence (Ξ ^K _K

0

, Γ ^K _K

0

) _K converges in law to (Ξ K

0

, Γ K

0

) as K → ∞. Using Skorokhod representation, we also assume that this convergence holds a.s. Following the proof of [EK09, Theorem 4.1 p.354], for all but countably many K < K ₀ , (τ _K ^K ) _K converges a.s. to τ K . Indeed, from [JS13, Proposition 2.11, Chapter VI], the hitting time τ _K ^K is a continuous function of the process Ξ ^K _K

0

, except for discontinuity points of Ξ ^K _K

0

. This set of points is at most countable, see [JS13, Lemma 2.10 b), Chapter VI].

Let us use Lemma 3.6 with g = 0 and f ∈ C ^1,b (X ) such that f (M, ·) = 0. Writing f (S, ·) = f ∈ C ^1,b ([0, a ∞ )),

M ^K (t) = 1 λ _K

( Z

[0,a

∞

)

f (a)Y _K ^K

0

(t ∧ τ _K ^K , {S}, da) − Z

[0,a

∞

)

f (a)Y _K ^K

0

(0, {S}, da) )

− Z t∧τ

_K^K

0

Z

X

H(Ξ ^K _K

₀

(s), r, a)Γ ^K _K

₀

(ds, dr, da) + 1

λ _K

Z t∧τ

_K^K

0

Z

[0,a

∞

)

(γ S (a) − β S (a))f (a)Y _K ^K

0

(s, {S}, da)ds,

is a square integrable martingale, where

H(x, S, a) = ∂ _a f (a) − α _S (x, a)f(a), H(x, M, a) = α _M (x, a)f (0). (16) Step 1. Let us prove that

K→∞ lim Z t∧τ

_K^K

0

Z

X

H(Ξ ^K _K

₀

(s), r, a)Γ ^K _K

₀

(ds, dr, da)

= Z t∧τ

_K

0

Z

X

H(Ξ K

₀

(s), r, a)Γ K

₀

(ds, dr, da) in probability Assumption 3.2 and Lemma 3.5 guarantee that

C = sup

K E

"

Z t∧τ

_K^K

0

Z

X

V(a)(1 + α _r (a, K))Γ ^K _K

0

(ds, dr, da)

#

< +∞.

Furthermore, Fatou Lemma ensures that E

"

Z t∧τ

_K^K

0

Z

X

V(a)(1 + α r (a, K))Γ _K

₀

(ds, dr, da)

#

≤ C.

The fact that V(a) → ∞ as a → a ∞ and sup

x∈K,a∈[0,a

∞

)

|H(x, r, a)|

1 + α r (a, K) < ∞ ensure by Markov inequality that for any ε > 0,

b→a lim

∞

sup

K P

Z t∧τ

_K^K

0

Z

{S,M }×(b,a

∞

)

H(Ξ ^K _K

₀

(s), r, a)Γ ^K _K

₀

(ds, dr, da) ≥ ε

!

= 0

b→a lim

∞

P

Z t∧τ

K

0

Z

{S,M}×(b,a

∞

)

H(Ξ K

₀

(s), r, a)Γ K

₀

(ds, dr, da) ≥ ε

!

= 0.

To conclude the first step, we need to prove that for any b ∈ (0, a ∞ ),

K→∞ lim Z t∧τ

_K^K

0

Z

{S,M}×[0,b]

H(Ξ ^K _K

₀

(s), r, a)Γ ^K _K

₀

(ds, dr, da)

= Z t∧τ

_K

0

Z

{S,M}×[0,b]

H(Ξ K

₀

(s), r, a)Γ K

₀

(ds, dr, da) a.s.

Indeed, we observe that |∆Ξ ^K _K

0

(s, r, a)| ≤ _K ¹ , and the limiting process Ξ K

0

is continuous and the convergence of (Ξ ^K _K

0

) _K is uniform on [0, T ]. Adding that by convergence of (Γ ^K _K

0

) to Γ K

₀

, we have

sup

K≥1

Γ ^K _K

₀

([0, T ∧ τ _K ^K ] × {S, M } × [0, b]) < +∞, we get that

K→∞ lim Z t∧τ

_K^K

0

Z

{S,M }×[0,b]

H(Ξ ^K _K

₀

(s), r, a)Γ ^K _K

₀

(ds, dr, da)

− Z t∧τ

K

0

Z

{S,M }×[0,b]

H(Ξ K

0

(s), r, a)Γ ^K _K

₀

(ds, dr, da) = 0 a.s.

Finally the fact that H is continuous and bounded on K × {S, M } × [0, b] ensures that

K→∞ lim Z t∧τ

_K^K

0

Z

{S,M }×[0,b]

H(Ξ K

₀

(s), r, a)Γ ^K _K

₀

(ds, dr, da)

− Z t∧τ

K

0

Z

{S,M }×[0,b]

H(Ξ K

₀

(s), r, a)Γ K

₀

(ds, dr, da) = 0 a.s, which ends the proof.

Step 2. We can now prove that that for any t ≥ 0 and f ∈ C ^1,b ([0, a ∞ )), we have Z t∧τ

_K

0

Z ∞ 0

H(Ξ K

₀

(s), r, a)Γ K

₀

(ds, dr, da) = 0 a.s. (17) where we recall that H is defined in (16). Indeed, (13) ensures that

λ ⁻¹ _K Z

[0,a

∞

)

f (a)Y _K ^K

0

(t, {S}, da) − Z

[0,a

∞

)

f (a)Y _K ^K

0

(0, {S}, da)

(18)

≤ Ckf k ∞

λ K

sup

t≤T

Y _K ^K

₀

(t, {S}, [0, a ∞ )), which tends to 0, in probability, as K → ∞. Similarly, in probability,

K→∞ lim 1 λ _K

Z t∧τ

_K^K

0

Z

[0,a

∞

)

(γ _S (a) − β _S (a))f(a)Y _K ^K

0

(s, {S}, da)ds = 0.

Combining this three last convergence results, we obtain that M ^K converges in law to M , given, for all t ≥ 0, by

M(t) = − Z t∧τ

K

0

Z

X

H(Ξ K

0

(s), r, a)Γ K

0

(ds, dr, da).

Process M remains a martingale. It is also a.s. Lipschitz because function H is bounded.

Consequently, it is null. It proves (17) and ends step 2.

Step 3. Using the previous step, let us prove that for r ∈ {S, M } Γ K

₀

(ds, {r}, da) = γ K

₀

(s, {r}, [0, a ∞ )) p _r (Ξ K

0

(s), a)

R

[0,a

∞

) p r (Ξ K

0

(s), w)dw daΛ _r (ds) a.s., (19) where γ K

₀

(s, {r}, ·) is a measure on [0, a _∞ ) for any s ≥ 0 and Λ _S a measure on R + . Thanks to [Kur92, Lemma 1.4], there exist s 7→ γ K

0

(s, {r}, ·) is a measurable application from [0, T ] to the space of probabilities on [0, a ∞ ), and Λ _S is a measure on R + such that

Γ K

0

(ds, {S}, da) = γ K

0

(s, {S}, da)Λ _S (ds).

As (17) holds for every t ≥ 0, focusing on functions f such that f(0) = 0, we obtain a.s.

and for Λ _S -almost all s ≤ t ∧ τ K , Z ∞

0

H(Ξ K

0

(s), S, a)γ K

0

(s, {S}, da) = 0.

In conclusion, for every f ∈ C ^1,b ([0, a ∞ )) such that f (0) = 0 and for Λ S -almost all s ≤ t ∧ τ K , we almost surely have

Z

[0,a

∞

)

(∂ _a f _S (a) − α _S (Ξ K

₀

(s), a)f (a))γ K

₀

(s, {S}, da) = 0. (20) Let us show now that this functional equation imposes the form of γ K

₀

through the solutions of the associated Poisson Equation. We proceed with a fix realization of the process and the results hold a.s. Consider s ≤ t∧τ _K . For any test function g ∈ C _c ¹ ([0, a ∞ ))

such that Z ∞

0

g(v)p S (Ξ K

0

(s), v)dv = 0, the function f defined by

f : a 7→ p _S (Ξ K

₀

(s), a) ⁻¹ Z a

0

g(v)p _S (Ξ K

₀

(s), v)dv

is well-defined for each fixed s and belongs to C ^1,b (X ). This function verifies f (0) = 0 and is solution of the Poisson equation:

∀a ∈ [0, a ∞ ), ∂ _a f(a) − α _S (Ξ K

₀

(s), a)f (a) = g(a) a.s.

By (20), it yields

Z

[0,a

∞

)

g(a) γ K

0

(s, {S}, da) = 0.

We extend this identity to g ∈ C ¹ ([0, a ∞ )) such that R ∞

0 g(v)p _S (Ξ K

₀

(s), v)dv = 0 by an approximation argument. We can then apply this identity to g : a 7→

h(a) − R

[0,a

∞

) h(v)p S (Ξ K

0

(s), v)dv for any h ∈ C ¹ ([0, a ∞ )). Using that γ K

0

(s, {S}, da) is a probability measure, we obtain that p _S (Ξ K

₀

(s), ·) is the density of the measure γ K

0

(s, {S}, ·) with respect to Lebesgue measure. It proves (19) for r = S and the case r = M is obtained similarly.

Step 4. We can now conclude. Using (17) with f ≡ 1 yields for every t ≥ 0, Z t∧τ

_K

0

Z

[0,a

∞

)

α S (a, Ξ K

0

(s))Γ K

0

(ds, {S}, da) = Z t∧τ

_K

0

Z

[0,a

∞

)

α M (a, Ξ K

0

(s))Γ K

0

(ds, {M }, da).

This implies the following equality of measures Z

[0,a

∞

)

α _S (a, Ξ K

0

(s))Γ K

0

(ds, {S}, da) = Z

[0,a

∞

)

α _M (a, Ξ K

0

(s))Γ K

0

(ds, {M }, da).

Integrating (19) over [0, a ∞ ) and using the previous equality, we obtain γ(s, {S}, [0, a ∞ ))

R ∞

0 p _S (Ξ K

₀

(s), w)dw Λ _S (ds) = γ(s, {M}, [0, a ∞ )) R ∞

0 p _M (Ξ K

₀

(s), w)dw Λ _M (ds).

Finally, we have

γ (s, {r}, [0, a ∞ ))Λ r (ds) = R

[0,a

∞

) p r (s, w)dw

p(Ξ K

0

(s)) Γ K

0

(ds, {S, M }, [0, a ∞ )), and

Γ(ds, {r}, da) = p _r (Ξ K

0

(s), a)

p(Ξ K

₀

(s)) Γ K

₀

(ds, {S, M }, [0, a ∞ )) da.

It ends the proof.