Rinaldo M Colombo

Magali Lécureux-Mercier

Nonlocal Crowd Dynamics Models for Several Populations

Keywords: Hyperbolic conservation laws, nonlocal flow, pedestrian traffic. 2010 MSC: 35L65

This paper develops the basic analytical theory related to some recently introduced crowd dynamics models. Where well posedness was known only locally in time, it is here extended to all of R + . The results on the stability with respect to the equations are improved. Moreover, here the case of several populations is considered, obtaining the well posedness of systems of multi-D non-local conservation laws. The basic analytical tools are provided by the classical Kružkov theory of scalar conservation laws in several space dimensions.

Introduction

From a macroscopic point of view, a crowd can be described through its density ρ, with ρ ∈ L1 (R 2 ; R), and assuming that ρ satisfies a continuity equation of the form ∂ t ρ + div (ρ V ) = 0 .

(1.1)

The vector V is in general a function of the space coordinate x ∈ R 2 and of the density ρ. The latter dependence may well be also of functional type since, in general, it is realistic to assume that V depends on ρ through some sort of weighted space average of ρ.

A first example of a model of this kind was presented in [6, Section 4]. There, it is assumed that pedestrians follow prescribed paths but adjust their speeds to the density they evaluate near to their positions. This amounts to postulate a speed law of the form:

V = v(ρ * η) #» v (x) .
(1.2)

The integral curves of the vector field #» v are the trajectories followed by the pedestrians. For instance, #» v (x) is the unit tangent at x to the geodesic curve joining x to the destination of the pedestrian at x. The convolution ρ * η stands for R 2 η(xξ) ρ(t, ξ) dξ, for a suitable non-negative smooth kernel η. It represents the average density measured, or felt, by the pedestrian at time t in position x. Below, we extend the results in [START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF] proving the global in time existence of the solutions to (1.1)- (1.2). Moreover, we complete the stability estimates with an estimate on the dependence of the solutions from v and #» v , see Theorem 2.2. The resulting model (1.1)-(1.2) enjoys further remarkable analytical properties. Indeed, a standard conservation law generates a semigroup which is not differentiable with respect to the initial data see [3, Section 1] for an explicit example. On the contrary, under suitable conditions, the semigroup generated by (1.1)-(1.2) turns out to be differentiable with respect to the data. This allows to obtain, rigorously, necessary conditions for optimality in various control problems based on (1.1)-(1.2), see Section 2 below and [6, Section 4].

Assuming that pedestrians adapt their path to the crowd density they meet lead to the model presented in [START_REF] Colombo | A class of non-local models for pedestrian traffic[END_REF]. There, the speed law

V = v(ρ) #» v (x) + I(ρ) .
(1.3) is considered. Again, #» v is the unit vector field describing the preferred paths. But, contrary to (1.2), here pedestrians may deviate from it, due to the nonlocal term I, which can be assumed, for instance, of the form

I(ρ) = -ε ∇(ρ * η) 1 + ∇(ρ * η) 2 .
(1.4)

Again, (ρ * η)(t, x) is the average density felt by the pedestrian at (t, x), so that (2.1) states that each individual is ready to leave the preferred path in order to avoid regions where the crowd density increases. The denominator in (1.4) is a normalization factor, so that the modulus of V is essentially controlled by the function v in (1.3), a smooth non increasing function that vanishes at the maximal density. We refer to [START_REF] Colombo | A class of non-local models for pedestrian traffic[END_REF] for further justifications of the choices leading to (1.1)- (1.3). Below, we show through numerical integrations that the formation of lanes, first noted in [START_REF] Colombo | A class of non-local models for pedestrian traffic[END_REF], is present also in the present multi-populations setting.

For both problems (1.1)-(1.2) and (1.1)-(1.3), we then consider the case of several, say n, populations. By this, we mean that different groups of pedestrians are considered, distinguished for instance by their destination. This amounts to consider systems of the form

∂ t ρ i + div (ρ i V i ) = 0 , i = 1, . . . , n (1.5)
where, in general, V i depends on the densities of all populations: V i = V i (ρ 1 , . . . , ρ n ). To extend all the above well posedness and stability results, we rely here essentially on [START_REF] Colombo | Stability and total variation estimates on general scalar balance laws[END_REF], with the improvements in [START_REF] Lécureux-Mercier | Improved stability estimates on general scalar balance laws[END_REF]. We recall that systems of the form (1.5) are considered also in [START_REF] Crippa | Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow[END_REF], where measure theoretic techniques are exploited. The interaction among different populations is considered also in [START_REF] Appert-Rolland | Two-way multi-lane traffic model for pedestrians in corridors[END_REF] through a macroscopic model and in [START_REF] Cristiani | Multiscale modeling of granular flows with application to crowd dynamics[END_REF] by means of a multiscale model. For a general review about crowd dynamics models we refer to [START_REF] Bellomo | On the modelling of traffic and crowds -a survey of models, speculations, and perspectives[END_REF].

The next section deals with the n-populations version of (1.1)-(1.2). Then, Section 3 is devoted to the analogous extension of (1.1)-(1.3), presenting also a sample numerical integration. All proofs are collected in Section 4.

Throughout, we state and prove every result in R d , for a dimension d ∈ N, d > 0, since the 2D case contains the same difficulties as the general d-dimensional situation. By R + we denote the interval [0, +∞[.

A Differentiable Model

The natural generalization of (1.2) to the case of n populations is

V i = v i (ρ 1 * η 1 + . . . + ρ n * η n ) #» v i (x) , (2.1)
so that (ρ i * η i )(x) is an average of the values attained by ρ i in B(x, 1). The map v i is the usual speed law, typically required to be non increasing since at higher densities the mean traffic speed is lower. Below, only the regularity of v is used. The vector #» v i (x) is the direction of the pedestrian belonging to the i-th population and situated at x ∈ R 2 . The presence of boundaries, obstacles or other geometric constraints can be described through #» v i , see for instance [START_REF] Colombo | On the continuum modeling of crowds[END_REF]. This section is devoted to the well posedness of (1.5)-(2.1), extending the results in [START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF] not only for what concerns the number of populations, but also obtaining global in time existence and more complete stability estimates. Before stating the main result, we rigorously specify what we mean by solution.

Definition 2.1. Let T > 0. Fix ρ o ∈ L ∞ (R d ; R n ). A weak entropy solution to (1.5)-(2.1) on [0, T ] is a bounded measurable map ρ ∈ C 0 [0, T ]; L 1 loc (R d ; R n ) whose i-th component ρ i , for all i = 1, . . . , n, is a Kružkov solution to the problem    ∂ t ρ i + div ρ i V i (t, x) = 0 ρ i (0, x) = ρ i o (x)
where

V i (t, x) = v i ρ 1 (t) * η 1 + . . . + ρ n (t) * η n #» v i (x) .
For the definition of Kružkov solution, see [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] or [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften[END_REF]Paragraph 6.2]. Above, the convolution products in the arguments of v i are intended in the sense

ρ i (t) * η i (x) = R d ρ i (t, ξ) η i (x -ξ) dξ i = 1, . . . , n .
The next Theorem summarizes various results in [START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF], particularized to (1.1)-(1.2).

Theorem 2.2. Fix d, n ∈ N with d, n > 0. Assume the following conditions:

(v) v i ∈ (C 2 ∩ W 2,∞ )(R; R) for i = 1, . . . , n. ( #» v ) #» v i ∈ (C 2 ∩ W 2,1 )(R d ; S 1 ) for i = 1, . . . , n. (η) η i ∈ (C 2 ∩ W 2,∞ )(R d ; [0, 1]) and η i L 1 = 1 for i = 1, . . . , n.
Then, there exists a semigroup S :

R + × (L 1 ∩ L ∞ ∩ BV)(R d ; R n ) → (L 1 ∩ L ∞ ∩ BV)(R d ; R n ) such that: 1. For all ρ o ∈ (L 1 ∩ L ∞ ∩ BV)(R d ; R n ), for all t ≥ 0, the orbit t → S t ρ o is the unique solution to (1.1)-(2.1) in the sense of Definition 2.1 with initial datum ρ o . Furthermore, the map t → S t ρ o is in C 0 R + ; L 1 (R d ; R n ) . 2. For all ρ o ∈ (L 1 ∩ L ∞ ∩ BV)(R d ; R n ), if ρ i o ≥ 0 for i = 1 . . . , n, then (S t ρ o ) i ≥ 0 for all t > 0.
3. There exists a constant L such that for all

ρ o ∈ (L 1 ∩ L ∞ ∩ BV)(R d ; R n ), the corre- sponding solution satisfies for all t ∈ R + TV ρ(t) ≤ TV (ρ o ) + L t ρ o L ∞ e Lt and ρ(t) L ∞ ≤ ρ o L ∞ e Lt .
4. Fix a positive M . Then there exists functions

L, A η , A v , A #» v ∈ C 0 (R + ; R + ) such for all ρ o,1 , ρ o,2 in L 1 (R d ; R n ) with max ρ o,i L 1 , ρ o,i L ∞ , TV (ρ o,i ) ≤ M , for all v 1 , v 2 satisfying (v), for all #» v 1 , #» v 2 satisfying ( #» v )
and for all η 1 , η 2 satisfying (η), the corresponding solutions ρ 1 , ρ 2 satisfy, for all t ∈ R + ,

ρ 1 (t) -ρ 2 (t) L 1 ≤ 1 + tL(t) ρ o,1 -ρ o,2 L 1 +t A η (t) η 1 -η 2 W 1,∞ + t A v (t) v 1 -v 2 W 1,∞ +t A #» v (t) #» v 1 -#» v 2 L ∞ + #» v 1 -#» v 2 W 1,1 .
5. More regular initial data imply more regular solutions, in the sense that

ρ o ∈ (W 1,1 ∩ L ∞ )(R d ; R n ) =⇒ ∀t ∈ R + , ρ(t) ∈ W 1,1 (R d ; R n ) , ρ o ∈ W 1,∞ (R d ; R n ) =⇒ ∀t ∈ R + , ρ(t) ∈ W 1,∞ (R d ; R n ) .
Furthermore, there exists a positive constant C such that

ρ(t) W 1,1 ≤ (1 + Ct)e Ct ρ o W 1,1 and ρ(t) W 1,∞ ≤ (1 + Ct)e Ct ρ o W 1,∞ . 6. If v, #» v and η are of class C 3 , then ρ o ∈ (W 2,1 ∩ L ∞ )(R d ; R n ) =⇒ ∀t ∈ R + , ρ(t) ∈ W 2,1 (R d ; R n ) , ρ o ∈ W 2,∞ (R d ; R n ) =⇒ ∀t ∈ R + , ρ(t) ∈ W 2,∞ (R d ; R n ) ,
and for a suitable non-negative constant C, we have the estimates

ρ(t) W 2,1 ≤ e Ct (1 + Ct) 2 ρ o W 2,1 , ρ(t) W 2,∞ ≤ e Ct (1 + Ct) 2 ρ o W 2,∞ . 7. If v is of class C 4 , for any initial data ρ o ∈ (W 2,∞ ∩ W 2,1 )(R d ; R n ), σ o ∈ (W 1,1 ∩ L ∞ )(R d ; R n
) and for all time t ∈ R + the semigroup S is strongly

L 1 Gâteaux differ- entiable in the direction σ o . The derivative DS t (ρ o )(σ o ) of S t at ρ o in the direction σ o is DS t (ρ o )(σ o ) = Σ ρo t (σ o )
, where Σ ρo is the linear semigroup generated by the Kružkov solution to

   ∂ t σ i + div σ i V i (ρ) + ρ i DV i (ρ)(σ) = 0 (t, x) ∈ I × R d σ i (0, x) = σ i o (x) x ∈ R d (2.2)
where ρ(t) = S t ρ o for all t ∈ R + and V i is defined in (2.1).

Note that the linear problem (2.2) is the equation that is obtained through a merely formal linearization of (1.5)-(2.1).

The above regularity results allow to state and prove the following necessary condition for optimality.

Proposition 2.3. Let f ∈ C 1,1 (R n ; R + ), ψ ∈ L ∞ (R + × R d ; R + )
× (L 1 ∩ L ∞ )(R d ; R n ) → (L 1 ∩ L ∞ )(R d ; R n ) the semigroup generated by (1.5)-(2.1). Introduce the integral cost functional J(ρ o ) = R d f (S t ρ o ) ψ(t, x) dx .
(2.3)

Then, J is strongly L ∞ Gâteaux differentiable in any direction σ o ∈ (W 1,1 ∩ L ∞ )(R d ; R n ). Moreover, let Σ : R + ×(W 1,1 ∩L ∞ )(R d ; R n ) → (W 1,1 ∩L ∞ )(R d ; R n
) be the linear semigroup generated by (2.2). Then,

DJ(ρ o )(σ o ) = R d f ′ (S t ρ o ) Σ ρo t (σ o )ψ(t, x) dx . If ρ ⋆ ∈ (L 1 ∩ L ∞ )(R d ; R n ) solves the problem find ρ ⋆ ∈ (L 1 ∩ L ∞ )(R d ; R n ) such that J(ρ ⋆ ) = min ρo∈(L 1 ∩L ∞ )(R d ;R n ) J(ρ)
then, for all

σ o ∈ (L 1 ∩ L ∞ )(R d ; R n ), R d f ′ (S t ρ ⋆ ) Σ ρ⋆ t σ o ψ(t, x) dx = 0 . (2.4)
The proof directly follows from [6, Proposition 2.12 and Theorem 4.2] and is hence omitted. Remark that (1.5)-(2.1)provides an environment where optimal control problems can be considered. On the other hand, no uniform upper bound in L ∞ is available.

Pattern Formation

In the case of n populations trying to avoid each other, we are led to consider (1.5) with

V i = v i (ρ i ) #» v i (x) + I i (ρ 1 , . . . , ρ n ) for i = 1, . . . , n , (3.1)
where I 1 , . . . , I n are suitable nonlocal functionals. According to (3.1), the velocity V i of the i-th population is the product of a scalar crowding factor v i (ρ i ) with a vector #» v i (x) + I i (ρ 1 , . . . , ρ n ), which is the sum of a preferred direction #» v i (x) and a deviation I i (ρ 1 , . . . , ρ n ). The scalar v i (ρ i ) approximately gives the modulus of the speed. A further standard condition on v i typically required in the engineering literature is that v i be weakly decreasing. However, this assumption is here not exploited. The unit vector field #» v i can be, for instance, the vector tangent at x to the geodesic that the individuals in the i-th population would follow to get to their destination, if unaffected by any other individual. The term I i (ρ 1 , . . . , ρ n ) describes how the i-th population deviates from its preferred trajectory due to the interaction among individuals, both of the same and of different populations. In general, it is a nonlocal functional, since its value at any position x depends on the population densities averaged over a neighborhood of x. The present setting generalizes the model in [START_REF] Colombo | A class of non-local models for pedestrian traffic[END_REF]. Below, we first address the main analytical properties of (1.5)-(3.1), such as the existence of solutions, their continuous dependence from the initial data and their stability with respect to #» v i and I i . Then, a numerical integrations shows further qualitative properties of the solutions to (1.5)-(3.1).

Denote by R > 0 a given maximal density. Throughout, we also denote the flux of the i-th population by q i (ρ

i ) = ρ i v i (ρ i ), for all ρ i ∈ [0, R].
Our starting point is the rigorous definition of solution to (1.5)-(3.1), analogous to Definition 2.1.

Definition 3.1. Fix the initial datum (ρ 1 o , . . . , ρ n o ) ∈ (L 1 ∩ L ∞ )(R d ; [0, R] n ). A map ρ ∈ C 0 [0, T ]; L 1 (R d ; [0, R] n ) is a weak entropy solution to (1.5)-(3.1) corresponding to the initial condition (ρ 1 o , . . . , ρ n o ) if, for i = 1, .
. . , n, ρ i is a Kružkov solution to the Cauchy problem for the scalar conservation law

   ∂ t ρ i + div ρ i v i (ρ i ) V i (t, x) = 0 ρ i (0, x) = ρ i o (x)
where

V i (t, x) = #» v i (x) + I i ρ 1 (t), . . . , ρ n (t) (x) .
For the definition of Kružkov solution we refer to [13, Definition 1].

Theorem 3.2. Assume the following conditions:

(v) For i = 1, . . . , n, v i ∈ C 2 ([0, R]; R + ) satisfies v i (R) = 0. ( #» v ) For i = 1, . . . , n, #» v i ∈ (C 2 ∩ W 1,∞ )(R d ; R d ) and div #» v i ∈ W 1,1 (R d ; R d×d ).
(I) There exists a constant C I > 0 such that the functional

I i : L 1 (R d ; [0, R] n ) → C 2 (R d ; R d ) satisfies, for i = 1, . . . , n, ∀ρ ∈ L 1 (R d ; [0, R] n )        ∇I i (ρ) L ∞ (R d ;R d ) ≤ C I ρ L 1 (R d ;R n ) , ∇ div I i (ρ) L 1 (R d ;R d×d ) ≤ C I ρ L 1 (R d ;R n ) . ∀ρ, ρ ′ ∈ L 1 (R d ; [0, R] n )        I i (ρ) -I i (ρ ′ ) L ∞ (R d ;R d ) ≤ C I ρ -ρ ′ L 1 (R d ;R n ) , div I i (ρ) -I i (ρ ′ ) L 1 (R d ;R) ≤ C I ρ -ρ ′ L 1 (R d ;R n ) .
Then, there exists a semigroup S :

R + × (L 1 ∩ BV)(R d ; [0, R] n ) → (L 1 ∩ BV)(R d ; [0, R] n ) such that 1. For all ρ o ∈ (L 1 ∩ BV)(R d ; [0, R] n ), the orbit t → S t ρ o is

For all

ρ o ∈ (L 1 ∩ BV)(R d ; [0, R] n ), the map t → S t ρ o is in C 0 R + ; L 1 (R d ; [0, R] n ) . 3. For all ρ o ∈ (L 1 ∩ BV)(R d ; [0, R] n
), the following estimate holds:

TV (S t ρ o ) ≤ TV (ρ o ) e κot + d W d e κot q L ∞ ([0,R]) (C I + div #» v L ∞ ) t , (3.2)
where W d is defined in (4.1).

Fix

M > 0. Let v 1 , v 2 satisfy (v), #» v 1 , #» v 2 satisfy (v) and I 1 , I 2 satisfy (I). Then, there exist b, c ∈ C 0 (R + ; R + ) such that for all ρ o,1 , ρ o,2 ∈ L 1 (R d ; [0, R] d ) with TV (ρ o,i ) ≤ M and for all t ∈ R + S t ρ o,1 -S t ρ o,2 L 1 ≤ (1 + t e t b(t) ) ρ o,1 -ρ o,2 L 1 +t c(t) q 1 -q 2 W 1,∞ + #» v 1 -#» v 2 L ∞ + div ( #» v 1 -#» v 2 ) L 1 .
and the functions b, c depend on

d, C I , R, q 1 W 1,∞ , #» v 1 W 1,∞ , div #» v 1 W 1,1 , ∇η L ∞ and on M .
Detailed expressions for the functions b and c are available, together with the proof, in Paragraph 4.2.

This theorem also provides a kind of maximum property since each density remains bounded by R. However, it does not guarantee that the sum ρ 1 (t) + . . . + ρ n (t) remains bounded by R.

Preliminary to any use of (1.5)-(3.1) is the choice of a specific I. First, we consider the following lemma that eases the construction of operators that satisfy (I). Lemma 3.3. Let N : R d → R d be defined by

N (u) = u 1 + u 2 .
If I satisfies (I), then so does N • I.

The proof consists in long but elementary computations and is hence omitted. By Lemma 3.3, it is immediate to check that (1.4) is satisfied by the operator I defined in (1.4) in the case of one population, as soon as η ∈ C 2 c (R d ; R + ) and η L 1 = 1, considered in [START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF]. When more populations are present, it is natural to consider different kinds of interactions. For instance, the population ρ 1 might deviate from its preferred path #» v 1 due to the population ρ 2 pushing in the direction #» v 2 , thus leading to consider the operator

I 12 (ρ) = ρ 2 v(ρ 2 ) #» v 2 * η 1 + ρ 2 v(ρ 2 ) #» v 2 * η 2 , (3.3)
Under assumption (v) on v and with η ∈ C 2 c (R d ; R + ) such that η L 1 = 1, also I 12 as defined in (3.3) satisfies (I).

When these or other operators are to be considered together, then the following lemma makes the verification of (I) immediate. Lemma 3.4. Let Î and Ǐ satisfy (I). Fix any two functions α, β

∈ (C 2 ∩ W 2,∞ )(R d ; R). Then, also α Î + β Ǐ satisfies (I).
The proof is elementary and hence omitted.

Differently from the model in Section 2, the semigroup generated by (1.5)-(3.1) is not proved to be differentiable with respect to the initial data. On other hand, it develops solutions with an apparently rich structure. Moreover, as stated in Theorem 3.2, an upper bound in L ∞ is available.

Numerical Integration

The study of the qualitative properties of the solutions to (1.5)-(3.1) is in general not amenable to purely analytical tools. Numerical integrations, besides being of interest from the point of view of the applications, show interesting pattern formations. In the case (1.1)-(1.3) of a single population, the formation of queues was already noted in [START_REF] Colombo | A class of non-local models for pedestrian traffic[END_REF].

The algorithm used is the Lax-Friedrichs method with dimensional splitting. A uniform grid (x i , y j ) for i = 1, . . . , n x and j = 1, . . . , n y is introduced and the density ρ is approximated through the values ρ ij on this grid. At every time step, the convolution in I(ρ) is computed through products of the type A ih ρ hk B kj , where the matrices A and B depend only on η.

In both the integrations, we use the same geometry and the same numerical parameters. More precisely, the space available to the pedestrians is the rectangle R × [-3, 3], while the numerical domain is [-8, 8] × [-4, 4]. Pedestrian may exit along the segments {-8} × [-3, 3] and {8} × [-3, 3]. The mesh size is ∆x = ∆y = 0.025.

The preferred path of each pedestrian is the sum g + δ. The vector g is tangent to the geodesic towards the pedestrian's target or 0 when the pedestrians would not move. The vector δ describes the discomfort of pedestrian when walking too near to a wall. It is perpendicular to the walls and pointing towards the interior of the room. Analytically, it also ensures the invariance of the room, see [START_REF] Colombo | A class of non-local models for pedestrian traffic[END_REF] for more details. Its maximum modulus is δ max along the walls at |x 2 | = ∆ and decreases linearly towards the room interior, vanishing at a distance δ r from the walls. In the present integration we set

δ max = 0.8 , δ r = 0.75 . (3.4)

Two Groups of People Crossing

A classical situation considered in the engineering literature is that of two groups of people moving in opposite directions and crossing each other. Typically, lanes, also called paths or trails in the engineering literature, are formed, see for instance [START_REF] Helbing | Self-organizing pedestrian movement[END_REF][START_REF] Daamen | Experimental research of pedestrian walking behavior[END_REF] or [START_REF] Appert-Rolland | Two-way multi-lane traffic model for pedestrians in corridors[END_REF] for a one dimensional description. They consist of people going in the same direction. The model (1.1)-(2.1) captures this phenomenon. Indeed, we consider (1.5)-(2.1) in the following setting:

                 ∂ t ρ 1 + div   ρ 1 v(ρ 1 ) #» v 1 (x) -ε 11 ∇(ρ 1 * η) 1+ ∇(ρ 1 * η) 2 -ε 12 ∇(ρ 2 * η) 1+ ∇(ρ 2 * η) 2   = 0 ∂ t ρ 2 + div   ρ 2 v(ρ 2 ) #» v 2 (x) -ε 21 ∇(ρ 1 * η) 1+ ∇(ρ 1 * η) 2 -ε 22 ∇(ρ 2 * η) 1+ ∇(ρ 2 * η) 2   = 0 (3.5)
where the ρ 1 population moves to the right and the ρ 2 population move to the left. Above, we assume that each pedestrians wants to avoid entering regions with increasing densities and that the repulsion towards pedestrians of the other population is bigger than that due to the proper population. For the sake of simplicity, we use the same kernel η and the same speed v in both equations. We consider the specific situation defined by

#» v 1 = 1 0 + δ , η(x 1 , x 2 ) = 1 -(2 x 1 ) 2 3 1 -(2 x 2 ) 2 3 χ [-0.5,0.5] 2 (x 1 , x 2 ) , #» v 2 = -1 0 + δ , v(ρ) = 4 (1 -ρ) , ε 11 = 0.3 , ε 12 = 0.7 , ε 21 = 0.7 , ε 22 = 0.3 . (3.6)
As initial datum we choose

ρ 1 o (x 1 , x 2 ) = 0.9 • χ [-6.4,-3.2]×[-2.4,2.4] (x 1 , x 2 ) , ρ 2 o (x 1 , x 2 ) = 0.7 • χ [3.2,6.4]×[-2.4,2.4] (x 1 , x 2 ) . (3.7)
The resulting numerical integration in Figure 1 shows several lanes forming. First, when most of the populations are still separated, the lanes in the two populations are rather symmetric, in spite of the different initial densities Then, when the interaction between ρ 1 and ρ 2 gets Time = 0.000 Above, the population ρ 1 moving to the right and, below, ρ 2 moving to the left. Note the lanes that are formed. First, due to the self-interaction within each populations and then due to the crossing between the two populations. The latter lanes of different populations do not superimpose.

relevant, lanes sharply bend their trajectory so that they are not superimposed. In this way, pedestrians lower how much they block each other. After the interaction, the density is so low that no more patterns arise.

Part of an Audience Leaving a Room

A sample situations developing various interesting features is the following. Two populations are initially uniformly distributed in the same region. At time t = 0, the first populations starts moving towards an exit, on the right in Figure 2), while the second moves only to let the first one pass. We describe this situation through (1.5)-(3.1). The preferred path of ρ 1 is tangent to #» v 1 = g + δ, where g is the geodesic vector field directed toward the right exit and δ is the discomfort, as above. The ρ 1 population then deviates from this trajectory only to avoid the other population. On the contrary, the preferred path of ρ 2 is tangent to #» v 2 = δ, since this population moves only to avoid the walls and the ρ 1 population. All this leads to the system

                 ∂ t ρ 1 + div   ρ 1 v(ρ 1 ) #» v 1 (x) -ε ∇(ρ 2 * η) 1+ ∇(ρ 2 * η) 2   = 0 , ∂ t ρ 2 + div   ρ 2 v(ρ 2 ) #» v 2 (x) -ε ∇(ρ 1 * η) 1+ ∇(ρ 1 * η) 2   = 0 . (3.8)
We consider the specific situation defined by

#» v 1 = 1 0 + δ , η(x 1 , x 2 ) = 1 -(2 x 1 ) 2 3 1 -(2 x 2 ) 2 3 χ [-0.5,0.5] 2 (x 1 , x 2 ) , #» v 2 = δ , v(ρ) = 4 (1 -ρ) , ε = 0.3 . (3.9)
with initial data

ρ i o = 0.5 χ [-6.4, -3.2]×[-2.4, 2.4] , i = 1, 2 , (3.10)
The resulting numerical integration is in Figure 2. Note that the initial distributions of Time = 0.000 Figure 2: Numerical integration of (3.8) with data and parameters as in (3.4)-(3.9). Above, the population ρ 1 and, below, ρ 2 . Note first the formation of small clusters separating the two populations, then lanes and, finally, a sort of fingering.

both populations are uniform. Very quickly, patterns start to form. First, clustering: the different populations separate forming separated peaks of high density. Then, population ρ 1 starts moving significantly to the right along lanes that follow paths in regions of low ρ 2 density. Finally, the two populations are almost separated, with a sort of fingering remaining in the region when they are superimposed. Concerning the initial clustering, we underline the analogy with [START_REF] Cristiani | Multiscale modeling of granular flows with application to crowd dynamics[END_REF], where a population is described through a measure possibly containing also an atomic part. Concerning the latter fingering, we borrowed here this term from the entirely different case of, for instance, thin films, see [START_REF] Levy | The motion of a thin liquid film driven by surfactant and gravity[END_REF]).

Technical Details

Below, for any

ρ ∈ L 1 (R d ; R n ), ρ L 1 (R d ;R n ) stands for n i=1 ρ i L 1 (R d ;R)
. Similarly, TV (ρ) = n i=1 TV (ρ i ). More generally, if not otherwise stated, the norm of a vector v in R n is the 1-norm, i.e. v = n i=1 v i . The following constant enters several estimates below:

W d = π/2 0 (cos θ) d dθ . (4.1)
The analytical tools below are based on the classical Kružkov work [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF], see also [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften[END_REF]Chapter VI]. The stability estimates are taken from [START_REF] Colombo | Stability and total variation estimates on general scalar balance laws[END_REF][START_REF] Lécureux-Mercier | Improved stability estimates on general scalar balance laws[END_REF].

First, we study the following Cauchy problem for a scalar non-linear conservation law:

∂ t ρ + div q(ρ) V (t, x) = 0 ρ(0) = ρ o . (4.2) Lemma 4.1. Assume that q ∈ C 2 (R + ; R + ) satisfies q(0) = 0 , V ∈ C 0 (R + × R d ; R d ) satisfies div V (t) ∈ W 1,1 (R d ; R d ) V (t) ∈ W 1,∞ (R d ; R d ) for all t ∈ R + , ρ o ∈ (L 1 ∩ L ∞ )(R d ; R + ) . (4.3)
Then, there exists a unique weak entropy solution ρ

∈ C 0 R + , L 1 (R d ; R + ) to (4.2). If furthermore ρ o ∈ BV(R d ; R), then ρ(t) ∈ BV(R d ; R) for all t ∈ R + and, denoting R T = ρ(t) L ∞ ([0,T ]×R d ) , TV ρ(t) ≤ TV (ρ o )e κot + C d e κot q L ∞ ([0,R T ]) t 0 R d ∇ div V (τ, x) dx dτ , (4.4)
where

κ o = (2d + 1) q ′ L ∞ ([0,R T ]) ∇V L ∞ ([0,T ]×R d ) and C d = d W d , with W d defined in (4.4). Let now v 1 , v 2 , V 1 , V 2 and ρ o,1 , ρ o,2 satisfy (4.3). Call ρ 1 , ρ 2 the solutions to ∂ t ρ 1 + div ρ 1 v 1 (ρ 1 ) V 1 (t, x) = 0 ρ 1 (0) = ρ o,1
and

∂ t ρ 2 + div ρ 2 v 2 (ρ 2 ) V 2 (t, x) = 0 ρ 2 (0) = ρ o,2 (4.5)
Then, renaming

R T = max{ ρ 1 L ∞ ([0,T ]×R d ) , ρ 2 L ∞ ([0,T ]×R d ) }, ρ 1 (t) -ρ 2 (t) L 1 ≤ ρ o,1 -ρ o,2 L 1 +t e κot q ′ 2 L ∞ ([0,R T ]) V 1 -V 2 L ∞ + q ′ 1 -q ′ 2 L ∞ ([0,R T ]) V 1 L ∞ × TV (ρ o,1 ) + C d q 1 L ∞ ([0,R T ]) t 0 R d ∇ div V 1 (τ, x) dx dτ + q 1 L ∞ ([0,R T ]) t 0 R d div V 1 (τ, x) -V 2 (τ, x) dx dτ + q 1 -q 2 L ∞ ([0,R T ]) t 0 R d div V 2 (τ, x) dx dτ .
Proof. We consider the following steps separately.

Existence. Let f (t, x, ρ) = q(ρ) V (t, x). Then, we have f ∈ C 0 (R + × R d × R; R d ) and f (t, • , • ) ∈ C 2 (R d × R; R d ) for any t ∈ R + .
Through elementary computations the following implications can be checked:

∂ ρ f (t, x, ρ) = q ′ (ρ) V (t, x) ⇒ ∂ ρ f is bounded on [0, T ] × R d × [-A, A] div f (t, x, ρ) = q(ρ) div V (t, x) ⇒ div f ∈ L ∞ ([0, T ] × R d × [-A, A], R) ∂ ρ div f (t, x, ρ) = q ′ (ρ) div V (t, x) ⇒ ∂ ρ div f ∈ L ∞ ([0, T ] × R d × [-A, A])
for all A ∈ R + . Hence, [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]Theorem 4] can be applied and the existence of solutions follows.

Maximum principle. Since q(0) = 0, ρ ≡ 0 is solution to (4.2). The maximum principle of Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]Theorem 3] then ensures that ρ o ≥ 0 implies ρ(t) ≥ 0 for all t ≥ 0.

BV bound. The BV bound follows from [14, Theorem 2.2], which can be applied since

∇∂ ρ f = q ′ (ρ) ∇V (t, x), so that ∇∂ ρ f ∈ L ∞ ([0, T ] × R d × [-A, A]). Moreover, for any A ≥ 0, T 0 R d ∇ div f (t, x, • ) L ∞ ([-A,A]) dx dt = q(ρ) L ∞ ([-A,A]) T 0 R d ∇ div #» v (t, x) dx dt <+∞. Let κ o = (2d + 1) q ′ L ∞ ([0,R T ]) ∇V L ∞ ([0,T ]×R d ) .
Then, for any t ≥ 0, we have:

TV ρ(t) ≤ TV (ρ o )e κot + C d e κot q L ∞ ([0,R T ]) t 0 R d ∇ div V (τ, x) dx dτ , with C d = d W d ,
T 0 R d div (f 1 -f 2 )(t, x, • ) L ∞ ([-A,A]) dx dt = q 1 (ρ 1 ) L ∞ ([-A,A]) T 0 R d div V 1 (τ, x) -div V 2 (τ, x) dx dt + q 1 -q 2 L ∞ ([-A,A]) T 0 R d div V 2 (τ, x) dx dt < +∞ . Then, with R T = max{ ρ 1 L ∞ ([0,T ]×R d ) , ρ 2 L ∞ ([0,T ]×R d ) }, we get ρ 1 (t) -ρ 2 (t) L 1 ≤ ρ o,1 -ρ o,2 L 1 +t e κot q ′ 2 L ∞ ([0,R T ]) V 1 -V 2 L ∞ + q ′ 1 -q ′ 2 L ∞ ([0,R T ]) V 1 L ∞ × TV (ρ o,1 ) + C d q 1 L ∞ ([0,R T ]) t 0 R d ∇ div V 1 (τ, x) dx dτ + q 2 L ∞ ([0,R T ]) t 0 R d div V 1 (τ, x) -V 2 (τ, x) dx dτ + q 1 -q 2 L ∞ ([0,R T ]) t 0 R d div V 1 (τ, x) dx dτ , completing the proof.
Corollary 4.2. In the same setting as Lemma 4.1, if q(ρ) = q 1 (ρ) = q 2 (ρ) = ρ then q ′ (ρ) = 1, q ′′ (ρ) = 0 and the stability estimate reduces to

ρ 1 (t) -ρ 2 (t) L 1 ≤ ρ o,1 -ρ o,2 L 1 +t e κot V 1 -V 2 L ∞ × TV (ρ o,1 ) + C d ρ 1 L ∞ ([0,T ]×R d ) t 0 R d ∇ div V 1 (τ, x) dx dτ + ρ 2 L ∞ ([0,T ]×R d ) t 0 R d div V 1 (τ, x) -V 2 (τ, x) dx dτ .

Proofs related to Section 2

Proof of Theorem 2.2. 1.

Let ρ o ∈ (L 1 ∩ L ∞ ∩ BV)(R d ; R n ) and let N 1 = ρ o L 1 .
Let us introduce a given time T > 0 and the sphere

X N 1 = ρ ∈ C 0 [0, T ]; L 1 (R d ; R n ) : for all t ∈ [0, T ] , ρ(t) L 1 ≤ N 1 ,
equipped with the distance induced by the norm

• L ∞ ([0,T ];L 1 (R d ;R n )) . Consider first a fixed i ∈ {1, . . . , n}. Choose any r 1 , r 2 ∈ X N 1 and denote ρ i 1 , ρ i 2 ∈ L ∞ [0, T ]; L 1 (R d ; R
) the solutions of the Cauchy problems (4.2) with respectively, for k ∈ {1, 2},

q k (ρ) = ρ , V k (t, x) = v i (r 1 k * η 1 + . . . + r n k * η n ) . (4.6)
Thanks to the properties of v i ∈ C 2 and η the hypotheses on V k in Lemma 4.1 are satisfied. Hence, we get existence and uniqueness of a weak entropy solution, Besides, the L 1 bound on the solution ρ(t) L 1 = ρ o L 1 for any t ≥ 0 ensures that the solution of (4.2)-(4.6) belongs to X N 1 .

Noting furthermore that

V i ∈ L ∞ ([0, T ] × R d ; R d ) , ∇V i ∈ L ∞ ([0, T ] × R d ; R d×d ) ∩ L ∞ [0, T ]; L 1 (R d ; R d×d ) , ∇ 2 V i ∈ L ∞ [0, T ]; L 1 (R d ; R d×d×d )
we can apply [6, Corollary 5.2 and Lemma 5.3] or more directly [14, Theorem 2.2 and Theorem 2.6].

Remark that N 1 ≥ sup t∈[0,T ] r(t) L 1 , so that we have the estimate

div V i (t) L ∞ ≤ (v i ) ′ L ∞ r(t) L 1 ∇η L ∞ #» v i L ∞ + v i L ∞ ∇ #» v i L ∞ ≤ v i W 1,∞ #» v i W 1,∞ (N 1 ∇η L ∞ + 1) .
Thus, with

K 1 = v W 1,∞ #» v W 1,∞ (N 1 ∇η L ∞ + 1) , (4.7)
we obtain by [6, Corollary 5.2]

ρ i (t) L ∞ ≤ ρ i o L ∞ e K 1 t .
Summing over i = 1, . . . , n, we obtain

ρ(t) L ∞ ≤ ρ o L ∞ e K 1 t .
We want now to apply Lemma 4.1 or [14, Theorem 2.2]. Note that, with the notations of [START_REF] Lécureux-Mercier | Improved stability estimates on general scalar balance laws[END_REF], we have

κ o = (2d + 1) ∇V i L ∞ ≤ (2d + 1)K 1 , ∇ div V i L 1 ≤ (v i ) ′′ L ∞   n j=1 ρ j L 1 ∇η j L ∞   2 #» v i L 1 + (v i ) ′ L ∞ j ρ j L 1 ∇ 2 η j L ∞ #» v i L 1 +2 (v i ) ′ L ∞ j ρ j L 1 ∇η j L ∞ ∇ #» v i L 1 + v i L ∞ ∇ 2 #» v i L 1 ≤ v i W 2,∞ #» v i W 2,1 N 2 1 ∇η 2 L ∞ + 2N 1 ∇η W 1,∞ + 1 . Denoting K 2 = v W 1,∞ #» v W 1,1 N 2 1 ∇η 2 L ∞ + 2 N 1 ∇η W 1,∞ + 1 , (4.8)
we obtain

TV ρ i k (t) ≤ TV (ρ i o )e (2d+1)K 1 t + te K 1 t dW d K 2 ρ i o L ∞ .
Let us now apply Corollary 4.2 or [14, Theorem 2.6]. We obtain

ρ i 1 -ρ i 2 L 1 ≤ Kte Kt r 1 -r 2 L ∞ ([0,T ];L 1 ) TV (ρ i o ) + (t dW d K 2 + 2 + N 1 ∇η L ∞ ) ρ i o L ∞ where K = v ′ W 1,∞ η W 1,∞ ( #» v L ∞ + #» v W 1,1 ) . Note that K 1 , K 2 and K are constants depending on v W 2,∞ , #» v L ∞ , #» v W 2,1 , η W 2,∞
, N 1 and d and not on the initial condition ρ o . We do not need here precision on the constant so, with a C large enough, not depending on ρ o and T , we have, denoting F (t) = C t e Ct TV (ρ o ) + C(t + 1) ρ o L ∞ and summing over i = 1, . . . , n,

ρ(t) L ∞ ≤ ρ o L ∞ e Ct , TV ρ(t) ≤ TV (ρ o )e Ct + Cte Ct ρ o L ∞ , (ρ 1 -ρ 2 )(t) L 1 ≤ F (t) r 1 -r 2 L ∞ ([0,T ];L 1 ) .
Choose T so that F (T ) = 1/2. Then, the map Q : X N 1 → X N 1 , defined by Q(r) = ρ, is a contraction. Banach Fixed Point Theorem ensures local in time existence and uniqueness of weak entropy solutions to (1.1)-(2.1).

In order to get global in time existence, we iterate the previous procedure. Starting from time T n , we obtain

(ρ 1 -ρ 2 )(t) L 1 ≤ r 1 -r 2 L ∞ ([0,t];L 1 ) C(t -T n )e C(t-Tn) × TV (ρ o )e CTn + CT n e CTn ρ o L ∞ + C (1 + t -T n ) e CTn ρ o L ∞ .
Iteratively, we choose T n+1 > T n such that

C (T n+1 -T n ) e CT n+1 TV (ρ o ) + C 1 + T n+1 ) ρ o L ∞ = 1 2 .
Since the sequence T n grows to +∞, we have global in time existence, proving point 1.

2.

The positivity of the solution directly follows from the fact that ρ i ≡ 0 is a solution and from Kružkov Maximum Principle [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]Theorem 3].

3. To prove these estimates, we apply [6, Corollary 5.2] and [14, Theorem 2.2 and Theorem 2.6] to each equation of the system, which is possible since the coupling is only present in the nonlocal term. Let i ∈ {1, . . . , n}. Denote K 1 as in (4.7),

N 1 = ρ o L 1 = n j=1 ρ j o L 1 and V i = v i (ρ 1 * η 1 + . . . + ρ n * η n ) #» v i (x), we have ρ(t) L ∞ ≤ ρ o L ∞ e K 1 t .
To prove the estimate on the total variation, apply [14, Theorem 2.2]. Denoting K 2 as in (4.8), we obtain

TV ρ i (t) ≤ e (2d+1)K 1 t TV (ρ i o ) + te (2d+1)K 1 t dW d K 2 ρ i o L ∞ .
Summing over i = 1, . . . , n we obtain the desired bound on TV ρ(t) = n i=1 TV ρ i (t) . 4. To prove the L 1 stability of the solution with respect to initial conditions and parameters, we apply [START_REF] Lécureux-Mercier | Improved stability estimates on general scalar balance laws[END_REF]Theorem 2.10]. Denote by ρ i k , for k = 1, 2, the solutions to the Cauchy problems

   ∂ t ρ i k + div ρ i k V i k (t, x) = 0 , ρ i k (0, • ) = ρ i o,k .
where

V i k (t, x) = v i k (ρ 1 k * η 1 k + . . . + ρ n k * η n k ) #» v i k (x) . Note that div V i k ∈ L ∞ [0, T ]; L 1 (R d ; R d ) for k = 1, 2
, so that the necessary hypotheses hold. Moreover,

div (V i 1 -V i 2 )(t) L 1 ≤ (v i 1 ) ′ -(v i 2 ) ′ L ∞ #» v i 1 L 1 ρ o,1 L 1 ∇η 1 L ∞ + v i 1 -v i 2 L ∞ div #» v i 1 L 1 + (ρ 1 -ρ 2 )(t) L 1 η 1 L ∞ + ρ o,2 L 1 η 1 -η 2 L ∞ × #» v i 1 L 1 (v i 1 ) ′′ L ∞ ρ o,1 L 1 ∇η 1 L ∞ + div #» v i 1 L 1 (v i 1 ) ′ L ∞ + (ρ 1 -ρ 2 )(t) L 1 ∇η 1 L ∞ + ρ o,2 L 1 ∇η 1 -∇η 2 L ∞ #» v i 1 L 1 (v i 2 ) ′ L ∞ + div ( #» v i 1 -#» v i 2 ) L 1 v i 2 L ∞ + #» v i 1 -#» v i 2 L 1 (v i 2 ) ′ L ∞ ρ o,2 L 1 ∇η 2 L ∞ , and
(V i 1 -V i 2 )(t) L ∞ ≤ v i 1 -v i 2 L ∞ #» v i 1 L ∞ + v i 2 L ∞ #» v i 1 -#» v i 2 L ∞ + (ρ 1 -ρ 2 )(t) L 1 η 1 L ∞ + ρ o,2 L 1 η 1 -η 2 L ∞ (v i 1 ) ′ L ∞ #» v i 1 L ∞ .
Let us introduce

α = v ′′ 1 L ∞ η 1 L ∞ ρ o,1 L 1 ∇η 1 L ∞ #» v 1 L 1 + v ′ 2 L ∞ ∇η 1 L ∞ #» v 1 L 1 + v ′ 1 L ∞ η 1 L ∞ div #» v 1 L 1 , β = v ′′ 1 L ∞ ρ o,2 L 1 ρ o,1 L 1 ∇η 1 L ∞ #» v 1 L 1 + v ′ 1 L 1 ρ o,2 L 1 div #» v 1 L 1 + v ′ 2 L ∞ ρ o,2 L 1 #» v 1 L 1 , γ = div #» v 1 L 1 + ρ o,1 L 1 ∇η 1 L ∞ #» v 1 L 1 , δ = v ′ 2 L ∞ ρ o,2 L 1 ∇η 2 L ∞ + v 2 L ∞ , and
α ′ = v ′ 1 L ∞ η 1 L ∞ , β ′ = v ′ 1 L ∞ ρ o,2 L 1 #» v 1 L ∞ , γ ′ = #» v 1 L ∞ , δ ′ = v 2 L ∞ ,
these coefficients being chosen so that

div (V i 1 -V i 2 )(τ ) L 1 ≤ α (ρ 1 -ρ 2 )(τ ) L 1 + β η 1 -η 2 W 1,∞ + γ v 1 -v 2 W 1,∞ +δ #» v 1 -#» v 2 W 1,1 , (V i 1 -V i 2 )(τ ) L ∞ ≤ α ′ (ρ 1 -ρ 2 )(τ ) L 1 + β ′ η 1 -η 2 L ∞ + γ ′ v 1 -v 2 L ∞ +δ ′ #» v 1 -#» v 2 L ∞ .
Hence, we get

(ρ i 1 -ρ i 2 )(t) L 1 ≤ ρ i o,1 -ρ i o,2 L 1 + e (2d+1)K 1 t TV (ρ o,1 ) + tdW d K 2 ρ o,1 L ∞ t 0 (V i 1 -V i 2 )(τ ) L ∞ dτ + max ρ 1 (t) L ∞ , ρ 2 (t) L ∞ t 0 div (V i 1 -V i 2 )(τ ) L 1 dτ ≤ ρ i o,1 -ρ i o,2 L 1 + A ρ (t) t 0 (ρ i 1 -ρ i 2 )(τ ) L 1 dτ + t A v (t) v 1 -v 2 W 1,∞ +t A η (t) η 1 -η 2 W 1,∞ + t A #» v (t) #» v 1 -#» v 2 W 1,1 + #» v 1 -#» v 2 L ∞ ,
where A ρ , A v , A η , A #» v are smooth, positive, and increasing, functions of t depending on

d, v W 2,∞ , #» v W 1,1 , #» v L ∞ , η W 1,∞ , TV (ρ o,1 ), max ρ o,1 L ∞ , ρ o,2 L ∞ , ρ o,1 L 1 and ρ o,2 L 1 . More precisely, denoting f (t) = e (2d+1)K 1 TV (ρ o,1 ) + tdW d K 2 ρ o,1 L ∞ , R = max( ρ o,1 L ∞ , ρ o,2 L ∞ ) , K = max k∈{1,2} v k W 1,∞ #» v k W 1,∞ ( ρ o,k L 1 ∇η k L ∞ + 1) , we have A ρ (t) = α ′ f (t) + αRe Kt , A η (t) = β ′ f (t) + βRe Kt , A v (t) = γ ′ f (t) + γRe Kt , A #» v (t) = δ ′ f (t) + δRe Kt .
We conclude applying the Gronwall Lemma, obtaining

(ρ 1 -ρ 2 )(t) L 1 ≤ ρ o,1 -ρ o,2 L 1 + tA η (t) η 1 -η 2 W 1,∞ + tA v (t) v 1 -v 2 W 1,∞ +tA #» v (t) #» v 1 -#» v 2 L ∞ + #» v 1 -#» v 2 W 1,1 × 1 + t f (t)α ′ + Re Kt α e tf (t)α ′ +te Kt Rα ,
and the stability estimate at 4. follows.

5. These properties follow through standard computations from the representation formula given in [6, Lemma 5.1].

6. As above, we deduce this further regularity property and the W 2,1 estimate from the representation formula in [6, Lemma 5.1].

7. Let ρ ∈ C 0 R + ; L 1 (R d ; R n ) be a solution to the initial problem (1.5)-(2.1) such that for all t ≥ 0, ρ(t) ∈ W 2,∞ ∩ W 2,1 (R d ; R n ). Consider now the linearized equations          ∂ t σ 1 + div ρ 1 (v 1 ) ′ (ρ ⋆ η) σ ⋆ η + σ 1 v 1 (ρ ⋆ η) #» v 1 (x) = 0 , . . . ∂ t σ n + div ρ n (v n ) ′ (ρ ⋆ η) σ ⋆ η + σ n v n (ρ ⋆ η) #» v n (x) = 0 , where ρ = (ρ 1 , . . . , ρ n ), η = (η 1 , . . . , η n ), σ = (σ 1 , . . . σ n ) and ρ ⋆ η = ρ 1 * η 1 + . . . + ρ n * η n .
To prove the existence and uniqueness of weak entropy solutions to this linearized problem, we use the technique that proved to be effective for the initial value problem: let M 1 = σ o L 1 , let s 1 , s 2 ∈ X M 1 are fixed functions; we fix the nonlocal term and study the Cauchy problems

   ∂ t σ i k + div σ i v i (ρ ⋆ η) #» v i (x) = -div ρ i (v i ) ′ (ρ ⋆ η) s k ⋆ η #» v i (x) σ i o,k = σ i o for k = 1, 2 i = 1, . . . , n
We study the map T : X M 1 → X M 1 defined by T (s) = σ. This application is well defined thanks to Kružkov Theorem [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] and thanks to [6, Lemma 5.1]. Indeed, denoting

f (t, x, u) = u v i (ρ ⋆ η) #» v i (x) , F (t, x, u) = -div ρ i (v i ) ′ (ρ ⋆ η) s 1 ⋆ η #» v i (x) , g(t, x, u) = u v i (ρ ⋆ η) #» v i (x) , G(t, x, u) = -div ρ i (v i ) ′ (ρ ⋆ η) s 2 ⋆ η #» v i (x) , we have ∂ u f, ∂ u g ∈ L ∞ ([0, T ]× R d × [-U, U ]), F -div f, G-div g ∈ L ∞ ([0, T ]× R d × [-U, U ]), ∂ u (F -div f ), ∂ u (G -div g) ∈ L ∞ ([0, T ] × R d × [-U, U ]
). Hence Kružkov hypotheses are satisfied. To apply [14, Theorem 2.6], we have now to check that

∇∂ u f ∈ L ∞ ([0, T ] × R d × [-U, U ]) , T 0 R d ∇(F -div f ) L ∞ ([-U,U ]) dx dt < +∞ , T 0 R d F -G -div (f -g) L ∞ ([-U,U ]) dx dt < +∞ .
We have

∇F (t, •, u) L 1 ≤ ρ(t) W 2,∞ v ′ W 1,∞ s 1 L 1 η W 2,∞ #» v W 2,1 × 9 + 6 ρ(t) L 1 ∇η L ∞ + ρ(t) 2 L 1 ∇η 2 L ∞ + ρ(t) L 1 ∇ 2 η L ∞ , ∇ div f (t, •, u) L 1 ≤ |u| K 2 , (F -G)(t, •, u) L 1 ≤ ρ(t) W 1,∞ v ′ W 1,∞ η W 1,∞ #» v W 1,1 3 + ρ(t) L 1 ∇η L ∞ × (s 1 -s 2 )(t) L 1 , div(f -g)(t, •, u) L 1 = 0 , ∂ u (f -g)(t, •, u) L ∞ = 0 .
where K 2 is defined as in (4.8). Using the estimate on ρ(t) W 1,∞ obtained in 4. and denoting

K 4 = ρ o W 1,∞ v ′ W 1,∞ η W 1,∞ #» v W 1,1 3 + ρ o L 1 ∇η L ∞ , (4.9)
we obtain

σ i 1 (t) -σ i 2 (t) L 1 ≤ t 0 div ρ i (v i ) ′ (ρ ⋆ η) (s 1 -s 2 ) ⋆ η #» v I L 1 dt ≤ t K 4 (1 + Ct)e Ct s 1 -s 2 L ∞ ([0,t],L 1 ) .
For T small enough, we obtain that T is a contraction, proving the local in time existence and uniqueness of solutions by Banach Fixed Point Theorem. We then extend to +∞ the time of existence by iteration of the process. Denote now by ρ, respectively ρ h , the solution to the initial problem (1.5)-(2.1) with initial conditions ρ o , respectively ρ o + h σ o . Moreover, call σ the solution to the linearized equation (2.2) with initial condition σ o and define z h = ρ + h σ. If σ is smooth enough we can write for z h the equation

∂ t z i h + div z i h v i (ρ ⋆ η) + h(v i ) ′ (ρ ⋆ η) σ ⋆ η #» v i (x) = h 2 div σ i (v i ) ′ (ρ ⋆ η) σ ⋆ η #» v i (x) .
We want now to estimate (ρ hz h )(t) L 1 /h. In order to do that, we use [START_REF] Lécureux-Mercier | Improved stability estimates on general scalar balance laws[END_REF]Theorem 2.6].

As it is similar to the estimate in the proof of [6, Theorem 2.10], we omit it.

Proofs related to Section 3

Lemma 4.3. In the same setting as Lemma 4.1, if furthermore for a given R > 0, we have

q(R) = 0, then ρ o ∈ [0, R] ⇒ ∀t ≥ 0 , ρ(t) ∈ [0, R] .
Proof. Since q(0) = q(R) = 0, ρ ≡ 0 and ρ ≡ R are solutions to (4.2). The maximum principle of Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] then ensures that 0 ≤ ρ ≤ R implies 0 ≤ ρ(t) ≤ R for all t ≥ 0.

Proof of Theorem 3.2. Consider the following steps separately.

Existence. Fix an arbitrary positive time T , whose precise value will be chosen later. Let (r 1 , . . . , r n ), (s 1 , . . . , s

n ) ∈ C 0 [0, T ]; L 1 (R d ; [0, R] n ) .
For any i ∈ {1, . . . , n}, we define V i (t, x) = #» v i (x) + I i r 1 (t), . . . , r n (t) (x) , W i (t, x) = #» v i (x) + I i s 1 (t), . . . , s n (t) (x) .

As v satisfies (v) and I satisfies (I), for any r, s ∈ C 0 [0, T ], L 1 (R d , [0, R] n ) , we have for any t ≥ 0, V (t), W (t) ∈ (C 2 ∩ W 1,∞ )(R d , R d ) and div V (t), div W (t) ∈ W 1,1 (R d , R). Then, according to Lemma 4.1, there exists ρ, σ ∈ C 0 [0, T ]; L 1 (R d ; R n + ) , that are the weak entropy solutions to the systems of decoupled equations

     ∂ t ρ 1 + div ρ 1 v 1 (ρ 1 ) V 1 (t, x) = 0 , . . . ∂ t ρ n + div ρ n v n (ρ n ) V n (t, x) = 0 ,      ∂ t σ 1 + div σ 1 v 1 (σ 1 ) W 1 (t, x) = 0 , . . . ∂ t σ n + div σ n v n (σ n ) W n (t, x) = 0 , with initial condition ρ(0) = σ(0) = ρ o ∈ (L 1 ∩ L ∞ )(R d ; [0, R] n ).
Note furthermore that q i (R) = 0; thus, thanks to Lemma 4.3, that if ρ is a solution of the above equation, then ρ o ∈ [0, R] implies for all t ≥ 0, ρ(t) ∈ [0, R]. Hence, we have invariance of the interval [0, R].

Consider the map T :

   C 0 [0, T ]; L 1 (R d ; [0, R] n ) -→ C 0 [0, T ]; L 1 (R d ; [0, R] n ) r → ρ    .
Use Lemma 4.1 and define κ o = max i (2d + 1)

(q i ) ′ L ∞ ([0,R]) ∇V i L ∞ ([0,T ]×R d )
, we get:

ρ i (t) -σ i (t) L 1 ≤ t e κot (q i ) ′ L ∞ ([0,R]) V i -W i L ∞ ([0,t]×R d ) × TV (ρ i o ) + C d q i L ∞ ([0,R]) t 0 R d ∇ div V i (τ, x) dx dτ + q i L ∞ ([0,R]) t 0 R d div V i (τ, x) -W i (τ, x) dx dτ .
Using hypothesis (I), we obtain

ρ i (t) -σ i (t) L 1 ≤ t C I r -s L ∞ ([0,t];L 1 (R d ;R n ))
× t e κot (q i ) ′ L ∞ ([0,R])

C d q i L ∞ ([0,R]) ∇ div #» v i (x) L 1 + C I r L ∞ ([0,t],L 1 (R d ;R n )) +e κot (q i ) ′ L ∞ ([0,R]) TV (ρ i o ) + q i L ∞ ([0,R])
.

Hence, for T small enough, we can apply the Banach Fixed Point Theorem. Now, using the total variation estimate (4.4), a standard iteration procedure allows to obtain global in time existence. Total Variation Estimate. To prove the estimate on the total variation, apply Lemma 4.1 and use a procedure entirely similar to that exploited in the proof of Theorem 2.2.

Stability. We use [START_REF] Lécureux-Mercier | Improved stability estimates on general scalar balance laws[END_REF]Proposition 2.10] to obtain

ρ 1 (t) -ρ 2 (t) L 1 ≤ ρ o,1 -ρ o,2 L 1 + e κot TV (ρ o,1 ) + t C d q 1 L ∞ (C I + ∇ div #» v 1 L 1 ) × q ′ 1 L ∞ ([0,R]) t 0 I ρ 1 (τ ) -I ρ 2 (τ ) L ∞ (R d ) dτ +t q ′ 1 L ∞ ([0,R]) #» v 2 -#» v 1 L ∞ + t(C I + #» v 2 L ∞ ) q ′ 1 -q ′ 2 L ∞ ([0,R]) + q 1 L ∞ ([0,R]) t 0 div I ρ 1 (τ ) -div I ρ 2 (τ ) L 1 (R d ) dτ +t q 1 L ∞ ([0,R]) div #» v 2 -div #» v 1 L 1 + t(C I + div #» v 2 L 1 ) q 1 -q 2 L ∞ ([0,R]) ≤ ρ o,1 -ρ o,2 L 1 +t (C I + #» v 2 L ∞ )e κot × TV (ρ o,1 ) + tC d q 1 L ∞ ([0,R]) (C I + ∇ div #» v 1 L 1 ) q ′ 1 -q ′ 2 L ∞ ([0,R]) +(C I + div #» v 2 L 1 ) q 1 -q 2 L ∞ ([0,R]) +t e κot q ′ 1 L ∞ ([0,R]) TV (ρ o,1 ) + t C d q 1 L ∞ ([0,R]) (C I + ∇ div #» v 1 L 1 ) #» v 2 -#» v 1 L ∞ + q 1 L ∞ ([0,R]) div ( #» v 2 -#» v 1 ) L 1
+C I e κot q ′ 1 L ∞ ([0,R]) TV (ρ o,1 ) + tC d q 1 L ∞ ([0,R]) (C I + ∇ div #» v 1 L 1 )

+ q 1 L ∞ ([0,R])

× t 0 ρ 1 (τ ) -ρ 2 (τ ) L 1 dτ .
Let us denote

a(t) = t (C I + #» v 2 L ∞ )e κot TV (ρ o,1 ) + t C d q 1 L ∞ ([0,R]) (C I + ∇ div #» v 1 L 1 ) × q ′ 1 -q ′ 2 L ∞ ([0,R]) +(C I + div #» v 2 L 1 ) q 1 -q 2 L ∞ ([0,R]) + q 1 L ∞ ([0,R]) div ( #» v 2 -#» v 1 ) L 1 +e κot q ′ 1 L ∞ ([0,R]) TV (ρ o,1 ) + t C d q 1 L ∞ ([0,R]) (C I + ∇ div #» v 1 L 1 ) × #» v 2 -#» v 1 L ∞ , b(t) = C I e κot q ′ 1 L ∞ ([0,R]) TV (ρ o,1 ) + t C d q 1 L ∞ ([0,R]) (C I + ∇ div #» v 1 L 1 ) + q 1 L ∞ ([0,R]) .
so that, by integration, we get ρ 1 (t)ρ 2 (t) L 1 ≤ 1 + t e t b(t) ρ o,1ρ o,2 L 1 + a(t) ,

  the unique solution to (1.1)-(3.1) with initial datum ρ o in the sense of Definition 3.1.

Figure 1 :

 1 Figure 1: Numerical integration of (3.5) with data (3.7) and parameters as in (3.4)-(3.6).Above, the population ρ 1 moving to the right and, below, ρ 2 moving to the left. Note the lanes that are formed. First, due to the self-interaction within each populations and then due to the crossing between the two populations. The latter lanes of different populations do not superimpose.

  and assume that the problem (1.1)-(2.1) satisfies the assumptions at 6. in Theorem 2.2. Denote by S : R +

  and W d as defined in (4.1).

	Stability estimate. The stability estimate follows from [14, Theorem 2.6]. We have, for
	any A ≥ 0,

Department of Mathematics, Brescia University, Via Branze 38,

Brescia, Italy 2 Technion, Israel Institute of Technology, Amado Building, 32000 Haifa, Israel

Acknowledgment:

The second author was partially supported by the GNAMPA 2011 project Non Standard Applications of Conservation Laws.

which completes the proof.


López Perez

Pedreño Ma

Maria A Pedreño
email: mpedreno@um.es

New method to enhance ajmalicine production in Catharanthus roseus cell cultures based on the use of cyclodextrins

Keywords: Ajmalicine, Catharanthus roseus, cell culture, cyclodextrin, elicitation, methyl jasmonate

The joint use of cyclodextrins and methyljasmonate, when accompanied by a short exposure to UV, enhanced extracellular ajmalicine accumulation to 1040 ± 26.6 mg/l in suspension cultured cells of Catharanthus roseus. The success of this strategy is due to the use of cyclodextrins, which not only induce ajmalicine biosynthesis but also promote adduct formation. This removes ajmalicine from the medium, reduces feedback inhibition and ajmalicine degradation, and allows its accumulation in the culture medium at elevated concentrations.

Introduction

Catharanthus roseus synthesizes a range of indole alkaloids that have pharmaceutical applications. Serpentine and ajmalicine have anti-hypertensive activities and are used to combat heart arrhythmias and to improve blood circulation in the brain (van der Heijden et al. 2004). Global demand for these compounds is increasing and new methods are needed to enhance their production. Plant cell cultures could be considered an alternative for producing indole alkaloids rather than extracting them from the whole plant. One of the strategies to enhance metabolite production using suspension cultured cells (SCC) is to use elicitors.

Random methylated β-cyclodextrins ( -CDs) are modified cyclic oligosaccharides derived from starch with seven glucose residues linked by α(14) glycosidic bonds. They can solubilize hydrophobic molecules by virtue of their hydrophobic interior. -CDs are true elicitors since they act by inducing defence responses, including phytoalexin biosynthesis (Zamboni et al. 2009). Their action resides in their chemical structure, which is similar to that of the alkylderived pectic oligosaccharides released from the cell walls when a fungal infection occurs [START_REF] Bru | Modified cyclodextrins are chemically defined glucan inducers of defense responses in grapevine cell cultures[END_REF].

Plant cell cultures can also be elicited by jasmonates, resulting in increased accumulation of defense-related compounds (Xu & Dong 2008). Treatment of C. roseus SCC with methyl jasmonate (MJ) promoted ajmalicine extracellular accumulation [START_REF] Cwt | Enhancement of ajmalicine production in Catharanthus roseus cell cultures with methyl jasmonate is dependent on timing and dosage of elicitation[END_REF]). In addition, UV-B light induces both the formation of indole alkaloids and signalling events which enhance the production of catharantine in SCC (Ramani & Chelliah 2008).

On the other hand, the production of high levels of indole alkaloids is dependent on both tissue and cellular compartimentation. [START_REF] Neumann | Indole alkaloids formation and storage in cell suspension cultures of Catharanthus roseus[END_REF] demonstrated that indole alkaloids were localized in the vacuoles of C. roseus. Therefore, the addition of sequestering agents, like -CDs, to cell cultures might promote their secretion to the culture medium and improve their production by mimicking the effect of storage in the vacuoles or cell walls.

The objective of this study was to analyze the effect of these different inducing factors on ajmalicine production in C. roseus SCC.

Materials and Methods

Chemical reagents

-CDs were from Wacker Chemie. Other chemicals were standard commercial products of analytical grade.

Plant Materials

C. roseus SCC cv. First Kiss Apricot were obtained from young leaf-derived callus by inoculating small friable callus pieces into 250 ml flasks containing 100 ml LS medium [START_REF] Linsmaier | Organic Growth Factor Requirements of Tobacco Tissue Cultures[END_REF] supplemented with 0.4 mg thiamine/l, 100 mg myo-inositol/l, 30 g sucrose/l, 0.19 mg naphthalene acetic acid/l and 0.22 mg 2,4dichlorophenoxyacetic acid/l. The SCC were incubated in the dark at 25°C and shaken at 110 rpm.

Elicitor treatments

Elicitation experiments were performed in triplicate on 14 day old C. roseus SCC. Cells were transferred into 250 ml flasks and suspended in 100 ml of fresh medium supplemented with either β-CDs or MJ (in ethanol) or a combination of both. In other cases, elicitation was started with MJ at different concentrations in combination with 50 mM β-CDs and/or under UV-light at short and long exposure times in the presence or in the absence of the above chemical elicitors.

Flasks were opened in a laminar flow hood and exposed to UV-C light (254 nm, 10 μW/cm 2 ) for 15 or 120 min.

Extraction, identification and quantification of ajmalicine

Ajmalicine was extracted three times from extracellular media by phase partitioning using ethyl acetate after adjusting pH to 10. The organic phase was collected and evaporated at 40 ºC in vacuum. The solid residue was dissolved in 1ml of methanol, filtered and analyzed by HPLC-DAD (Jasco LC-NetII/ADC) as described Zhao et al. (2000) using a Teknokroma Mediterranea Sea 18 column (150 x 4.6 mm, 3µ). Ajmalicine was identified and quantified by comparison with its commercial standard.

Results and discussion

Characterization of C. roseus cell growth

C. roseus growth was characterized by measuring g dry weight/l, pH and conductivity of the spent media and growth kinetic parameters. Fig. 1 shows a typical S-shaped growth curve with a maximum of 10 g dry wt/l at 14 days and depletion time, which represents the estimated time at which the scarcity of nutrients would cause stress and cell death, at 12 days.

Effect of cell density on ajmalicine production in C. roseus SCC elicited with -CDs and MJ individually or in combination.

100 μM MJ was used because this is considered as the optimal dose to induce metabolites in various plant species, including C. roseus (Lee-Pearson et al. 2004), while 50 mM -CDs is considered sufficiently high not to limit metabolite production [START_REF] Bru | Modified cyclodextrins are chemically defined glucan inducers of defense responses in grapevine cell cultures[END_REF]).

Ajmalicine production using low cell densities (100 g fresh wt/l or 5 g dry wt/l) was significantly lower than when high cell densities (200 g fresh wt/l) were used, in all conditions tested (Fig. 2). Our results represent an improvement over those of [START_REF] Lee | The effect of inoculum density and conditioned medium on the production of ajmalicine and catharanthine from immobilized Catharanthus roseus cells[END_REF], who studied the effect of cell-inoculum density on both fresh weight and ajmalicine production using C. roseus cells immobilized on alginate beads. They found that once the maximum fresh weight concentration was reached (450 g fresh wt/l) the maximum production of ajmalicine was over 120 mg/l. In our experiments, using 200 g fresh wt/l, the maximum value of extracellular ajmalicine reached in the presence of -CDs (which act in a similar way to the amberlite XAD-7 resin used by the above authors) was 216 ± 18 mg/l. [Note that at a high cell density, the combined treatment of MJ and β-CDs led to much higher ajmalicine accumulation than the sum of the individual additions, so that the effect can be regarded as synergistic. This effect would be due to the presence of elicitors which both act as inducers of ajmalicine biosynthesis while the action of β-CDs also promotes the adsorption and recovery of ajmalicine from the medium and reduces feedback inhibition and degradation.] A similar improvement was described by Zhao et al. (2001), who observed a synergistic effect on indole alkaloid accumulation in C. roseus SCC elicited with fungal preparations and chemicals. However, their highest ajmalicine levels were always below 100 mg/l.

Ajmalicine production is dependent on MJ dose, time of elicitation and scale-up

As shown in Fig. 3, the maximum ajmalicine production was reached when cell suspensions were incubated in the presence of 50 mM β-CDs and 100 μM MJ. Extracellular ajmalicine increased linearly until 192 h (= 930 ± 16 mg/l), and then decreased for the rest of the experiment (on day 14 = 740 ± 8 mg/l) -SEE Fig. 4). However, in the control culture medium, ajmalicine was hardly detected (data not shown). [START_REF] Cwt | Enhancement of ajmalicine production in Catharanthus roseus cell cultures with methyl jasmonate is dependent on timing and dosage of elicitation[END_REF] also studied the relationship between MJ dose and ajmalicine production using different MJ concentrations with resin (Amberlite XAD-7HP). Maximum ajmalicine accumulation (8.1 ± 0.8 mg/l) was obtained with 100 μM MJ on day 15, a 240% increase over the controls. These values are well below those obtained using MJ and /or β-CDs in our experiments at a high cell density (Figs. 2 and3) but are of the same order or above those observed when a lower cell density was used (Fig. 2: 0.7 ± 3.1 mg/l in the presence of MJ, and 11.9 ± 5 mg/l in the presence of MJ and β-CDs). In conclusion, the enhancement of extracellular ajmalicine production is not only dependent on the MJ dose but also on the cell density used and time of elicitation.

The effect of cell biomass and flask size on ajmalicine production was studied, using a cell suspension volume of 100, 200 and 400 ml in flasks of 250, 500 and 1000 ml of capacity, respectively (results not shown). Ajmalicine production slightly decreased as the flask size increased probably as a result of the amount of dissolved O 2 and CO 2 , ethylene production, nutrient supply and cell suspension volume.

Effect of UV light exposure and elicitors on ajmalicine production

As shown in Fig. 5, ajmalicine production was dependent on UV-light exposure time, since long UV-light exposure (120 min) caused a reduction in ajmalicine accumulation in all treatments. Short UV-light had no stimulatory effect when used with MJ. However, with β-CDs, short exposure to UV-light led to a significant increase in ajmalicine production. At a similar short exposure time, the use of β-CDs and MJ led to a further increase in ajmalicine accumulation. Thus, in addition to the synergistic effect observed on ajmalicine accumulation provoked by the joint presence of β-CDs and MJ, there was also synergism when these elicitors were used in combination with short UV-light exposure. Similarly, Ramani & Chelliah (2008) observed an increase in the production of catharanthine and vindoline from C. roseus SCC when cells were irradiated with UV-B light for 5 min.

In conclusion, to achieve high ajmalicine production levels using a productive C. roseus cell line, the best operating conditions were elicitation using a combination of β-CDs, MJ and short UV-light exposure. β-CD was fixed at 50 mM and the optimal MJ dose for the production of ajmalicine was 100 μM. Short exposure to UV-light led to a significant increase in ajmalicine while long UV-light irradiation was clearly detrimental to ajmalicine accumulation. Moreover, ajmalicine production increased as cell density increased. The enhancement of extracellular ajmalicine production was not only dependent on the culture growth stage and elicitor dose but also on cell density used and the duration of elicitation. The combined use of elicitors during a relatively short time could well be considered as a novel way for the industrial production of ajmalicine, since its accumulation is hardly affected as a result of scaling-up.

Figure captions

Fig. 1

 1 Fig. 1 Time course of C. roseus cell growth measured as g dry weight/l. The initial cell density was 2 g dry weight/l.

Fig. 2

 2 Fig. 2 Effect of cell density on extracellular ajmalicine production in C. roseus SCC. Cell suspensions were treated with 50 mM β-CDs and 100 μM MJ individually or in combination. Results were evaluated 96 h after treatment. Values are given as the mean ± SD of one experiment with three replicates.

Fig. 3 Fig. 4

 34 Fig. 3 Effect of different MJ concentrations (25, 75, 100, 200 and 300 µM) on extracellular ajmalicine production in C. roseus SCC treated with 50 mM β-CDs. Results were evaluated 96 h after treatment. Values are given as the mean ± SD of one experiment with three replicates. Fig. 4 Time course of ajmalicine production in C. roseus SCC. Cell suspensions were treated with 50 mM β-CDs and 100 μM MJ individually or in combination. Ajmalicine was extracted from the spent medium after different incubation periods. Values are given as the mean ± SD of one experiment with three replicates.

Fig. 5 Fig

 5 Fig. 5 Effect of UV-light exposure time (15 and 120 min) on extracellular ajmalicine production in C. roseus SCC elicited with 50 mM β-CDs and 100 μM MJ individually or in combination. Results were evaluated 192 h after treatment. Values are given as the mean ± SD of three replicates

Acknowledgements

Almagro L. held a grant from the MICINN. We thank José Martínez Parra and Ana Belén Sabater-Jara for the identification of indole alkaloids and Mariana Sottomayor for providing alkaloid standards. This work has been partially supported both by (BIO BVA 07 01 0003) and II PCTRM 2007-10 (08799/PI/08).

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