Bruno Jaffuel

The Critical Barrier for the Survival of the Branching Random Walk with Absorption

We study a branching random walk on R with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. [4] determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a particle in a given generation, we add a second order term an 1/3 to the position of the barrier for the n th generation and find an explicit critical value a c such that the process dies when a < a c and survives when a > a c . We also obtain the rate of extinction when a < a c and a lower bound for the population when it survives.

Introduction

We study a discrete-time branching random walk on R. The population forms a well-known Galton-Watson tree T , and some extra information is added: to each individual u ∈ T we attach a displacement ξ u ∈ R from the position of her parent. We set the initial ancestor ̺ at the origin, hence the individual u has position

V (u) = ̺<v≤u ξ v = |u| i=1 ξ u i ,
where |u| is the generation of u and u i the ancestor of u in generation i. We define an infinite path u through T as a sequence of individuals u = (u i ) i∈N such that ∀i ∈ N, |u i | = i and u i < u i+1 .

We denote their collection by T ∞ . Now we explain how the displacements ξ u , u ∈ T are distributed. A simple choice, with very nice properties would be to take them i.i.d. but actually everything still works in a more general setting. All individuals still reproduce independently and the same way, but we allow correlations in the number and displacements of the children of every single individual.

If we write c(u) for the set of children of u, our requirement is that the point processes {ξ v , u ∈ c(u)}, (with u running over all the potential individuals of the random tree T ) are i.i.d.

We define a barrier as a function ϕ : N → R. In the branching random walk with absorption, the individuals u such that V (u) > ϕ(|u|), i.e. born above the barrier are removed: they are immediately killed and do not reproduce.

Kesten [START_REF] Kesten | Branching Brownian motion with absorption[END_REF], Derrida and Simon [START_REF] Derrida | The survival probability of a branching random walk in presence of an absorbing wall[END_REF], [START_REF] Derrida | Quasi-stationary regime of a branching random walk un presence of an absorbing wall[END_REF], Harris and Harris [START_REF] Harris | Survival probabilities for branching Brownian motion with absorption[END_REF] have studied the continuous analog of this process, the branching Brownian motion with absorption. The understanding of what happens in the continuous setting, more convenient to handle from technical point of view, greatly helps us in the discrete one. In particular, we borrow here some ideas from Kesten [START_REF] Kesten | Branching Brownian motion with absorption[END_REF]. Biggins et al. [START_REF] Biggins | A branching random walk with barrier[END_REF] introduced the branching random walk with an absorbing barrier in order to answer questions about parallele simulations. Pemantle [START_REF] Pemantle | Search cost for a nearly optimal path in a binary tree[END_REF] and Gantert et al. [START_REF] Gantert | Asymptotics for the survival probability in a killed branching random walk[END_REF] also studied this model.

A natural question that arises is whether the process survives. This obviously depends on the walk as well as on the barrier. The case of the linear barriers has been solved by Biggins et al. [START_REF] Biggins | A branching random walk with barrier[END_REF].

Before stating their result, we need to introduce some notation:

We denote the intensity mesure of this point process by µ, and its Laplace-Stieljes transform by Φ:

Φ(t) = E   |u|=1 e -tξu   = R e -tz µ(dz).
We assume that the expected number of children Φ(0) is finite and that negative displacements occur, i.e. that µ((-∞, 0)) > 0.

We also define Ψ = log Φ, this is a strictly convex function that takes values in (-∞, +∞].

We call critical the case where This can also be written Ψ(1) = 0 and Ψ ′ (1) = 0.

Theorem 1.1 (Biggins et al. [START_REF] Biggins | A branching random walk with barrier[END_REF]). In the critical case, we have:

P (∃u ∈ T ∞ , ∀i ≥ 1, V (u i ) ≤ iε) = 0 if ε ≤ 0, > 0 if ε > 0.
The aim of this article is to refine this result by replacing the linear barrier i → iε with a more general barrier i → ϕ(i).

Given a barrier ϕ we do not know in general whether P (∃u ∈ T ∞ , ∀i ≥ 1, V (u i ) ≤ ϕ(i)) = 0 or not. Theorem 1.1 leads us to focus on barriers such that ϕ(i) i → 0. Here is the main result we will prove in this paper.

Theorem 1.2. We assume:

σ 2 := Φ ′′ (1) = E   |u|=1 ξ 2 u e ξu   < +∞. Let a c = 3 2 (3π 2 σ 2 )
1/3 . Then we have:

P ∃u ∈ T ∞ , ∀i ≥ 1, V (u i ) ≤ ai 1/3 = 0 if a < a c , > 0 if a > a c .
Unfortunately, we have not been able to conclude in the case a = a c , nor give a necessary and sufficient condition on a general barrier for a line of descent to survive below it.

While proving Theorem 1.2, we actually obtain stronger results. The two following propositions together imply the theorem.

Proposition 1.3 (surviving population). If a > a c , then the equation a = b + 3π 2 σ 2
2b 2 has two solutions in b, let b a be the one such that b a > 2ac 3 . For any ε > 0, for any N ∈ N large enough, we have with positive probability :

∀k ≥ 1, #{u ∈ T N k : ∀i ≤ N k , (a -b a )i 1/3 ≤ V (u i ) ≤ ai 1/3 } ≥ exp N k/3 (b a -ε) . Proposition 1.4 (rate of extinction). If a < a c , then there exists some constant c > 0 such that 1 n 1/3 log P ∃u ∈ T n , ∀i ≤ n, V (u i ) ≤ ai 1/3 → -c.
The constant c, which depends on a, is determined in Section 5.

About general barriers

Let ϕ : N → R be a barrier. We define a + = lim sup n→∞ ϕ(n) n 1/3 and a -= lim inf n→∞ ϕ(n) n 1/3 . We deduce from theorem 1.2 that there is extinction when a + < a c and survival when a -> a c . Making some modifications to the computations of Section 3, we can prove the following result :

Theorem 1.5. Assume a + ≥ a c . The equation a + = b + 3π 2 σ 2 2b 2 admits a unique solution b = 2ac 3 if a + = a c , and two solutions if a > a c . Let b a + ≥ 2ac 3 be the larger solution. If a -< 3π 2 σ 2 2b 2 a +
, then there is extinction.

We notice that 3π 2 σ 2 2b 2 a + ≤ 3π 2 σ 2 2b 2 ac = ac 3 < a c . When a + ≥ a c , a -> 3π 2 σ 2 2b 2 a +
, there is not always survival. For example, if ϕ(n) equals a + n 1/3 for n even and a -n 1/3 for n odd, then a + does not matter, it is easy to see that there is extinction if a -< a c : staying below this barrier is almost as difficult as for the barrier n → a -n 1/3 . The trouble comes from the fact that ϕ(n) n 1/3 is too often close to a -. Actually the condition in Theorem 1.5 is sharp in the sense that, if we choose some

a + ≥ a c and a -> 3π 2 σ 2 2b 2 a +
, we can construct a barrier ϕ satisfying lim sup n→∞ ϕ(n)

n 1/3 = a + and lim inf n→∞ ϕ(n) n 1/3 = a -such that the process survives. It suffices to take ϕ(n) = a -n 1/3 if n ∈ {N k : k ∈ N} and ϕ(n) = a + n 1/3
else, for some integer N big enough, depending on a + and a -. The proof of this is essentially identical to the proof of the lower bound contained in Section 4.

The reduction to the critical case

It is possible to apply these results to certain non critical branching random walks. We analyze in which cases this reduction is possible in Appendix A. Let t > 0 such that Φ(t) < +∞ and Φ ′ (t) < +∞. We define a new branching random walk by changing the position of

the individual z into V (z) = tV (z) + Ψ(t)|z| for all z ∈ T .
Then, with obvious notation, a straightforward computation gives :

Φ(1) = E   |u|=1 e -e ξu   = 1; Φ ′ (1) = -E   |u|=1 ξ u e -e ξu   = t Φ ′ (t) Φ(t) -Ψ(t) = tΨ ′ (t) -Ψ(t).
When we can find t * > 0 such that :

(1.1)

t * Ψ ′ (t * ) -Ψ(t * ) = 0,
then the new branching random walk is critical and we can apply Theorem 1.2, provided σ 2 := E |u|=1 ξ 2 u e -e ξu = t * 2 Φ ′′ (t * )-Ψ(t * ) 2 is finite. This last condition, which is equivalent to Φ ′′ (t * ) < ∞ will always be fulfilled when t * < ζ.

The strict convexity of Ψ implies that, when it exists, t * is unique.

The existence of t * is discussed in Appendix A.

The rest of the paper is organized as follows:

Section 2 introduces the tools we will use in the proof of our main results. Section 3 is devoted to the proof of the upper bound in Proposition 1.4, which contains the first part of Theorem 1.2.

In Section 4, we prove Proposition 1.3 which implies the second part of Theorem 1.2. In Section 5, we complete the proof of Proposition 1.4. We skip many details of technical arguments already exposed in Section 4 to obtain the lower bound and go back over some results of Section 3 in order to prove that the two bounds agree.

2 Some preliminaries

Many-to-one lemma

Since E |u|=1 e -ξu = 1 , we can define the law of a random variable X such that for any measurable nonnegative function f ,

E[f (X)] = E   |u|=1 e -ξu f (ξ u )   . Then E[X] = E
|u|=1 ξ u e -ξu so that X is centered by hypothesis. We denote

σ 2 := E[X 2 ] = Φ ′′ (1)
and make the additional assumption that σ 2 < +∞.

Let (X i ) i∈N * be a i.i.d. sequence of copies of X. Write for any n ∈ N, S n := 0<i≤n X i .

S is then a mean-zero random-walk starting from the origin.

We can now state the many-to-one lemma:

Lemma 2.1 (Biggins-Kyprianou [START_REF] Biggins | Seneta-Heyde norming in the branching random walk[END_REF]). For any n ≥ 1 and any measurable function

F : R n → [0, +∞), E   |u|=n e -V (u) F (V (u i ), 1 ≤ i ≤ n)   = E [F (S i ), 1 ≤ i ≤ n)] .

Mogul'skii's estimate

Here we state a particular case (we set the normalizing sequence) of Mogul'skii's estimate:

Theorem 2.2 (Mogul'skii [START_REF] Mogul'skii | Small deviations in the space of trajectories[END_REF]). Let S be a random walk starting from 0, with zero mean and finite variance σ 2 . Let g 1 and g 2 be continuous functions [0, 1] → R such that

(2.1) g 1 (0) ≤ 0 ≤ g 2 (0) and ∀t ∈ [0, 1], g 1 (t) < g 2 (t).
We consider the measurable event

E j = ∀i ≤ j, g 1 ( i j ) ≤ S i j 1/3 ≤ g 2 ( i j ) .
Then

lim j→∞ 1 j 1/3 log P(E j ) = - π 2 σ 2 2 1 0 dt [g 2 (t) -g 1 (t)] 2 .
Corollary 2.3. The upper bound in Theorem 2.2 is still valid with the second condition in equation (2.1) replaced by ∀t ∈ [0, 1], g 1 (t) ≤ g 2 (t).

Proof. Let g 1 , g 2 and E j be as in the statement of the corollary. For any ε > 0, replace g 2 by g 2 := g 2 + ε. For any integer j ≥ 1, g 1 and g 2 allow us to define:

E j := ∀i ≤ j, g 1 ( i j ) ≤ S i j 1/3 ≤ g 2 ( i j ) .
Now we apply Theorem 2.2 to this pair of functions:

lim sup j→∞ 1 j 1/3 log P( E j ) ≤ - π 2 σ 2 2 1 0 dt [ g 2 (t) -g 1 (t)] 2 .
Using the fact that E j ⊂ E j , we obtain lim sup j→∞ 1 j 1/3 log P(E j ) ≤ -

π 2 σ 2 2 1 0 dt [ε + g 2 (t) -g 1 (t)] 2 .
Now we make ε tend to 0 and conclude with the monotone convergence theorem.

Corollary 2.4. Under the conditions of Theorem 2.2, we have, for any ε > 0,

lim j→∞ 1 j 1/3 log P(E j ∩ {g 2 (1) -ε ≤ S j j 1/3 ≤ g 2 (1)}) = - π 2 σ 2 2 1 0 dt [g 2 (t) -g 1 (t)] 2 .
Proof. Let g 1 , g 2 and E j be as above. Fix 0 < ε < g 2 (1)g 1 (1) and A > 0. From g 1 and g 2 , we define

g 1 by ∀t ∈ [0, 1], g 1 (t) = max{g 1 (t), g 2 (1) -ε + (t -1)A}.
g 1 and g 2 define some pipe E j such that for any j ≥ 1,

E j ⊂ E j ∩ {g 2 (1) -ε ≤ S j j 1/3 ≤ g 2 (1)}.
We apply Theorem 2.2 to this new pair of functions to get lim inf

j→∞ 1 j 1/3 log P(E j ∩ {g 2 (1) -ε ≤ S j j 1/3 ≤ g 2 (1)}) ≥ - π 2 σ 2 2 1 0 dt [g 2 (t) -g 1 (t)] 2 .
Now we make A go to infinity in the last inequality and use the monotone convergence theorem to obtain the lower bound:

lim j→∞ 1 j 1/3 log P(E j ∩ {g 2 (1) -ε ≤ S j j 1/3 ≤ g 2 (1)}) ≥ - π 2 σ 2 2 1 0 dt [g 2 (t) -g 1 (t)] 2 .
The upper bound is an obvious consequence of Theorem 2.2, so the corollary is proved.

3 Upper bound for the survival probability

Splitting the survival probability

Fix a > 0.

Obviously,

P ∃u ∈ T ∞ , ∀i, V (u i ) ≤ ai 1/3 = lim n→∞ P ∃u ∈ T n , ∀i ≤ n, V (u i ) ≤ ai 1/3 . Now on, n ≥ 1 is fixed.
We now set a second barrier i → ai 1/3b i,n (with b i,n > 0 for 1 ≤ i ≤ n yet to be determined) below the first one i → ai 1/3 : if a particle crosses it, then its descendance will be likely to stay below the first one until generation n.

Let H(u) be the first time a particle u ∈ T n crosses this second barrier (H(u) = ∞ if the particle stays between the barriers until time n). We split the sum accordingly:

(3.1) P ∃u ∈ T n , ∀i ≤ n, V (u i ) ≤ ai 1/3 ≤ R ∞ + n j=1 R j ,
where

R j = P ∃u ∈ T n , H(u) = j, ∀i ≤ n, V (u i ) ≤ ai 1/3 for j = 1, . . . , n, ∞.
By Chebyshev's inequality and then Lemma 2.1, we get

R ∞ ≤ E u∈Tn 1I {∀i≤n,ai 1/3 -b i,n ≤V (u i )≤ai 1/3 } ≤ E e Sn 1I {∀i≤n,ai 1/3 -b i,n ≤S i ≤ai 1/3 } ≤ e an 1/3 P ∀i ≤ n, ai 1/3 -b i,n ≤ S i ≤ ai 1/3 . (3.2) For 1 ≤ j ≤ n, R j ≤ E   v∈T j 1I {∀i<j,ai 1/3 -b i,n ≤V (v i )≤ai 1/3 ,V (v)<aj 1/3 -b j,n}   ≤ E e S j 1I { ∀i<j,ai 1/3 -b i,n ≤S i ≤ai 1/3 ,V (S j )<aj 1/3 -b j,n} ≤ e aj 1/3 -b j,n P ∀i < j, ai 1/3 -b i,n ≤ S i ≤ ai 1/3 . (3.3) 3.2 Asymptotics for R ∞ Fix ε > 0.
In order to apply Corollary 2.3, we set b i,n := n 1/3 g( i n ) for some continuous function g : [0, 1] → [0, +∞). Then we have

(3.4) ∃N ≥ 1, ∀n ≥ N, 1 n 1/3 log P ∀i ≤ n, ai 1/3 -b i,n ≤ S i ≤ ai 1/3 ≤ - π 2 σ 2 2 1 0 dt g(t) 2 + ε.
Putting together equations (3.2) and (3.4), we get

R ∞ ≤ exp n 1/3 [a - π 2 σ 2 2 dt g(t) 2 + ε] .
Since ε is arbitrary small, we get

(3.5) lim sup n→∞ log R ∞ n 1/3 ≤ -s 1 , where (3.6) s 1 := -a + π 2 σ 2 2 1 0 dt g(t) 2 .

Asymptotics for R j

Now suppose j = j(n) := αn + 1 for some rational α ∈ [0, 1) and for values of n such that αn is an integer. Define for any t ∈ [0, 1], g α (t) := α -1/3 g(αt). We can now apply Corollary 2.3 so that for j(n) large enough, we have

1 (j -1) 1/3 log P ∀i < j, ai 1/3 -(j -1) 1/3 g α ( i j -1 ) ≤ S i ≤ ai 1/3 ≤ - π 2 σ 2 2 1 0 dt g α (t) 2 + ε, 1 (αn) 1/3 log P ∀i < j, ai 1/3 -n 1/3 g( i n ) ≤ S i ≤ ai 1/3 ≤ - π 2 σ 2 2α 1/3 α 0 du g(u) 2 + ε. (3.7)
Putting together equations (3.3) and (3.7), we get

(3.8) R α,n := R j (n) ≤ exp n 1/3 [aα 1/3 -g(α) - π 2 σ 2 2 α 0 dt g(t) 2 + ε] .
Unfortunately, these inequalities hold only for n greater than some n 0 which depends on α (and such that αn is an integer). Thus we can only apply Theorem 2.2 to a finite number of values of α at the same time.

Since α → g(α) and α → α 0 dt g(t) 2 are uniformly continuous on the compact [0, 1], we can choose For any 1 ≤ j ≤ n, we consider α := j-1 n and 0

N ≥ 1 such that ∀α 1 < α 2 ∈ [0, 1], α 2 -α 1 ≤ 1 N ⇒ | π 2 σ 2 2 α 2 α 1 dt g(t) 2 | < ε . We apply N -1 times Theorem 2.
≤ k < N such that k N ≤ α < k+1 N . Using (3.8) for α = k N and the fact that m → P ∀i < m, ai 1/3 -n 1/3 g( i n ) ≤ S i ≤ ai 1/3 is nonincreasing, we have R α,n ≤ P ∀i ≤ kn N , ai 1/3 -n 1/3 g( i n ) ≤ S i ≤ ai 1/3 exp n 1/3 [aα 1/3 -g(α)] ≤ exp n 1/3 [aα 1/3 -g(α) - π 2 σ 2 2 e α 0 dt g(t) 2 + ε] ≤ exp n 1/3 [aα 1/3 -g(α) - π 2 σ 2 2 α 0 dt g(t) 2 + 2ε] . (3.9)
As a consequence, (3.10) lim sup

n→∞,n∈N N 1 n 1/3 log n j=1 R j (n) ≤ -s 2 + 2ε.
where (3.11)

s 2 := min 0≤α≤1 -aα 1/3 + g(α) + π 2 σ 2 2 α 0 dt g(t) 2 .
Remark 3.1. This is enough to prove the extinction when a < a c in Theorem 1.2 but does not rigorously leads to the upper bound of Proposition 1.4 because of the restriction n ∈ NN (with N depending on ε). This will be fixed in Section 5.1.

Combining (3.10) with (3.5) and (3.1), we obtain lim sup

n→∞,n∈N N 1 n 1/3 log P ∃u ∈ T n , ∀i ≤ n, V (u i ) ≤ ai 1/3 ≤ -s + 2ε,
where s := min(s 1 , s 2 ).

Choice of g for the upper bound

Set a > 0 and s > 0. We are looking for a function g such that s > 0 when a < a c . Taking ε small enough and N large enough, the existence of such a function implies extinction and ends the proof the first part of Theorem 1.2.

We add the constraint g(1) = 0. Taking α = 1, we see from (3.11) and (3.6) that this implies s 2 ≤ s 1 and, as a result, s = s 2 . We choose g in such a way that the quantity -aα 1/3 + g(α)

+ π 2 σ 2 2 α 0 dt g(t)
2 which appears in (3.11) does not depend on α. Hence g is defined as the solution of the equation:

(3.12) ∀t ∈ [0, 1], -at 1/3 + f (t) + π 2 σ 2 2 t 0 du f (u) 2 = s,
where s is some positive constant, the value of which is to be set later.

Equivalently, this can be written f (0) = s and ∀t ∈ (0, 1),

(3.13) f ′ (t) = a 3 t -2/3 - π 2 σ 2 2f (t) 2 .
By the Picard-Lindelöf theorem, such an ordinary differential equation admits a unique maximal solution f defined on an interval [0, t max ) with t max ∈ (0, +∞]. And if t max < +∞, then f has limit 0 or +∞ when t goes to t max .

Remark 3.2. The fact that f ′ (0) does not exist here is not troublesome at all since the proof of the theorem, using Picard iterates, actually relies on equation (3.12).

In order to prove that there exists s such that t max = 1 and lim t→1 f (t) = 1, we get a close look to the differential equation.

First we state three simple results specific to this differential equation.

Proposition 3.3 (invariance property). Let λ > 0 and f a continuous function [0, t 0 ) → (0, +∞). Define f λ : (0, λ -1 t 0 ) → (0, +∞) by f λ (t) = λ -1/3 f (λt).
Then f satisfies equation (3.13) on (0, t 0 ) if and only if f λ does on (0, λ -1 t 0 ).

Proof. Assume that f satisfies equation (3.13) for any 0 < t < t 0 . Then for any 0 < t < λ -1 t 0 ,

f ′ λ (t) = λ 2/3 f ′ (λt) = λ 2/3 a 3 (λt) -2/3 - π 2 σ 2 2f (λt) 2 = a 3 t -2/3 - π 2 σ 2 2f λ (t) 2 .
This means that f λ also satisfies equation (3.13) for any 0 < t < λ -1 t 0 .

Conversely, assume that f λ satisfies equation (3.13) on (0, λ -1 t 0 ). We notice that if λ ′ > 0, then (f λ) λ ′ = f λλ ′ . We take λ ′ = λ -1 . Hence (f λ) λ ′ = f also satifies equation (3.13) for any 0 < t < (λλ ′ ) -1 t 0 = t 0 . Proposition 3.4. Set 0 < a 1 < a 2 and s > 0. Let f 1 and f 2 be functions [0, t max ) → (0, +∞) such that

∀0 < t < t max , ∀i ∈ {1, 2}, -at 1/3 + f i (t) + π 2 σ 2 2 t 0 du f i (u) 2 = s. Then, for all 0 ≤ t < t max , f 1 (t) ≤ f 2 (t).
Proof. It suffices to prove that, if 0 ≤ t start , 0 < a 1 < a 2 and 0 < x 1 ≤ x 2 , then there exist t next > t start such that there are functions f 1 and f 2 : [t start , t next ) → (0, +∞) such that

∀t start ≤ t < t next , ∀i ∈ {1, 2}, -a i (t 1/3 -t 1/3 start ) + f i (t) + π 2 σ 2 2 t tstart du f i (u) 2 = x i ; then, for any t start ≤ t < t next , f 1 (t) ≤ f 2 (t).
We choose t next such that the Picard interates f n i defined, for i ∈ {1, 2}, by : 2 , exist and converge on [t start , t next ). The limits f i are solutions of the integral equations for i ∈ {1, 2}.

∀t start ≤ t < t next , f 0 i (t) = x i ; ∀n ∈ N, ∀t start ≤ t < t next , f n+1 i (t) = f n i (t start ) + a i (t 1/3 -t 1/3 start ) - π 2 σ 2 2 t tstart du f n i (u)
It is easy to prove by induction on n that

∀n ∈ N, ∀t start ≤ t < t next , f n 1 (t) ≤ f n 2 (t).
Letting n tend to infinity gives us the desired conclusion.

Proposition 3.5. Let f as above. Then we are in one of the following cases: (A) t max = +∞ and f (t) → +∞ as t → +∞;

(B) t max < +∞ and f (t) → 0 as t → t max .

Proof. First notice that for any 0 < t < t max , f (t) ≤ s + at 1/3 . A consequence of this inequality is that if t max < +∞, then the limit of f when t goes to t max can only be 0. Now, suppose that t max = +∞ but that f does not go to infinity. Then there are M > 0 and a sequence (t n ) n ≥ 1 with lim n t n = +∞ such that for any n ≥ 1, f (t n ) ≤ M. We can choose n such that a 3 t

-2/3 n -π 2 σ 2 2M 2 < 0.
Then it is easy to see that f decreases after t n . Indeed, for t ≥ t n ,

f ′ (t) = a 3 t -2/3 - π 2 σ 2 2f (t) 2 ≤ a 3 t -2/3 n - π 2 σ 2 2M 2 < 0.
To be rigorous, we must consider

t * := inf t ≥ t n , f ′ (t) > a 3 t -2/3 n -π 2 σ 2 2M 2
and notice that if we assume t * < +∞, then f decreases in a neighborhood of t * and the inequality still holds on this neighborhood, which contradicts the definition of t * .

We have proved that f ′ (t) is less than a negative constant for t ≥ t n , which implies that f reaches zero in finite time.

Assume we are in the second case of Proposition 3.5. We set λ := t -1 max and define the function f λ as in Proposition 3.3 (with t 0 = t max ). We choose g = f λ and set g(1) = 0 so that g is continuous over [0, 1] and satifies (3.13) for all t ∈ (0, 1). Remark 3.6. A consequence of Proposition 3.3 is that the choice of the value s of f (0) does not matter at all. If we replace s > 0 with another s > 0, we then replace λ with λ = λ e s s 3 and finally get the same g.

So we only have to prove that, when a < a c , we are in case (B) of Proposition 3.5, and we will deduce the upper bound in Theorem 1.2. This is contained in the following: Proposition 3.7. Let f be the solution of equation (3.13) with initial condition f (0) = 1.

(i) If a > a c , then t max = +∞ and f (t) ∼ bt 1/3 as t → +∞ with b defined by b > 2ac 3 and a = b + 3π 2 σ 2 2b 2 .

(ii) If a = a c , then t max = +∞ and f (t) ∼ 2ac 3 t 1/3 as t → +∞.

(iii) If a < a c , then t max < +∞ and f (t) → 0 as t → t max .

In the proof of the proposition, we will need the following lemma:

Lemma 3.8. Assume that f is a solution on [0, +∞) of the differential equation and that:

lim sup t→+∞ f (t) t 1/3 ≤ b.
Then we have

lim sup t→+∞ f (t) t 1/3 ≤ b ′ := a - 3π 2 σ 2 2b 2 .
Proof of Lemma 3.8. Let ε > 0. By hypothesis, for any t greater than some t 0 , we have

f (t) ≤ (b + ε)t 1/3
. For some real constants c 0 and c ′ 0 and any t ≥ t 0 , we have, by equation (3.12):

f (t) ≥ c 0 + at 1/3 - π 2 σ 2 2(b + ε) 2 t t 0 du u 2/3 = c ′ 0 + a - 3π 2 σ 2 2(b + ε) 2 t 1/3 . Hence lim sup n f (t) t 1/3 ≤ a - 3π 2 σ 2 2(b + ε) 2 .
Letting ε tend to 0 ends the proof of the lemma.

Iterating Lemma 3.8, we obtain: Lemma 3.9. Assume that f is a solution on [0, +∞) of the differential equation and that for all lim sup t→+∞ f (t)

t 1/3 ≤ b 0 . We define the sequence (b n ) n∈N recursively by b n+1 := a-3π 2 σ 2 2b 2 n . Then ∀n ≥ 1, lim sup t→+∞ f (t) t 1/3 ≤ b n .
Proof of Proposition 3.7. (i) Assume a ≥ a c and let b such that a = b + 3π 2 σ 2 2b 2 . Define, for 0 ≤ t ≤ t max , f 0 (t) := bt 1/3 . Then f 0 satisfies equation (3.13) as f does, with initial condition

f 0 (0) = 0 < f (0) = s. Hence ∀0 ≤ t ≤ t max , f (t) ≥ f 0 (t).
This implies t max = +∞. Now let h = ff 0 . Then, by equation (3.12), we have, for t ≥ 0,

h(t) = s + (a -b)t 1/3 - t 0 π 2 σ 2 du 2f (u) 2 = s + (a -b - 3π 2 σ 2 2b 2 )t 1/3 + t 0 π 2 σ 2 du 2 1 f 0 (u) 2 - 1 f (u) 2 . Since a = b + 3π 2 σ 2 2b 2 , h(t) = t 0 π 2 σ 2 2 du f 0 (u) 2 - 1 f (u) 2 ≤ t 0 π 2 σ 2 2 2h(u) du f 0 (u) 3 .
We apply Gronwall's lemma and obtain, for any 0 < t 0 < t,

(3.14) h(t) ≤ h(t 0 ) exp t t 0 π 2 σ 2 du b 3 u = h(t 0 )( t t 0 ) π 2 σ 2 b 3 . Notice that π 2 σ 2 b 3 = 1 3 a 3 c b 3 .
Then if a > a c and b > 2ac 3 , the exponent in the right-hand side of (3.14) will be less than 1 3 . (ii) Now assume a = a c and b = 2ac 3 . This is the same as when a > a c , except that the exponent in the right-hand side of (3.14) is exactly 1 3 , which means that for some constant b

0 > 2ac 3 , ∀t ≥ t 0 , f 0 (t) ≤ f (t) ≤ b 0 t 1/3 .
Now apply Lemma 3.9. The result follows from that lim n b n = 2ac 3 . (iii) Assume a < a c and t max = +∞. Then, by (ii) and Proposition 3.4, we have that any b 0 > 2ac 3 , for t large enough, lim sup

t→+∞ f (t) t 1/3 ≤ b 0 .
Now we apply Lemma 3.9. If b 0 is close enough to 2ac 3 , we will have b 1 < 2ac 3 and b n → -∞ as n goes to infinity, which is absurd. We conclude that the hypothesis t max = +∞ is false, which proves the proposition.

4 Lower bound for the survival probability

Strategy of the estimate

The basic idea is to consider only the population between two barriers (below i → ai 1/3 but above i → (ab)i 1/3 ), estimate the first two moments of the number of individuals in generation n and then to use the Paley-Zygmund inequality to get the lower bound.

Unfortunately, Mogul'skii's estimate causes the apparition of a factor e o(n 1/3 ) in the estimates of the moments of the surviving population at generation n, so we will not be able to prove directly that the population survives with positive probability.

Here is how to overcome this difficulty:

Set λ > 0 such that e λ ∈ N and (v k ) k≥1 a sequence of positive integers. We consider the population surviving below the barrier i → ai 1/3 : any individual that would be born above this barrier is removed and consequently does not reproduce. For any k ∈ N, we pick a single individual z at position V (z) in generation e λk and consider the number Y k (z) of descendants she eventually has in generation e λ(k+1) .

We get a lower bound for Y k (z) by considering, instead of z, a virtual individual z in the same generation e λk but positionned on the barrier at V (z) := ae λk/3 ≥ V (z). The number and displacements of the descendants of z are exactly the same as those of z. Obviously, for any u > z, V ( u) = V (u) + ae λk/3 -V (z) ≥ V (u). Hence the descendants of z are more likely to cross the barrier and be killed, which means that Y k ( z) ≤ Y k (z).

In order to apply Mogul'skii's estimate, we add a second absorbing barrier i → (ab)i 1/3

for some b > 0 and kill any descendant of z that is born below it. This way, we obtain that, almost surely,

Z k ≤ Y k ( z) ≤ Y k (z),
where

Z k := # u ∈ T e λn : u > z, ∀e λk < i ≤ e λ(k+1) , (a -b)i 1/3 ≤ V ( u i ) ≤ ai 1/3 .
It is clear that Z k depends on z but its law and in particular

A k := P(Z k ≥ v k ) ≤ P(Y k (z) ≥ v k ) do not.
We define, for any n ≥ 1:

P n := P ∀1 ≤ k ≤ n, # u ∈ T λk e : ∀i ≤ e λk , V (u i ) ≤ ai 1/3 ≥ v k-1 .
If 1 ≤ n 0 ≤ n, then we have:

P n+1 ≥ P n (1 -(1 -A n ) v n-1 )
.

By induction, we obtain:

P n ≥ P n 0 n-1 k=n 0 (1 -(1 -A k ) v k-1 ) ≥ P n 0 n k=n 0 1 -e -v k-1 A k .
This makes Proposition 1.3 a consequence of the following lemma:

Lemma 4.1. If a > a c , and λ is large enough and such that e λ ∈ N, then

(4.1) ∞ k=0 e -v k A k+1 < +∞.
Fix θ ∈ (0, 1), for example θ = 1 2 . The Paley-Zygmund inequality, with

v k := θE[Z k ] gives (4.2) A k ≥ (1 -θ) 2 (E[Z k ]) 2 E[Z 2 k ]
.

Estimate of E[Z k ]

We set k ≥ 0 and consider only the descendance of an initial ancestor u 0 starting at time e λk at position ae λk/3 over ℓ k := e λ(k+1)e λk generations. The individuals of generation i are killed and have no descendance if they are out of the interval:

I i := [(a -b)i 1/3 , ai 1/3 ].
With (S i ) 0≤i≤ℓ k the zero-mean random walk starting from 0 defined in Lemma 2.1,

E[Z k ] = E   u>u 0 ,|u|=e λ(k+1) 1I {∀e λk <i≤e λ(k+1) ,V (u i )∈I i}   = E e S ℓ k 1I { ∀i≤ℓ k ,S i ∈I e λk +i } (4.3) ≥ e ae λ(k+1)/3 -b e λ(k+1) P (∀i ≤ ℓ k , S i ∈ I e λk +i ) . (4.4)
Define for some continuous function g : [0, 1] → (0, +∞) and 0 ≤ i ≤ ℓ k :

(4.5) g 2 (t) := a t + e λk ℓ k 1/3 , g(t) := b t + e λk ℓ k 1/3 , g 1 (t) := g 2 (t) -g(t).
Then, the lower bound in Corrolary 2.4 implies that ∀ε > 0,

∃k 0 ≥ 1, ∀k ≥ k 0 , (4.6) E[Z k ] ≥ exp ℓ 1/3 k g 2 (1) - π 2 σ 2 2 1 0 dt g(t) 2 -ε . 4.3 Estimate of E[Z 2 k ]
We split the double sum over u, v ∈ T according to the generation j of u j = u ∧ v ∈ T the lowest common ancestor of u and v:

(4.7) E[Z 2 k ] = E     u>z,v>z |u|=|v|=e λ(k+1) 1I {∀e λk <i≤e λ(k+1) ,V (u i )∈I i ,V (v i )∈I i}     = ℓ k j=0 B k,j , where (4.8)
B k,j := E   u>z,|u|=e λ(k+1) 1I { ∀e λk <i≤e λ(k+1) ,V (u i )∈I i} v>u j ,|v|=e λ(k+1) 1I { ∀e λk +j<i≤e λ(k+1) ,V (v i )∈I i}   .
Thanks to Lemma 2.1, we have:

h k,j (x) := E   v>u j ,|v|=e λ(k+1) 1I {∀e λk +j<i≤e λ(k+1) ,V (v i )∈I i} V (z) = x   = E e S ℓ k -j 1I {∀0<i≤ℓk-j,x+Si∈I e λk +i }
≤ e a(e λ(k+1)/3 -(e λk +j) 1/3 )+b e λk +j P (∀0 < i ≤ ℓ kj, x + S i ∈ I e λk+1 +i ) . (4.9) Note that by equation (4.8),

B k,j ≤ sup x∈I e λk +j h k,j (x)E [Z k ] .
We proceed to estimate sup

x∈I e λk +j P (∀0 < i ≤ ℓ k -j, x + S i ∈ I e λk+1 +i ) .
Because Theorem 2.2 can only be applied with a specified initial position, we will not apply it directly with the functions specified in equations (4.5) but we keep b e λk +i := ℓ 1/3 k g( i ℓ k ) for the same continuous g : [0, 1] → (0, +∞) as previously in this section and g 2 (t) :=

a t + e λk ℓ k 1/3 = a t + 1 e λ -1 1/3
. The upper bound (4.9) gives:

(4.10) h k,j (x) ≤ e ℓ 1/3 k " g 2 (1)-g 2 ( j ℓ k )+g( j ℓ k ) " P (∀0 < i ≤ ℓ k -j, x + S i ∈ I e λk+1 +i ) .
For any ε > 0 we set M ∈ N. We then split the interval I e λk +j = ℓ

1/3 k [g 2 ( j ℓ k )-g( j l k ), g 2 ( j ℓ k )] into M intervals J m k,j := ℓ 1/3 k [g 2 ( j ℓ k ) -m+1 M g( j ℓ k ), g 2 ( j ℓ k ) -m M g( j ℓ k )], 0 ≤ m < M. For y ∈ 1 ℓ 1/3 k J m k,j = [g 2 ( j ℓ k ) -m+1 M g( j ℓ k ), g 2 ( j ℓ k ) -m M g( j ℓ k )], P ∀0 < i ≤ ℓ k -j, ℓ 1/3 k y + S i ∈ I e λk +i = P ∀0 < i ≤ ℓ k -j, g 2 ( j ℓ k ) -g( j ℓ k ) ≤ y + S i ℓ k ≤ g 2 ( j ℓ k ) ≤ P ∀0 < i ≤ ℓ k -j, m + 1 -M M g( j ℓ k ) ≤ S i ℓ k ≤ m M g( j ℓ k ) . (4.11)
Now let j =: ℓ k α, for a fixed α ∈ (0, 1) such that e λk 0 α ∈ N for some k 0 ∈ N, and apply Theorem 2.2 using a trick similar to the one we used when we estimated R j (n) in the preceding section, by replacing any boundary function f by the corresponding f α defined by f α (t) := (1α) -1/3 g(α + (1α)t). We use in particular the equality

(1 -α) 1/3 1 0 dt g α (t) 2 = 1 α dt g(t) 2 .
This way, we obtain that, for any 0 ≤ m ≤ M, as k → ∞:

1 ℓ 1/3 k log P ∀0 < i ≤ ℓ k -j, m + 1 -M M g( j ℓ k ) ≤ S i ℓ k ≤ m M g( j ℓ k ) → - π 2 σ 2 2 M 2 (M -1) 2 1 α dt g(t) 2 .
Then for any 0 ≤ m ≤ M, we have for k larger than some k α ∈ N:

1 ℓ 1/3 k log P ∀0 < i ≤ ℓ k -j, m + 1 -M M g( j ℓ k ) ≤ S i ℓ k ≤ m M g( j ℓ k ) ≤ - π 2 σ 2 2 M 2 (M -1) 2 1 α dt g(t) 2 +ε.
Now we set the value M. We choose it large enough to have:

π 2 σ 2 2 M 2 (M -1) 2 -1 1 α dt g(t) 2 ≤ ε.
Plugging the last two inequalities into (4.11), we obtain for any 0 ≤ m ≤ M,

1 ℓ 1/3 k log sup x∈J m k,j P (∀0 < i ≤ ℓ k -j, x + S i ∈ I e λk+1 +i ) ≤ - π 2 σ 2 2 1 α dt g(t) 2 + 2ε.
Since I e λk +j = 0≤m<M J m k,j , this implies:

1 ℓ 1/3 k log sup x∈I e λk +j P (∀0 < i ≤ ℓ k -j, x + S i ∈ I e λk+1 +i ) ≤ - π 2 σ 2 2 1 α dt g(t) 2 + 2ε.
Combining this with equation (4.6) and inequality (4.10), we obtain for k ≥ max(k 0 , k α ):

1 ℓ 1/3 k log B k,j (E[Z k ]) 2 ≤ a -(e λk + j) 1/3 + e λk/3 + ℓ 1/3 k g(α) + π 2 σ 2 2 α 0 dt g(t) 2 + 3ε .
The constant k α depends on α, but with an argument similar to the one we used in the previous section, (notice that in the case α = 1 the last inequality still holds, take some N = e λk 1 for some k 1 ∈ N and apply Theorem 2.2 N -1 times for α = n N , 0 < n < N), we obtain that for k large enough, for any 0

< α = j ℓ k ≤ 1: log B k,αℓ k (E[Z k ]) 2 ≤ ℓ 1/3 k g 2 (0) -g 2 (α) + g(α) + π 2 σ 2 2 α 0 dt g(t) 2 + 3ε .
With ℓ k = e λ ℓ k-1 , we get that for k large enough, for any 0 < α = j ℓ k ≤ 1:

1

ℓ 1/3 k log B k,αℓ k (E[Z k ]) 2 v k ≤ ℓ 1/3 k g 2 (0) -g 2 (α) + g(α) + π 2 σ 2 2 α 0 dt g(t) 2 + 4ε +ℓ 1/3 k-1 g 2 (0) -g 2 (1) + 1 0 dt g(t) 2 + ε .
We combine this with equations (4.1), (4.2) and (4.7) and choose ε small enough to obtain:

(4.12) max 0≤α≤1 G λ (α) < 0 ⇒ P ∃u ∈ T ∞ , ∀i ≥ 1, V (u i ) ≤ ai 1/3 > 0 where G λ (α) := g 2 (0) -g 2 (α) + g(α) + π 2 σ 2 2 α 0 dt g(t) 2 + e -λ/3 g 2 (0) -g 2 (1) + π 2 σ 2 2 1 0 dt g(t) 2 .

Choice of g and λ for the lower bound

We denote

∀t ∈ [0, 1], f (t) := t + 1 e λ -1 1/3 .
We have g 2 = af . We choose for the width of the pipe the function g := bf . Then

G λ (α) = af (0) + (b -a)f (α) + π 2 σ 2 2b 2 α 0 dt f (t) 2 + e -λ/3 af (0) -af (1) + π 2 σ 2 2b 2 1 0 dt f (t) 2 .
Since f (1) = e λ/3 f (0) and f ′ = 1 3 f -2 , this becomes:

G λ (α) = b + 3π 2 σ 2 2b 2 -a f (α) + e -λ/3 af (0) - 3π 2 σ 2 2b 2 f (0) .
Assuming a > a c , we can choose b such that b

+ 3π 2 σ 2 2b 2 < a. Since f is increasing on [0, 1], max 0≤α≤1 G λ (α) = G λ (0) = f (0) b + 3π 2 σ 2 2b 2 -a + e -λ/3 a - 3π 2 σ 2 2b 2 .
This value is negative for sufficiently large λ (that we can choose such that we also have e λ ∈ N), which, in view of (4.12), completes the proof.

The extinction rate

Throughout this section, we assume a < a c .

Removing the condition n ∈ N N

In Section 3, we assumed that n was always a multiple of N. There were two reasons for taking such a restriction over the values of N : The first one is that it avoided heavier notation with integer parts. The second one is that it allowed to obtain equation (3.7) faster.

This restriction does not matter if we only want to prove Theorem 1.2, but in order to prove rigorously Proposition 1.4, we have to remove it. This can be achieved at the cost of some extra ε in equation (3.7).

Let α ∈ (0, 1], ε > 0 and, for any n ∈ N, j = j(n) := ⌊αn⌋ + 1. Let g and g be some continuous functions [0, 1] → [0, +∞). We define, for any t ∈ [0, 1], g α (t) := α -1/3 g(αt). We can now apply Corollary 2.3 so that for n large enough, we have

1 (j -1) 1/3 log P ∀i < j, ai 1/3 -(j -1) 1/3 g α ( i j -1 ) ≤ S i ≤ ai 1/3 ≤ - π 2 σ 2 2 1 0 dt g α (t) 2 + ε.
Using the definition of g α , the left-hand side of this inequality becomes:

LHS := 1 (j -1) 1/3 log P ∀i < j, ai 1/3 - j -1 α 1/3 g( αi j -1 ) ≤ S i ≤ ai 1/3 ;
and the right-hand side:

RHS := - π 2 σ 2 2α 1/3 α 0 du g(u) 2 + ε.
What we want is to obtain an inequality of the form LHS ≤ RHS + ε, where

LHS := 1 (j -1) 1/3 log P ∀i < j, ai 1/3 -n 1/3 g( i n ) ≤ S i ≤ ai 1/3 ; and RHS := - π 2 σ 2 2α 1/3 α 0 du g(u) 2 + ε.
It is sufficient to have LHS ≤ LHS and RHS ≤ RHS. The first inequality holds as soon as we have

∀i < j, n 1/3 g( i n ) ≤ j -1 α 1/3 g( αi j -1
).

Since 0 ≤ αi j-1 -i n ≤ 1 αn and j-1 αn ≥ 1 -1 αn , we set:

∀t ∈ [0, 1], g(t) := (1 -η) -1/3 max max(t-η,0)≤u≤t g(u),
where η > 0 is arbitrary small. This works when n ≥ 1 ηα . Since this is to be applied with α ∈ { 1 N , 2 N , . . . , N -1 N , 1} for some (large) fixed N, we have just proved that LHS ≤ LHS holds when n is large enough.

According to our choice of g, RHS ≤ RHS holds for η close enough to zero.

Consequently inequality (3.7) (with one extra ε) still holds without the condition n ∈ NN, and it is easy to verify that all the arguments and results of Section 3 are still valid.

Upper bound

It follows from the computations of Section 3 that, for any continuous function g :

[0, 1] → [0, +∞) such that g(0) = 1, lim sup n 1 n 1/3 log P ∃u ∈ T n , ∀i ≤ n, V (u i ) ≤ ai 1/3 ≤ -c g , where c g := min 0≤t≤1 g(t) + π 2 σ 2 2 t 0 du g(u) 2 -at 1/3 .
The best choice for g is the one described in the end of Section 3: it is the solution of the integral equation (3.12) with s = c g such that g(1) = 0 (or equivalently, t max = 1). We can make this choice thanks to Proposition 3.3 and Proposition 3.7(iii).

Lower bound

We directly apply the Paley-Zygmund inequality to the number

W n of individuals u ∈ T n such that. ∀i ≤ n, ai 1/3 -n 1/3 g( i n ) ≥ V (u) ≤ ai 1/3 .
Following the computations of Section 4, we obtain lim inf

n 1 n 1/3 log P ∃u ∈ T n , ∀i ≤ n, V (u i ) ≤ ai 1/3 ≥ lim inf n 1 n 1/3 log (P(W n ≥ 1) ≥) ≥ -d g ,
where

d g := max 0≤t≤1 g(t) + π 2 σ 2 2 t 0 du g(u) 2 -at 1/3 .
The optimal g would be exactly the same as in the upper bound, except that we are forced to take approximations because g must be strictly positive on [0, 1]. Since this optimal g is such that g(t) + π 2 σ 2 2 t 0 du g(u) 2at 1/3 does not depend on t, we have proved

c := sup g c g = inf g d g .
This completes the proof of Proposition 1.4. Proof. For t > 0 very small, tΨ ′ (t) -Ψ(t) < 0, since by convexity of Ψ, either Ψ ′ (0) ∈ R or Ψ ′ (0) = -∞. ∀t > 0, tΨ ′ (t) -Ψ(t) = 0 by hypothesis, and so by continuity, it is always negative. As a consequence, for any 0 < t < ζ, F ′ (t) = tΨ ′ (t) -Ψ(t) t 2 < 0;

A Extension of the results to the non critical case

F ′′ (t) = t 2 Ψ ′′ (t) + 2(Ψ(t) -tΨ ′ (t)) t 3 > 0;
which ends the proof of the lemma.

We are now ready to determine whether a branching random walk can be applied the reduction of Section 1. where ε(t) := E |u|=1 1I ξu>x min e t(x min -ξu) decreases to 0 as t increases to infinity. Like in case b), for t large enough, log (µ({x min }) + ε(t)) < 0 and F (t) < -x min .

In these three cases, we have proved that F is not decreasing, and we conclude thanks to Lemma A.1: there is some t * ∈ (0, ζ) such that equation (1.1) hold, since otherwise the lemma would imply that F is decreasing, which is false. d) When x min > -∞ and µ({x min }) ≥ 1, we still have Ψ(t) = -tx min + log (µ({x min }) + ε(t)) ≥ -x min , where ε(t) := E |u|=1 1I ξu>x min e t(x min -ξu) decreases to 0 as t increases to infinity. By convexity, Ψ ′ increases, and so converges to -x min as t goes to +∞.

Finally, for any t > 0, we proved that Ψ ′ (t) < -x min whereas Ψ(t) ≥ -tx min , hence tΨ ′ (t) < Ψ(t).

Φ( 1

 1

  2 in order to obtain equation (3.8) for simultaneously α = 1 N , 2 N , . . . , N -1 N for n ≥ n 0 for some n 0 ≥ 1 and with n ∈ NN. Observe that equation (3.8) trivially holds for α = 0.

A. 1 F 1 .

 11 When the reduction to the critical case is possibleWe assume Φ(0) < +∞ and ζ := sup{t : Φ(t) < +∞} > 0. We define, for 0 < t < ζ, If equation(1.1) does not hold for any t * , then ∀0 < t < ζ, tΨ ′ (t) < Ψ(t) and F is decreasing and convex.

2 .F

 2 It easy to construct examples for any of the cases studied below. A.1.1 Case ζ < +∞ a) If ζ < +∞ and Φ(ζ) = +∞, then, by Fatou's lemma, lim t→ζ Φ(t) = +∞. b) If ζ < +∞, Φ(ζ) < +∞ and Φ ′ (ζ) = +∞. In these two cases, it is easy to deduce from the lemma that we can find some t * ∈ (0, ζ) such that equation (1.1) holds. When ζ < +∞, Φ(ζ) < +∞ and Φ ′ (ζ) < +∞, it depends on the sign of ζΨ ′ (ζ) -Ψ(ζ): c) If ζΨ ′ (ζ)-Ψ(ζ) > 0, then by continuity we can find some t * ∈ (0, ζ) such that equation (1.1) holds.d) If ζΨ ′ (ζ) -Ψ(ζ) < 0, then it is impossible to find such a t * (because t → tΨ ′ (t) -Ψ(t)is increasing) and consequently the reduction to the critical case does not work. e) If ζΨ ′ (ζ) -Ψ(ζ) = 0, then t * = ζ works, but we still have to check whether σ 2 is finite or not.It is easy to construct examples fitting any of these five cases.A.1.2 Case ζ = +∞In this case, we can be much more precise and tell whether we can find some t * ∈ (0, ζ) such that equation (1.1) holds directly from the intensity measure µ of the point process {ξ u , |u| = 1} Define x min = inf{x ∈ R, µ((-∞, x)) > 0} the minimum of the support of µ (and -∞ if µ is not lower bounded). It is clear that lim t→+∞ F (t) = -x min . If x min > -∞ we will consider µ({x min }) the mass of the eventual atom of µ in x min . We can now state :Proposition A.2.There is some t * ∈ (0, ζ) such that equation (1.1) holds if and only ifx min > -∞ or µ({x min }) < 1.Proof. We distinguish four cases: a) If x min = -∞, then lim +∞ F = -x min = +∞. Consequently F can not be decreasing. b) If x min > -∞ and µ({x min }) = 0. We still have lim +∞ F = -x min . Almost surely, for all u ∈ T 1 , ξ u < x min , hence |u|=1 e t(x min -ξu) → 0 as t → +∞. By the monotone convergence theorem lim t→+∞ E |u|=1 e t(x min -ξu) = 0. This implies that for t large enough, x min .c) If x min > -∞ and 0 < µ({x min }) < 1, we can write F (t) = -x min + 1 t log (µ({x min }) + ε(t))


Cecile Fauvelot
email: cecile.fauvelot@univ-perp.fr

Francesca Bertozzi

Federica Costantini

Laura Airoldi

Marco Abbiati

Cécile Fauvelot

Dr Cécile Fauvelot

Lower genetic diversity in the limpet Patella caerulea on urban coastal structures compared to natural rocky habitats

Keywords: Patella caerulea, genetic diversity, microsatellites, Adriatic Sea, null alleles, coastal urbanization, artificial rocky habitats

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Human-made structures (such as sea walls, breakwaters, groynes, dykes and other rock armoured urban structures) are increasingly built in marine coastal habitats for a variety of purposes. A recent review of the status of European coastlines [START_REF] Airoldi | Loss, status and trends for coastal marine habitats of Europe[END_REF] has shown that, nowadays, 22000 km 2 of the European coastal zone are covered in concrete or asphalt, and that urban artificial surfaces have increased by nearly 1900 km 2 between 1990 and 2000 alone. Similar examples occur in other parts of the world -e.g. California [START_REF] Davis | Artificial armoured shorelines: sites for open-coast species in a southern California bay[END_REF], Australia [START_REF] Connell | Urban structures as marine habitats: an experimental comparison of the composition and abundance of subtidal epibiota among pilings, pontoons and rocky reefs[END_REF] and Japan [START_REF] Koike | The countermeasures against coastal hazards in Japan[END_REF] -where hundreds of kilometres of coasts are hardened to some extent.

In most instances, artificial hard structures are built in areas which otherwise have soft sediment habitats (e.g. breakwaters on sandy shores, Figure 1). These artificial substrata may alter native softbottom assemblages [START_REF] Martin | Ecological impact of coastal defence structures on sediment and mobile fauna: Evaluating and forecasting consequences of unavoidable modifications of native habitats[END_REF]) and promote the establishment of non-native hard-bottom species [START_REF] Bulleri | Artificial marine structures facilitate the spread of a non-indigenous green alga, Codium fragile spp. tomentosoides, in the north Adriatic Sea[END_REF][START_REF] Moschella | Low-crested coastal defence structures as artificial habitats for marine life: Using ecological criteria in design[END_REF] creating unnatural changes in species composition, abundance and diversity (Airoldi et al. 2005a,b;[START_REF] Bulleri | Role of recruitment in causing differences between intertidal assemblages on seawalls and rocky shores[END_REF]. This suggests that the expansion of urban structures may be one of the major drivers of biotic homogenization [START_REF] Mckinney | Urbanization as a major cause of biotic homogenization[END_REF]. Numerous benthic organisms dwelling on artificial structures rely on a pelagic larval phase to disperse and colonize new habitats. As a consequence, the introduction of artificial hard structure may provide new substrates for invasive species but may also generate novel ecological corridors for native hard bottom species by increasing the connectivity among isolated (e.g. by stretches of sandy habitats) and differentially adapted populations. While the spread of aquatic invasive species through human mediated introductions has received wide consideration (see [START_REF] Roman | Paradox lost: genetic diversity and the success of aquatic invasions[END_REF] for a review), surprisingly, limited attention has been paid to the possible role of marine urban coastal structures in connecting discrete populations of native hard bottom species [START_REF] Dethier | Colonization and connectivity of habitat patches for coastal marine species distant from source populations[END_REF] and in locally modifying genetic diversity in populations inhabiting artificial structures, recently made available for colonization. Genetic diversity within a population can affect the productivity, growth and stability, as well as inter specific interaction within community, and ecosystem-level processes [START_REF] Hughes | Ecological consequences of genetic diversity[END_REF].

Importance of genetic diversity in adaptation processes is well documented and crucial for species survival in highly variable environment or those subject to rapid anthropogenic changes (see [START_REF] Reusch | Ecosystem recovery after climatic extremes enhanced by genotypic diversity[END_REF] for an example). Moreover, recent studies have shown that increasing genetic diversity within species can have positive effects on coexistence of competing species [START_REF] Vellend | The consequences of genetic diversity in competitive communities[END_REF].

The aim of the present study is to explore whether urban coastal structures can affect the genetic diversity and structure of hard-bottom species. We tested this hypothesis along the coastlines of the Adriatic Sea. In this region, extensive and uncontrolled urbanization during the past century has caused the proliferation of hundreds of kilometers of hard coastal artificial structures, which are now particularly abundant along the Italian sandy shores (Figure 1, see below "Study area and species"). We focussed on the limpet Patella caerulea, one of the most common and numerically abundant intertidal species found on both artificial structures and natural rocky shores in this region.

Limpets have a key role in structuring intertidal and shallow subtidal rocky shores assemblages, and factors affecting their distribution can cause significant changes in these systems [START_REF] Jenkins | Regional scale differences in the determinism of grazing effects in the rocky intertidal[END_REF]. We used a comparative spatial framework (artificial hard structures versus natural rocky shores) and microsatellite molecular markers to examine genetic diversity and structure of samples of the limpet Patella caerulea. Based on a hierarchical sampling design replicated in several locations (Figure 2), we specifically tested 1) for possible differences in the genetic structuring of populations between natural shores and artificial structures, and 2) whether the genetic diversity of populations on artificial structures was reduced compared to populations on natural reefs, as could be expected from the recent founding of artificial substrates, or alternatively enhanced through the increased number of human mediated introduction vectors.

MATERIAL AND METHODS

Study area and species

Within the Adriatic Sea, the Italian coast consists of a sandy flat coastal system almost uninterrupted, in contrast to the prevailingly rocky shores of the Balkans. Along the Italian shoreline, natural hard-bottom habitats are scarce and represented by isolated rocky promontories (from North East to South: Sistiana/Miramare, Gabicce, Conero and Gargano, see Figure 2).

Human-made structures (mainly rock-armoured breakwaters, but also groynes, seawalls and harbour jetties) have proliferated on these sandy coasts along hundreds of kilometers of coast (Figure 1 and 2), with most coastal defense structures built since the 80ies [START_REF] Cencini | Physical processes and human activities in the evolution of the Po delta, Italy[END_REF]. The artificial structures included in the present study were offshore detached breakwaters, built with large blocks of quarried rock (mainly limestone), and set on shallow sediments. The assemblages and main ecological characteristics of urban coastal structures in this regions are described in [START_REF] Bacchiocchi | Distribution and dynamics of epibiota on hard structures for coastal protection[END_REF], Airoldi et al. (2005b), [START_REF] Bulleri | Artificial marine structures facilitate the spread of a non-indigenous green alga, Codium fragile spp. tomentosoides, in the north Adriatic Sea[END_REF]. Information on the geomorphology, hydrology and environmental characteristics of the Adriatic Sea can be found in [START_REF] Poulain | Adriatic Sea surface circulation as derived from drifter data between 1990 and 1999[END_REF].

The limpet Patella caerulea, is common along the Adriatic coastline. It is patchily distributed and tends to be up to three times more abundant on artificial structures than on natural rocky shores, reaching on some structures peak densities above 600 ind.m -2 (Airoldi et al. unpublished data). P. caerulea is a sedentary species, and colonises new isolated habitats, such as those provided by artificial urban structures, by means of dispersing larvae. Its spawning period is ranging from September to April, with a peak in mid winter [START_REF] Bacci | Correlations between characters and environmental conditions in Patella of caerulea group[END_REF]Airoldi et al. unpublished data). Limpets are long-lived broadcast spawners. After a brief embryonic period, offspring hatch as free-swimming trochophores [START_REF] Buckland-Nicks | ) Atlas of Marine Invertebrate Larvae[END_REF]. Little information is available on life history and effective dispersal of P. caerulea. Larval duration and behaviour is not known and in the closely related species Patella vulgata the larval period is up to 12 days long with a precompetency period of 4 days [START_REF] Dodd | Artificial fertilisation, larval development and metamorphosis in Patella vulgata L. and Patella caerulea L[END_REF].

Sampling

Limpets were sampled at mid intertidal levels (10 to 30 cm above Mean Low Water Level).

Sampling was conducted repetitively in different locations following a hierarchical design (Figure 2). At each of the selected locations where natural rocky shores occurred (Trieste, Ortona, Gallipoli and Split) limpets were collected from 2 natural sites approximately 2 km apart. We included the two sampling sites of Split and Gallipoli in order to acquire a genetic picture of P. caerulea populations in prevailing natural rocky shores. In Trieste and Ortona, where both artificial and natural rocky substrata occur, limpets were also sampled on artificial structures at 2 sites, to provide a comparative framework to test for genetic differences between artificial and natural substrata.

Artificial structures were few 100m apart from natural rocky coasts, and were spaced about 2 km apart, similarly to the natural sites. One additional sampling was carried out following the same design in Cesenatico where only artificial structures are present and the closest natural rocky shores is > 40 km apart. Sampling was carried out during the summers 2002 (for Cesenatico, Trieste and Ortona) and 2004 (for Gallipoli and Split). For each sampling site, either on natural rocks or on artificial structures, 21 to 50 specimens of P. caerulea were randomly collected, for a total of 549 limpets. Specimens collected were generally larger than 15 mm, thus not including juveniles. Live specimens were transported to the laboratory, foot muscle were cut and stored at -80°C until processing.

Microsatellite isolation and genotyping

A dinucleotide-enriched partial genomic library has been constructed using the FIASCO protocol [START_REF] Zane | Strategies for microsatellite isolation: a review[END_REF]. Genomic DNA was extracted from frozen foot muscle tissue of a single individual using the CTAB extraction procedure [START_REF] Winnepenninckx | Extraction of high molecular weight DNA from molluscs[END_REF] as described in [START_REF] Costantini | Fine-scale genetic structuring in Corallium rubrum (L.): evidences of inbreeding and limited effective larval dispersal[END_REF]. Following extraction, DNA was simultaneously digested with MseI, ligated to MseI-adaptors (5'-TACTCAGGACTCAC -3'/ 5' -GACGATGAGTCCTGAG -3') and amplified with MseI adaptor specific primers (5'-GATGAGTCCTGAGTAA(CATG)-3': hereafter referred as MseI-N). The 20l PCR reaction contained 1x PCR buffer (Promega), 1.5mM MgCl 2 , 120ng primer MseI-N, 0.2mM of each dNTP, 0.4 units Taq polymerase (Promega) and 5 l of a 1/10 dilution of the digested-ligated product. PCRs were carried out in a GeneAmp® PCR System 2700 (Applied Biosystems): 94 °C 30 s, 53 °C 1 min, 72°C 1 min for 20 cycles. Amplified DNA was hybridised with a biotinylated probe (AC) 17 (denaturation of 3 min at 95°C followed by a 15 min annealing at room temperature), selectively captured using streptavidine-coated beads (Roche) and separated by a magnetic field. DNA was eluted from the beads-probe with TE 1x buffer (Tris-HCl 10 mM, EDTA 1mM, pH 8) at 95 C° for 5 min, precipitated with sodium acetate and ethanol, reamplified by 30 cycles of PCR using the MseI-N primer under the conditions described above, and cloned using the TOPO-TA cloning kit (Invitrogen) following the manufacturer's protocol.

Recombinant clones were screened by PCR amplification with M13 forward-reverse primers and sequenced using the BigDye Terminator Cycle Sequencing kit (Applied Biosystem) and resolved on a ABI 310 Genetic Analyser (Applied Biosystem). About 200 colonies were screened and sequenced for the presence of simple sequence repeats.

Analyses revealed the occurrence of repeats in 55 clones. After excluding loci with too short flanking regions, primers for more than 40 loci were designed using the PRIMER 3 program (Rozen and Skaletsky 1998). Primer pairs were then optimised for PCR amplification testing over a range of annealing temperatures and MgCl 2 concentrations. Excluding loci that failed to amplify or resulted in monomorphic patterns, five polymorphic dinucleotide microsatellite remaining loci were reliably amplified in all tested individuals (Table 1).

For all collected P. caerulea, DNA was isolated as described above and samples were screened for variation at the five loci newly isolated and optimized. The 20l PCR reaction contained about 50ng of genomic DNA, 1.0-1.5mM MgCl 2 (Table 1), 0.5M of each primer, 0.2mM of each dNTP, 10mM Tris-HCl (pH 9), 50mM KCl, 0.1% Triton X-100 and 1U of Taq polymerase (Promega). PCR reactions were performed on a GeneAMP PCR System 2700 (Applied Biosystems): denaturation for 3 min at 94°C, followed by 30 cycles of 30s at 94°C, 30s at 55°C, and 30s at 72°C, and a final holding at 72°C for 5 min. Amplified fragments were run on an ABI310 automated Genetic Analyser (Applied Biosystems), using forward primers 5'-labelled with 6-FAM, HEX or TAMRA (MGW Biotech) and the ROX HD400 (Applied Biosystems) as internal standard. Genotyping of individuals was performed by allele sizing using the GENESCAN Analysis Software v. 2.02 (Applied Biosystems).

Data analysis

Observed heterozygosity (H O ) and unbiased gene diversity (H S , Nei 1987) were calculated within each population for each locus and overall loci in GENETIX (Belkhir et al. 2004), and multilocus allelic richness (Ar, El Mousadik and Petit 1996) was computed in FSTAT v.2.9.3 (Goudet 1995[START_REF] Goudet | FSTAT, a program to estimate and test gene diversities and fixation indices Version 2.9.3[END_REF]). Significant differences in genetic diversity (H O , H S , and Ar) among groups of samples (related to natural versus artificial habitats) were tested using a permutation procedure (10000 iterations) in FSTAT. Linkage disequilibrium between loci, and deviations from Hardy-Weinberg (HW) expectations were tested using Fisher's exact tests based on Markov chain procedures in GENEPOP v.3.4 [START_REF] Raymond | GENEPOP: a population genetic software for exact tests and ecumenicism[END_REF] as implemented for online uses (http://genepop.curtin.edu.au/). Significance levels for multiple comparisons of loci across samples were adjusted using a standard Bonferroni correction [START_REF] Rice | Analysing tables of statistical tests[END_REF]). The presence of null alleles was examined by estimating null allele frequencies for each locus and sample following the Expectation Maximization (EM) algorithm of [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] using FREENA [START_REF] Chapuis | Microsatellite null alleles and estimation of population differentiation[END_REF].

In order to reveal the presence (if any) of genetic bottleneck signatures in the 14 samples populations, we used the M ratio of number of alleles k divided by the allelic size range r, averaged across all loci in each sample [START_REF] Garza | Detection of reduction in population size using data from microsatellite loci[END_REF]. This ratio calculated over all loci for each sample using the program M_P_VAL [START_REF] Garza | Detection of reduction in population size using data from microsatellite loci[END_REF] is intended to quantify gaps in the allele size frequency distribution resulting from loss of alleles through bottlenecking. The observed values of M average over all loci were then compared to the equilibrium distribution of M simulated according to the method described in [START_REF] Garza | Detection of reduction in population size using data from microsatellite loci[END_REF], and given values of theta, ps (proportion of one-step mutations) and g (average size of non one-step mutations) set to 2, 0.8 and 3.5 respectively [START_REF] Garza | Detection of reduction in population size using data from microsatellite loci[END_REF]. If the observed value of M is lower than the critical value of M, Mc, (defined such that only 5% of the simulation values fall below), it is taken as evidence that the sample is from a population that had experienced a recent bottleneck/founding. Genetic divergences among samples were estimated using the F ST estimates of [START_REF] Weir | Genetic signature of recent glaciations on populations of a near-shore marine fish species (Syngnathus leptorhynchus)[END_REF] and following the so-called ENA method described in [START_REF] Chapuis | Microsatellite null alleles and estimation of population differentiation[END_REF] since the presence of null alleles was found (see results). The null allele frequencies are estimated based on Hardy-Weinberg equilibrium, the genotypes are adjusted based on the null allele frequencies, and the ENA method provides unbiased F ST estimates based on the adjusted data set. These calculations were conducted using FREENA [START_REF] Chapuis | Microsatellite null alleles and estimation of population differentiation[END_REF]. Owing to deviation from Hardy-Weinberg equilibrium, genotypic differentiation among samples was tested with an exact test (Markov chain parameters: 1000 dememorizations, followed by 1000 batches of 1000 iterations per batch), and the P-value of the log-likelihood (G) based exact test [START_REF] Goudet | Testing differentiation in diploid populations[END_REF] was estimated in GENEPOP.

Significant threshold values were adjusted with a sequential Bonferroni correction [START_REF] Rice | Analysing tables of statistical tests[END_REF]) that corrects for sampling error associated with multiple tests.

In order to examine the partition of the genetic variance among limpet samples based on the type of habitat to further test the impact of artificial reefs in possibly increasing populations' connectivity, an analysis of molecular variance (AMOVA) [START_REF] Excoffier | Analysis of molecular variance inferred from metric distances among DNA haplotypes: application to human mitochondrial DNA restriction data[END_REF]) implemented in ARLEQUIN version 3.1 [START_REF] Excoffier | Arlequin ver. 3.0: An integrated software package for population genetics data analysis[END_REF]) was conducted on the original dataset (i.e. not adjusted for null alleles).

RESULTS

For the five reliably amplified and analyzed microsatellite loci, number of alleles ranged from 13 for Pc11 to 30 for Pc38. Over all microsatellite loci, highly significant multilocus deviations from HW proportions were observed in all 14 samples (Table 2). At single loci, nearly all comparisons (66 out of 70) showed heterozygote deficiencies, from which 46 showed significant heterozygote deficiencies after Bonferroni corrections. In particular, three of the five scored microsatellites (Pc15, Pc36 and Pc73) showed strong heterozygote deficiencies in all 14 samples (i.e. equally affecting all samples), suggesting the presence of null alleles, while the two others (Pc11 and Pc38) were in HW equilibrium in nearly all samples (though H O < H S in nearly all cases). Assuming HWE, estimated null allele frequencies (R) ranged among loci from 0 to 0.66 (Table 2). The number of expected null homozygotes within samples based on HW equilibrium (N*R 2 ) was significantly higher than the average number of observed null homozygotes for Pc15, Pc38 and Pc73 (paired t-test, P = 0.004, P = 0.026 and P < 0.001 respectively), suggesting that although null alleles are present in the dataset, they were overestimated.

Over all loci, allelic richness within samples based on a minimum sample size of 12 diploid individuals (i.e. the number of genotypes at Pc36 in Split1) ranged from 8.09 in Tri1A to 9.42 in Ort1N (Table 2, Figure 3A) and gene diversity from 0.76 in Tri1A to 0.86 in Spl2N (Table 2).

Allelic richness (Figure 3B) and gene diversity were both significantly higher in natural habitat (average and standard deviation: Ar = 8.98 ± 0.29; H S = 0.836 ± 0.01) than in artificial habitat (average Ar = 8.40 ± 0.25; H S = 0.805 ± 0.02; p-values associated with the permutation procedure: P = 0.0008 and P = 0.0004, respectively). Among the four sampled sites for which both natural and artificial habitats were sampled (Tri1, Tri2, Ort1 and Ort2), allelic richness based on a minimum sample size of 22 individuals (Figure 3C) and gene diversity were both significantly higher in natural (Ar = 10.97 ± 0.55, H S = 0.838 ± 0.01) than in artificial habitats (Ar = 10.16 ± 0.06, H S = 0.805 ± 0.03; permutation procedure, P = 0.014 for both tests). Conducting the same comparisons, but only with Pc11 and Pc38 (the two loci in HWE), results are similar with both allelic richness (based on 21 individuals) and gene diversity being significantly higher in natural habitat (average and standard deviation: Ar = 11.77 ± 0.99; H S = 0.846 ± 0.02) than in artificial habitat (average Ar = 10.71 ± 0.27; H S = 0.824 ± 0.01; p-values associated with the permutation procedure: P = 0.006 and P = 0.005, respectively).

None of the sampled populations experienced a recent bottleneck since none of the M ratios for individual sites fell under the lower 5% of the distribution of simulated M values. The M ratio for the samples ranged from 0.67 for Gal1 to 0.87 for Gal2 (Mc = 0.57 for a sample size of 40 individuals, 5 loci and given the three parameters used for the simulations). Over all 14 samples, the multilocus F ST estimate was low (0.0094) though the overall genotypic differentiation was significant (P < 0.0001). Pairwise F ST estimates among the 14 samples ranged from 0 to 0.032 (between Tri1A and Gal1N) and significant genotypic differentiations among samples after sequential Bonferroni corrections were found in 16 out of 91 comparisons (Table 3).

The sample Tri1A appeared the most differentiated from all other samples with pairwise F ST estimates of 0.013-0.032. Over all the five sampling locations (i.e. when pooling samples according to the sampling location), multilocus F ST estimate decreased to 0.0051 (P < 0.0001).

The AMOVAs conducted on the dataset not adjusted for the presence of null alleles showed a weak but significant differentiation among limpet samples (F ST = 0.016, P < 0.001). The nuclear variance attributed to the type of habitat was not significant either across all samples (Variance component = 0.003, P = 0.152) or including only the sites where both natural and artificial sites were sampled (Variance component = -0.007, P = 0.972).

DISCUSSION

Urban coastal structures offer suitable substrata for the colonization of P. caerulea, up to the point that at some sites (e.g. Cesenatico) this limpet is three times more abundant on recently built urban coastal structures than on nearby natural rocky shores (Airoldi et al., unpublished data). At the same time, the present results showed that the genetic diversity within populations of P. caerulea is significantly smaller on artificial structures than on natural reefs. No evidence of genetic differentiation between artificial and natural substrates was found at the five neutral molecular markers studied, and a subtle genetic structure was found over all Adriatic samples.

Hardy-Weinberg equilibrium and null alleles

Nearly all loci at all sites showed heterozygote deficiencies, with extremely strong deficiencies observed at three loci (Pc15, Pc36 and Pc73). Null alleles were present at these three loci, as revealed by the occurrence of null homozygotes (i.e. non amplifying individuals at some loci), but their occurrence appeared overestimated assuming HW equilibrium within samples. Though likely overestimated null allele frequencies in our study are high (up to 0.66), they fall in the range of null allele frequencies presented in [START_REF] Dakin | Microsatellite null alleles in parentage analysis[END_REF] based on 74 microsatellite loci from a wide range of organisms, notably with large effective population sizes [START_REF] Chapuis | Microsatellite null alleles and estimation of population differentiation[END_REF].

Although null alleles lead to underestimated genetic diversity within samples [START_REF] Paetkau | The molecular basis and evolutionary history of a microsatellite null allele in bears[END_REF], it is a minor source of error in estimating heterozygosity excess for the detection of bottlenecks [START_REF] Cornuet | Description and power analysis of two tests for detecting recent population bottlenecks from allele frequency data[END_REF] and in parental assessments [START_REF] Dakin | Microsatellite null alleles in parentage analysis[END_REF].

Moreover, though estimates of differentiation and the probability of detecting genetic differences among populations both diminished when locus heterozygosities are high and data corrected for null alleles [START_REF] O'reilly | Inverse relationship between FST and microsatellite polymorphism in the marine fish, walleye pollock (Theragra chalcogramma): implications for resolving weak population structure[END_REF][START_REF] Peijnenburg | Spatial and temporal genetic structure of the planktonic Sagitta setosa (Chaetognatha) in European seas as revealed by mitochondrial and nuclear DNA markers[END_REF]present results), in the presence of null alleles, F ST estimates are unbiased in the absence of population structure [START_REF] Chapuis | Microsatellite null alleles and estimation of population differentiation[END_REF]. This is likely the case in our study since we found that adjusting our data set according to the presence of null alleles did not alter our conclusions regarding the low levels of genetic structure (overall mutlilocus F ST estimated from adjusted data set in FreeNA = 0.009; overall mutlilocus F ST estimated from original data set in ARLEQUIN = 0.016). Heterozygote deficiencies have already been observed in P. caerulea populations analysed using allozymes with no null homozygotes observed [START_REF] Mauro | Artificial environmental conditions can affect allozymes genetic structure of the marine gastropod Patella caerulea[END_REF]. Consistency between microsatellite and allozyme data suggest that heterozygote deficiencies may be partially explained by a Wahlund effect (i.e. fine scale genetic patchiness), a common feature in limpets, as well as in other marine invertebrates (e.g. [START_REF] Côrte-Real | Population differentiation and taxonomic status of the exploited limpet Patella candei in the Macaronesian islands (Azores, Madeira, Canaries)[END_REF][START_REF] Costantini | Fine-scale genetic structuring in Corallium rubrum (L.): evidences of inbreeding and limited effective larval dispersal[END_REF][START_REF] Hurst | Comparison of allozyme and mitochondrial DNA spatial differentiation in the limpet Patella vulgata[END_REF][START_REF] Johnson | Pattern beneath the chaos: the effect of recruitment on genetic patchiness in an intertidal limpet[END_REF][START_REF] Pérez | Development of microsatellite loci for the black-footed limpet, Patella depressa, and cross-amplification in two other Patella species[END_REF]. Such localised genetic heterogeneity could result from spatial or temporal heterogeneity in the genetic composition of recruits, or from postsettlement selection [START_REF] Johnson | Pattern beneath the chaos: the effect of recruitment on genetic patchiness in an intertidal limpet[END_REF].

Genetic structure of Adriatic P. caerulea populations P. caerulea population genetic analysis at five neutral molecular markers revealed a weak but significant structure in the Adriatic Sea, mostly associated with the distinctiveness of one Trieste sample (Tri1A). The mean multilocus F ST estimate was very low over all the 14 sample sites (0.009, P < 0.0001), comparable to what was found by [START_REF] Mauro | Artificial environmental conditions can affect allozymes genetic structure of the marine gastropod Patella caerulea[END_REF] using allozymes across the same region and similar spatial scales (0.007, P >0.05). Also, a lack of significant differentiation of the Trieste sample from Sicily samples was observed using allozymes [START_REF] Mauro | Artificial environmental conditions can affect allozymes genetic structure of the marine gastropod Patella caerulea[END_REF]. Therefore, the significant genetic differentiation observed between Tri1A and most of the samples may rather be due to a lower genetic diversity in this sample as compared to all others (Table 2, [START_REF] Chapuis | Microsatellite null alleles and estimation of population differentiation[END_REF] or a sampling bias associated with a Wahlund effect, also suggested by the observed heterozygote deficits (see above).

The fact that we observed only a slight significant genetic differentiation between samples located along the Italian coats, and no significant differentiation between the East and West Adriatic coasts suggests that P. caerulea forms here a large unique population. This pattern further suggests that P. caerulea planktonic larvae allow enough dispersion to cause genetic homogeneity across the study area. P. caerulea may therefore differ in life history traits compared to other limpets, e.g. P. vulgata, P. candei, P. rustica [START_REF] Côrte-Real | Population differentiation and taxonomic status of the exploited limpet Patella candei in the Macaronesian islands (Azores, Madeira, Canaries)[END_REF][START_REF] Sá-Pinto | Patterns of colonization, evolution and gene flow in species of the genus Patella in the Macaronesian Islands[END_REF] for which structured genetic variation has been observed at similar spatial scales. An alternative explanation could be related to the geological history of the Adriatic Sea. During the Last Glacial Maximum (about 18.000ya) the sea level was about 100 m below the actual mean water level, and most of the Adriatic Sea bed was dried [START_REF] Dondi | Recent geological evolution of the Adriatic Sea[END_REF][START_REF] Thiede | A glacial Mediterranean[END_REF]. The sea water invaded the Adriatic during the last 10.000 years and the colonization by the marine flora and fauna is very recent.

Genetic similarities in Adriatic samples of P. caerulea may reflect past founder effects linked with the colonization of the Adriatic Sea after the Pleistocene glaciation. Indeed, several studies have recently stressed the relevance of palaeoecological events in determining the genetic patterns in marine populations (e.g. [START_REF] Fauvelot | Reductions in the mitochondrial DNA diversity of coral reef fish provide evidence of population bottlenecks resulting from Holocene sea-level change[END_REF][START_REF] Imron | Pleistocene isolation and recent gene flow in Haliotis asinina, an Indo-Pacific vetigastropod with limited dispersal capacity[END_REF][START_REF] Virgilio | Deep phylogenetic splits and disjunct haplotype distribution suggest cryptic speciation in Hediste diversicolor (Polychaeta: Nereididae)[END_REF]Wilson 2006).

Consequently, observed genetic patterns of P. caerulea in the Adriatic Sea likely reflect the interaction between historical events (long-term barriers followed by range expansion associated with Pleistocene sea level changes) and contemporary processes (gene flow modulated by life history and oceanography).

Genetic diversity of P. caerulea populations on artificial and natural substrates

One of the main outcomes of our study was the lower genetic diversity in populations from artificial structures compared to those from natural habitats. Indication of important effects of artificial substrata on the genetic structure of this limpet also comes from a previous study of [START_REF] Mauro | Artificial environmental conditions can affect allozymes genetic structure of the marine gastropod Patella caerulea[END_REF], which found significant differences in the genetic structures of P. caerulea between artificial structures and natural rocky shores at two enzymatic systems out of twelve under study (AAT* and SOD-1*), though no differences in genetic diversity were observed among samples.

Altered genetic patterns and diversity may be expected in small, isolated, recently founded populations [START_REF] Bradshaw | Low genetic diversity in the bottlenecked population of endangered non-native banteng in northern Australia[END_REF][START_REF] Mcelroy | Extinction-colonization dynamics structure genetic variation of spottedsunfish (Lepomis punctatus) in the Florida Everglades[END_REF][START_REF] Spencer | Experimental evaluation of the usefulness of microsatellite DNA for detecting demographic bottlenecks[END_REF] or in small founding populations of introduced species (Allendorf and Lundquist 2003; but see Roman and Darling Urban structures and other artificial substrata are often uncritically claimed as reasonable mimics of natural hard-bottom habitats and valuable replacements for the habitats that they damage. Our results contribute to the growing body of evidence showing that although artificial structures attract and support species typical of hard bottoms, they are not analogues of natural rocky habitats (see among others [START_REF] Bulleri | Role of recruitment in causing differences between intertidal assemblages on seawalls and rocky shores[END_REF][START_REF] Glasby | Urban structures as marine habitats[END_REF][START_REF] Moreira | Seawalls do not sustain viable populations of limpets[END_REF][START_REF] Moschella | Low-crested coastal defence structures as artificial habitats for marine life: Using ecological criteria in design[END_REF].

They can alter not only the identity and nature of marine coastal landscapes and the distribution of species, but also the genetic diversity of populations at local to regional scales. This is particularly important because the management of sea walls and similar artificial structures is generally carried out at local scales, without careful consideration of possible effects at larger spatial scales (Airoldi et al. 2005a). Future work should attempt to characterize more deeply how the type, quality and spatial arrangement (e.g. location relative to natural habitats and other artificial habitats) of fragmented artificial urban substrates affect the dispersal, distribution and genetic structure of species at a regional landscape scale, and the implications of these changes on the functioning of coastal marine systems at all spatial scales.

FIGURE LEGENDS

Gal1N

-0 . 0 0 1

Figure 1 :

 1 Figure 1: Aerial view of the urban structures along the coasts of the Adriatic Sea (photo by Benelli,

Figure 2 :Figure 3 :

 23 Figure 2: Location of sampling areas of Patella caerulea in the Adriatic Sea. Within each of the

Figure 1 :

 1 Figure 1: Aerial view of the urban structures along the coasts of the Adriatic Sea (photo by Benelli,

Figure 2 :

 2 Figure 2: Location of sampling areas of Patella caerulea in the Adriatic Sea. Within each of the

Figure 3 :

 3 Figure 3: Mean allelic richness per locus (Ar) based on five analysed microsatellite loci (A) within

Table 1 :

 1 Primer sequences, repeat motif and amplification details for the five microsatellite loci specifically developed for Patella caerulea. Concentrations of MgCl 2 are given in mM.

	Locus Accession	Primer sequences (5'-3')	Repeat motif	MgCl 2 Cycles
		no.
	Pc11	AY727872	F : TTACGAAGCCCCAACTTCAC	(AC) 3 GC(AC) 7	1.5	30
			R : AAGCCAGGGATAATGACACG
	Pc15	AY727873	F : CCTTCTTCATGGGGACTTCA	(TG) 12 (TATG) 4 (TG) 12	1.5	30
			R : GCCCCAAAAACAATAGGGAT
	Pc36	AY727874	F : GAACTAGCCGTGCCAATATGAT	(CT) 16	1	28
			R : GGTCGCTTCTGAGAAATGAAAT
	Pc38	AY727875	F : GCTAATCTTTCAACGTATTTTT	(AG) 18 (AC) 6	1.5	30
			R : GGTGTGGCTTGGAGATA
	Pc73	AY727876	F : TGAAACAATATTCGCTGCTAGG	(AC) 11 CA(AC) 3 CC(AC) 5	1	27
			R : GCCCCAACGTAAAAATAACAGA

Table 2 :

 2 Genetic diversity within Patella caerulea samples. n: total number of individuals genotyped, N : number of genotypes per locus ; Ar: allelic richness per locus and mean allelic richness per locus computed over all loci; H S : gene diversity (Nei 1987); H O : observed heterozygosity; R: null alleles frequency (Demspter estimator); significant deviations from Hardy-Weinberg equilibrium are indicated by an asterix following the H O .

		Trieste				Cesenatico	Ortona				Split		Gallipoli
		Tri1A Tri1N Tri2A Tri2N	Ces1A Ces2A	Ort1A Ort1N Ort2A Ort2N	Spl1N Spl2N	Gal1N Gal2N
	n	40	40	40	36	50	40	48	44	47	48	22	48	21	25
	Pc11
	N 40	40	39	34	50	40	48	44	47	48	22	48	21	25
	Ar 5.80	5.61	6.40	6.48	6.32	6.64	6.21	7.42	6.44	6.95	4.53	6.62	5.62	7.23
	H S 0.74	0.76	0.79	0.76	0.72	0.76	0.76	0.81	0.75	0.79	0.72	0.82	0.70	0.79
	H O 0.58	0.68	0.72	0.56	0.60	0.65	0.77	0.75	0.79	0.83	0.64	0.63*	0.48	0.52*
	R 0.08	0.03	0.08	0.19	0.05	0.06	0.00	0.02	0.00	0.00	0.07	0.12	0.12	0.15
	Pc15
	N 38	34	34	27	47	34	39	37	37	38	18	44	16	19
	Ar 5.60	8.31	6.30	6.40	6.77	6.27	6.79	7.26	6.13	7.27	5.76	7.54	8.56	5.93
	HS 0.51	0.78	0.77	0.78	0.70	0.69	0.74	0.78	0.76	0.80	0.74	0.81	0.83	0.79
	H O 0.13*	0.21*	0.21*	0.26*	0.26*	0.26*	0.28*	0.27*	0.35*	0.29*	0.11*	0.39*	0.31*	0.21*
	R 0.33	0.45	0.44	0.50	0.33	0.41	0.44	0.43	0.43	0.46	0.50	0.32	0.48	0.51
	Pc36
	N 29	35	22	29	35	32	36	36	33	39	12	31	20	17
	Ar 9.39	11.23	11.28	10.05	11.54	9.87	10.08	10.99	9.57	10.07	10.00	10.72	8.96	9.78
	HS 0.86	0.90	0.90	0.89	0.90	0.85	0.89	0.89	0.88	0.87	0.84	0.90	0.85	0.86
	HO 0.38	0.17*	0.23*	0.28*	0.26*	0.28*	0.25*	0.42*	0.36*	0.36*	0.42*	0.13*	0.30*	0.18*
	R 0.48	0.47	0.66	0.47	0.55	0.47	0.52	0.40	0.51	0.43	0.61	0.63	0.34	0.59
	Pc38
	N 40	40	39	34	49	40	46	44	46	48	22	48	21	23
	Ar 11.16	12.41	11.02	10.47	10.50	10.46	11.40	11.46	11.66	11.95	13.04	12.24	14.65	13.31
	H S 0.87	0.90	0.84	0.85	0.89	0.87	0.88	0.89	0.90	0.90	0.91	0.91	0.92	0.89
	HO 0.85	0.80	0.72	0.94	0.73*	0.75	0.70*	0.84	0.83	0.79	0.82	0.79	0.86	0.65*
	R 0.03	0.05	0.10	0.07	0.11	0.07	0.14	0.04	0.07	0.05	0.02	0.06	0.03	0.21
	Pc73
	N 32	34	34	32	38	35	42	40	39	45	17	31	19	18
	Ar 8.52	7.74	7.99	9.91	8.56	7.46	7.65	9.99	8.28	7.68	10.14	8.24	8.94	7.64
	H S 0.81	0.84	0.79	0.87	0.83	0.83	0.83	0.87	0.84	0.82	0.89	0.84	0.86	0.81
	H O 0.06*	0.12*	0.24*	0.25*	0.11*	0.17*	0.36*	0.15*	0.21*	0.18*	0.18*	0.13*	0.16*	0.06*
	R 0.55	0.50	0.44	0.42	0.56	0.45	0.37	0.45	0.47	0.40	0.53	0.62	0.45	0.60
	Multilocus
	Ar 8.09	9.06	8.60	8.66	8.74	8.14	8.42	9.42	8.42	8.79	8.69	9.07	9.35	8.78
	HS 0.76	0.84	0.82	0.83	0.81	0.80	0.82	0.85	0.83	0.83	0.82	0.86	0.83	0.83
	HO 0.40*	0.39*	0.42*	0.46*	0.39*	0.42*	0.47*	0.49*	0.51*	0.49*	0.43*	0.41*	0.42*	0.32*

Table 3 :

 3 Genetic differentiation of Patella caerulea among 14 sample sites obtained from the analysis of five microsatellite loci. Pairwise F ST estimates[START_REF] Weir | Genetic signature of recent glaciations on populations of a near-shore marine fish species (Syngnathus leptorhynchus)[END_REF] computed following the ENA method[START_REF] Chapuis | Microsatellite null alleles and estimation of population differentiation[END_REF]. Values in italics indicate significant genotypic differentiation of the samples at the 5% threshold and those in bold indicate significant genotypic differentiation of the samples after sequential Bonferroni correction of the 5% threshold.

		Tri1N Tri2A Tri2N Ces1A Ces2A Ort1A Ort1N Ort2A Ort2N Spl1N Spl2N Gal1N Gal2N
	Tri1A	0.019	0.031	0.029	0.013	0.014	0.020	0.024	0.019	0.021	0.025	0.029	0.032	0.022
	Tri1N		0.006	0.013	0.007	0.007	0.011	0.004	0.006	0.009	-0.005	0.015	0.008	0.003
	Tri2A			0.014	0.025	0.015	0.021	0.010	0.011	0.014	0.010	0.017	0.013	0.000
	Tri2N				0.012	0.004	0.004	0.004	0.007	0.009	0.012	0.007	0.004	0.004
	Ces1A					0.000	0.003	0.007	0.004	0.005	0.013	0.017	0.008	0.010
	Ces2A						-0.002	0.002	0.000	0.003	0.012	0.011	0.002	0.007
	Ort1A							0.003	0.004	0.003	0.016	0.010	0.004	0.007
	Ort1N								0.000	0.001	0.005	0.007	0.004	0.003
	Ort2A									-0.002	0.007	0.010	-0.001	0.002
	Ort2N										0.013	0.011	0.002	0.002
	Spl1N											0.009	0.011	0.005
	Spl2N												0.007	0.005

ACKNOWLEDGMENTS

This work was supported by the EU projects DELOS (EVK3-CT-2000-00041) and EUMAR (EVK3CT2001-00048) and by the project MedRed (Bilateral Projects Italy-Israel, Italian Ministry of the Environment). We thank F. Bacchiocchi, C. Papetti, and M. Virgilio for help at various stages of the work. We are grateful to M. Coleman and anonymous reviewers whose comments greatly improved this manuscript.

2007

). However, we did not find evidence of recent bottlenecks in populations sampled on artificial substrates and P. caerulea is a native species in the study area. A related study on the gastropods Nucella lapillus [START_REF] Colson | Rapid recovery of genetic diversity of dogwhelk (Nucella lapillus L.) populations after local extinction and recolonization contradicts predictions from life-history characteristics[END_REF] did not show reduced genetic diversity in recently colonized/recolonized populations. This discrepancy between studies could be related to differences in life-history traits between P. caerulea and N. lapillus, including differences in dispersal abilities, invasiveness, population turnover, and/or reproductive success [START_REF] Johnson | Pattern beneath the chaos: the effect of recruitment on genetic patchiness in an intertidal limpet[END_REF]. Further, N. lapillus was sampled in natural habitats solely. Therefore, the lower genetic diversity observed in P. caerulea from artificial structures could also be related to the impact of artificial urban structures themselves. Indeed, in the Adriatic Sea, as well as in other geographical regions, there is growing evidence that artificial structures support assemblages that differ significantly in composition, structure, reproductive output, patterns of recruitment and population dynamics from assemblages on nearby natural rocky habitats (e.g. [START_REF] Bulleri | Role of recruitment in causing differences between intertidal assemblages on seawalls and rocky shores[END_REF][START_REF] Bulleri | Intertidal assemblages on artificial and natural habitats in marinas on the north-west coast of Italy[END_REF][START_REF] Glasby | Urban structures as marine habitats[END_REF][START_REF] Moschella | Low-crested coastal defence structures as artificial habitats for marine life: Using ecological criteria in design[END_REF][START_REF] Perkol-Finkel | Can artificial reefs mimic natural reef communities? The roles of structural features and age[END_REF]). These findings suggest important functional and ecological differences between these two types of habitats. For example, in Sydney Harbour (Australia) it has been shown experimentally that the reproductive output of populations of the limpet Siphonaria denticulata was significantly smaller on seawalls compared to natural shores, with possible important implications for the self-sustainability of local populations [START_REF] Moreira | Seawalls do not sustain viable populations of limpets[END_REF]. Also, variations in competition interactions on rocky shores and artificial structures have been observed among Mediterranean limpets [START_REF] Espinosa | Effects of competition on an endangered limpet Patella ferruginea (Gastropoda: Patellidae): Implications for conservation[END_REF]. All these processes may act on propagule pressure [START_REF] Lockwood | The role of propagule pressure in explaining species invasions[END_REF]) through small inoculum size (i.e the number of viable settlers), creating a filter from the amount of genetic diversity found in source populations, further causing genetic diversity to decrease, but maintaining genetic homogeneity between newly colonized and source populations [START_REF] Roman | Paradox lost: genetic diversity and the success of aquatic invasions[END_REF].