Houda Hani

Moez Khenissi

ON A FINITE DIFFERENCE SCHEME FOR BLOW UP SOLUTIONS FOR THE CHIPOT-WEISSLER EQUATION by

Keywords: Nonlinear parabolic equation, Chipot-Weissler equation, finite time blow up, finite difference scheme, numerical solution, nonlinear gradient term

In this paper, we are interested in the numerical analysis of blow up for the Chipot-Weissler equation u t = ∆u + |u| p-1 u -|∇u| q with Dirichlet boundary conditions in bounded domain when p > 1 and 1 ≤ q ≤ 2p p + 1 .

To approximate the blow up solution, we construct a finite difference scheme and we prove that the numerical solution satisfies the same properties of the exact one and blows up in finite time.

Introduction

In this paper, we study the numerical approximation of solutions which achieve blow up in finite time of the Chipot-Weissler equation

u t = ∆u + |u| p-1 u -|∇u| q in Ω × R + , (1.1)
with Dirichlet boundary conditions u(x, t) = 0 , x ∈ ∂Ω and t > 0, (1.2) and initial data u(x, 0) = u 0 (x) ; x ∈ Ω,

where Ω is a regular bounded domain in R d , and p, q are fixed finite parameters.

This problem represents a model in population dynamics which is proposed by Souplet in [START_REF] Ph | Souplet, finite time blow up for a nonlinear parabolic equation with a gradient term and applications[END_REF], where (1.1)- (1.3) describes the evolution of the population density of a biological species, under the effect of certain natural mechanisms; u(x, t) denote the spatial density of individuals located near a point x ∈ Ω at a time t ≥ 0. The evolution of this density depends on three types of mechanisms: displacements, births and deaths. The reaction term represents the rate of births. If we suppose that the individuals can be destroyed by some predators during their displacements, then the dissipative gradient term represents the density of predators.

We are concentrated on solution u of (1.1)-(1.3) which blow up in the L ∞ norm in the following sence : There exists T * < ∞, called the blow up time such that the solution u exists on [0, T * ) and lim t→T * u(t) L ∞ = +∞. Numerous articles have been published concerning the problem of global existence or nonexistence of solutions to nonlinear parabolic equations. Problem (1.1)-(1.3), has been widely analyzed from a mathematical point of view, on the profile, blow up rates, asymptotic behaviours and self similar solutions (see for example: [START_REF] Snoussi | asymptotically self similar global solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF], [START_REF] Ph | blow up rates for nonlinear heat equations with gradient terms and for parabolic inequalities[END_REF], [START_REF] Ph | exact self similar blow up of solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF], [START_REF] Quittner | on global existence and stationary solutions for two classes of semilinear parabolic problems[END_REF] and [START_REF] Zaag | Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term[END_REF]), but to our knowledge, there are no results concerning its numerical approximation.

Let us first have a look at the theoretical analysis of this problem. The quasilinear parabolic equation (1.1) was introduced in 1989 by Chipot and Weissler [START_REF] Chipot | some blow up results for a nonlinear parabolic problem with a gradient term[END_REF], in order to investigate the effect of a damping term on global existence or nonexistence of solutions. They proved that under appropriate conditions on q, p and d, there exists a suitable initial value u 0 so that the corresponding solution of (1.1)-(1.3) blows up in a finite time. More precisely: Theorem 1.1. - [START_REF] Chipot | some blow up results for a nonlinear parabolic problem with a gradient term[END_REF] : Let p > 1, 1 ≤ q ≤ 2p p + 1 and u 0 ∈ W 3,s (Ω) for s sufficiently large, u 0 not identically zero. Suppose in addition that:

1. u 0 = 0 on ∂Ω.

2. ∆u 0 -|∇u 0 | q + |u 0 | p = 0 on ∂Ω.

3. u 0 ≥ 0 in Ω. 4. ∆u 0 -|∇u 0 | q + u p 0 ≥ 0 in Ω.

E(u

0 ) = 1 2 ∇u 0 2 2 - 1 p + 1 u 0 p+1 p+1 ≤ 0.
6. If q < 2p p + 1 then u 0 p+1 is sufficiently large.

7. If q = 2p p + 1 then p is sufficiently large.

Then the corresponding solution of (1.1)-(1.3) blows up in finite time, in the L ∞ norm.

The obvious difficulty with this result is the existence of such initial data.

Souplet and Weissler have proved in [START_REF] Ph | self similar sub-solutions and blow up for nonlinear parabolic equations[END_REF] that finite time blow up occurs in W 1,s norm (s sufficiently large), for large data whenever p > q, more precisely we have :

Theorem 1.2.
- [START_REF] Ph | self similar sub-solutions and blow up for nonlinear parabolic equations[END_REF] : Assume p > q and let ψ ∈ W 1,s 0 (Ω) for s sufficiently large with ψ ≥ 0, (ψ = 0).

1.

There exists some λ 0 = λ 0 (ψ) > 0 such that for all λ > λ 0 , the solution of (1.1)-(1.3) with initial data φ = λψ blows up in finite time in W 1,s norm. 2. There is some C > 0 such that

T * (λψ) ≤ C (λ |ψ| ∞ ) p-1 , λ → ∞.
The phenomenon of blow up in finite time for nonlinear parabolic equations has been extensively studied for last decades. Several papers contain numerous references on blow up results, see for example [START_REF] Friedman | blow up solutions of nonlinear parabolic equations[END_REF], [START_REF] Levine | the role of critical exponents in blow up theorems[END_REF], [START_REF] Fujita | on the blowing up of solutions of the Cauchy problem for u t = ∆u + u 1+α[END_REF] and [START_REF] Hayakawa | on nonexistence of global solutions of some semilinear parabolic equations[END_REF]. There has been many works in the past concerning numerical computation of solutions of nonlinear parabolic equation (see [START_REF] Hamada | on the finite difference approximation for a parabolic blow up problem[END_REF], [START_REF] Cho | A finite difference scheme for blow up solutions for nonlinear wave equations[END_REF], [START_REF] Nakagawa | blowing up on a finite difference solution to u t = u xx + u 2[END_REF] and [START_REF] Chen | asymptotic behaviours of blowing up solutions for finite difference analogue of u t = u xx + u 1+α[END_REF]) but without the gradient term. By studing various papers, we found that many interesting numerical problems for the Chipot-Weissler equation are left unsolved, and we would like to solve some of them in this and other forcoming works. The results of this paper are used to study the properties of the numerical solution associated to (1.1)-(1.3) and hence we can assimilate the dissipatif role of the gradient term [START_REF] Hani | asymptotic behaviours of solutions for finite difference analogue of the Chipot-Weissler equation[END_REF].

Although the details are explained in the subsequent sections, we outline here the main ideas of this study. Let us recall a result of Chen where he considered the Fujita equation u t = u xx + u 1+α and proved that some numerical solutions can blow up in finite time at more than one point [START_REF] Chen | asymptotic behaviours of blowing up solutions for finite difference analogue of u t = u xx + u 1+α[END_REF]. The following questions may naturally arise:

1. What happens numerically, if we add a gradient term in the Fujita equation? 2. Which conditions on the reaction term and the gradient term to provide or prevent blow-up? 3. Can we compare the blowing-up rates of the equations with and without gradient term?

In this paper, we develop a numerical scheme in order to approximate the solutions of the nonlinear Chipot-Weissler equation in Ω = [-1, 1], and we show that the finite difference solution blow up in finite time if a certain condition is assumed. Next, we study the numerical blow up set and the asymptotic behaviours of our numerical solution near the blow up point, we also try to give an approximation of the blowing up time.

Our paper is organized as follows: In the next section, we present some properties of the exact solution. In section 3, we construct a finite difference scheme and we prove that if the initial data is positive, monotone and symmetric then the numerical solution is also positive, monotone and symmetric. In section 4, we shall prove that the difference solution blows up in x = 0 the middle of the interval [-1, 1]. In section 5, we give some numerical results to illustrate our analysis. In the last section, we present some interesting questions which will be solved in the future study.

Properties of the exact solution

We consider the semilinear equation

u t = u xx + |u| p-1 u -|u x | q in ]0, T [×] -1, 1[, (2.1)
with initial data

u(0, x) = u 0 (x) for x ∈ [-1, 1] (2.2)
and Dirichlet boundary conditions

u(t, -1) = u(t, 1) = 0 for t ∈ [0, T ]. (2.3)
where p > 1, 1 ≤ q ≤ 2p p + 1 and T < T * .

Here if t ∈ [0, T ] then u(t, .) ∞ := max x |u(t, x)| < ∞. For the sake of simplicity, we assume that the initial function u 0 satisfies the following conditions: (A1) u 0 is continuous, nonconstant and nonnegative in

[-1, 1]. (A2) u 0 is spatially symmetric about x = 0. (A3) u 0 is strictly monotone increasing in [-1, 0]. (A4) u 0 is large in the sense that u 0 ∞ >> 1.
(A5) u 0 (-1) = u 0 (1) = 0. These properties will be preserved by our numerical scheme and make computations easier. Under these conditions, it is known from [START_REF] Cheblik | blowup of positive solutions of a semilinear parabolic equation with a gradient term[END_REF] that the solution blows up only at the central point, that is

∃ T * < +∞ such that lim t→T * u(t, 0) = +∞ but lim t→T * u(t, x) < ∞ when x = 0. 2.1. Regularity. -It is known from [1] that Theorem 2.1. -[1] : For Ω =] -1, 1[ and r > max(1, q -1), problem (2.1)-(2.
3) generates a local semiflow on W 1,r 0 (Ω) + := u ∈ W 1,r (Ω); u ≥ 0 in Ω and u = 0 on ∂Ω and for any u 0 ∈ W 1,r 0 (Ω) + there exists a unique maximal solution u ∈ C [0, T * ) , W

∀x ∈ [-1, 1], u(x, t) = u(-x, t) ∀t ∈ [0, T * ).
Proof. -Let u(x, t) be the solution of (2.1)-(2.3). We define, for all t ≥ 0, the function

v(x, t) =    u(-x, t) x ∈ [-1, 0] u(x, t) x ∈ [0, 1].
We shall prove that v is a solution of (2.1)-

(2.3) in [-1, 1].
-In [0, 1], we have v(x, t) = u(x, t). Then for all x ∈]0, 1[ and t ≥ 0, v satisfies,

     ∂v ∂t (x, t) - ∂ 2 v ∂x 2 (x, t) -|v(x, t)| p + ∂v ∂x (x, t) q = 0 v(1, t) = 0. (2.4) Then v is a solution of (2.1) in [0, 1[. -In ] -1, 0], we have v(x, t) = u(-x, t).
Then for x ∈] -1, 0] and t ≥ 0 we get

∂v ∂t (x, t) - ∂ 2 v ∂x 2 (x, t) -|v(x, t)| p + ∂v ∂x (x, t) q = ∂u ∂t (-x, t) - ∂ 2 u ∂x 2 (-x, t) -|u(-x, t)| p + ∂u ∂x (-x, t) q .
Since -x ∈ [0, 1[ and u(-x, t) is a solution in [0, 1[, we deduce then that

     ∂v ∂t (x, t) - ∂ 2 v ∂x 2 (x, t) -|v(x, t)| p + ∂v ∂x (x, t) q = 0 v(-1, t) = 0.
(2.5)

Using (A2), we obtain v(x, 0) = u 0 ∀x ∈ [-1, 1]. (2.6)
Finally, by (2.4), (2.5), (2.6) and by unicity of the solution of (2.1)-( 2.3), we

get u = v in [-1, 1].
This finishes the proof of the symmetry.

2.4. Monotony. - Theorem 2.4. -If the initial data satisfies (A3), then the exact solution u of (2.1)-(2.3) is increasing in [-1, 0[, that is: ∀x ∈ [-1, 0[ and t ≥ 0 we have ∂u ∂x (x, t) ≥ 0.
Proof. -Let u(x, t) be the solution of (2.1)-(2.3) in [-1, 0[, and v(x, t) = ∂u ∂x (x, t).

For all x ∈ [-1, 0[ and t > 0, we have

∂u ∂t (x, t) - ∂ 2 u ∂x 2 (x, t) -u p (x, t) + ∂u ∂x (x, t) q = 0. Then ∂v ∂t (x, t) - ∂ 2 v ∂x 2 (x, t) -pu p-1 (x, t)v(x, t) + q ∂v ∂x sign(v) |v| q-1 (x, t) = 0. (2.7) Let v -= max(0, -v).
Multiplying (2.7) by v -(x, t), we get

∂v ∂t v -- ∂ 2 v ∂x 2 v --pu p-1 vv --q ∂v ∂x v -|v| q-1 = 0 ⇒ - ∂v - ∂t v -+ ∂ 2 v - ∂x 2 v -+ pu p-1 (v -) 2 + q ∂v - ∂x v -v -q-1 = 0.
Using symmetry and integrating over [-1, 1], we get

1 -1 ∂v - ∂t v -dx - 1 -1 ∂ 2 v - ∂x 2 v -dx -p 1 -1 u p-1 (v -) 2 dx - 1 -1 ∂v - ∂x (v -) q dx = 0 ⇒ 1 2 d dt 1 -1 (v -) 2 dx - ∂v - ∂x v - 1 -1 - 1 -1 ∂v - ∂x 2 dx -p 1 -1 u p-1 (v -) 2 dx - 1 -1 ∂v - ∂x (v -) q dx = 0. But ∂v - ∂x v - 1 -1 = ∂v - ∂x (1, t)v -(1, t) - ∂v - ∂x (-1)v -(-1, t) = 2v -(1, t) ∂v - ∂x (1, t) = -2v -(1, t) ∂v ∂x (1, t). (2.8)
In the other hand, from (2.2), we have

u p (1, t) = ∂u ∂t (1, t) = 0, then by (2.1) ∂v ∂x (1, t) = ∂ 2 u ∂x 2 (1, t) = ∂u ∂x (1, t) ≥ 0,
and then we can deduce from (2.8) that

∂v - ∂x v - 1 -1 < 0. So 1 2 d dt 1 -1 (v -) 2 dx + 1 -1 ∂v - ∂x 2 dx -p 1 -1 u p-1 (v -) 2 dx - 1 -1 ∂v - ∂x (v -) q dx = ∂v - ∂x v - 1 -1 ≤ 0.
Using the symmetry propriety, we get

d dt 0 -1 (v -) 2 dx + 2 0 -1 ∂v - ∂x 2 dx -2p 0 -1 u p-1 (v -) 2 dx -2 0 -1 ∂v - ∂x (v -) q dx ≤ 0 ⇒ 1 2 d dt 0 -1 (v -) 2 dx ≤ - 0 -1 ∂v - ∂x 2 dx + p 0 -1 u p-1 (v -) 2 dx + 0 -1 ∂v - ∂x (v -) q dx .
We refer to proposition 2.2 in [START_REF] Chipot | some blow up results for a nonlinear parabolic problem with a gradient term[END_REF], we can see that u and ∇u are bounded before blow up, then there exists M, N > 0 such that

u p-1 (x, t) ≤ M for all x ∈ [-1, 1] and 0 < t < T, |∇u(x, t)| ≤ N for all x ∈ [-1, 1] and 0 < t < T.
We use Young's inequality, then for all ǫ > 0, there exists

C ǫ > 0, such that 0 -1 ∂v - ∂x (v -) q dx ≤ ǫ 0 -1 ∂v - ∂x 2 dx + C ǫ 0 -1 (v -) 2q dx.
Which implies that

0 -1 ∂v - ∂x (v -) q dx ≤ ǫ 0 -1 ∂v - ∂x 2 dx + C ǫ N 2q-2 0 -1 (v -) 2 dx.
Then we get

1 2 d dt 0 -1 (v -) 2 dx ≤ - 0 -1 ∂v - ∂x 2 dx + M 0 -1 (v -) 2 dx + ǫ 0 -1 ∂v - ∂x 2 dx +C ǫ N 2q-2 0 -1 (v -) 2 dx = (ǫ -1) 0 -1 ∂v - ∂x 2 dx + M 1 0 -1 (v -) 2 dx,
where M 1 is a constant depending on N, M and C ǫ .

For ǫ sufficiently small we get

d dt 0 -1 (v -) 2 dx ≤ M 1 0 -1 (v -) 2 dx.
Integrationg over [0, t], for t ≤ T , we get

0 -1 (v -) 2 (x, t)dx ≤ 0 -1 (v -) 2 (x, 0)dx + M t 0 0 -1 (v -) 2 (x, s)dxds.
From (A3), we have

v -(x, 0) = ( ∂u ∂x ) -(x, 0) = ∂u 0 ∂x - (x) = 0.
Then using Gronwall lemma, we deduce that ∀x ∈ [-1, 0[, t ≥ 0 v -(x, t) = 0, which achieve the proof of Theorem 2.4.

Full discretization

We consider the semilinear parabolic equation

       u t = u xx + |u| p-1 u -|u x | q in ]0, T [×] -1, 1[, u(0, x) = u 0 (x) for x ∈ [-1, 1], u(t, -1) = u(t, 1) = 0 for t ∈ [0, T ]. (3.1)
In this section, we construct a finite difference scheme whose solution approximate the solution of (3.1) and satisfies the same properties proved above. Throughout this paper, we use the following notations in the list below:

1. τ : size parameter for the variable time mesh τ n .

2. h : positive parameter for which λ := τ h 2 < 1 16 kept fixed. 3. t n : n-th time step on t > 0 determined as:

       t 0 = 0 t n = t n-1 + τ n-1 = n-1 k=0 τ k ; n ≥ 1. 4. x j : j-th net point on [-1, 1] determined as:        x 0 = -1 x j = x j-1 + h n ; j ≥ 1 and n ≥ 0 x Nn+1 = 1.
5. u n j : approximate of u(t n , x j ). 6. τ n : discrete time increment of n-th step determined by

τ n = τ min(1, u n -p+1 ∞ ). 7. h n : discrete space increment of n-th step determined by h n = min(h, (2 u n -q+1 ∞ ) 1 2-q ). 8. N n = 1 h n -1.
Under the assumption that a spatial net point x m coincides with the middle point x = 0 (we can easily achieve this by taking N n + 1 = 2m), we will prove that

u n m = max 0≤j≤Nn+1 |u n j | = u n ∞ and u n m-1 = max j =m |u n j |.
By using the notations above, our difference equation is introduced by: For j = 1, ..., N n and n ≥ 0:

           u n+1 j -u n j τ n = u n+1 j+1 -2u n+1 j + u n+1 j-1 h 2 n + (u n j ) p - 1 (2h n ) q |u n j+1 -u n j-1 | q-1 |u n+1 j+1 -u n+1 j-1 | u 0 j = u 0 (x j ) u n 0 = u n Nn+1 = 0. (3.2)
We denote by U n := (u n 0 , ..., u n Nn+1 ) t .

Remark 3.1. -The convergence of solution of (3.2) to the solution of (3.1) is proved in [START_REF] Hani | asymptotic behaviours of solutions for finite difference analogue of the Chipot-Weissler equation[END_REF].

Properties of the discrete solution

In this section, we prove that the difference solution has the same properties as the exact solution.

3.1. Positivity:-Let u be the classical solution of (3.1). By the maximum principal [START_REF] Chipot | some blow up results for a nonlinear parabolic problem with a gradient term[END_REF], we know that if the initial condition is nonnegative, then u is nonnegative too. We shall prove now the same result, as Theorem 2.2, for the discrete solution of (3.2).

Lemma 3.2. -Suppose U 0 satisfies (A1)-(A5). Let U n be the solution of (3.2), then we have U n ≥ 0 for all n ≥ 0.

Proof. -In view of assumption (A1), we see that U n ≥ 0 holds for n = 0. Supposing that it holds for some n ≥ 0, we have to show that U n+1 ≥ 0. For all j = 1, ..., N n , define

(u n+1 j ) -:= max j (0, -u n+1 j
) and (u n+1 j

) + := max j (0, u n+1 j
).

If we multiply the equation of (3.2) by (u n+1 j ) -we obtain

u n+1 j -u n j τ n (u n+1 j ) -= u n+1 j+1 -2u n+1 j + u n+1 j-1 h 2 n (u n+1 j ) -+ (u n j ) p (u n+1 j ) - - 1 (2h n ) q |u n j+1 -u n j-1 | q-1 |u n+1 j+1 -u n+1 j-1 |(u n+1 j ) -.
We use

u n+1 j = (u n+1 j ) + -(u n+1 j ) -and (u n+1 j ) + .(u n+1 j ) -= 0.
Then we have

- (u n+1 j ) --(u n j ) - τ n (u n+1 j ) -- 1 h 2 n (u n+1 j+1 ) + (u n+1 j ) -+ 1 h n (u n+1 j+1 ) --(u n+1 j ) - h n (u n+1 j ) - - 1 h 2 n (u n+1 j-1 ) + (u n+1 j ) -- 1 h n (u n+1 j ) --(u n+1 j-1 ) - h n (u n+1 j ) -+ 1 2 u n j+1 -u n j-1 2h n q-1 u n+1 j+1 -u n+1 j-1 h n (u n+1 j ) - = 1 τ n (u n j ) + (u n+1 j ) -+ (u n j ) p (u n+1 j ) - ≥ 0
We multiply by (-1) and we sum for j = 1, ..., N n , we obtain

Nn j=1 (u n+1 j ) --(u n j ) - τ n (u n+1 j ) -+ 1 h 2 n Nn j=1 (u n+1 j+1 ) + (u n+1 j ) -- 1 h n Nn j=1 (u n+1 j+1 ) --(u n+1 j ) - h n (u n+1 j ) - + 1 h 2 n Nn j=1 (u n+1 j-1 ) + (u n+1 j ) -+ 1 h n Nn j=1 (u n+1 j ) --(u n+1 j-1 ) - h n (u n+1 j ) - ≤ 1 2 Nn j=1 1 (2h) q |u n j+1 -u n j-1 | q-1 |u n+1 j+1 -u n+1 j-1 |(u n+1 j ) -. But Nn j=1 (u n+1 j-1 ) + (u n+1 j ) -= Nn j=1 (u n+1 j ) + (u n+1 j+1 ) -,
and

Nn j=1 (u n+1 j ) --(u n+1 j-1 ) - h n (u n+1 j ) -= Nn j=1 (u n+1 j+1 ) --(u n+1 j ) - h n (u n+1 j+1 ) -+ ((u n+1 1 ) -) 2 h n . Then Nn j=1 (u n+1 j ) --(u n j ) - τ n (u n+1 j ) -+ 1 h 2 n Nn j=1 (u n+1 j+1 ) + (u n+1 j ) -- 1 h n Nn j=1 (u n+1 j+1 ) --(u n+1 j ) - h n (u n+1 j ) - + 1 h 2 n Nn j=1 (u n+1 j-1 ) + (u n+1 j ) -+ 1 h n Nn j=1 (u n+1 j ) --(u n+1 j-1 ) - h n (u n+1 j ) - = Nn j=1 (u n+1 j ) --(u n j ) - τ n (u n+1 j ) -+ 1 h 2 n Nn j=1 (u n+1 j+1 ) + (u n+1 j ) -+ (u n+1 j ) + (u n+1 j+1 ) - + Nn j=1 (u n+1 j+1 ) --(u n+1 j ) - h n 2 + (u n+1 1
) - h n 2 .
We use M := U n ∞ < +∞ before blow-up, we can write that

|u n j+1 -u n j-1 | 2h n ≤ M h n , which implies that Nn j=1 (u n+1 j ) --(u n j ) - τ n (u n+1 j ) -+ 1 h 2 n Nn j=1 (u n+1 j+1 ) + (u n+1 j ) -+ (u n+1 j ) + (u n+1 j+1 ) - + Nn j=1 (u n+1 j+1 ) --(u n+1 j ) - h n 2 + (u n+1 1 ) - h n 2 ≤ 1 2 Nn j=1 1 (2h) q |u n j+1 -u n j-1 | q-1 |u n+1 j+1 -u n+1 j-1 |(u n+1 j ) - ≤ 1 2 M h n q-1 1 h n Nn j=1 (u n+1 j+1 ) + (u n+1 j ) -+ (u n+1 j ) + (u n+1 j+1 ) -+ 1 2 M h n q-1 ((u n+1 1 ) -) 2 h n + 1 2 M h n q-1 Nn j=1 |(u n+1 j+1 ) --(u n+1 j ) -| h n ((u n+1 j+1 ) -+ (u n+1 j ) -).
We define now the operator

(D(U n+1 ) -) j := (u n+1 j+1 ) --(u n+1 j ) - h n for j = 1, ..., N n
and we denote by X = Nn j=1 (X j ) 2 , then we have

1 h 2 n - 1 2 M h n q-1 1 h n Nn j=1 (u n+1 j+1 ) + (u n+1 j ) -+ (u n+1 j ) + (u n+1 j+1 ) - + Nn j=1 (u n+1 j ) --(u n j ) - τ n (u n+1 j ) -+ D(U n+1 ) -2 + 1 h 2 n - 1 2 M h n q-1 1 h n (u n+1 1 ) -2 ≤ 1 2 M h n q-1 Nn j=1 (u n+1 j+1 ) --(u n+1 j ) - h n (u n+1 j+1 ) -+ (u n+1 j ) -.
Using the definition of h n , we get

h n = min h, (2 u n -q+1 ∞ ) 1 2-q ≤ 2 u n -q+1 ∞ 1 2-q = 2M -q+1 1 2-q ,
which implies that

1 h 2 n - 1 2 M h n q-1 1 h n > 0,
and we obtain Nn j=1

(u n+1 j ) --(u n j ) - τ n (u n+1 j ) -+ D(U n+1 ) -2 ≤ 1 2 M h n q-1 Nn j=1 (u n+1 j+1 ) --(u n+1 j ) - h n (u n+1 j+1 ) -+ (u n+1 j ) -.
For the second term, we use Young's inequality to obtain that, for all ǫ > 0 1 2

M h n q-1 Nn j=1 (u n+1 j+1 ) --(u n+1 j ) - h n (u n+1 j+1 ) -+ (u n+1 j ) - ≤ ǫ Nn j=1 |(u n+1 j+1 ) --(u n+1 j ) -| h n 2 + 1 4ǫ M h n 2(q-1) Nn j=1 (u n+1 j+1 ) -+ (u n+1 j ) -2 ≤ ǫ D(U n+1 ) -2 + 1 4ǫ M h n 2(q-1) Nn j=1 (u n+1 j+1 ) -+ (u n+1 j ) -2 ≤ ǫ D(U n+1 ) -2 + 1 ǫ M h n 2(q-1) (U n+1 ) -2 .
If we take ǫ = 1 2 then we obtain

Nn j=1 (u n+1 j ) --(u n j ) - τ n (u n+1 j ) -+ D(U n+1 ) -2 ≤ 1 2 D(U n+1 ) -2 +2 M h n 2(q-1) (U n+1 ) -2 ,
and hence we get

Nn j=1 (u n+1 j ) --(u n j ) -(u n+1 j ) -≤ 2τ n M h n 2(q-1) (U n+1 ) -2 .
We use

(a -b)a = a 2 2 - b 2 2 + (a -b) 2 2 ,
then we have

1 2 Nn j=1 (u n+1 j ) -2 - 1 2 Nn j=1 (u n j ) -2 + 1 2 Nn j=1 ((u n+1 j ) --(u n j ) -) 2 ≤ 2τ n M h n 2(q-1) (U n+1 ) -2 which implies 1 -4τ n M h n 2(q-1) (U n+1 ) -2 ≤ (U n ) -2 .
We use that λ = τ h 2 < 1 16

, we can verify that

1 -4τ n M h n 2(q-1)
> 0 and finally we have

(U n+1 ) -2 ≤ 1 1 -4τ n M hn 2(q-1) (U n ) -2 .
By U n ≥ 0, we have (U n ) -= 0, which implies that (U n+1 ) -= 0, this gives U n+1 ≥ 0.

Monotony:-

The following result, analogue of Theorem 2.4, establishes monotony for the difference solution: Proof. -We will prove monotony by applying the similar argument as the nonnegativity to v n j = u n j+1u n j for n ≥ 0 and 1 ≤ j ≤ m -1. Let

V n = (v n 1 , v n 2 , ..., v n m-1 ) t .
It is easy to see that for j = 1, ..., m -1, v n j satisfies

v n+1 j -v n j τ n = v n+1 j+1 -2v n+1 j + v n+1 j-1 h 2 n + (u n j+1 ) p -(u n j ) p (3.3) - 1 (2h n ) q |v n j+1 -v n j | q-1 |v n+1 j+1 -v n+1 j | -|v n j -v n j-1 | q-1 |v n+1 j -v n+1 j-1 | .
In view of assumption (A3), we see that V n ≥ 0 holds for n = 0. Supposing that it holds for some n ≥ 0, we have to show that V n+1 ≥ 0. We use that

(u n j+1 ) p -(u n j ) p > 0 and 1 (2h n ) q |v n j -v n j-1 | q-1 |v n+1 j -v n+1 j-1 | > 0.
We multiply equation (3.4) by (v n+1 j

) -and we sum for j = 1, ..., m -1, we obtain

m-1 j=1 (v n+1 j ) --(v n j ) - τ n (v n+1 j ) -+ 1 h 2 n m-1 j=1 (v n+1 j+1 ) + (v n+1 j ) - - 1 h n m-1 j=1 (v n+1 j+1 ) --(v n+1 j ) - h n (v n+1 j ) -+ 1 h 2 n m-1 j=1 (v n+1 j-1 ) + (v n+1 j ) - + 1 h n m-1 j=1 (v n+1 j ) --(v n+1 j-1 ) - h n (v n+1 j ) -- m-1 j=1 | v n j+1 -v n j 2h n | q-1 | v n+1 j+1 -v n+1 j 2h n |(v n+1 j ) - = - m-1 j=1 (v n j ) + (v n+1 j ) -- m-1 j=1 ((u n j+1 ) p -(u n j ) p )(v n+1 j ) - - m-1 j=1 | v n j+1 -v n j 2h n | q-1 | v n+1 j+1 -v n+1 j 2h n |(v n+1 j ) - ≤ 0, so that m-1 j=1 (v n+1 j ) --(v n j ) - τ n (v n+1 j ) -+ 1 h 2 n m-1 j=1 (v n+1 j+1 ) + (v n+1 j ) - - 1 h n m-1 j=1 (v n+1 j+1 ) --(v n+1 j ) - h n (v n+1 j ) -+ 1 h 2 n m-1 j=1 (v n+1 j-1 ) + (v n+1 j ) - + 1 h n m-1 j=1 (v n+1 j ) --(v n+1 j-1 ) - h n (v n+1 j ) - ≤ 1 2 M h n q-1 m-1 j=1 | v n+1 j+1 -v n+1 j h n |(v n+1 j ) -.
We use the same calculations as the proof of Lemma 3.1, we see that

m-1 j=1 (v n+1 j ) --(v n j ) - τ n (v n+1 j ) -+ m-1 j=1 (v n+1 j+1 ) --(v n+1 j ) - h n 2 ≤ 1 2 M h n q-1 m-1 j=1 | (v n+1 j+1 ) --(v n+1 j ) - h n |(v n+1 j ) -+ ((v n+1 m ) -) 2 h n .
We denote by X * := m-1 j=1 (X j ) 2 and for all ǫ > 0 we have

m-1 j=1 (v n+1 j ) --(v n j ) - τ n (v n+1 j ) -+ D(V n+1 ) -2 * ≤ ǫ D(V n+1 ) -2 * + 1 4ǫ M h n 2(q-1) (V n+1 ) -2 * + 1 h n (V n+1 ) -2 * .
If we take ǫ = 1 2 we obtain

m-1 j=1 (v n+1 j ) --(v n j ) - τ n (v n+1 j ) -+ D(V n+1 ) -2 * ≤ 1 2 D(V n+1 ) -2 * + 1 2 M h n 2(q-1) (V n+1 ) -2 * + 1 h n (V n+1 ) -2 * . Then we can deduce that m-1 j=1 ((v n+1 j ) --(v n j ) -)(v n+1 j ) -≤ τ n 1 h n + 1 2 M h n 2(q-1) (V n+1 ) -2 * ,
which implies that

(V n+1 ) -2 * -(V n ) -2 * + 1 2 m-1 j=1 ((v n+1 j ) --(v n j ) -) 2 τ n 1 h n + 1 2 M h n 2(q-1) (V n+1 ) -2 * .
And hence we get

1 -2τ n 1 h n + 1 2 M h n 2(q-1) (V n+1 ) -2 * ≤ (V n ) -2 * .
We use that λ = τ h 2 < 1 16 we can verify that

1 -2τ n 1 h n + 1 2 M h n 2(q-1) > 0.
And finally by V n ≥ 0, we can deduce that (V n+1 ) -= 0, this gives V n+1 ≥ 0.

We do the same thing to obtain that v n j = u n ju n j+1 ≥ 0 for n ≥ 0 and m ≤ j ≤ N n . Proof. -For n ≥ 0, let λ n := τ n h 2 n . For j = 1, ..., N n the first equation of (3.2) can be rewritten as

-λ n u n+1 j-1 +(1+2λ n )u n+1 j -λ n u n+1 j+1 + τ n (2h n ) q |u n j+1 -u n j-1 | q-1 |u n+1 j+1 -u n+1 j-1 | = u n j +τ n (u n j ) p .
In view of the assumption (A2), we see that u 0 m-i = u 0 m+i for all i = 1, ..., m -1. Supposing that it holds for some n ≥ 0, then for i = 1, ..., m -1,

u n m-i = u n m+i . (3.4)
We have to show that

u n+1 m-i = u n+1 m+i for i = 1, ..., m -1. Let i ∈ {1, ..., m -1}, for j = m -i we have -λ n u n+1 m-i-1 + (1 + 2λ n )u n+1 m-i -λ n u n+1 m-i+1 (3.5)
+ τ n (2h n ) q |u n m-i+1 -u n m-i-1 | q-1 |u n+1 m-i+1 -u n+1 m-i-1 | = u n m-i + τ n (u n m-i ) p . Let α n i,i = τ n (2h n ) q |u n m-i+1 -u n m-i-1 | q-1 ; α n i,j = 0 for i = j and α n = (α n i,j ) i,j . (3.6) Then (3.6)
can be rewritten as

A 1 U n+1 + α n |B 1 U n+1 | = V n 1 , (3.7)
where

A 1 =   A 11 0 0 0   with A 11 =              -λ n 0 • • • • • • • • • 0 1 +
• • • 0 -λ n 1 + 2λ n -λ n              B 1 =   B 11 0 0 0   with B 11 =              -1 0 • • • • • • • • • 0 0 . . . . . . 1
• • • 0 1 0 -1              and V n 1 = v n 1j j with v n 1j = u n j + τ n (u n j ) p if 0 ≤ j ≤ m and v n 1j = 0 if j > m.
We do the same thing for i = 1, ..., m -1 and j = m + i, we obtain

-λ n u n+1 m+i-1 + (1 + 2λ n )u n+1 m+i -λ n u n+1 m+i+1 (3.8) + τ n (2h n ) q |u n m+i+1 -u n m+i-1 | q-1 |u n+1 m+i+1 -u n+1 m+i-1 | = u n m+i + τ n (u n m+i ) p .
Using (3.4) and (3.6), then (3.9) can be rewritten as

A 2 U n+1 + α n |B 2 U n+1 | = V n 2 , (3.9)
where

A 2 =   0 0 0 A 22   with A 22 =              -λ n 1 + 2λ n -λ n 0 • • • 0 0 . .
0 • • • • • • • • • 0 -λ n              B 2 =   0 0 0 B 22   with B 22 =              -1 0 1 0 • • • 0 0 . .
• • • • • • • • • 0 -1              and V n 2 = v n 2j j with v n 2j = u n j + τ n (u n j ) p if m + 1 ≤ j ≤ N n and v n 2j = 0 if j ≤ m.
We sum (3.7) and (3.9) we obtain

AU n+1 + α n (|B 1 U n+1 | + |B 2 U n+1 |) = V n , with A =   A 11 0 0 A 22   and V n =                      0 u n 1 + τ n (u n 1 ) p . . . u n m-1 + τ n (u n m-1 ) p u n m + τ n (u n m ) p u n m+1 + τ n (u n m+1 ) p Lemma 4.2.
-For a large initial data, we have u n m >> 1 for all n ≥ 0. Moreover, u n+1 m ≥ u n m for all n ≥ 0. Proof. -For n = 0, we have u 0 m >> 1 because of (A.3). Supposing that it holds for some n ≥ 0, we have to show that u n+1 m >> 1. In the equation of (3.2), if we take j = m and we use symmetry we obtain

(1 + 2λ n )u n+1 m = u n m + 2λ n u n+1 m-1 + τ n (u n m ) p , where λ n := τ n h 2 n
, and then we have

u n+1 m ≥ 1 + τ n (u n m ) p-1 1 + 2λ n u n m . (4.1)
Using the recurrence hypothesis we get

1 + τ n (u n m ) p-1 = 1 + τ and λ n = τ 2 2 2-q (u n m ) p-q 2-q . Then 1 + τ n (u n m ) p-1 1 + 2λ n = 1 + τ 1 + τ 2 1-2 2-q (u n m ) -p+ q 2-q = 1 + τ 1 + τ 2 -q 2-q (u n m ) -2p+q (p+1) 2-q
.

For 1 ≤ q ≤ 2p p + 1

, we have

-r := -2p + q(p + 1) 2 -q ≤ 0, (4.2)
which implies that

u n+1 m ≥ 1 + τ 1 + τ (2 -q 2-q (u n m ) -r ) u n m . (4.3)
Now we have to show that 1 + τ

1 + τ (2 -q 2-q (u n m ) -r ) ≥ 1.
Using u n m >> 1 and r ≥ 0, we have

2 -q 2-q (u n m ) -r = 1 2 q 2-q (u n m ) r < 1. Then 1 + τ > 1 + τ 2 -q 2-q (u n m ) -r , hence 1 + τ 1 + τ (2 -q 2-q (u n m ) -r ) > 1,
and then, by u n m >> 1

u n+1 m ≥ 1 + τ 1 + τ (2 -q 2-q (u n m ) -r ) u n m >> 1. ( 4
u n+1 m ≥ 1 + τ 1 + τ 2 -q 2-q (u n m ) -2p+q(1+p) 2-q u n m ≥ 1 + τ 1 + τ 2 -q 2-q (u 0 m ) -2p+q(1+p) 2-q u n m ,
which implies by iterations

u n m ≥ 1 + τ 1 + τ 2 -q 2-q (u 0 m ) -2p+q(1+p) 2-q n u 0 m .
For a large initial data, we have

1 + τ 1 + τ 2 -q 2-q (u 0 m ) -2p+q(1+p) 2-q > 1,
this implies that u n m → +∞ as n → +∞, which achieve the proof of Theorem 4.1.

Numerical simulation

In this section, we present some numerical simulation that illustrate our results. As it is shown in figure 1, we take u 0 (x) = 10 3 sin( π 2 (x + 1)), which satisfies the conditions (A1)-(A5). We take p = 3, q = 1.3 < 2p p + 1

. Figures 2, 3 and4 show the evolution of the numerical solution for different iterations. One can see that, numerically, the solution blows up in x = 0 and the growth of the solution leads to the reduction of h n and hence increasing the number of points of discretisation.

In T able 1, we present some results about the decreasing of h n and the increasing of N n (the number of points of discretisation of the interval [-1, 1]) in each iteration. Initially, simulation started with a discrete space step h n = 0.138, a discrete time step τ n = 10 -4 , a number of points of discretisation of the interval [-1, 1], N n = 15 and a maximum value M = 10 3 , after 350 iterations, we have increasing of the maximum value which leads to the decreasing of the discrete space step and discrete time step. From figure 5, we observe the evolution of the maximum point (blow-up point) x = 0, it gives an idea about the blow up rates given in [START_REF] Ph | Recent results and open problems on parabolic equations with gradient nonlinearities[END_REF]. We show in figure 6, that the solution decays with a small initial data u 0 (x) = sin( π 2 (x+1)), and hence blowing up can not occur, this was proved theoretically in [START_REF] Hesaaraki | blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in[END_REF]. We can see that the solution without gradient term showen in figure 8 blows up more rapidely than the solution of the Chipot-Weissler equation showen in figure 7, this proves the damping effect of the gradient term.

Concluding remarks

In this paper we have developped a numerical scheme in order to approximate the blow-up solution of the Chipot-Weissler equation, we have showed that we have blow up in x m = 0.

Our goal in another work, is to use our scheme to study the competition between the gradient term which fights against blow up and the reaction term which may cause blow up in finite time as in the Fujita equation (without gradient term). In particuler, we would like to answer some questions in the future study:

1. Can we determine the numerical blow-up set exactly? 2. What can we say about the asymptotic behaviours of the numerical solution near the blow-up set? 3. Can we give an approximation about the blow up time? 4. Let consider the equation u t = ∆u + a |u| pb |∇u| q . What conditions should be satisfied by a and b to reproduce blowing-up phenomena?

Lemma 3 . 3 .

 33 -Under the assumption (A1)-(A5), for U n solution of (3.2) n j < u n j+1 for j = 1, ..., m -1 and n ≥ 0 0 < u n j+1 < u n j for j = m, ..., N n and n ≥ 0

3. 3 . 4 Lemma 3 . 4 .

 3434 Symmetry:-The last property of the difference solution is the symmetry, analogue of Theorem 2.-Under the assumption (A1)-(A5), for U n solution of (3.2) we have u n m-i = u n m+i for all i = 1, ..., m -1 and n ≥ 0.

Figure 1 .

 1 Figure 1. initial data: u 0 (x) = 1000 sin( π 2 (x + 1))

Figure 2 .

 2 Figure 2. Evolution of the numerical solution in 50 iterations.

Figure 3 .

 3 Figure 3. Evolution of the numerical solution in 200 iterations.

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Evolution of the numerical solution in 300 iterations.

Figure 7 . 1 -Figure 8 .

 718 Figure 7. The shape of the numerical blow-up solution with gradient term in (t, x, u)-space

Table 1 .

 1 .10 3 4.190.10 3 7.245.10 3 1.317.10 4 1.607.10 4 1.980.10 4 3.203.10 4 Reduction of h n , τ n and increasing of the number of points of discretisation.

	Iteration	1	110	145	200	260	280	300	350
	M 2.928τn 10 3 10 -4 10 -6	5.10 -7	2.10 -9	2.10 -10	10 -10	4.10 -11	10 -11
	hn	0.13 0.87.10 -1 0.75.10 -1 0.59.10 -1 0.46.10 -1 0.42.10 -1 0.38.10 -1 0.31.10 -1
	Nn	15	23	27	33	41	47	51	63

Acknowledgment:

The present paper is an outgrowth of the first author's thesis' under the guidance of the second author to him he is highly acknowledged.

If we use monotony, we have

To ensure unicity of the solution, we should prove that

But it is easy to verify that for

Now, it is easy to see that if

is a solution of (3.10) then

is also a solution of (3.10). Then by uniqueness, we deduce that U n+1 = W n+1 which achieve the proof of symmetry.

Blow up theorem

In this section we will prove that the solution of the numerical problem blows up for all p > 1 and 1 ≤ q ≤ 2p p + 1 . To prove the theorem, we need the next lemma:


Dr Emilie Genin

Dr Zhenghong Gao

Dr Juan A Varela

Jonathan Daniel

Dr Talia Bsaibess

Dr Isabelle Gosse

Dr Laurent Groc

Dr Laurent Cognet
email: lcognet@u-bordeaux1.fr

Mireille Blanchard-Desce
email: mireille.blanchard-desce@u-bordeaux1.fr

« Hyper-bright» Near-Infrared Emitting Fluorescent Organic Nanoparticles for Single Particle Tracking

Keywords: luminescence, biocompatibility, photostability, single NP, imaging in cellular

environment Single molecule imaging is powerful tool for revealing the dynamic behavior of molecules in complex environments. In general, imaging and tracking a biomolecule of interest requires labeling it with a marker giving optical contrast to be detected by an optical microscope.

Common markers can be luminescent [1] or non luminescent [2] nano-objects and can allow single particle tracking (SPT) the motion of biomolecules both for in vivo and in vitro applications. Ideally, optical probes should be bright, stable in physiological and cellular environments, have large photostability with low toxicity and good biocompatibility (i.e. do not disturb or destroy healthy cells). Small aromatic organic fluorophores [3] and autofluorescent proteins have been the earliest single molecule fluorescent probes used in living cells. [4] They can generate bright fluorescence, but most of the time they experience fast photobleaching and photo-blinking and are thereby limited to short time imaging. Inorganic luminescent nanoparticles, such as quantum dots, are popular probes due to their high brightness and photostability, [1] but raise biodegradability issues. In addition, all these probes commonly have emission in the visible range, where the luminescence signal can be readily obscured by the background fluorescence from organisms (i.e. originating from endogenous chromophores). Actually, the optimal imaging spectral window (both for excitation and emission) lies in the near Infra-red (NIR) region, where biological samples are the most transparent. In that context, the search and synthesis of bright, biocompatible nanoparticles with NIR emission appears of major importance for the development of high-performance sensors. NIR-emitting inorganic quantum dots are one option; however, they commonly contain heavy metals and thereby show toxicity, resulting in cells death. [5] Single wall carbon nanotubes are another promising option but still require extensive synthesis and preparation steps to assess biocompatibility. [6] Fluorescent organic nanoparticles (FONs) have advantages such as their ease of preparation, reduced cost, as well as their high versatility, and possible degradability and possible reduced toxicity (depending on their organic components). [7] They have motivated numerous studies for lighting and imaging purposes. [8] However, the synthesis of bright and biocompatible FONs with stable NIR emission over time and compatible with physiological conditions is still in its infancy. This is due to the relative scarcity of NIRemitting dyes suitable for the elaboration of FONs and retaining NIR luminescence in water.

Most molecular NIR emitters such as cyanine dyes include an extended planar delocalized πsystem, favorable to NIR luminescence of isolated molecules, but which favors π-π stacking in aggregates or crystals. Such intermolecular interactions may result in fluorescence quenching, calling for subtle molecular engineering strategies to be implemented to prevent molecular packing deleterious to fluorescence. Moreover intermolecular excited-state processes such as energy and electron transfer enabled in close-packed assemblies, can lead to fluorescence quenching. Hence, the design of high-performance NIR emitting dyes dedicated to the elaboration of FONs retaining NIR emission in water and having robust structures and stable physicochemical properties (including in biological environments) remains challenging.

In this communication, we present our approach to specially-engineered synthetic NIR emitting dye allowing the fast preparation of hyperbright, NIR-emitting FONs, which we named HIFONs. These innovative fully-organic nanoparticules are shown to exhibit excellent chemical and colloidal stability as well as robust luminescence (also in cell environments) and good photostability and biocompatibility. Thanks to the exceptional combination of these specific features, the HIFONs could be used for single NP imaging in bioenvironments as well as for SPT in water.

Our strategy was based on tailored "push-pull" chromophores built from an electrondonating group (D) connected via a π-conjugated system to an electron-withdrawing group (A). [7] Such chromophores display intense, low-energy intramolecular charge transfer (ICT) transition in the near UV-visible range depending on the nature of the D/A pair and conjugated system. In particular suitable combination allows to tune the ground-state structure and polarization close to the "cyanine-limit" where the electron are delocalized along the conjugated system and the energy gap is minimum while the transition dipole is maximum. [9] These characteristics result in both intense and red-shifted excitation bands. In addition, the electronic delocalization results in lower conformational flexibility of the conjugated πsystem which is favorable to luminescence as some of the vibrational deactivation processes are repressed. In order to shift the excitation (and emission) to red and NIR region, a powerful strategy is to increase the π-conjugated system. [10] Our choice was to focus on push-pull chromophores built from an oligothiophene π-conjugated system (Figure 1) since this connector allows efficient electronic communication, which leads to large radiative decay rates, while limiting non-radiative decay compared to polyenes (due to lower degrees of conformational flexibility) thus leading to luminescent compounds. [11] We selected a pair of strong donor (D=aminophenyl) and acceptor (A=diethylthiobarbituric) groups so as to favor strong ICT and electron delocalization. [10] As large ground-state polarization is expected from design, resulting in large electrostatic (dipole-dipole) interactions in condensed phases, additional molecular engineering was required to prevent antiparallel association (which is favored in the case of rod-like chromophores) and subsequent fluorescence quenching. [12] In order to hinder the formation of tight antiparallel dimers, two tbutyl-biphenyl substituents were grafted on the donating nitrogen atom (Scheme 1) which act as fins, shifting the overall shape of the push-pull structure from a rod-like to a three-dimensional one (Figure 1). These substituents are also conveying increased solubility of chromophore 3 in organic solvents, while decreasing its solubility in pure water. This difference in solubility is critical for the easy and rapid production of organic nanoparticles using the simple reprecipitation process. [13] The synthesis of dye 3 was easily achieved starting from aldehyde 1 by means of a three-step sequence involving a bromination reaction followed by a Suzuki-Miyaura crosscoupling and a Knoevenagel condensation (Scheme 1). Dye 3 was obtained with 52 % overall yield as a pure dark solid which was fully characterized by 1 H and 13 C NMR spectroscopies, elemental analysis and mass spectrometric techniques. The experimental procedures details and related compounds characterizations are reported in the Experimental Section. Dye 3 shows intense absorption bands located in the UV and visible region when dissolved in organic solvents (see Figure S1) and a definite fluorescence emission varying from orange to red or NIR depending on the polarity of its environment (see Figure S2). FONs made from dye 3 were easily obtained by fast addition of a mM solution of 3 in THF into pure water at room temperature, leading almost instantaneously to limpid and bright NIR-emitting (λ em max =800 nm) fluorescent organic nanoparticles suspensions (HIFONs) in water. The formation of organic nanoparticles was further attested by Transmission Electron Microscopy (TEM) which was used to evaluate the morphology and size distribution of the FONs. TEM images showed that FONs were mainly spherical (Figure 1). The size distribution is close to a unimodal distribution centered on 40 nm with a FWHM of 22 nm, indicating a narrow and homogeneous distribution of the diameters (See Figure S3). The sizes obtained from TEM observations were in a very good agreement with diameters obtained from Dynamic Light Scattering (DLS) which yields an average hydrated diameter of 53 nm (PdI=0.19). All these observations suggested that FONs had uniform nearly spherical and homogeneous dimensions.

Interestingly, both the chemical and colloidal stability of HIFONs were found to be Whereas dye which is obtained from a Knoevenagel condensation, is observed to undergo retro-Knoevenagel reaction when dissolved in polar organic solvents containing water traces (such as acetone or DMSO, Figure S5), such a reaction is suppressed in HIFONs in water (see Figure S4). This demonstrates the absence of reactivity of surface dyes in HIFONs with surrounding water (or exchange with bulk), emphasizing that the chemical integrity of the HIFONs is preserved over time. Furthermore, the colloidal stability is found to be remarkable with morphology and size remaining stable after several days. Importantly the absorption or photoluminescence characteristics of FONs suspension in water were found to be stable over long period of times (see Figures S4), indicating that HIFONs made from dye 3 retain both their chemical structure and morphological features but also their giant brightness (with an estimated value of 2.10 7 M -1 cm -1 in deionized water) over time. Indeed this brightness is about 3 to 4 orders of magnitude larger than those of prototypical water-soluble NIR emitting organic dyes (such as cyanine or ATTO dyes, see Table S2), validating the appellation HIFONs. Interestingly the HIFONs are found to be stable in different pH conditions (Figures S6) and their luminescence is found to be enhanced in the presence of PBS buffer (see Figure S7).

Taking advantage of the robustness of the newly obtained HIFONS, we next studied their photoluminescence (PL) properties. We employed a home-made wide-field fluorescence microscope to image the FONs in different aqueous suspensions. HIFONs were diluted either in water or PBS 1X buffer to prepare an aqueous HIFONs suspension (∼50 pM) for single particle observations. A drop (~20 µL) of solutions was sandwiched between two glass slides and mounted on the optical microscope. PL images showed that individual FONs could be detected in both media when excited at 568 nm with a laser. This establishes that HIFONs have stable structures and PL properties in water as well as in highly salty environments. In both cases, HIFONs displayed bright signals and no evidence of aggregation occurring during imaging. Figure 2 presents the PL intensity distribution measured on 521 individual FONs.

Consistently with TEM analysis, the PL intensity distribution is close to a unimodal distribution centered on 1100 counts with a FWHM of 870 counts, indicating again a narrow and homogeneous distribution of the FON diameters with uniform PL properties in aqueous phases. Under these excitation conditions, individual HIFONs displayed a bi-exponential photobleaching decay behavior, with typical 1/e times of ~ 6 s (50%) and ~ 40 s (50 %). This behavior likely originates from the combined effect of individual dye photophysics and multidye interactions within HIFONs. Owing to their bright emission, individual HIFONs can be easily detected for more than 25 s (Figure 2) while imaging time of a single NIR emitting organic dye (such as cyanine dyes) is limited to typically a few seconds. [1a] This good signal stability allowed us to track the movement of individual HIFONs in water, which after an automated trajectory reconstruction process indicated again a monodisperse size distribution, as revealed by the diffusion coefficients histogram corresponding to the analyzed Brownian movements (Figure 2). The possibility of detecting individual HIFONs in a fluorescence microscope makes them real candidates to use as labels for single molecule studies in biological systems. The stability and toxicity of the HIFONs in biological environment was hence tested by incubating fibroblasts with HIFONs dispersed in cellular growth medium. As illustrated in Figure 2, HIFONs are internalized and retain their remarkable stability and PL luminescence in cell environment (even after hours). To further attest their bicompatibility, we performed two viability assays evaluating viable mitochondria and plasma membrane permeability, yielding no toxic effects of HIFONs for the tested concentration and incubation times (see Figure S8).

In i) NBS (2 eq.), THF, rt, 18 h, 93%; ii) 4-tert-butylbenzeneboronic acid (2.4 eq.), Pd(dppf)Cl 2 (20 %), K 2 CO 3 (5 eq.), toluene, methanol, 70 o C, 18 h, 70%; iii) 1,3-diethyl-2-thiobarbituric acid (2 eq.), EtOH, reflux, 18h, 80%; iv) fast addition of mM 3 solution in THF in pure water

  Scheme 1. Multi-step synthesis of chromophore 3 and subsequent one-pot synthesis of HIFONs by reprecipitation process.

Figure 1 .

 1 Figure 1. left: Molecular engineering of NIR chromophore 3 for the elaboration of hyperbright near IR emitting FONs (HIFONS), right: absorption and emission spectra of HIFONs made from 3 in water suspension and typical TEM image.

Figure 2 .

 2 Figure 2. Left column: fluorescence properties and tracking of single HIFONs. (a) the distribution of fluorescence signals of HIFONs, inset shows a typical wide-field fluorescence image; (b) the photobleaching behavior of a single HIFON, black squares present normalized fluorescence, red line presents fitted fluorescence by a bi-exponential function; (c) the distribution of diffusion coefficients of HIFONs, inset shows the trajectories of HIFONs in water. Right column: fluorescence imaging of single HIFONs cultured with living mouse embryonic fibroblasts for 24 h. (d) Bright field image, (e) wide-field fluorescence, and (f) the merged image.

Acknowledgements

MBD gratefully acknowledges Conseil Régional d'Aquitaire for financial support (Chaire d'Accueil grant) and Univ. Bordeaux 1 for a fellowship to TB. LG, JV, LC, ZG are supported by CNRS, Conseil Régional d'Aquitaine and Agence Nationale de la Recherche. JV is supported by Marie Curie Training Program. We thank the Bordeaux Imaging Center-Imagerie Electronique where TEM analysis was performed.

Supporting Information

Supporting Information is available online from the Wiley Online Library.


Andriamparany Rakotomavo

Dynamics of mangrove forests in the Mangoky River delta, Madagascar, under the influence of natural and human factors

Keywords:

Introduction

, p. 201). For the next periods and from FAO (2007) mangrove areas decreased of 15 000 ha between 1990 and 2000 -corresponding to an annual loss of 1500 ha -and of 15 000 ha from 2000 to 2005, i.e. 3000 ha/year loss on this period. In a more recent paper dealing with a comprehensive assessment of mangrove forest areas in Madagascar, Giri and Muhlhausen (2008) confirmed that mangrove deforestation rate varied both spatially and temporally in this country, and estimated that Malagasy mangrove areas slightly increased from 1975 to 1990 (5.6%), then decreased from 1990 to 2005, with a global loss of 7% of mangrove forests from 1975 to 2005.

 confirmed that areas of mangroves are decreasing all over the world at an alarming rate, possibly more rapidly than any other type of tropical forests. However, in their study area (Sunderbans, Bangladesh and Indian coasts), these authors observed a very active natural dynamics, linked to sedimentological processes, and an over-exploitation of

Mangroves in Madagascar cover 327 000 ha to 340 300 ha. Several authors have studied these mangroves, but we do not yet have a complete knowledge of these ecosystems. The present study is an attempt to respond to this concern by analyzing dynamics and changes in the extent of the mangrove forest in the Mangoky delta (SW Madagascar) between 1951 and2000. The comparison between base maps and Landsat TM images shows that the mangrove area remained relatively stable, even increasing slightly (+189 ha) between 1951and 1979. From 1979to 1994, its area decreased from 21 426 ha to 11 706 ha (À9720 ha). It then increased again (+84 ha) between 1994 and 2000. The balance was a decrease of 41.74% between 1951 and 2000. These figures may be compared with national (À15% in 56 years) and world (À35% for the last 20 years) statistics. Three main reasons are given to explain these changes: (1) the hydrological and sedimentological behavior of the Mangoky River; (2) the mechanism of plant succession, which depends on tides and on the characteristics of the substratum and (3) human activities and logging, which have increased since the 1980s in the Mangoky delta. resources of mangrove forest, but a relative stability of mangrove surface area.

The various physiognomic types of mangrove stands, from the coastal fringe to the bare and salted areas (''tannes''), through the real mangrove forests, participate to the coastline protection and make drainage processes more regular. Their ecological and biological roles are unique (wood provision, water storage, provision of nurseries and habitat for several species of invertebrate and vertebrate animals) [START_REF] Rasolofo | Conservation et utilisation rationnelle des fore ˆts de mangrove de l'Ame ´rique Latine et de l'Afrique[END_REF]. [START_REF] Mumby | Mangroves enhance the biomass of coral reef fish communities in the Caribbean[END_REF] pointed out the particular importance of mangroves in the structure of fish communities. Therefore, ''depletion starts being perceptible in Malagasy mangroves: decreasing fish catches, rapid deforestation, coastline erosion, extension of tannes, disorganization of the river network'' [START_REF] Rasolofoharinoro | A remote sensing based methodology for mangrove studies in Madagascar[END_REF].

As a signatory of the Ramsar Convention, Madagascar, within the framework of its coastal zones sustainable development policy, adheres to the principle of ''Integrated Coastal zone'' (ICZM), commonly used in international agreements and treaties such as those elaborated on by UNEP 1 and IPCC. 2 In order to implement an ICZM, an ''environmental diagnosis'' is first necessary. Moreover, [START_REF] Dugan | Wetlands management: a critical issue for conservation in Africa[END_REF] proposed that this diagnosis should be taken into account in a regional policy for the conservation of African humid zones.

Unfortunately, very little reliable and recent scientific data exist on mangroves in Madagascar [START_REF] Diop | Conservation et utilisation rationnelle des fore ˆts de mangrove de l'Ame ´rique Latine et de l'Afrique[END_REF][START_REF] Lebigre | Natural spatial dynamics of mangals through their margins: diagnostic elements[END_REF], especially regarding regeneration or dynamics. However, some local studies should be mentioned, particularly in relation to the southwestern [START_REF] Bigot | Contribution a `l'e ´tude e ´cologique des peuplements halophiles de la re ´gion de Tule ´ar. II. La plage et le cordon dunaire[END_REF][START_REF] Derijard | Contribution a `l'e ´tude du peuplement des se ´diments sablovaseux et vaseux intertidaux compacte ´s ou fixe ´s par la ve ´ge ´tation de la re ´gion de Tule ´ar[END_REF][START_REF] Weiss | Etude phytosociologique des mangroves de la re ´gion de Tule ´ar, Madagascar. Laboratoire de Phytosociologie et cartographie[END_REF], the northwestern [START_REF] Ioniarilala | Les impacts e ´cologiques de la crevetticulture industrielle sur l'e ´cosyste `me mangrove, cas de la re ´gion de Besalampy[END_REF][START_REF] Pasqualini | Mangrove mapping in North-Western Madagascar using SPOT-XS and SIR-C radar data[END_REF][START_REF] Rasolofoharinoro | A remote sensing based methodology for mangrove studies in Madagascar[END_REF], the eastern and the western parts of the island (various studies carried out within the framework of the CNRE-ORSTOM Mangrove Program in the 1990s).

According to [START_REF] Lebigre | Proble ´matique de recherches sur les marais maritimes de Madagascar en vue de leur protection et leur ame ´nagement[END_REF], ''Malagasy mangroves are very poorly understood and we do not have in hand the elements needed for a good diagnosis.'' This assertion is still true particularly in the case of the ''very large mangroves of Mangoky'' [START_REF] Kiener | Ecologie, biologie et possibilite ´s de mise en valeur des mangroves malgaches[END_REF], one of the marine swamps for which a prior investigation was necessary. With the exception of the study by [START_REF] Oliva | Le delta de Mangoky et sa re ´gion (Madagascar). Cartographie par te ´le ´de ´tection spatiale[END_REF], which used remote sensing, these mangroves are still unexplored and there is much work still to be carried out in order to produce an ecological and socio-economic diagnosis.

The present study is an attempt to give a partial response to these scientific needs: (1) by determining the area and the present status of the mangrove forest in the Mangoky delta, (2) by analyzing the changes that have occurred between 1951 and 2000, and (3) by deducing the natural and human causes of this evolution.

Materials and methods

The study area

Located on the west coast of Madagascar, south of Morondava, the study area lies between 43826 0 and 43846 0 East, and 21815 0 and 21835 0 South. This area constitutes the Mangoky River delta (Fig. 1). The climate is tropical semi-arid, with a mean annual rainfall of about 400 mm distributed over 30-50 days/year between November and March, and decreasing from the north to the south. The tidal range on the west coast is about 3.5 m.

The whole delta extends over 57 280 ha, of which 11 790 ha were mangrove forest in 2000. The delta is a part of the wide Mangoky watershed (55 750 km 2 ), the rivers of which originate in the Madagascar Central Highlands. The substratum is mainly made of sand and alluvium on a post-Miocene geological formation. Older alluvial layers are essentially sandy clay, covered with coarse rubified sand. Recent alluvium is silty and micaceous, making it excellent for growing crops.

The forest mangroves of the Mangoky delta are made up of six main tree species: Avicennia marina (Forsk.) Vierh., Xylocarpus granatum Ko ¨nig, Ceriops tagal (Perr.) C.B. Rob., Rhizophora mucronata Lamk, Bruguiera gymnorrhiza (L.) Lamk., and Sonneratia alba J. Smith. Two other species (Lumnitzera racemosa Willd. and Heritiera littoralis Buch.-Ham.) are present in the area with a very localized distribution.

Data and image processing

The following data were used to analyze the evolution of land cover in the Mangoky delta:

-Landsat TM satellite imagery from 19 June 1994, -Landsat ETM+ satellite imagery from 27 June 2000, It is important to note that satellite images and aerial photographs were all taken between June and August, i.e. during the dry season. The two images from 1994 and 2000 were processed using IDRISI 1 3.2 software.

The 2000 image, geometrically corrected by FTM and georeferenced using the base map BD 500, was used to georeference the 1994 image. The RMS errors computed from more than 10 homologous points on images and maps were 30 m in x and 51 m in y. We worked on 55 750 km 2 windows extracted from 1994 and 2000 Landsat images, corresponding to the whole Mangoky delta, and where all types of land use and mangrove facies were represented.

Color composites were made using band 1-5 of Landsat images (visible, near and mid-infrared) with the following combinations: bands 3-2-1 (natural-color composition), 4-3-2 and 4-5-1 (falsecolor composition). Band 6 (thermal infrared) peculiarly interesting for the discrimination of land cover classes in the dry tropical forests [START_REF] Southworth | An assessment of Landsat TM band 6 thermal data for analysing land cover in tropical dry forest regions International[END_REF] and generally not used for mangrove forest analyses [START_REF] Giri | Monitoring mangrove forest dynamics of the Sundarbans in Bangladesh and India using multi-temporal satellite data from 1973 to 2000[END_REF] has not been taken into account in this study. Band 7, associated with bands 1-5, was used for Principal Component Analysis (PCA) and to produce, through a non-supervised classification, a pre-map with 11 unsupervised land use classes [START_REF] Ramirez-Garcia | Mangrove vegetation assessment in the Santiago River Mouth, Mexico, by means of supervised classification using Landsat TM imagery[END_REF].

After fieldwork (analysis of training areas and vegetation plots identified from the pre-map, see below) supervised classifications were performed (confusion matrix, accuracy probabilities) and 11 supervised-classes were established.

Thematic analysis was carried out with the aim of characterizing the evolution of the mangrove areas and the river network in 1951, 1979, 1994 and 2000, by comparing base maps and satellite data. This was carried out using the GIS Arc View 1 and it consisted of:

-Superimposing maps and images, -Extracting from the study area two representative zones, A and B, to indicate changes in the land cover on a larger scale over the whole delta. The choice of zones A and B was controlled by their accessibility, their location within the delta (A to the north, B to the south) and the fact that they formed part of both the coastline and the land, -Exporting from IDRISI to Arc View the classified images and computing the various units of land cover, -Recording changes in the morphology of river courses and ponds, and deducing from these observations the causes and mechanisms of changes in the delta.

Analysis of mangrove stand structure and measurement of ecological parameters

Structural analyses have been conducted on the field with the following design:

-First, considering the 2 years time gap between the acquisition of satellite images (2000) and fieldwork ( 2002), and to be sure not to take into account new established stands in analyses, we selected only forest stands more than five years old as training areas. To estimate mangrove stand age, we have both considered mean stand height and mean diameter of individuals and knowledge of the local inhabitants about mangrove stand development.

-In each of the three main representative types of mangrove stands of the study area, i.e. pure A. marina stands, pure Rh. mucronata stands and mixed Rhizophora-Avicennia (70-40%), three plots have been chosen corresponding to different densities of mangrove tree populations, respectively low density (less than 3000 trees ha À1 ), medium density (3000-8000 trees ha À1 ) and high density (more than 8000 trees ha À1 ). These nine 100 m  100 m plots, representing one density class in each forest type, were located with a GPS so they could be used as training areas and control zones in image processing. -In each of these plots, four 10 m  10 m quadrats were selected at random, and all individual trees were recorded. Tree density, diameter at breast height (or 30 cm above the uppermost intersection of the prop roots for Rhizophora when intersections where up to 1.30 m) and height were measured, and then basal areas were computed [START_REF] Fromard | Structure, above-ground biomass and dynamics of mangrove ecosystems: new data from French Guiana[END_REF]. Thirty-six 10 m  10 m quadrats in the whole were then used for structural analysis.

Due to local and material constraints, only few ecological measurements were made directly on the field, i.e. conductivity or total suspended solid concentration (SSC) in water bodies, through a multiparameter water quality portable meter, from the water surface to 1.20 m depth. Sediment cores samples were taken for analyses in laboratory (CNRE Madagascar, National Center for Environmental Research) and complete results will appear in a next document [START_REF] Rakotomavo | Etat des lieux et dynamique de la mangrove du delta de Mangoky (Sud-Ouest de Madagascar). The `se de Doctorat[END_REF].

Results

Image analysis

Our main aim is to analyze land cover change within the Mangoky delta-and more peculiarly changes in mangrove communities, at a regional scale. So, it is necessary to evaluate the accuracy level of imagery analysis processes, which allow to extrapolate from local level (field plots) to regional scale.

Principal component analysis applied to bands 1-5 and 7 of Landsat ETM+ (2000) showed that the three first components explained 99.34% of the total variance of the data (Table 1). From this analysis, we defined, through unsupervised classification, 11 land use classes, including three mangrove classes where training areas and field sample plots have been located, i.e. pure A. marina forest, pure Rh. mucronata forest, mixed forest with A. marina and Rh. mucronata.

By the mean of classification error matrix (Table 2) and calculation of omission and inclusion probabilities (Table 3), assessment of supervised classification was established. Classification error matrix showed that 83% of sampled pixels were well classified, with a total accuracy of 84.4%. Regarding mangrove pixel classification, we obtained respectively 83.8%, 85% and 86.7% of well-classified pixels for pure A. marina stands, Rh. mucronata stands, and mixed stands. Finally we noted that accuracy rates were greater than 80% for producer and user.

Mangrove status in 2000

Considering field observations, map analyses and above image processes, we established structural and surface characteristics of mangrove stands in study area.

In 2000, the Mangoky River delta supported 11 790 ha of mangrove, divided into three main types as followed (Fig. 2):

-Mixed forest with A. marina and Rh. mucronata (around 4550 ha), -Pure A. marina forest (around 5116 ha), -Pure Rh. mucronata forest (around 2124 ha).

Forest density varied according to the forest type (Table 4). Avicennia forest had 5005 trees ha À1 (of which 37% regrowth with a height of less than 30 cm). The mixed forest containing Avicennia and Rhizophora plants had 7188 trees ha À1 (of which 33% regrowth with a height of less than 30 cm, most of which being A. marina, Ceriops tagal and Rh. mucronata). The Rhizophora forest contained 8500 trees ha À1 .

The mean height of mature living trees was 6.22 m in the Avicennia forest. Their mean diameter and basal area were 12 cm and 18.98 m 2 ha À1 , respectively. Their height was 5.19 m in the Rhizophora forest, with a diameter and basal area of 5 cm and 17.34 m 2 ha À1 , respectively. In the mixed forest, trees were 6.10 m high on average, having a diameter of 7 cm and a basal area of 14.27 m 2 ha À1 .

Evolution of the area of mangrove distribution from 1951 to 2000

By comparing the base maps and satellite images, changes in the area of mangrove distribution have been assessed (Table 5, Fig. 3).

-There was a slight increase of 6%, i.e. 1189 ha, mangrove area going up from 20 237 ha in 1951 to 21 426 ha in 1979, corresponding to a mean increase of 42 ha year À1 over 28 years. -From 1979 to 1994, nearly 9720 ha of mangrove forest disappeared. In other words, the forest area dropped from 21 426 ha to 11 706 ha within 15 years, which is a mean loss of 648 ha year À1 over this period. Through a visual comparison of the 1979 and 1994 maps, it can be seen that most of the loss occurred in the inland part of the delta. The decreasing of mangrove areas was followed by the appearance of ''tannes'', specifically in the northern part of the delta (Fig. 2). -Between 1994 and 2000, the dynamics was different, with a very slight increase of mangrove areas, from 11 706 ha to 11 790 ha.

Overall, there was a loss of 42% recorded in the mangroves of the Mangoky River delta, between 1951 and 2000.

Mangoky River dynamics

The lower course of the Mangoky River was clearly shown to have moved to the south, deserting the previous location of its mouth near Ambohibe (Fig. 1). The previous course appears on the 1951 and 1979 base maps, and consequently on more recent thematic documents based on these maps [START_REF] Faramalala | Carte des Formations Ve ´ge ´tales de Madagascar[END_REF][START_REF] Humbert | Notice de la carte de Madagascar[END_REF]. [START_REF] Salomon | Le sud-ouest de Madagascar[END_REF]

had already

Discussion

The mangrove dynamics in the Mangoky River delta can be explained by various interlinked factors: (i) shifting of the lower course of the Mangoky River, (ii) pioneering and regression mechanisms in tidal zones, and, together with these natural causes, and (iii) human activities, which also play an important role in modeling the landscape in the delta.

Hydrological and sedimentological dynamics leading to alluvial deposits on back mangrove forests

According to [START_REF] Andriamasinoro | Mobilite ´: re ´ponses des socie ´te ´s paysannes face a l'instabilite ´des conditions naturelles d'un milieu deltaı ¨que. Exemples de quelques villages du delta de la Tsiribihina, centre ouest de Madagascar[END_REF], ''the river appears to be the basic element in building the delta landscape. The abundance or lack of clear water conditions hydrogeomorphological units and mangrove species distribution''. The hydrodynamic behavior of the Mangoky River is directly related to the mechanism of alluvial deposits, as this enables mud accumulation, the filling up of channels, the shifting of watercourses and the formation of tidal channels in the delta.

So, due to different slopes upstream and downstream (0.7-3 m km À1 ) as well as the mean water speed in some parts (3 m s À1 ), floods, sediment deposits and hydromorphological processes can be spectacular [START_REF] Andriamasinoro | Mobilite ´: re ´ponses des socie ´te ´s paysannes face a l'instabilite ´des conditions naturelles d'un milieu deltaı ¨que. Exemples de quelques villages du delta de la Tsiribihina, centre ouest de Madagascar[END_REF]. Water flowing downwards spreads slowly on the gentle slopes of the delta and then meanders and branches out, rarely flowing in a straight line, while depositing tons of alluvium. In September 2002 (i.e. end of the dry season) we measured 48-209 mg l À1 of sediments in some of the branches of the Mangoky River. These values increased during the rainy (or flood) season up to 4 Â 10 3 mg l À1 , compared with 120 mg l À1 during times of medium flow [START_REF] Chaperon | Fleuves et rivie `res de Madagascar[END_REF]. According to the same author, 5.5-19.2 million tons of sediments were recorded annually in the Mangoky River between 1951 and 1955. This sediment load corresponds to a mean erosion of 0.15 mm in the whole Mangoky watershed and granulometric analyses showed that they were made up of 20% clay, 20% silt, about 60% fine sand and at least 2% coarse sand.

These hydrological and sedimentological phenomena contributed to the mangrove dynamics, through two opposite processes:

-They facilitated the spread of mangrove seeds and seedlings, allowing mangrove regrowth on the sides of the river (SW and NW Ambohibe, Beresaka, Andranopasy and Ankilifaly (Figs. 2 and4). -They contributed to the decay and to the death of mangrove stands in some inland zones by the obstruction of channels.

Massive deposition of sediments led to branches or main courses of rivers shifting. Between 1979 and 2000, the number of main river branches fell from three to one, the two northern Mangoky tributaries disappearing when they were filled with alluvium (Figs. 2 and5).

General regression in back mangrove forests and development of tanne areas

Figs. 2 and 5 illustrate the regression in back mangrove forests between 1951 and 2000. On the 2000 map, only some patches of A. marina and Rh. mucronata are recorded, as they can survive in the environmental conditions of Zone B (Fig. 5). Between 1951 and 2000, about 768 ha of mangrove, i.e. 38% of the original area disappeared in this zone. In 2000, 60% of the remaining forest consisted of Avicennia, with 35% being a mixed Avicennia/ Rhizophora forest. The previous mangrove forest was probably replaced by tannes. These changes in mangrove stand structure, i.e. decreasing but persistence of pure Avicennia stands, rarefaction or disappearance of other species, could have been caused by salinity increase and tannification processes, added to anthropogenic activities.

In order to explain these phenomena, Ranaivoson (1998) stated ''that while river deposits accumulate seaward, mangrove forest extends downstream, whereas the soil rises towards the inland regions. This soil is only flooded during spring tides and then gives way to a tanne, i.e. prolonged dryness and excess salt due to evaporation lead to decaying of the mangroves''. In such conditions, mangrove establishment becomes difficult, even unlikely. Only in some barren locations that become less saline due to sandy clay deposits, can halophytic plants such as Sporobolus spp. and Arthrocnemum spp. progressively grow.

Conductivity measurements made directly in different water bodies showed values from 46.6 to 60.9 ms cm À1 for Mangoky tributaries -i.e. around 30-39.7% in terms of salinity -and from 59.5 to 78.4 ms cm À1 for mangrove creeks, i.e. 38.7-51.2%. Higher values correspond to mangrove zones where A. marina, salt-tolerant species, is largely dominant and where Rh. mucronata, saltintolerant one, is clearly decaying. [START_REF] Rasolofoharinoro | A remote sensing based methodology for mangrove studies in Madagascar[END_REF] have previously established for mangrove stands in Madagascar that Rhizophora trees are limited in their distribution by mean water salinities exceeding 30%, and that Avicennia trees could become stunted when water salinity approched 60%. In Australia, [START_REF] Ball | Mangrove species richness in relation to salinity and waterlogging: a case study along the Adelaide River Floodplain, Northern Australia[END_REF] established that mangrove species richness analyzed along a river floodplain was minimal both in high salinity and drying environments and was the greatest in moderatily saline sites.

Elster's study in mangroves of Colombia (2000) confirmed that hypersalinization, increasing sediment deposits and lower water levels were the main causes of mangrove death. [START_REF] Hoffman | Habitat restoration in Tampa Bay[END_REF] and [START_REF] Komiyama | Microphotography, soil hardness and survival of mangrove (Rhizophora apiculata) seedlings planted in an abandoned tin-mining area[END_REF] reported the role of floods in Florida, and [START_REF] Cintron | Mangroves of arid environments in Puerto Rico and adjacent islands[END_REF] and [START_REF] Jimenez | The structure and function of dry weather mangroves on the Pacific coast of Central America, with emphasis on Avicennia bicolor forests[END_REF] stated the importance of salinity and the flowing of rivers in Puerto Rico and on the Pacific coast of Central America. According to these studies, it was observed that river floods, one of the necessary conditions for allowing mangroves to thrive, cannot penetrate anymore into some parts of the Mangoky delta. In places receiving such a deficient supply of water during tidal periods, mangrove stands were decaying, in relation with salinization and drying processes.

Plant succession mechanism allowing mangrove expansion seaward

Oliva and [START_REF] Oliva | Le delta de Mangoky et sa re ´gion (Madagascar). Cartographie par te ´le ´de ´tection spatiale[END_REF], using Landsat MSS remote sensing data (from 1973) on the Mangoky delta confirmed anecdotal evidence of mangrove expansion seaward and young mangrove trees appearing in the delta. The spontaneous expansion of mangrove forest is not exceptional in delta and estuary environments where a large amount of sediment is regularly deposited. [START_REF] Ioniarilala | Les impacts e ´cologiques de la crevetticulture industrielle sur l'e ´cosyste `me mangrove, cas de la re ´gion de Besalampy[END_REF] recorded in NW Madagascar an annual net increase in mangrove forest of 14 ha between 1957 and 1996. This result is different from that obtained in our study area (about 42 ha year À1 for the whole delta between 1951 and 1979). However, both cases illustrate the natural vegetation dynamics in an area influenced by river courses.

Generally, A. marina benefits more than other species from the growth of mudflats and is considered as a pioneer species in these environments. This was also observed by [START_REF] Weiss | Etude phytosociologique des mangroves de la re ´gion de Tule ´ar, Madagascar. Laboratoire de Phytosociologie et cartographie[END_REF] in S and SW Madagascar. We noticed in Zone A (Fig. 4) the spontaneous expansion of a mangrove forest in a region where human pressure was very low: the mangrove area increased from 1315 ha to 1394 ha between 1951 and 2000, i.e. a mean increase of 1.7 ha year À1 over 49 years. As shown in Fig. 4, nearly five-sixths of the area occupied by water in 1951 had been invaded by mangrove forest in 2000. It can also be seen (Fig. 4) that 90% and 50% of the two mudflats in 1951 (to the west and to the north of Ankilifaly, respectively) had been invaded by A. marina and/or Rh. mucronata by 2000.

Other studies carried out in different parts of Madagascar [START_REF] Andriamasinoro | Mobilite ´: re ´ponses des socie ´te ´s paysannes face a l'instabilite ´des conditions naturelles d'un milieu deltaı ¨que. Exemples de quelques villages du delta de la Tsiribihina, centre ouest de Madagascar[END_REF][START_REF] Ioniarilala | Les impacts e ´cologiques de la crevetticulture industrielle sur l'e ´cosyste `me mangrove, cas de la re ´gion de Besalampy[END_REF], in Vietnam [START_REF] Hong | Mangroves of Vietnam[END_REF], as well as in other African and Asian countries, emphasize the same succession mechanism after the deposition of sediment. That is, Avicennia spp. first becomes established in places with large tidal ranges that are supplied with large amounts of alluvium that have high salinity. In some places that are more exposed to tides, Sonneratia spp. is the pioneer species. After the substratum has become elevated due to sediment deposition, other species such as Rhizophora spp. appear on the landward fringe, where conditions are calmer and where ecological ranges are lower, specifically relating to salinity and flooding.

Fieldwork confirms that there is no transition between A. marina and Rh. mucronata stands. The boundary between the two types is generally sharp, except for a thin fringe of Avicennia containing a few Rhizophora trees (0.1-1 tree m À2 ) mixed through it. The whole forest is generally closed and dead trees give way to regeneration, as we noticed to the north of Ankilifaly and in Ambohibe. Recruitment is greater than the number of dead trees in the coastal fringe. The balance in these places showed no loss between 1951 and 2000, as the pioneering mechanism is rather efficient.

Heavy pressure by humans since the early eighties

Mangrove forest regression between 1979 and 2000 is also the result of heavy pressure from human habitation. With a loss in area of about 45% over 15 years (1979)(1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994), the Mangoky delta mangrove had a deforestation rate higher than the rate for all of Madagascar during the period 1980-1990 and for the whole world for the last two decades [START_REF] Valiela | Mangrove forests: one of the world's threatened major tropical environments[END_REF].

In the delta, mangrove forest supplies wood to fishermen living on the coast as well as to the farmers in inland areas. Following the restoration of hydro-agricultural works, the 1980s were marked by masses of migrants arriving and clearing the delta again. Slashand-burn practices (hatsaky), grazing cows and small ruminants, and an increasing need for firewood and timber had a negative impact on the extent of the mangrove forest. In fact, due to a strong decrease in the supply of inland species for human use (i.e., Dalbergia spp., Cedrelopsis grevei Baill., Commiphora spp., Tamarindus spp.), because they are generally protected either by local tradition or by forest regulation, mangrove forests become a useful alternative resource for inland people [START_REF] Rakotomavo | Strate ´gies d'utilisation des ressources ve ´ge ´tales chez les Vezo et les Masikoro du delta de Mangoky[END_REF]. So, since the 1980s, mangrove species have become much more valued. People enter mangrove stands looking for the species that are most convenient for cooking and building. Rhizophora and Bruguiera are the most sought-after genera.

Therefore, according to [START_REF] Lebigre | Proble ´matique de recherches sur les marais maritimes de Madagascar en vue de leur protection et leur ame ´nagement[END_REF], ''each species has its own qualities: B. gymnorrhiza has a straight trunk that gives standard poles 5 m high. It is more sought after than Rh. mucronata, which has the same characteristics. H. littoralis is used for timber beams. On the other hand, A. marina provides better firewood.'' Therefore, searching for new stands containing Rhizophora or other interesting species has become unavoidable since the 1990s. For example, a family needs one cart of firewood monthly, i.e., nearly 2.4-2.6 tons year À1 . For such a quantity, it is necessary to clear-cut yearly at least half a hectare of Rhizophora forest more than 6 m high, with a mean diameter of more than 6 cm and a stand density up than 500 trees ha À1 .

Conclusion

Through Landsat TM image processing supported by fieldwork, mangrove maps of the Mangoky River delta were carried out. Three vegetation types were recognized in the delta: a mixed forest containing A. marina and Rh. mucronata (4550 ha), a forest of pure A. marina (5116 ha) and a forest of pure Rh. mucronata (2124 ha).

In addition, a comparison between several base maps and satellite images from various dates using GIS showed clearly, for the first time, a spectacular shifting of the Mangoky River mouth between 1951 and 2000. This phenomenon is not rare in deltas, but no cartographic document had recorded this change up to now. The large mangrove forest on the delta is consequently affected by this shifting of the river. Because the northern delta was not sufficiently irrigated by the river, back mangrove trees decayed and gave place to large tannes. While the delta contains pioneer mangroves as well as decaying mangroves, calculating mangrove area changes, however, shows a negative balance between 1951 and 2000. This significant mangrove regression over the last 30 years is a feature that the Mangoky delta has in common with other coastal areas in Madagascar (south, NW and N-NW). Nevertheless the mangrove loss in the Mangoky delta, 41.7% between 1951 and 2000, seems to be higher than in other parts of Madagascar, as established by [START_REF] Valiela | Mangrove forests: one of the world's threatened major tropical environments[END_REF], which reported a loss of 15% for the whole country from 1921 to 1997, and a loss of more 35% for the world's mangrove area during the last two decades.

In the case of the Mangoky delta, periods of regression (1979-1994 and 1994-2000) and periods of expansion were found, with an important loss of 10 000 ha between 1979 and 1994. This last figure suggests that human influence is an important cause of the destruction. The arrival of masses of migrants in the 1980s seems to have caused a sudden regression of the mangrove forest, but this phenomenon is likely to stabilize from the 1990s onwards.

Finally, the mangrove forest dynamics in the Mangoky delta appears as correlated to the delta functioning. The main threat comes from human activities, which have to change in order to avoid environmental disturbance.

Fig. 1 .

 1 Fig. 1. Location of the study area.

Fig. 2 .

 2 Fig. 2. Mangoky delta in 2000, from Landsat TM.

Fig. 3 .

 3 Fig. 3. Mangrove forest changes in Mangoky delta between 1951 and 2000.

Fig. 4 .

 4 Fig. 4. Mangrove pioneering in Zone A.

Fig. 5 .

 5 Fig. 5. Mangrove regression in Zone B.

Table 1

 1 Principal components (PC) matrix (eigenvectors) and percentage of variance for the six used bands of Landsat ETM+(2000). Evaluation of the supervised classification.

	Components	Band 1	Band 2	Band 3	Band 4		Band 5	Band 7	% variance
	1st PC		0.117		À0.320	0.225	À0.050		0.415		0.812	85.51
	2nd PC		0.195		À0.381	0.386	À0.049		0.572	À0.581	10.38
	3rd PC		0.369		À0.481	0.323	À0.250		À0.683		0.002	3.45
	4th PC		0.381		0.670	0.617	0.144		À0.026		0.060	0.38
	5th PC		0.638		0.198	À0.474	À0.544		0.180	À0.008	0.20
	6th PC		0.509		À0.183	À0.301	0.785		À0.026	À0.001	0.08
	Table 2
	Classification error matrix for the supervised classification.
	Classes	Rz		RA	Av	Ta	Mr	Wt	Sd	Df	Gs	Ts	Cp	P Real + inclusion
	Rz		190 861	3	56	310	30 099	3667	700	3	3469	15	246	229 429
	RA		1	20 491	2806	33	477	56	0	0	1	0	3	23 868
	Av		67	2415	41 628	2577	1362	11	0	0	2	5	180	48 247
	Ta		166	85	2971	45 647	3471	128	2127	3	55	41	1352	56 046
	Mr	24 064	513	2064	3560	383 780	11 896	5909	2035	35 662	1131	2625	473 239
	Wt	4603	106	9	671	18 903	78 487	460	8	757	1	52	104 057
	Sd		545	0	7	2651	8409	128 40 967	204	2507	287	144	55 849
	Df		0	0	0	1	1315	35	22 77 655	5936	1750	44	86 758
	Gs	3612	1	2	41	53 768	327	2194	6172 177 057	803	82	244 059
	Ts		167	5	38	152	2180	25	93	3566	531 23 423	3200	33 380
	Cp		487	3	81	816	1316	28	22	34	33	837 424 251	427 908
	P Real + inclusion 224 573	23 622	49 662	56 459	505 080	94 788 52 494 89 680 226 010 28 293 432 179 1 782 840
	Classes	Producer's	Omission	User's	Inclusion
		accuracy (%)	probability (%)	accuracy (%)	probability (%)
	Rz	82.77	17.23		80.77	19.23
	RA	86.57	13.43		85.66	14.34
	Av	83.37	16.63		85.89	14.11
	Ta	80.23	19.77		80.84	19.16
	Mr	67.31	32.69		73.62	26.38
	Wt	81.74	18.26		74.05	25.95
	Sd	77.33	22.67		72.54	27.46
	Df	85.91	14.09		88.95	11.05
	Gs	74.9	25.1		68.56	31.44
	Ts	82.46	17.54		69.69	30.31
	Cp	97.59	2.41		98.87	1.13
	Total accuracy = 84.4%
	Table 4
	Area, density and regrowth percentage (seedlings less than 30 cm high) for the 3
	mangrove forest types, in Mangoky delta.
	Mangrove stands		Surface area	Density	Regrowth
				in 2000 (ha)	(trees/ha)	(%)
	Avicennia marina and		4550	7188	33
	Rhizophora mucronata
	Pure Avicennia marina		5116	5005	37
	Pure Rhizophora mucronata	2124

Rz = Rhizophora mucronata; RA = Rhizophora mucronata and Avicennia marina; Av = Avicennia marina; Ta = bare and/or herbaceous tanne (on tidal area)-grass savanna (on mainland); Mr = foreshore and/or marsh; Wt = water; Sd = sand and/or bare soil; Df = dry forest; Gs = grass savanna; Ts = tree savanna; Cp = crops Table

3

Rz = Rhizophora mucronata; RA = Rhizophora mucronata and Avicennia marina; Av = Avicennia marina; Ta = bare and/or herbaceous tanne (on tidal area)-grass savanna (on mainland); Mr = foreshore and/or marsh; Wt = water; Sd = sand and/or bare soil; Df = dry forest; Gs = grass savanna; Ts = tree savanna; Cp = crops

Table 5

 5 Landcover changes in Mangoky delta between 1951 and 2000 (areas in hectares).

	Landcover types	1951	1979	1994	2000
	Mangrove	20 237	21 426	11 706	11 790
	Continental dry forest	23 374	15 891	59 920	13 144
	Tree savanna	43 717	49 950		42 166
	Bare soil, sand, tree	18 865	18 926	34 567	39 093
	savanna, tannes,
	settlements, crops

Acknowledgements

The first author is a doctoral researcher, funded by the SCAC (Service de Coope ´ration et d'Action Culturelle) of the France Embassy in Madagascar, through the CAMPUS/CORUS Project no. 98 518 120.

Authors are also grateful to Marie-France Bellan, who participated in remote sensing processing and mapping. We also thank two anonymous reviewers for their very useful criticism and suggestions for improvement.