Matthieu Alfaro
email: matthieu.alfaro@umontpellier.fr

Rémi Carles
email: remi.carles@math.cnrs.fr

Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

We consider the heat equation with a logarithmic nonlinearity, on the real line. For a suitable sign in front of the nonlinearity, we establish the existence and uniqueness of solutions of the Cauchy problem, for a well-adapted class of initial data. Explicit computations in the case of Gaussian data lead to various scenarii which are richer than the mere comparison with the ODE mechanism, involving (like in the ODE case) double exponential growth or decay for large time. Finally, we prove that such phenomena remain, in the case of compactly supported initial data. Contents 1. Introduction 1 2. Steady states 5 3. Cauchy problem 6 4. Large time behavior: Gaussian data 8 5. Large time behavior in the case of more general data 10 References

Introduction

In this work we consider the nonnegative solutions u(t, x) of the heat equation with a logarithmic nonlinearity, namely (1.1)

∂ t u = ∂ xx u + λ u ln u 2 , t > 0, x ∈ R,
where λ > 0 is a given parameter. Our primary goal is to investigate the large time behavior of the solutions, since this study reveals mechanisms which seem interesting to us. Equation (1.1) shares some similarities with bistable equations modelling an Allee effect in population dynamics, but does not seem to correspond clearly to any model proposed in e.g. biology or chemistry. On the other hand, (1.1) has challenging aspects from the mathematical point of view. Our results may be extended to the multidimensional case, leading to a more technical setting. We have chosen to stick to the one-dimensional case to simplify the presentation, thus highlighting the main mechanisms.

Associated with (1.1) is the following energy

(1.2) E[u](t) := 1 2 R (∂ x u) 2 (t, x)dx + R λ 2 u 2 1 -ln u 2 (t, x)dx.
Formally, solutions to (1.1) satisfy

dE[u] dt = - R (∂ t u) 2 (t, x)dx ≤ 0.
Many features make (1.1) interesting from a mathematical point of view. First, the nonlinearity is not Lipschitzean, which causes difficulties already at the level of the local Cauchy problem. Also, the second term in the energy (1.2) has no definite sign, which makes a priori estimates a delicate issue. Next, (1.1) supports the existence of Gaussian solutions. Last, the Cauchy problem may exhibit superexponential growth or decay.

1.1. The Cauchy problem. Such a logarithmic nonlinearity has been introduced in Physics in the context of wave mechanics and optics [START_REF] Birula | Wave equations with logarithmic nonlinearities[END_REF]6]. From a mathematical point of view, the Cauchy problem for logarithmic Schrödinger equations and logarithmic wave equations have been studied in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF][START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]: in the case of the logarithmic Schrödinger equation, it is shown that a unique, global weak solution can be constructed in a subset of H 1 (in any space dimension), whichever the sign of λ. For the three dimensional wave equation and a suitable sign for λ, a similar result is available. Due to the lack of regularity of the nonlinearity, solutions are constructed by compactness methods, and uniqueness is a rather unexpected property: in the case of Schrödinger equation, it is a consequence of an elegant estimate in complex analysis noticed in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF], while for the three dimensional wave equation, it follows from fine properties of the wave equation and a general result concerning the trace (see [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF] or [START_REF] Haraux | Nonlinear evolution equations-global behavior of solutions[END_REF]).

In the context of the heat equation like (1.1), the presence of a logarithmic nonlinearity has been considered in [START_REF] Chen | Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity[END_REF], in the case of a bounded domain Ω, with Dirichlet boundary conditions. They construct global solutions in H 1 0 (Ω), and exhibit some classes of solution growing or decaying (at least) exponentially in time, thanks to variational arguments (potential well method). On the other hand, it seems very delicate, if possible, to construct a solution to (1.1) by compactness methods on the whole line R. Also, uniqueness is missing in the Cauchy theory developed in [START_REF] Chen | Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity[END_REF]. We will see that this issue can be overcome by changing functional spaces in which the Cauchy problem is studied. Definition 1.1 (Notion of solution). Let u 0 be continuous and bounded, with continuous and bounded derivative, and bounded and piecewise continuous second derivative. A (global) solution to (1.1) starting from u 0 is a function u : [0, ∞) × R → R which is continuous and bounded on [0, T ] × R, for which u t , u x and u xx exist and are continuous on (0, T ] × R, such that u(t, x) solves (1.1) on (0, T ] × R (for any T > 0), and u |t=0 = u 0 . In addition, we require that u(t, x) is uniformly bounded as |x| → ∞ for t ∈ [0, T ]. Proposition 1.2 (Global well-posedness for (1.1)). Let u 0 ≥ 0 be as in Definition 1.1. Then (1.1) has a unique solution u starting from u 0 , in the sense of Definition 1.1.

1.2. Superexponential growth vs. decay. In [START_REF] Ji | Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity[END_REF], the presence of the logarithmic nonlinearity is motivated as some limiting case for a nonlinearity of the form λu 1+ε in the limit ε → 0 (in the growth regime u → ∞), still in a bounded domain Ω ⊂ R N with Dirichlet boundary conditions. The authors show in particular that time periodic solutions are highly unstable, in the sense that a small perturbation of the initial data can lead to double exponential growth or double exponential decay in time, see [START_REF] Ji | Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity[END_REF]Theorem 1.1]. Some of our results are qualitatively similar (superexponential growth or decay), with the Gaussian steady state (2.1) acting as a separation comparable to the time periodic solutions in [START_REF] Ji | Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity[END_REF]. Nevertheless, let us mention two main differences. First, our results are valid on the whole line R, and not (only) on a bounded domain. On the other hand, we provide in Section 5 initial data leading to superexponential growth or decay but that cannot be handled by [START_REF] Ji | Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity[END_REF]Theorem 1.1]. Roughly speaking, as can be seen from the proof, initial data of [START_REF] Ji | Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity[END_REF]Theorem 1.1] are multiples of the separating time periodic solution -which is comparable to the present Remark 4.4-whereas initial data in Section 5 are allowed to "cross" the separating Gaussian steady state (2.1) (see also Corollary 3.10).

Let us recall that, in his seminal work [START_REF] Fujita | On the blowing up of solutions of the Cauchy problem for ut = ∆u + u 1+α[END_REF], Fujita considered solutions u(t, x) to the nonlinear heat equation

(1.3) ∂ t u = ∆u + u 1+p , t > 0, x ∈ R N ,
supplemented with a nonnegative and nontrivial initial data. For p > 0 solutions of the underlying ordinary differential equation (ODE) problem -namely dn dt = n 1+p , n(0) = n 0 > 0 -blow up in finite time. The dynamics of the partial differential equation (1.3) is more complex and rich. Precisely, there is a critical exponent p F := 2 N , referred to as the Fujita exponent, such that: If 0 < p ≤ p F then any solution blows up in finite time, like those of the ODE. On the other hand, if p > p F there is a balance between diffusion and reaction. Solutions with large initial data blow up in finite time whereas solutions with small initial data are global in time and go extinct as t → ∞. Those facts are proved in [START_REF] Fujita | On the blowing up of solutions of the Cauchy problem for ut = ∆u + u 1+α[END_REF], except the critical case p = p F which is studied in [START_REF] Hayakawa | On nonexistence of global solutions of some semilinear parabolic differential equations[END_REF] when N = 1, 2, in [START_REF] Kobayashi | On the growing up problem for semilinear heat equations[END_REF] when N ≥ 3, and in [START_REF] Weissler | Existence and nonexistence of global solutions for a semilinear heat equation[END_REF] via a direct and simpler approach.

Concerning equation (1.1), the underlying ODE problem

(1.4) dn dt = 2λn ln n, n(0) = n 0 > 0,
is globally solved as

(1.5) n(t) = e (ln n0)e 2λt . As t → ∞, n(t) → 0 if 0 < n 0 < 1 (extinction) whereas n(t) → ∞ if n 0 > 1 (blow up in infinite time).
Hence the dynamics of the ODE (1.4) already shares some similarities with the Fujita supercritical regime p > p F for the PDE (1.3). The dynamics of the PDE (1.1) is much richer than the mechanism of (1.4), and our main goal is to understand its long time behavior for initial data u 0 "crossing" the equilibrium 1.

Notice also that the composition of exponential functions in (1.5) is a strong indication that possible extinction or growth phenomena are strong, and can thus hardly be captured numerically. In practice, the superexponential growth may appear like a blow-up phenomenon, while the superexponential decay may be understood like a finite time extinction.

When possible, a second goal is to estimate these rates of convergence.

To give a flavor of the results established in the sequel, recall that the authors in [START_REF] Chen | Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity[END_REF] consider (1.1) (possibly in multidimension) on a bounded domain, with Dirichlet boundary conditions. By variational arguments, they exhibit classes of initial data whose evolution under (1.1) leads to (at least) exponential decay in L 2 , and another class of initial data whose evolution leads to unboundedness of the L 2 norm in large time. As a consequence of our analysis on the whole line R, we actually provide more precise information on those phenomena for the equation on a bounded domain, say (α, β).

Proposition 1.3 (Growth/decay rates in a bounded domain). Let α < β and Ω = (α, β). Consider the mixed problem

(1.6)      ∂ t u = ∂ xx u + u ln(u 2 ), t > 0, x ∈ Ω, u |∂Ω = 0, t > 0, u |t=0 = u 0 .
There are nonnegative initial data u 0 ∈ C 1 c (Ω) such that (1.6) has a unique solution, whose L 2 and L ∞ norms decay at least like a double exponential in time,

∃C, η > 0, u(t) L 2 (Ω) ≤ |Ω| 1/2 u(t) L ∞ (Ω) ≤ Ce -ηe 2t .
There are nonnegative initial data u 0 ∈ C 1 c (Ω) such that (1.6) has a unique solution, whose L 2 and L ∞ norms grow at least like a double exponential in time,

∃C, η > 0, |Ω| 1/2 u(t) L ∞ (Ω) ≥ u(t) L 2 (Ω) ≥ Ce ηe 2t .
1.3. Changing the sign of the nonlinearity. Let us observe that, for λ > 0, the problem

(1.7) ∂ t u = ∂ xx u -2λ u ln u, t > 0, x ∈ R,
is of different nature. Indeed for the underlying ODE,

n ′ (t) = -2λn(t) ln n(t),
the equilibrium 0 is (very) unstable, while 1 is stable. Hence, by the comparison principle, solutions are a priori bounded between 0 and max(1, u 0 L ∞ ). Moreover, by comparison with Fisher-KPP equations, much can be said on the long time behavior of the Cauchy problem. For instance, consider a nontrivial compactly supported initial data 0 ≤ u 0 ≤ 1. For any r > 0, we can construct

g r : [0, 1] → R concave with g r > 0 on (0, 1), g r (0) = g r (1), r = g ′ r (0) > 0 > g ′ r (1)
, which is referred as to a Fisher-KPP nonlinearity, and such that g r (u) ≤ -2λu ln u. By the comparison principle, we deduce that u r (t, x) ≤ u(t, x) ≤ 1, where u r is the solution of ∂ t u r = ∂ xx u r + g r (u r ), starting from u 0 . But it is known [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF] that the spreading speed of this Fisher-KPP equation, with compactly supported data, is c

* r := 2 g ′ r (0) = 2 √ r, meaning that if c > c * r then u r (t, x) → 0 uniformly in {|x| ≥ ct} as t → ∞, if c < c * r then u r (t, x) → 1 uniformly in {|x| ≤ ct} as t → ∞.
Since this is true for any r > 0 we get that for any c > 0, u(t, x) → 1 uniformly in {|x| ≤ ct} as t → ∞, that is convergence to 1 with a superlinear speed.

The organization of the paper is as follows. In Section 2, we enquire on steady states, proving existence of a unique (Gaussian) nontrivial one. The well-posedness of the Cauchy problem is established in Section 3. The long time behavior (superexponential growth, decay or convergence to the steady state) is studied in Section 4 (Gaussian initial data and consequences), and Section 5 (more general data and consequences).

Steady states

It is readily checked that the only constant steady states of (1.1) are u ≡ 0 and u ≡ 1.

Proposition 2.1 (Steady state).

There is a unique (up to translation) nonnegative nontrivial steady state ϕ solving (1.1) and satisfying ϕ(±∞) = 0. It is the Gaussian given by

(2.1) ϕ(x) = e 1 2 e -λ 2 x 2 .
Proof. Let u = u(x) ≥ 0 be a nontrivial solution to (1.1), that is

(2.2) u ′′ (x) + 2λu(x) ln u(x) = 0, ∀x ∈ R,
with u(±∞) = 0. If u(x 0 ) = 0 for some x 0 ∈ R then u ≡ 0 from the strong maximum principle. Hence u > 0. Next, we multiply the equation by u ′ , integrate and infer that there is C ∈ R such that

(u ′ ) 2 (x) + λu 2 (x)(2 ln u(x) -1) = C, ∀x ∈ R.
From the above identity and since u(±∞) = 0, we deduce that u ′ (±∞) must exist in R and, thus, be equal to 0 (otherwise we cannot have u(±∞) = 0). Hence C = 0 and

(2.3) (u ′ ) 2 (x) = λu 2 (x)(1 -2 ln u(x)), ∀x ∈ R.
If x → 1 -2 ln u(x) never vanishes then this identity implies that u ′ has a constant sign, which contradicts u(±∞) = 0. Hence, there exists x 0 ∈ R such that 2 ln u(x 0 ) = 1, and thus u ′ (x 0 ) = 0 (from (2.3)), u ′′ (x 0 ) < 0 (from (2.2)). In the sequel, we work on [x 0 , +∞), the arguments being similar on (-∞,

x 0 ].
Assume that there is x 1 > x 0 such that u ′ (x 1 ) = 0. From (2.3), 2 ln u(x 1 ) = 1, and there must be a point x * ∈ (x 0 , x 1 ) where u reaches a minimum strictly smaller than e 1 2 , which contradicts (2.3). Hence u ′ < 0 on (x 0 , +∞). It therefore follows from (2.3) that -u ′ (x) = √ λu(x) 1 -2 ln u(x) for x ≥ x 0 . Separating variables we get

- √ λ(x -x 0 ) = u(x) u(x0) du u √ 1 -2 ln u = -1 -2 ln u(x), since u(x 0 ) = e 1 2
. We end up with u(x) = e 1 2 e -λ 2 (x-x0) 2 , which completes the proof.

Cauchy problem

As emphasized in the introduction, the Cauchy problem associated to (1.1) is not trivial, for two reasons:

• Local well-posedness: the nonlinearity is not Lipschitzean.

• Global well-posedness: the potential energy in (1.2) has no definite sign. The first aspect implies that constructing a solution certainly requires compactness arguments, and uniqueness is not granted. The second aspect shows that to have a solution defined for all t ≥ 0, it may be helpful that the first step yields this property "for free". This is the strategy adopted in [START_REF] Chen | Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity[END_REF], where, on a bounded domain Ω, with Dirichlet boundary conditions, the authors construct a solution in H 1 0 (Ω) by Galerkin approximation. However, uniqueness is not established in this case.

In this section, we prove Proposition 1.2, by showing that it fits perfectly into the framework of the PhD thesis of J. C. Meyer [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF]. Instead of working in spaces where the energy (1.2) is well defined, we adopt the approach of [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF], see also [START_REF] Meyer | The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF]. Consider more generally the Cauchy problem (3.1)

u t = u xx + f (u), 0 < t ≤ T, x ∈ R, u |t=0 = u 0 ,
so we can emphasize which are the suitable assumptions of the nonlinearity f described in [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF]. Notice that, in [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF][START_REF] Meyer | The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF], the standard examples, motivated by models from Chemistry, are of the form f (u) = ±(u p ) + , 0 < p < 1, and

f (u) = (u p ) + ((1 -u) q ) + , 0 < p, q < 1.
The generalization of Definition 1.1, as introduced in [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF], is the following.

Definition 3.1 (Notion of solution). Let u 0 be continuous and bounded, with continuous and bounded derivative, and bounded and piecewise continuous second derivative. A solution to (3.1) is a function u : [0, T ] × R → R which is continuous and bounded on [0, T ] × R, for which u t , u x and u xx exist and are continuous on (0, T ] × R, such that u(t, x) satisfies (1.1). In addition, we require that u(t, x) is uniformly bounded as |x| → ∞ for t ∈ [0, T ]. Notation 3.2. Following [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF][START_REF] Meyer | The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF], we denote by BPC 2 (R) the set of such initial data. Two notions are crucial, and correspond exactly to the type of logarithmic nonlinearity considered in the present paper. Definition 3.3 (Hölder continuity). Let α ∈ (0, 1). A function f : R → R is said to be α-Hölder continuous if for any closed bounded interval E ⊂ R, there exists a constant k E > 0 such that for all x, y ∈ E,

|f (x) -f (y)| ≤ k E |x -y| α .
A notion weaker than the standard notion of Lipschitz continuity turns out to be rather interesting, as we will see below.

Definition 3.4 (Upper Lipschitz continuity).

A function f : R → R is said to be upper Lipschitz continuous if f is continuous, and for any closed bounded interval E ⊂ R, there exists a constant k E > 0 such that for all x, y ∈ E, with

y ≥ x, f (y) -f (x) ≤ k E (y -x).
Essentially, this property suffices to have a comparison principle, hence a uniqueness result for (3.1).

Example 3.5. In the case of (1.1), f (u) = λu ln(u 2 ). First, f is α-Hölder continuous for any α ∈ (0, 1). Indeed, for y > x > 0, we have

|f (y) -f (x)| = 2λ (y -x) ln y + x ln 1 + y -x x ≤ 2λ|y -x|(| ln y| + 1), so that |f (y)-f (x)| |y-x| α ≤ 2λ|y -x| 1-α (| ln y| + 1) ≤ 2λ|y| 1-α (| ln y| + 1
), which remains bounded as y → 0. On the other hand, even though f is not Lipschitz continuous, we check that for λ > 0 (the case of interest in the present paper), f is upper Lipschitz continuous. Indeed, for x, y ∈ E bounded, with y > x > 0, Taylor formula yields

f (y) -f (x) = (y -x) 1 0 f ′ (x + θ(y -x)) dθ = 2λ(y -x) 1 0 (1 + ln) (x + θ(y -x)) dθ ≤ 2λ(y -x)2λ 1 + sup z∈E ln z .
The last factor remains bounded as x → 0. It would not be if the infimum was considered: f is not Lipschitz continuous. Definition 3.6 (Sub-and super-solutions). Let u, u : [0, T ] × R be continuous on [0, T ]×R and such that u t , u x , u xx , u t , u x , u xx exist and are continuous on (0, T ]× R. If

u t -u xx -f (u) ≤ 0 ≤ u t -u xx -f (u) , 0 < t ≤ T, x ∈ R, u(0, x) ≤ u 0 (x) ≤ u(0, x), ∀x ∈ R,
and u, u are uniformly bounded as |x| → ∞ for t ∈ [0, T ], then u is called a regular sub-solution, and u is called a regular super-solution to (3.1). Theorem 3.7 (Comparison; Theorem 7.1 from [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF][START_REF] Meyer | The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF]). Let f be upper Lipschitz continuous. If u and u and regular sub and super-solutions on [0, T ] × R, respectively, then u(t, x) ≤ u(t, x), ∀(t, x) ∈ [0, T ] × R.

Theorem 3.8 (Uniqueness; Theorem 7.2 from [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF][START_REF] Meyer | The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF]). Let f be upper Lipschitz continuous. Then, for any T > 0, (3.1) has at most one solution in [0, T ] × R.

The following statement is a slight modification from the original, where we add a uniqueness assumption to simplify the presentation. Theorem 3.9 (Existence; Theorem 8.1 and Corollary 8.6 from [START_REF] Meyer | Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF][START_REF] Meyer | The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations[END_REF]). Suppose that f is α-Hölder continuous for some α ∈ (0, 1), and let u 0 ∈ BPC 2 (R). Suppose that uniqueness holds for (3.1). Then (3.1) has a (unique) solution u : [0, T * [×R. In addition, either

T * = ∞, or u(t, •) L ∞ (R) is unbounded as t → T * .
Proof of Proposition 1.2. As emphasized above, the nonlinearity in (1.1) is both α-Hölder continuous (for any α ∈ (0, 1)) and upper Lipschitz continuous. Therefore, Theorem 3.8 implies uniqueness, and Theorem 3.9 yields a (unique) maximal solution u ∈ C([0, T * ) × R).

We conclude by showing that the solution is global (T * = ∞) thanks to a suitable a priori estimate. The solution of the ODE (1.4) starting from u 0 L ∞ , namely

u(t) = e ln u0 L ∞ e 2λt .
is a super-solution, while the zero function is obviously a sub-solution. Theorem 3.7 implies that 0 ≤ u(t, x) ≤ u(t), ∀t ∈ [0, T * ).

We conclude that T * = ∞, and the result follows.

Corollary 3.10 (Initial data comparable to 1). Let u 0 ∈ BPC 2 (R), u 0 ≥ 0, and ε ∈ (0, 1).

• If u 0 (x) ≥ 1+ε for all x ∈ R, then u grows at least like a double exponential in time:

u(t, x
) ≥ e ln(1+ε)e 2λt , ∀t ≥ 0, ∀x ∈ R.

• If u 0 (x) ≤ 1 -ε for all x ∈ R, then u decays at least like a double exponential in time:

u(t, x) ≤ e ln(1-ε)e 2λt , ∀t ≥ 0, ∀x ∈ R.
Proof. This corollary is a straightforward consequence of Proposition 1.2, the comparison principle (Theorem 3.7), and the ODE case (1.4)-(1.5).

Large time behavior: Gaussian data

Families of Gaussian solutions for nonlinear (and nonlocal) equations can be found in [START_REF] Vadim | A simple mathematical model of gradual Darwinian evolution: emergence of a Gaussian trait distribution in adaptation along a fitness gradient[END_REF], [START_REF]Explicit solutions for replicator-mutator equations: extinction versus acceleration[END_REF][START_REF] Alfaro | Replicator-mutator equations with quadratic fitness[END_REF], in the context of evolutionary genetics. In the case of a logarithmic nonlinearity, for the Schrödinger equation, it was observed in [6] that the flow preserves the Gaussian structures, and so the resolution of the partial differential equation boils down to the resolution of ordinary differential equations; see [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] for more details. It is not surprising that the same holds in the case of (1.1), and we have indeed: Proposition 4.1 (Gaussian solutions). Let b 0 > 0 and a 0 > 0 be given. The solution of (1.1) starting from the Gaussian

(4.1) u 0 (x) = b 0 e -a 0 2 x 2 ,
is the Gaussian given by

(4.2) u(t, x) = b(t)e -a(t) 2 x 2 := e ψ(t)e 2λt e -a(t) 2 x 2 ,
where

(4.3) ψ(t) = ln b 0 - a 0 2 ln λ -ln(a 0 + (λ -a 0 )e -2λt ) λ -a 0 , with the natural continuation ψ(t) = ln b 0 -1 2 (1 -e -2λt
) if a 0 = λ, and

(4.4) a(t) = λ a 0 e 2λt λ -a 0 + a 0 e 2λt .
Proof. We plug the ansatz (4.2) into equation (1.1), we identify the x 0 and the x 2 coefficients to obtain two ordinary differential equations. The first one is the logistic equation a ′ (t) = 2a(t)(λ -a(t)), whose solution, starting from a(0) = a 0 , is given by (4.4). The second one is b ′ (t) = 2λb(t) ln b(t) -a(t)b(t).

Denoting φ(t) := ln b(t) the above is recast

φ ′ (t) = 2λφ(t) -a(t),
=      a 0 -2(λ -a 0 ) ln a 0 + (λ -a 0 )e -2λt -ln λ if a 0 = λ 1 2 (1 -e -2λt ) if a 0 = λ,
which we plug into (4.5) to get (4.3).

Clearly, the sign of ψ ∞ := lim t→∞ ψ(t) decides between (superexponential) decay and growth of the Cauchy problem starting from a Gaussian data, the critical case ψ ∞ = 0 leading to convergence to the steady state. (i) If ψ ∞ < 0, then there is superexponential decay in the sense that

u(t, •) L ∞ = max x∈R u(t, x) ∼ t→∞ e 1 2 e ψ∞e 2λt , u(t, •) L 1 ∼ t→∞ 2π λ e 1 2 e ψ∞e 2λt .
(ii) If ψ ∞ = 0, then there is convergence to the steady state of Proposition 2.1 in the sense that

u(t, •) -ϕ L ∞ + u(t, •) -ϕ L 1 -→ t→∞ 0.
(iii) If ψ ∞ > 0, then there is superexponential growth in the sense that, for all R > 0,

min |x|≤R u(t, x) ∼ t→∞ e 1 2 e ψ∞e 2λt e -λ 2 R 2 , u(t, •) L 1 ∼ t→∞ 2π λ e 1 2 e ψ∞e 2λt .
Proof. One just has to use the asymptotic expansion ψ(t) = ψ ∞ + 1 2 e -2λt + O(e -4λt ) as t → ∞, and perform straightforward estimates.

In view of the comparison principle (Theorem 3.7) and of Corollary 4.2, we infer: Corollary 4.3 (Initial data comparable to a Gaussian). Let u 0 ∈ BPC 2 (R), u 0 ≥ 0, and ε ∈ (0, 1). Let a 0 , b 0 > 0, and denote again

ψ ∞ = ln b 0 - a 0 2 ln λ -ln a 0 λ -a 0 , with the natural continuation ψ ∞ = ln b 0 -1 2 if a 0 = λ. • If ψ ∞ < 0 and u 0 (x) ≤ b 0 e -a0x 2 /2 , then u decays at least like a double exponential in time, u(t, •) L ∞ ≤ 2e 1 2 e ψ∞e 2λt , u(t, •) L 1 ≤ 4π λ e 1 2 e ψ∞e 2λt , as t → ∞. • If ψ ∞ > 0 and u 0 (x) ≥ b 0 e -a0x

Large time behavior in the case of more general data

We have seen that the comparison with the constant initial datum equal to one leads to a strong dichotomy (Corollary 3.10). The same is true by comparison with an initial Gaussian, leading to a larger variety of initial data (Corollary 4.3 and Remark 4.4). Now, we enquire on initial data that can be compared neither to 1 nor to a Gaussian. In this direction, we can prove superexponential decay for small initial data. To do so, we need the following standard estimate, which stems from Young inequality applied to the formula

e t∂xx v 0 (x) = 1 √ 4πt R e -(x-y) 2 /(4t) v 0 (y)dy. Lemma 5.1. For any initial data v 0 ∈ L 1 (R) ∩ L ∞ (R), the solution of the Cauchy problem ∂ t v = ∂ xx v, v |t=0 = v 0 , satisfies v(t, •) L ∞ ≤ V (t) := min v 0 L ∞ , v 0 L 1 √ 4πt , for any t ≥ 0.
Theorem 5.2 (Superexponential decay for small data). For a nonnegative initial datum u 0 in BPC 2 (R), define m ∞ := u 0 L ∞ and m 1 := u 0 L 1 . Assume Then the solution of (1.1), starting from u 0 , is decaying superexponentially in the sense that (5.2) 0 ≤ u(t, x) ≤ min m ∞ , m 1 √ 4πt e ψ(t)e 2λt , where ψ(t) → ψ * ∞ < 0 as t → ∞.

Remark 5.3. The above criterion provides new initial data leading to superexponential decay. For instance, assume that u 0 has tails heavier than Gaussian (so that domination by a Gaussian cannot be used), that

(5.3) 1 < m ∞ < e +∞ 1
λe -2λs ln s ds , (so that domination by the ODE cannot be used), and that τ = 1. Then (5.1) holds true, hence (5.2). A typical example could be u 0 (x) = m ∞ e -α( √ 1+x 2 -1) , with (5.3) and α > 0 adjusted so that τ = 1.

Proof. Following [START_REF] García-Melián | Fujita exponents for evolution problems with nonlocal diffusion[END_REF] or [START_REF] Alfaro | Fujita blow up phenomena and hair trigger effect: the role of dispersal tails[END_REF], we look for a supersolution to (1.1) in the form g(t)v(t, x), where g(t) > 0 is to be determined (with g(0) = 1), and v(t, x) is the solution of the heat equation ∂ t v = ∂ xx v with u 0 as initial datum. A straightforward computation shows that to construct a supersolution, it is enough to have g ′ (t) g(t)

-2λ ln g(t) ≥ 2λ ln v(t, x).

By Lemma 5.1, it is therefore enough to have g(t) = e φ(t) where φ ′ (t) -2λφ(t) = 2λ ln V (t), φ(0) = 0, that is φ(t) = e 2λt t 0 2λe -2λs ln V (s)ds.

Observe that V (t) = m ∞ when t ≤ τ while V (t) = m1 √ 4πt when t ≥ τ . Cutting the above integral and performing straightforward computations, we end up with g(t) = e ψ(t)e 2λt , where ψ(t) := (ln m ∞ )(1 -e -2λτ ) + ln m 1 √ 4π (e -2λτ -e -2λt ) -t τ λe -2λs ln s ds, which tends to ψ * ∞ as t → ∞. It therefore follows from the comparison principle (Theorem 3.7) that u(t, x) ≤ g(t)v(t, x), which yields (5.2).

In the context of bounded solutions, typically for Lipschitz ignition or bistable nonlinearities, some threshold results between extinction and convergence to an equilibrium, say 1, are known to exist [START_REF] Zlatoš | Sharp transition between extinction and propagation of reaction[END_REF], [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF], [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reactiondiffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF]. For equation (1.1), we can prove a threshold result between decay and growth. We first need to construct compactly supported sub-solutions.

0 e

 0 whose solution, starting from φ(0) = ln b 0 , is(4.5) φ(t) = ln b 0 -t -2λs a(s)ds e 2λt .Next, using (4.4) we get t 0 e -2λs a(s)ds = t 0 λa 0 e -2λs a 0 + (λ -a 0 )e -2λs ds

Corollary 4 . 2 (

 42 Gaussian data: three scenarii). Let b 0 > 0 and a 0 > 0 be given. Define

( 4 . 6 )

 46 ψ ∞ := ln b 0 -a 0 2 ln λ -ln a 0 λ -a 0 ,with the natural continuation ψ ∞ = ln b 0 -1 2 if a 0 = λ. Denote by u(t, x) the Gaussian solution of Proposition 4.1.

Remark 4 . 4 . 1 2

 441 Observe that, if ψ ∞ = ln b 0 -a0 2 ln λ-ln a0 λ-a0 = 0, then initial data (1 -ε)b 0 e -a0x 2 /2 (0 < ε < 1), and (1 + ε)b 0 e -a0x 2 /2 (ε > 0), fall into the regime ψ ∞ < 0 (decay), and ψ ∞ > 0 (growth), respectively. Typical examples are (1 -ε)ϕ(x), (1 + ε)ϕ(x), where ϕ(x) = e e -λx 2 /2 is the steady state from Proposition 2.1.

( 5 1 √ 4π m ∞ 2 .

 512 .1)ψ * ∞ := ln m ∞ -+∞ τ λe -2λs ln s ds + e -2λτ ln √ τ < 0,where τ := m

  2 /2 , then u grows locally at least like a double exponential in time: for all R > 0,

	min |x|≤R	u(t, x) ≥	1 2	e	1 2 e ψ∞e 2λt e -λ 2 R 2 ,	u(t, •) L 1 ≥	π λ	e	1 2 e ψ∞e 2λt ,	as t → ∞.

Lemma 5.4 (High plateaux as sub-solutions). Let L > 0 and 0 < ε < L be given. Let Θ ∈ C ∞ ([-L, L]) ∩ BPC 2 (R) be such that Θ(x) = 0 for |x| ≥ L, Θ > 0 on (-L, L),

Then there is K 0 > 1 such that, for any K ≥ K 0 , the function Θ K := KΘ satisfies

Hence, Θ K is a sub-solution to (1.1).

Proof. By our assumption Θ(x) ∼ 1 6 γ(x + L) 3 , Θ ′′ (x) ∼ γ(x + L) for 0 < x + L ≪ 1, where we thus have

As result, there is δ > 0 such that (5.4) holds on (-L, -L + δ) and, by symmetry, on (L -δ, L).

which is nonnegative if K > 1 is large enough.

Theorem 5.5 (Threshold phenomena for compactly supported data). Let 0 < ε < L < L ′ be given. Select K ≥ K 0 > 1, where K 0 is given by Lemma 5.4. Let u 0 ∈ BPC 2 (R) be such that u 0 > 0 on (-L ′ , L ′ ) and u 0 ≡ 0 on (-∞, -L ′ ] ∪ [L ′ , +∞). For M > 0, we denote by u M (t, x) the solution of (1.1) starting from M u 0 .

(i) There is M decay > 0 such that, for any 0 < M < M decay , the solution u M (t, x) is decaying superexponentially in time. (ii) There is M growth > 0 such that, for any M > M growth , the solution u M (t, x) grows locally superexponentially in time, in the sense that

Proof. The first point is a consequence of Corollary 3.10, that is comparison with the ODE, provided we choose M decay = 1/ u 0 L ∞ . We now prove (ii). Since min -L≤x≤L u 0 (x) > 0, there is M growth > 0 such that, for all M ≥ M growth , M u 0 ≥ (K + 1)Θ. Next we take n(t) as the solution of the underlying ODE starting from n 0 := K+1 K , see (1.4) and (1.5), that is

Now we define

w(t, x) := Θ K (x)n(t) = KΘ(x)n(t).

We have w(0, •) = (K + 1)Θ ≤ M u 0 and

by Lemma 5.4. It therefore follows from the comparison principle that u M ≥ w. In particular, since Θ ≡ 1 on [-L + ε, L + ε], we get (5.5). Proposition 1.3, concerned with a bounded domain, is a straightforward consequence of the above result. One just has to use a translation in space c = 0 if necessary so that c + [-L ′ , L ′ ] ⊂ (α, β), and note that the quantities involved in Theorem 5.5 control the L 2 norm on Ω.


Adélaïde Roguet
email: adelaide.roguet@gmail.com

Grégory S Laigle

Claire Therial

Adèle Bressy

Frédéric Soulignac

Arnaud Catherine

Gérard Lacroix

Ludwig Jardillier

Céline Bonhomme

Thomas Z Lerch

Françoise S Lucas

Neutral community model explains the bacterial community assembly in freshwater lakes

Keywords: bacterioplankton, community structure, deterministic factors, neutral community model, lake, T-RFLP

One sentence summary: At both regional and local scales, the neutral community model appears to be relevant in explaining the bacterial assemblage structure in shallow

Introduction

One of the key issues in microbial ecology is to identify and quantify ecological processes that drive bacterial community assembly in aquatic environments. Freshwater bacterial communities appear to be shaped by environmental conditions or dispersal-related processes (e.g., [START_REF] Martiny | Microbial biogeography: putting microorganisms on the map[END_REF][START_REF] Lindström | Local and regional factors influencing bacterial community assembly[END_REF]. Taxa selection by local environmental conditions was suggested to be the main mechanism controlling aquatic bacterial biogeography (e.g., [START_REF] Beisner | The role of environmental and spatial processes in structuring lake communities from bacteria to fish[END_REF][START_REF] Langenheder | The role of environmental and spatial factors for the composition of aquatic bacterial communities[END_REF][START_REF] Van Der Gucht | The power of species sorting: local factors drive bacterial community composition over a wide range of spatial scales[END_REF]. Environmental factors, such as water temperature, osmotic conditions or nutrient availability, are known to shape the bacterial community at the regional and local scale (e.g., [START_REF] Muylaert | ) 19/20 between bacterial community composition and bottom-up versus top-down variables in four eutrophic shallow lakes[END_REF]Yannarell andTriplett 2004, 2005;[START_REF] Kent | Synchrony in aquatic microbial community dynamics[END_REF][START_REF] Lindström | Distribution of typical freshwater bacterial groups is associated with pH, temperature, and lake water retention time[END_REF][START_REF] Shade | ) 20/20 and phenology of bacterial communities in a eutrophic lake[END_REF][START_REF] Jones | Evidence for structuring of bacterial community composition by organic carbon source in temperate lakes[END_REF]. Inversely, dispersal related mechanisms were less often identified as having an influence on aquatic bacterial metacommunities (e.g., [START_REF] Logue | Species sorting affects bacterioplankton community composition as determined by 16S rDNA and 16S rRNA fingerprints[END_REF][START_REF] Östman | Regional invariance among microbial communities[END_REF][START_REF] Soininen | Disentangling the spatial patterns in community composition of prokaryotic and eukaryotic lake plankton[END_REF], which is likely due to an underestimation or a difficulty in taking into account these processes [START_REF] Lindström | Local and regional factors influencing bacterial community assembly[END_REF]. Dispersal-related mechanisms are represented by (i) dispersal limitation, i.e. the extent to which taxa reach another location [START_REF] Martiny | Microbial biogeography: putting microorganisms on the map[END_REF]), (ii) the mass effect that corresponds to a massive supply of an exogenous taxa that disturbs the composition of a local community [START_REF] Hubbell | The unified neutral theory of biodiversity and biogeography[END_REF][START_REF] Leibold | The metacommunity concept: a framework for multiscale community ecology[END_REF]) and (iii) the neutral model that describes the stochastic balance between the immigration, speciation, emigration and extinction of organisms [START_REF] Hubbell | The unified neutral theory of biodiversity and biogeography[END_REF][START_REF] Leibold | The metacommunity concept: a framework for multiscale community ecology[END_REF]).

To date, only a few studies using statistical approaches, such as variation partitioning analysis [START_REF] Borcard | Partialling out the spatial component of ecological variation[END_REF], have conjointly quantified the relative importance of local environmental factors and processes involved in the spatial dispersion of the bacterial taxa (e.g., [START_REF] Langenheder | The role of environmental and spatial factors for the composition of aquatic bacterial communities[END_REF][START_REF] Van Der Gucht | The power of species sorting: local factors drive bacterial community composition over a wide range of spatial scales[END_REF]. The neutral assembly theory has seldom been included in these studies (e.g., [START_REF] Drakare | Local factors control the community composition of cyanobacteria in lakes while heterotrophic bacteria follow a neutral model[END_REF][START_REF] Langenheder | Species sorting and neutral processes are both important during the initial assembly of bacterial communities[END_REF], although it can correctly explain on its own the bacterial community structure (BCS) in diverse aquatic environments at the regional and local scales (e.g., [START_REF] Sloan | Quantifying the roles of immigration and chance in shaping prokaryote community structure[END_REF][START_REF] Woodcock | Neutral assembly of bacterial communities[END_REF][START_REF] Ofiţeru | Combined niche and neutral effects in a microbial wastewater treatment community[END_REF].

In this study, we investigated the spatial distribution of freshwater bacterioplankton in a set of 49 shallow and artificial lakes located in the same hydrographical basin around Paris (France) for three consecutive years in summer. These ecosystems constitute a useful model to determine the relative importance of local environmental factors and spatial and neutral processes since they display large environmental gradients and strong variations in variables that could impact the distribution of bacteria (e.g., river or watershed connections) [START_REF] Catherine | Design and application of a stratified sampling strategy to study the regional distribution of cyanobacteria (Ile-de-France, France)[END_REF][START_REF] Catherine | Cost effective prediction of the eutrophication status of lakes and reservoirs[END_REF]. Our study had three specific aims. First, we sought to determine to what extent local environmental characteristics and spatial factors shaped the regional distribution of the bacterial community. Then, in an attempt to confirm or refute the role of environmental and spatial factors in explaining the observed BCS, we evaluated whether stochastic dynamics could accurately predict the metacommunity structure. Finally, because the different processes could also shape the BCS at the local scale, the importance of environmental, spatial and neutral processes was evaluated in a single lake by monthly monitoring over 2 years. Bacterioplankton community structure was assessed using terminal restriction fragment length polymorphism (T-RFLP) targeting a fragment of the 16S rRNA gene that allows screening of a large set of samples.

Materiel and methods

Study area and sampling

This study was conducted in the Paris area (Fig. 1), which is the most populated area in France (with 18% of the metropolitan French population) and covers about 12 000 km 2 (INSEE 2013). This region displays a large gradient of land use. In spite of large industrial towns and residential suburbs, half of the Paris area territory is used for agricultural purposes, while 26% is still covered by forests (INSEE 2012). Among the 248 water bodies larger than 5 ha referenced in the hydrological database Carthage 3.09® (IGN, Paris, France), 49 lakes (Fig. 1 and Supplementary Table S1) were selected using a random and stratified sampling strategy [START_REF] Catherine | Design and application of a stratified sampling strategy to study the regional distribution of cyanobacteria (Ile-de-France, France)[END_REF]. Brie y, the 49 lakes were chosen to represent an unbiased set of water bodies that reflects the whole range of environmental conditions (e.g., land use, hydrology, altitude and surface) found within the Paris area. All the 49 lakes have an artificial anthropogenic origin and can be considered as shallow according to the definition of [START_REF] Scheffer | Ecology of shallow lakes[END_REF]. The lakes were sampled yearly in late July from 2011 to 2013. Each sampling campaign was conducted in less than 15 days to reduce the variability caused by short-term changes in meteorological conditions and nutrient inputs.

For each lake, three equidistant sampling stations were selected. At each station, water samples were collected at three discrete depths (depending on the depth of the water column) using a Niskin bottle (General Oceanics Inc., Miami (FL), USA). All samples were then pooled to obtain an integrated sample.

4/20

Immediately after sampling, water was filtered through a 0.22 μm pore-size Sterivex GP filter (Millipore, Billerica (MA), USA) after pre-filtration through 50 μm pore-size nylon mesh. Sterivex filters were kept at 4 • C during transport and then stored at -20 • C. Among the 49 lakes, Créteil Lake (Figs 1 and 2) was monitored monthly at three different stations from December 2011 to December 2013 (25 sampling dates) in order to identify spatial and temporal variability in the BCS. This is a mesotrophic lake (Table 1) covering 40 ha in an urbanized area (Val-de-Marne, France). It is a former sandpit mainly supplied by alluvial groundwater. A storm sewer outlet drains an impermeable surface of 100 ha and releases its effluent into the lake. The water residence time of this lake is greater than 180 days. A horizontal and a vertical transect were studied (Fig. 2). Subsurface samples were obtained at three stations along the inlet-outlet axis of the lake (Storm outlet (S1), center of the lake (C1), and lake out ow (O1)). At the central station, the vertical axis of the lake was sampled at three depths (C1, C2 and C3). In addition, for all sampling dates, a pooled sample (M) was obtained as previously described to estimate the BCS variation in the integrated sample obtained in summer during the 49 lake campaigns. The average point M was the mix of the three stations at the three depths (S1, S2, S3, C1, C2, C3, O1, O2 and O3). All samples were stored at 4 • C in the dark until preparation (3-4h after collection). For each sample, 1 L of lake water was filtered through a Sterivex GP filter cartridge as previously described and then stored at -20 • C prior to analysis.

5/20

DNA extraction

For each sample, the membrane inside the Sterivex units was extracted under sterile conditions and cut into small pieces of approximately 1 mm 2 . All membrane pieces were pooled in a sterile tube and DNA was extracted using the FastDNA® SPIN Kit (QBiogene, Carlsbad (CA), USA) according to the manufacturer's instructions. Two modifications of this protocol were applied: bacterial cells were lysed in a FastPrep bead beater three times for 30 s at 4.0 m s -1 and an additional wash was performed on the SPIN filters.

T-RFLP analysis

A 1326 bp fragment of the 16S rRNA gene was amplified by PCR using the primer set 63F (5 ′ -CAGGCCTAACACATGCAAGTC-3 ′ ; labeled with 6-carboxy fluorescein) and 1392R (5 ′ -ACG GGCGGTGTGTACAAG-3 ′ ) targeting the bacterial domain [START_REF] Osborn | An evaluation of terminal-restriction fragment length polymorphism (T-RFLP) analysis for the study of microbial community structure and dynamics[END_REF]. Each 20 μL reaction contained 20 ng of template DNA, 0.2 μM of each primer (Microsynth, Balgach, Switzerland), 120 μM of each deoxynucleoside triphosphates (Promega, Madison (WI), USA), 1 mM of MgCl 2 , 0.1 mg mL -1 of bovine serum albumin (BSA), 1× GoTaq® Colorless buffer and 0.5 U of GoTaq DNA polymerase (Promega, Madison (WI), USA). Reactions were carried out in a T1 thermocycler (Biometra, Göttingen, 6/20

Germany) with the following cycle: initial denaturation at 94 • C for 2 min, 20 cycles of denaturation at 94 • C for 1 min, annealing at 55 • C for 1 min, extension at 72 • C for 2 min, and a final extension at 72 • C for 10 min. The amplification and the size of the amplicons were checked on a 1% agarose gel. Amplicons were digested using the restriction enzyme AluI (Promega, Madison (WI), USA) [START_REF] Osborn | An evaluation of terminal-restriction fragment length polymorphism (T-RFLP) analysis for the study of microbial community structure and dynamics[END_REF] with 10 U for 3 h at 37 • C. Restriction digests were desalted by ethanol precipitation in the presence of glycogen and subsequently resuspended in deionized formamide and size standard GeneScan 500 LIZ (Life Technologies, Carlsbad (CA), USA).

The terminal restriction fragments (T-RFs) were separated using the automated sequencer ABI Prism 3130 (Life Technologies, Carlsbad (CA), USA) at the IMRB facility (Institut Mondor de Recherche Biomédicale at University Paris-Est Créteil). The resolution was of ∼1 bp for fragments up to 500 bp. Raw data files containing peak information were tabulated in the Peak Scanner TM software v 1.0 (Life Technologies, Carlsbad (CA), USA). Only peaks between 60 and 500 bp were selected. TRFLP data were then processed and analysed with the on-line tool T-REX [START_REF] Culman | T-REX: software for the processing and analysis of T-RFLP data[END_REF]. Data were subjected to quality control procedures: noise filtering (peak area, standard deviation multiplier = 2) and T-RF alignment (clustering threshold = 0.5). T-RFs detected in only one sample were not taken into account. Relative abundances were calculated using peak areas.

Bacterial enumeration

Bacterial counts (heterotrophic bacteria and cyanobacteria) from the Créteil Lake samples were assessed by ow cytometry using a Becton Dickinson FACScan (BD Biosciences, Oxford, UK) instrument equipped with a 15 mW 488 nm laser. Pigmented microorganisms were identified based on the auto fluorescence of the chlorophyll a thanks to the orange fluorescence (FL2, 585/42 bandpass filter) vs red fluorescence (FL3, 650 nm longpass filter). Cells containing chlorophyll a were distinguished using side scatter (SSC) vs FL3 and distinct size classes were clustered to discriminate cyanobacteria from small and large picoeukaryotes.

A subsample of each sample was stained with the nucleic acid stain SYBr Green I [START_REF] Marie | Enumeration and cell cycle analysis of natural populations of marine picoplankton by flow cytometry using the nucleic acid stain SYBR Green I[END_REF]) in order to visualize populations of heterotrophic organisms using SSC vs green fluorescence (530/30 bandpass filter). Chlorophyll a-containing organisms were gated off plots of SSC vs FL1, having been identified in plots of FL3 vs FL1. Populations were enumerated using a syringe-pump calibrated to 0.5 and 1 μm microspheres (Polysciences, Eppelheim, Germany) according to [START_REF] Zubkov | Syringe pumped high speed flow cytometry of oceanic phytoplankton[END_REF].

Environmental and meteorological parameters

For the 49 lakes, the total nitrogen and phosphorus concentrations were assayed by colorimetry using a Cary 50 Scan spectrophotometer (Varian Inc., Palo Alto (CA), USA) respectively according to [START_REF] Rogora | A comparison between high-temperature catalytic oxidation and persulphate oxidation for the determination of total nitrogen in freshwater[END_REF] and the French standard (AFNOR NF T 90-023).

Secchi depth and vertical profiles of physicochemical parameters were determined for each lake at the three stations (49 lakes and Créteil Lake). Chlorophyll a (Chl a) concentration was determined using a FluoroProbe in situ fluorometer (BBE Moldaenke GmbH, Kiel, Germany). Conductivity, temperature, pH and oxygen profiles were measured using a submersible CTD profiler SBE 19 Seacat (Sea-Bird Electronics Inc., Bellevue (WA), USA).

Concentrations of total suspended solids (TSS) were quantified after filtration of 1 L of lake water on a precombusted tarred Whatman GF/F filter. Dissolved organic carbon (DOC) concentrations were measured 7/20 using a TOC-VCSN carbon analyzer (Shimadzu, Columbia (MD), USA).

Meteorological data (temperature and precipitation) were also collected during the Créteil Lake campaigns using a weather real-time transmitter WXT520 (Vaisala Inc., Boulder (CO), USA) placed on a sensing platform LakeESP (PME Inc., Vista (CA), USA).

Watershed land use assessment

Upstream watersheds were delineated for each of the 49 lakes based on the digital elevation model (DEM) BD ALTI® 15 m (IGN, Paris, France) using ArcGIS 10.0 (ESRI Inc., Redland (CA), USA). Then, they were adjusted to take into account physical barriers that potentially modify the water flow (e.g., roads or underground drainages). Land use of the adjusted watershed was classified into four categories (i.e. natural, agricultural, open peri-urban and dense urbanized areas) using the MOS® GIS 2012 database (IAURIF, Paris, France).

Data analysis

Drivers of bacterioplankton spatial distribution in the Paris area

Based on the 49 lakes dataset, we first checked if the intra-annual variability of the BCS among lakes differed significantly from inter-annual variability within each lake using the Mann-Whitney test based on the Bray-Curtis dissimilarity index. The beta diversity index or the dissimilarity between lakes was calculated for each sampling year based on the Sørensen index. Beta diversity was then partitioned into two components following the framework proposed by [START_REF] Baselga | Partitioning the turnover and nestedness components of beta diversity[END_REF] to quantify the fraction of dissimilarity explained by the T-RF replacement (turnover, based on Simpson's dissimilarity index) and from the random variations of the T-RF richness (nestedness) [START_REF] Azeria | Towards a better understanding of beta diversity: deconstructing composition patterns of saproxylic beetles breeding in recently burnt boreal forests[END_REF]). These parameters were estimated using the beta.pair function from the 'betapart' package [START_REF] Baselga | Partitioning beta diversity into turnover and nestedness components[END_REF].

Then, the relative importance of environmental and spatial factors was assessed by decomposing the total bacterial community variation (Peres-Neto et al. 2006) using a variance partitioning analysis (VPA). Among all the environmental parameters measured, only six non-collinear variables were retained for the statistical analysis, i.e. water temperature, conductivity, pH, DOC concentration, trophic status and dominant land use index of the watershed. Trophic status (TS) was determined for each lake according to OECD (1982). Dominant land use index was included to assess the putative role of land use on the bacterial community. It was assessed as the land use (among the four categories) with the highest percentage of occupation. Spatial factors comprised two components, (i) variables reflecting the dispersal of bacterial taxa, i.e. with the presence of storm sewer outlet into the lake and the type of link to the hydrographic network, and (ii) variables integrating the community structuring at the regional scale, evaluated using the eigenvectors derived from the principal components of neighbor matrices of spatial coordinates (PCNM) [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF][START_REF] Legendre | Statistical methods for temporal and space-time analysis of community composition data[END_REF]. The type of linkage to the hydrographic network was decomposed into three categories (which roughly characterized the lake water retention time): (i) isolated lakes, which were mainly filled by alluvial water; (ii) lakes crossed by rivers and located in the riverbeds; and (iii) connected lakes, which were linked to a river but only by a single connection (e.g., small bond, pipe) and thus received water from both groundwater and river. VPA estimated the proportion of BCS variation (adjusted R-squared (R 2 adj )) that can be attributed to local environmental characteristics [E] and spatial [S] components, environmental conditions without spatial component [E|S], spatial without 8/20 environmental component [S|E], variation explained by the interaction between both components [E∩S] and the unexplained variation (1 -[E + S]). The significance of the partial contribution of both components was also evaluated with a Monte Carlo permutation test (999 permutations under the reduced model). Analyses were conducted using the varpart function from the 'vegan' package [START_REF] Oksanen | Package "vegan". Community ecology package[END_REF]. Prior to VPA, forward selections were performed for both components according to [START_REF] Borcard | Numerical ecology with R[END_REF] and [START_REF] Legendre | Statistical methods for temporal and space-time analysis of community composition data[END_REF].

To assess the neutral assembly of the bacterial communities, we used the method developed by [START_REF] Sloan | Quantifying the roles of immigration and chance in shaping prokaryote community structure[END_REF] to fit the regional relative abundance of the T-RFs and their observed detection frequency. Contrary to Hubbell's discrete model [START_REF] Hubbell | The unified neutral theory of biodiversity and biogeography[END_REF], this continuous model is particularly suited for bacterial communities (i.e. with large population size) detected by fingerprinting methods [START_REF] Sloan | Quantifying the roles of immigration and chance in shaping prokaryote community structure[END_REF]. The parameter N T m depicts the relationship between detection probability and regional relative abundance, with N T corresponding to the size of the metacommunity and m to the immigration rate (the probability that a dead individual is replaced by an immigrant). By considering the metacommunity size to be roughly equal among regions, N T m estimates the dispersal connectivity between each lake community. This parameter was estimated using the best fit between detection frequency of T-RFs and their regional relative abundance, by minimizing the sum of squares of errors. The detection limit was fixed to 0.005, which corresponds approximately to the threshold of the T-RFLP method (i.e. electrophoregram peaks with more than 0.5% of the total peak area of the sample). The goodness of fit was evaluated using the determination coefficient R 2 .

Spatio-temporal variability of BCS in Créteil Lake

The beta diversity as well as the turnover and the nestedness index were calculated over the 2 years of survey, considering consecutive pairwise months. Impacts of the environmental factors on the three indexes were investigated using a linear model. Data from Créteil Lake were used to test for significant variability in BCS along the horizontal and vertical transects. The presence of a space-time interaction was first evaluated using a two-way ANOVA crossed design as described by [START_REF] Legendre | Community surveys through space and time: testing the space-time interaction in the absence of replication[END_REF]. Included in the 'STI' package [START_REF] Legendre | Package 'STI'. Space-time ANOVA models without replications[END_REF], this method is particularly recommended for data without replication. As no space-time interaction was identified (data not shown), the same model without interaction was then implemented (STI Model 2, STI2) to test the significance of spatial and temporal variability along the two transects. Moreover, the environmental factors (water temperature, cumulative precipitation on 1, 5 and 15 days, pH, saturated oxygen, TSS, Chl a, DOC and total phosphorous concentrations) shaping the BCS over the 2-year survey were then investigated on Hellinger-transformed data using redundancy analysis (RDA), considering the sampling date as covariable. The neutral community model was performed as previously described to evaluate the importance of stochastic dynamics on the spatial distribution (between S1, C1, C2, C3 and R1) of the bacterial assemblage for each of the 25 campaigns. The parameter N T was estimated from the bacterial counts. Moreover, on the average sample M, we used the model developed by [START_REF] Ofiţeru | Combined niche and neutral effects in a microbial wastewater treatment community[END_REF] to evaluate to what extent the neutral model could explain the temporal variations of the relative abundance for each T-RF.

All statistical analyses were conducted using the statistical environment R version 3.1.1 (R Development Core Team 2014).

Results

Spatial distribution of the BCS at the regional scale

A total of 295 T-RFs were detected by T-RFLP analysis. Overall, 43% of T-RFs were removed after quality control. The 167 remaining T-RFs were used for the statistical analyses with a median of 33 T-RFs per lake. The majority (69%) of the T-RFs were found every summer in at least one lake, while 24% of the T-RFs were detected twice and 7% once. Furthermore, if only two T-RFs were present in all the 49 lakes, none was restricted to a single lake.

Per lake, the BCS dissimilarity was on average 54% between the years 2011 and 2012, and 53% between 2012 and 2013. These variations were significantly smaller than the beta diversity, i.e. the inter-lake variability considering the same year of sampling (Mann-Whitney test, P < 0.001). Indeed, the beta diversity displayed 63, 59 and 54% of dissimilarity for 2011, 2012 and 2013, respectively. Overall, by partitioning the beta diversity for the three years of campaigns, we observed that the variations between lakes were dominated by T-RF replacements that explained respectively 76, 70 and 67% of the beta diversity, whereas the random shift of T-RFs richness explained only 24, 30 and 33% of the beta diversity.

Whatever the year of sampling, environmental conditions and spatial factors explained only a small portion of the BCS variations, as evaluated by the variance partitioning analysis (VPA) (Table 2). On average over the 3 years of monitoring, the environmental variables alone (mainly the trophic status) and the spatial processes alone (i.e. the type of connection to the hydrological network and the spatial structuring of the bacterial community characterized by PCNM variables) and their interaction explained respectively 3, 4 and 3% of the BCS variations. Consequently, a large amount of the variance (on average 90%) remained unexplained by these variables (Table 2). Symbols associated with adjusted R 2 correspond to the significance ( * P < 0.05; NS, non significant) of the partial contribution of factors tested alone in the presence of the other. Abbreviations: Env, environmental; TS, trophic status; DOC, dissolved organic carbon concentration; Hydro, type of linkage to the hydrographical network; PCNM (principal components of neighbor matrices), reflect the community structuring at the spatial scale using PCNM variable; SD, standard deviation.

10/20

The neutral model explained a large fraction of the relationship between the occurrence frequency of T-RFs and their relative abundance variations (Fig. 3), with 82, 75 and 70% of explained variance for 2011, 2012 and 2013 databases. However, assuming a similar N T in each case, large variations of the immigration rate m were observed over the 3 years of sampling. Indeed, the value of N T m increased from 13 in 2011 to 79 in 2013. Furthermore, since all the relationships seemed strongly constrained by two T-RFs (which were present during the three campaigns), simulations without these points were performed again. The same values of N T m but a decrease of about 3% of total explained variance were obtained (data not shown), suggesting that these two T-RFs did not bias the immigration rate estimates.

Spatial and temporal monitoring in Créteil Lake

In Créteil Lake, the 2-year monitoring of the bacterial community by T-RFLP allowed the detection of 311 T-RFs. As for the 49 lakes, about 47% of the T-RFs were removed after quality control. A total of 166 T-RFs were used for the statistical analysis with a median value of T-RFs per month close to the one obtained per lake for the 49 lakes (i.e. 37 T-RFs).

There was no significant heterogeneity in the BCS along the horizontal transect over the 2 years of survey (STI2, R 2 = 0.01, F = 0.296, P = 0.998). This result suggested that there was no obvious impact of the storm sewer effluent on the BCS at the surface water, as found in the analysis of the 49 lakes. Similarly, no vertical difference in the BCS was observed at the central point (STI2, R 2 = 0.01, F = 0.742, P = 0.713), even when the lake was stratified (8 out of the 25 campaigns, data not shown). Since no spatial variability was encountered, we then focused on the pooled samples (M) to analyse the temporal variation in the BCS.

Redundancy analysis of T-RFLP profiles revealed (R 2 adj = 14%) that the temporal variations in BCS were significantly explained by water temperature (P = 0.001). Moreover, quite important dissimilarities in BCS were observed from month to month, with values ranging from 23 to 77% (Fig. 4). No clear temporal pattern in the beta diversity fluctuations could be reported (Fig. 4). However, when decomposing the beta diversity into its turnover and nestedness components, we observed distinct temporal variations (Fig. 4) strongly linked to variations in water temperature (linear models: R 2 = 0.46, F = 20.220, P < 0.001 for turnover and R 2 = 0.38, F = 5.825, P = 0.004 for nestedness). Indeed, a positive relationship between the temperature and the turnover was observed (Supplementary Fig. S1a), indicating that T-RF replacement was higher in summer. In contrast, the negative relationship between temperature and nestedness (Supplementary Fig. S1b) suggested that the composition of T-RFs was strongly influenced by richness variations in winter.

At the local scale, the neutral community model significantly explained the relationship between the T-RFs' occurrence frequency and their relative abundance variations in 22 out of the 25 campaigns (Supplementary Fig. S2). Over the 22 campaigns, the neutral model explained on average 65% of the total variance with values ranging from 33 to 91%. Contrary to the temporal BCS variations, neither the total variance explained by the neutral model (r Pearson = -0.34, P = 0.116), nor the immigration probability 12/20 appeared to be related to the water temperature (r Pearson = -0.25, P = 0.267; Supplementary Fig. S2).

Regarding the temporal dynamics of the T-RFs' abundance on the average sample M, the neutral timeseries model proposed by [START_REF] Ofiţeru | Combined niche and neutral effects in a microbial wastewater treatment community[END_REF] could explain the relative abundance variations of only one T-RF (R 2 adj = 76%, P < 0.001). The lack of relevance of this model to our data may come from the presence of numerous zeros in the dataset corresponding to the T-RFs that were not detected during campaigns. Indeed, the single T-RF that fitted the neutral model was the only one detected several times in the 2 years. Besides, when performing the neutral time-series model on the average relative abundance of the five Créteil Lake points (S1, C1, C2, C3 and R1), four out of the nine T-RFs detected over all the 25 campaigns had a relative abundance variability significantly explained by the neutral model with an average R 2 adj of 34% (data not shown).

Discussion

The main goal of this study was to investigate not only the relative influence of deterministic processes (i.e. environmental conditions and spatial factors) but also the influence of neutral processes on the bacterioplankton metacommunity within a large set of shallow lakes. One of the original aspects of this study lies in the 2-year monthly monitoring of the Créteil Lake combined with the three repeated summer samplings of 49 contrasted lakes to take into account inter-annual variability. Indeed, metacommunity surveys in continental aquatic systems integrating more than a single sampling campaign are rare (e.g., [START_REF] Van Der Gucht | The power of species sorting: local factors drive bacterial community composition over a wide range of spatial scales[END_REF][START_REF] Logares | Biogeography of bacterial communities exposed to progressive long-term environmental change[END_REF]. Our results suggested that stochastic processes were relevant at both regional and local scales to explain the bacterioplankton metacommunity assembly.

Fingerprinting methodology considerations

Factors and processes influencing BCS were explored using a fingerprinting method that may create a bias by underestimating the contribution of rare members from the sampled bacterial community (Pedrós-Alió 2006). Thus, if non-detected taxa are generalists, spatial and temporal BCS variation would be buffered. Inversely, differences would be enhanced if non-detected taxa happened to be specialists of local environmental conditions [START_REF] Székely | The importance of species sorting differs between habitat generalists and specialists in bacterial communities[END_REF]. Although high-throughput sequencing yields a greater coverage of the bacterial diversity and may avoid this potential bias, comparable patterns in terms of diversity and structure were obtained on the same samples with fingerprinting methods including the TRFLP (e.g., [START_REF] Castro-Carrera | Rumen bacterial community evaluated by 454 pyrosequencing and terminal restriction fragment length polymorphism analyses in dairy sheep fed marine algae[END_REF][START_REF] Elsayed | Bacterial communities in batch and continuous-flow wetlands treating the herbicide S-metolachlor[END_REF][START_REF] Van Dorst | Community fingerprinting in a sequencing world[END_REF][START_REF] Thomson | Soil conditions and land use intensification effects on soil microbial communities across a range of European field sites[END_REF] whatever the primer set used. Thus it seems reasonable to validate our conclusions regarding the relative impact on BCS of environmental conditions, spatial factors and stochastic processes.

Minor influence of deterministic processes on the BCS at the regional scale

Our results indicate that environmental conditions and spatial factors play a minor role in shaping the BCS at the regional scale, as suggested by the low amount of total variance explained by these two factors. Several reasons could lead to this result. First, according to [START_REF] Lindström | Local and regional factors influencing bacterial community assembly[END_REF], the large proportion of the BCS variance that remains unexplained (90%) could be explained by the absence of 13/20 relevant variables taken into account in the statistical models. For example, [START_REF] Van Der Gucht | The power of species sorting: local factors drive bacterial community composition over a wide range of spatial scales[END_REF] showed in the de Maten reserve, a well studied system of connected ponds, that 89% of the total BCS variance was explained by environmental and spatial factors. Local factors used in the de Maten analysis included classical physicochemical parameters (e.g., conductivity, pH), but also biological variables (i.e. phytoplankton and zooplankton biomass) that could directly structure the BCS owing to top-down control. Second, since all lakes are different regarding the local parameters, it is possible that overall the statistical analysis does not detect any tendency. Third, the small influence of deterministic factors on the bacterial community assembly could also originate from the dominance of generalist taxa in these ecosystems. However, the relatively high beta diversity tends to invalidate this hypothesis. Finally, [START_REF] Gilbert | Partitioning variation in ecological communities: do the numbers add up?[END_REF] demonstrated that variance-partitioning analysis tends to underestimate the relative importance of the deterministic factors.

Although only a small amount of total variance was explained by these factors, several variables were identified as having a significant effect on the BCS. Among the environmental factors, the trophic status was the only variable systematically considered significant over the 3 years of sampling. This parameter has already been characterized in the literature to shape the BCS [START_REF] Lindström | Bacterioplankton community composition in five lakes differing in trophic status and humic content[END_REF][START_REF] Yannarell | Temporal patterns in bacterial communities in three temperate lakes of different trophic status[END_REF][START_REF] Yannarell | Within-and between-lake variability in the composition of bacterioplankton communities: investigations using multiple spatial scales[END_REF]. Among the spatial factors, a relative similar fraction of total BCS variations was weakly explained by the potential ability of bacteria to disperse (characterized by the type of linkage to the hydrographical network) and the spatial structure of the bacterial community at the regional scale (characterized by the PCNMs variables) (Supplementary Table S2). Furthermore, in this analysis, no significant direct influence of storm sewer discharges into the lakes was identified. Similarly, no influence of the dominant watershed land use (i.e. natural, agricultural, open or dense urban area) was detected, though previous literature suggested an impact of land use on BCS in rock biofilms or on bacterial population [START_REF] Scopel | Influence of nearshore water dynamics and pollution sources on beach monitoring outcomes at two adjacent Lake Michigan beaches[END_REF][START_REF] Mclellan | Distribution and fate of Escherichia coli in lake Michigan following contamination with urban stormwater and combined sewer overflows[END_REF][START_REF] Lear | Impact of catchment land use on bacterial communities within stream biofilms[END_REF].

Bacterial assembly shape by neutral processes at the regional scale

The neutral community model was very powerful in explaining the T-RF proportion observed in our panel of contrasted lakes (mean R 2 = 76%). This result suggested that the bacterial community was shaped by a stochastic balance between loss and gain of taxa [START_REF] Hubbell | The unified neutral theory of biodiversity and biogeography[END_REF][START_REF] Sloan | Quantifying the roles of immigration and chance in shaping prokaryote community structure[END_REF]. Although numerous microcosm or environmental experiments found evidence that bacterial assembly was jointly shaped by local factors and neutral processes [START_REF] Ayarza | Balance of neutral and deterministic components in the dynamics of activated sludge floc assembly[END_REF][START_REF] Langenheder | Species sorting and neutral processes are both important during the initial assembly of bacterial communities[END_REF][START_REF] Lee | Both species and neutral processes drive assembly of Roguet et al[END_REF][START_REF] Pholchan | Microbial community assembly, theory and rare functions[END_REF], our results are in an agreement with the study performed by [START_REF] Drakare | Local factors control the community composition of cyanobacteria in lakes while heterotrophic bacteria follow a neutral model[END_REF] on 13 Swedish lakes. Indeed, these authors also reported the predominance of neutral processes compared with local factors. Our results could be explained by the strong adaptability and plasticity of the bacterial compartment to environmental perturbations and gradients, reducing the apparent sensitivity to local factors (Östman et al. 2010). Although our data well fitted the neutral community model, it is difficult to infer the absence of the influence of deterministic factors on BCS [START_REF] Chisholm | Niche and neutral models predict asymptotically equivalent species abundance distributions in high-diversity ecological communities[END_REF], especially as the fraction of variance unexplained by the VPA could not be attributed to stochastic processes [START_REF] Vellend | Assessing the relative importance of stochasticity in ecological communities[END_REF]. Moreover, as discussed in [START_REF] Anderson | Navigating the multiple meanings of β diversity: a roadmap for the practicing ecologist[END_REF], the neutral and the deterministic mechanisms might coincidentally mirror the same patterns. Therefore, isolating the importance of both components separately could be perilous without experimental approaches.

14/20

Regarding the immigration rate, large inter-annual discrepancies were observed with an increase from 2011 to 2013. According to the neutral theory [START_REF] Hubbell | The unified neutral theory of biodiversity and biogeography[END_REF], this is consistent with the decrease in the beta diversity over the 3 years of sampling. The inter-annual variation of the immigration rate could also result from meteorological conditions that may have led to higher bacterial dispersion in 2013 (e.g., strong wind). However, these variations could also result from a difference in the metacommunity size and not from the immigration rate.

Furthermore in this study, we didn't identify any other important mechanism apart from the neutral processes that could influence the assembly of communities. Indeed, we were unable to clearly identify a mass effect in lakes linked to rivers as observed by [START_REF] Lindström | External control of bacterial community structure in lakes[END_REF] and [START_REF] Nelson | Contrasting the influences of stream inputs and landscape position on bacterioplankton community structure and dissolved organic matter composition in highelevation lake chains[END_REF]. According to the literature, this absence of effect could be due to the trophic status of the 49 lakes (mainly ranging from eutrophic to hypereutrophic levels). Indeed, a mass effect appears to be more effective in oligotrophic lakes [START_REF] Van Der Gucht | The power of species sorting: local factors drive bacterial community composition over a wide range of spatial scales[END_REF], whereas more productive lakes seem to be dominated by stochastic processes compared with deterministic processes [START_REF] Chase | Stochastic community assembly causes higher biodiversity in more productive environments[END_REF].

This survey showed the importance of the neutral dynamics on the bacterioplankton metacommunity assembly at the regional scale. However, this sole approach did not allow identification of spatial and temporal mechanisms that govern the BCS at the scale of a single ecosystem though these processes could have a direct influence on the dynamics at the regional scale. These issues were explored at the local scale within the Créteil Lake.

Influence of deterministic and neutral processes at the local scale

In Créteil Lake, we expected a spatial structure to exist as from the storm sewer outlet to the lake outlet as it may represent an important source of exogenous taxa [START_REF] Sercu | Storm drains are sources of human fecal pollution during dry weather in three urban southern california watersheds[END_REF]). In addition, lakes that have a long water retention time (>180 days for Créteil Lake) are characterized by significant dissimilarities in BCS between the lake water column and the inlet [START_REF] Lindström | External control of bacterial community structure in lakes[END_REF][START_REF] Nelson | Contrasting the influences of stream inputs and landscape position on bacterioplankton community structure and dissolved organic matter composition in highelevation lake chains[END_REF]. However, our 2-year monthly survey did not reveal any impact of the storm sewer effluent on the dominant bacterial taxa in the surface waters, even after strong rainfall events. The lack of intra-lake horizontal heterogeneity may originate from the moderate size of Créteil Lake and thus the small distance between the three sampling points. Indeed, important in-lake BCS spatial variation has generally been described in larger lakes (e.g., [START_REF] Yannarell | Within-and between-lake variability in the composition of bacterioplankton communities: investigations using multiple spatial scales[END_REF][START_REF] De Wever | Bacterial community composition in Lake Tanganyika: vertical and horizontal heterogeneity[END_REF][START_REF] Jones | Spatial and temporal scales of aquatic bacterial beta diversity[END_REF]) contrary to smaller lakes [START_REF] Jones | Spatial and temporal scales of aquatic bacterial beta diversity[END_REF]. In Créteil Lake, this result only applies to three surface locations and cannot be generalized to the whole lake or other compartments such as the sediments. In addition, the T-RFLP technique only assesses dominant taxa [START_REF] Bent | Measuring species richness based on microbial community fingerprints: the emperor has no clothes[END_REF] and we cannot exclude an impact of storm sewer outlet on less abundant taxa such as pathogens or fecal indicators. Regarding the vertical profile, no BCS stratification was observed at the central station, even when the water column was stratified for several days (from 3 to 30 days of stratification). These two results showed a relative spatial homogeneity of the dominant bacterioplankton community on the different sampled points that could result from neutral processes. Overall, the neutral community model explained a large proportion of the bacterial assemblage variance per campaign (R 2 mean = 65%). However, no distinct seasonal pattern of neutral model parameters (R 2 and N T m) was observed over the 2 years of survey, suggesting that the influence of stochastic processes was rather constant through the year.

15/20 Furthermore, the BCS within Créteil Lake displayed a marked temporal variation all along the sampling campaigns, essentially driven by the water temperature. Such importance of this deterministic parameter has already been reported in the literature in a wide range of aquatic habitats (e.g., [START_REF] Muylaert | ) 19/20 between bacterial community composition and bottom-up versus top-down variables in four eutrophic shallow lakes[END_REF][START_REF] Stepanauskas | Covariance of bacterioplankton composition and environmental variables in a temperate delta system[END_REF][START_REF] Jardillier | Bottom-up and top-down control of bacterial community composition in the euphotic zone of a reservoir[END_REF]). Moreover, this temporal analysis revealed that the taxa replacement was the main process shaping BCS in summer while random variation in T-RF richness occurred in winter. This pattern could result from the consequence of a trade-off between competitive abilities and resistance to predators over low and high productivity levels [START_REF] Leibold | Biodiversity and nutrient enrichment in pond plankton communities[END_REF][START_REF] Horner-Devine | Bacterial diversity patterns along a gradient of primary productivity[END_REF] or simply be due to purely random input of bacterial taxa, for example owing to the sediment resuspension or the sewer outlet input during rainy periods after a long period of dry weather. In addition to the importance of the water temperature on the BCS, some T-RFs displayed neutral time-series dynamics that explained up to 80% of their relative abundance variability. However, with our results, it is difficult to generalize the importance over time of the neutral processes on BCS since a large fraction of the T-RFs were not observed to be shaped by stochastic processes, indicating that bacterial groups were not all governed by the same processes (e.g., [START_REF] Barberán | Global phylogenetic community structure and βdiversity patterns in surface bacterioplankton metacommunities[END_REF][START_REF] Székely | The importance of species sorting differs between habitat generalists and specialists in bacterial communities[END_REF], probably due to specific ecological traits [START_REF] Philippot | The ecological coherence of high bacterial taxonomic ranks[END_REF].

This study showed that the bacterial assembly in the 49 lakes located in the Paris area appeared to be strongly governed by stochastic processes. Coupled to this approach, the finer monitoring performed on Créteil Lake revealed that neutral dynamics spatially structured the bacterial community. BCS was temporally governed by a deterministic factor, i.e. the water temperature, and also by stochastic processes for some T-RFs.

Further studies on artificial shallow lakes, performed on a balanced group in terms of trophic status and type of linkage to the hydrographical network, should be carried out in order to develop a more robust overview of the mechanisms shaping the bacterial community.

Figure 1 .

 1 Figure 1. Position of the 49 lakes in the Paris area (France). Créteil Lake is indicated by an open circle.

Figure 2 .

 2 Figure 2. Sampling strategies in the Créteil Lake. (A) Location of the three stations. (B) Representation of the two transects: the horizontal transect (S1, C1 and O1) and the vertical transect (C1, C2 and C3).

Figure 3 .

 3 Figure 3. Relationship between the detection frequency of T-RFs and their mean relative abundance from the three years of monitoring. Each point represents a different T-RF. The lines show the best fit to the neutral model with random immigration from a common source pool as described by[START_REF] Sloan | Quantifying the roles of immigration and chance in shaping prokaryote community structure[END_REF]. NTm values represent the meta-community size multiplied by the immigration rate. The percentages represent the R 2 value of the fit.

Figure 4 .

 4 Figure 4. Temporal variations of the bacterial community beta diversity, turnover and nestedness determined by comparing pairwise successive sampling campaign (n = 24).

Table 1 .

 1 Characteristics of Créteil Lake. Values are the means (standard deviations) of data obtained from December 2011 to December 2013.

	Parameter	Values
	Location	48°46.50'N 2°27.10'E
	Surface area (km 2 )	0.40
	Mean depth (m)	4.5
	Max depth (m)	6.0
	Trophic status	Mesotrophic
	Total phosphorous (μgP L -1 )	64.7 (41.8)
	Chlorophyll a (μg L -1 )	6.3 (4.3)
	Secchi depth (m)	2.4 (0.9)
	COD (mgC L -1 )	6.1 (0.6)
	COP (mgC L -1 )	0.8 (0.3)
	TSS (mg L -1 )	4.1 (1.5)
	pH	7.8 (0.5)
	Conductivity (μS cm -1 )	1509 (63)
	DOC, dissolved organic carbon; POC, particulate organic carbon.

Table 2 .

 2 Variation partitioning analysis of the bacterial community structure for the three years of sampling.

			Relative variance explained (adjusted R-squared)	Variables selected
	Year of sampling	Env. factors [E]	Spatial factors [S]	Env. factors alone	Spatial factors alone	Interaction [E∩S]	Residuals	Env. factors	Spatial factors
				[E|S]	[S|E]
	2011	0.05	0.06	0.03*	0.04*	0.02	0.91	TS, pH	Hydro, PCNM (no. 6)
	2012	0.08	0.11	0.03*	0.06*	0.05	0.86	TS, DOC	Hydro, PCNM (no. 2,4,7,9)
	2013	0.06	0.03	0.04*	0.01 NS	0.02	0.93	TS	Hydro, PCNM (no. 2,7,9)
	Mean	0.06	0.06	0.03	0.04	0.03	0.90
	(sd)	(0.02)	(0.04)	(0.01)	(0.03)	(0.02)	(0.04)

[START_REF] Chisholm | Niche and neutral models predict asymptotically equivalent species abundance distributions in high-diversity ecological communities[END_REF] (FEMS Microbiology Ecology doi: 10.1093/femsec/fiv125)

Acknowledgments

The authors thank Pierre Legendre for his helpful advices in the statistical analysis, Sophie Penisson for her support in decrypting the Sloan's neutral model equation and the two anonymous reviewers whose comments improved the manuscript. We warmly thank Ross Holland from Aqua Cytometry (UK) for his precious contribution in the bacterial count. We are grateful to the stakeholders and owners for their authorization to collect samples in each lake. We also thank the leisure base of Créteil Lake for lending us their boats over the 2 years of sampling. Finally we warmly thank all the colleagues who helped in sampling and quantifying the ecological parameters in the 49 lakes and Créteil Lake during these 3 years.

Funding

This work was supported by the French National Research Agency through the PULSE (Peri-Urban Lakes, Society and Environment) research project (ANR-10-CEPL-010).

Conflict of interest. None declared