Andrei R Akhmetzhanov
email: akhmetzhanov@gmail.com

Pierre Bernhard
email: pierre.bernhard@sophia.inria.fr

Frédéric Grognard
email: frederic.grognard@sophia.inria.fr

Ludovic Mailleret
email: ludovic.mailleret@sophia.inra.fr

Competition between foraging predators and hiding preys as a nonzero-sum differential game

In this work we investigate a (seasonal) preypredator model where the system evolves during a season whose length is fixed. Predators have the choice between foraging the food (eating preys) and reproducing (laying eggs at a rate proportional to their energy). Preys can either eat, which would maintain their population in the absence of predators, or hide from the predators but they then suffer a positive mortality rate. In this case the population size can decrease even faster than if they were not hiding and were foraged by the predators. In their own turn they lay eggs at a constant rate whether they are hiding or eating. Following Darwin's principle that the fittest population will survive we postulate that both populations must maximize the number of their offspring, which yields a nonzerosum differential game.

I. INTRODUCTION

Prediction of the population dynamics plays a significant role in pest management and other fields of biological control, which is the use of predators or more generally natural enemies of pests in order to eradicate its population. Therefore we should predict the changes in the size of the pest population and describe how it evolves along the seasons. The formulation of the prey-predator model which will describe the relevant properties of the system and will be close to the experimental data obtained in fields is an important problem faced by mathematical biologists.

The most classical representation of these interactions are ordinary differential equations (e.g. Lotka-Volterra model) and difference equations (e.g. Nicholson-Bailey model). In our case we will study seasonal insects whose dynamics are continuous during the season and discrete between seasons. Such a system can not be described by the classical models recalled above [START_REF] Geritz | On the mechanistic underpinning of discrete-time population models with complex dynamics[END_REF]. In the literature, the classical approach is to consider explicitly the continuous and discrete dynamics, what produces a hybrid or more precisely impulsive model [START_REF] Murdoch | Consumer-resource dynamics[END_REF]. We follow the notations of [START_REF] Pachepsky | Between discrete and continuous consumer-resource dynamics with synchronized reproduction[END_REF] and call such models semi-dicrete. Indeed we suppose that the natural process is subdivided into separate seasons and periods of hibernation when all species die and the initial population for the next season is determined by the number of eggs produced by the species in the past.

The model presented in the paper describes the specific form of prey-predator interaction within a season. Both populations of predators and preys are active and they can choose the appropriate tactic that gives the best response to the behavior of the opponent. Predators can either increase their energy by feeding on the preys or produce some offspring by laying eggs. However their reproduction rate depends on this reproductive energy which can be increased by feeding. If the preys can notice that the size of their population decreases very fast and it can become critically small, they can hide from the predators. But in this case they do not eat and their mortality rate becomes non-zero. We suppose also that the preys lay eggs at a constant rate all the time and it does not depend on their choice of the behavioral strategy either to hide or to eat. By eating they keep their mortality rate equal to zero. Such a model can be stated in terms of a nonzero-sum differential game such that both players (predators and preys) tend to maximize the amount of offspring they produce during a season. The work presented here actually follows a previous study in which the preys were not considered as capable of hiding and that reduced to an optimal control problem (one-player game) for the predators [START_REF] Akhmetzhanov | Long term preypredator dynamics induced by short term optimal predatory behavior[END_REF].

II. MAIN MODEL

A. Formulation of a nonzero-sum differential game

Let one consider a closed system of two species: predators and preys. The period of time when both populations are active is fixed at T . We will refer to T as a season. We make use of two variables to describe the population of predators: the reproductive energy p in average and the number of predators z. To describe the population density of preys we introduce the variable n. We suppose that both populations consist of two parts: a mature and an immature part. During the season, mature insects can invest in immatures by laying eggs. Between the seasons all matures die and immatures become matures for the next season. In this paper we are interested only in the within season dynamics.

At the beginning of the year all predators are small and p equals to zero. Since predators are laying eggs with a rate proportional to the value of the energy p, it seems intuitive that most of them will try and forage the preys at the beginning and reproduce at the end, once they have gathered enough energy. We consider the case where the predator has a choice between feeding on the preys (u = 1) and laying eggs (u = 0). From the other side the preys have a choice either to hide (v = 0) or to eat (v = 1). Here the variables u and v play the role of the controls.

From the previous considerations and after rescaling the states and time so as to limit the number of parameters, we can then assume that the within season dynamics can be written in the form

ṗ = -p + nuv, ṅ = -µn(1 -v) -nzuv (1)
where the parameter µ > 0 is the mortality rate of the preys. The value of z is supposed to be constant during the season which is more or less relevant to the real situations in nature.

The number of offspring produced by the predators and preys along the season depends on the current size of the population

J u = T 0 p(t)z(1 -u(t)) dt, J v = T 0 n(t) dt (2)
where the variable z can be omitted as a constant.

The goal of the mathematical analysis is to define the optimal (or rational) strategy of the players and predict the actual behavior in real-life situations. A major step in the understanding of such game problems with several players has been done by John Nash. One says that the profile of strategies constitute a Nash equilibrium if: whenever one of the players deviates from his initial strategy he will not increase his own payoff. In the case of two players the pair of controls

u = u * (p, n, t), v = v * (p, n, t)
is called a Nash equilibrium solution if the following inequalities hold

J u (p, n, τ, u, v * ) ≤ J u (p, n, τ, u * , v * ), J v (p, n, τ, u * , v) ≤ J v (p, n, τ, u * , v * )
for every alternative strategies u and v. Let one introduce the value functions Ũ and Ṽ for the predators and preys correspondingly. Roughly speaking the function Ũ (p, n, t) (or Ṽ (p, n, t)) give the equilibrium payoff expected by the player, if the game were to start at time t in the state (p(t), n(t)) = (p, n). Assume that both value functions exist. Under regularity conditions (see [5, p. 292]), the functions Ũ and Ṽ provide a solution to the system of Hamilton-Jacobi-Bellman (HJB) equations

- ∂ Ũ ∂τ + max 0≤u≤1 ∂ Ũ ∂p (-p + nuv * )+ ∂ Ũ ∂n (-µ(1 -v * )n -nzuv * ) + (1 -u)p = 0 - ∂ Ṽ ∂τ + max 0≤v≤1 ∂ Ṽ ∂p (-p + nu * v)+ ∂ Ṽ ∂n (-µ(1 -v)n -nzu * v) + n = 0
where u * and v * provide the maximum in the other equation, τ is the reverse time, τ = T -t. Regarding the terminal conditions Ũ (p(T ), n(T ), T ) = 0, Ṽ (p(T ), n(T ), T ) = 0

B. Transformation to 2D

One can show that all the data are homogeneous of degree one in the state variables, and on the other hand, one of the state variables is always positive. This is a particular case of Noether's theorem in the calculus of variations about problems whose data is invariant under a group of transformations. Therefore we can reduce the phase space of the problem on degree one and study only the trandsformed system. To do so let one change the value function Ũ and Ṽ to Ũ (p, n, τ ) = nU (x, τ ) and Ṽ (p, n, τ ) = nV (x, τ ) where the new variable x is introduced, x = p/n. In this case the system of HJB-equations transforms to

h u -π τ + max 0≤u≤1 π x (-x(1-µ(1-v * )-zuv * )+uv * )+ U (-µ(1 -v) -zuv * ) + (1 -u)x = 0 h v -ν τ + max 0≤v≤1 ν x (-x(1 -µ(1 -v) -zu * v) + u * v)+ V (-µ(1 -v) -zu * v) + 1 = 0
where we introduce the following notations for the conjugate variables π x = ∂U/∂x and π τ = ∂U/∂τ , ν x = ∂V /∂x and ν τ = ∂V /∂τ . The optimal behavior is the following

u * = Heav(A u ), A u = π x (xz + 1)v -U vz -x v * = Heav(A v ), A v = ν x (u(xz + 1) -µx) + V (µ -uz)
We can see that only two variables τ and x are left in the equations.

In order to solve the HJB equations, a characteristic system can be written as follows

x ′ = - ∂h u ∂π x = x(1 -µ(1 -v) -zuv) -uv, π ′ x = ∂h u ∂x + ∂h u ∂U π x = -π x + 1 -u, π ′ τ = ∂h u ∂τ + ∂h u ∂U π τ = -π τ (µ(1 -v) + zuv), ν ′ x = ∂h v ∂x + ∂h v ∂V ν x = -ν x , ν ′ τ = ∂h v ∂τ + ∂h v ∂V ν τ = -ν τ (µ(1 -v) + zuv), U ′ = -π x ∂h u ∂π x -π τ ∂h u ∂π τ = -U (µ(1 -v) + zuv) + (1 -u)x, V ′ = -ν x ∂h v ∂ν x -ν τ ∂h v ∂ν τ = -V (µ(1 -v) + zuv) + 1
where the prime denotes the derivative w.r.t. reverse time.

If we know the values of π i , ν i (i = x, τ ), we can obtain the value functions U and V directly from the system of HJB-equations

h i = 0 (i = u, v).
Terminal conditions at t = T give U (x, T ) = 0 and V (x, T ) = 0. This leads to the equality π x (T ) = ν x (T ) = 0. From a biological point of view we can also state that u(T ) = 0 and v(T ) = 1 since it would not makes sense for the predators to try and increase their reproductive energy just before dying. They should consume their energy for reproduction (u(T ) = 0) which allows the preys to safely eat (v(T ) = 1). This can also be shown through mathematical considerations. Then

π τ (T ) = (1 -u(T ))x(T ) = x(T ), ν τ (T ) = 1

III. PRIMARY SOLUTION

A. Appearance of the switching curve

At τ = 0 we have

x ′ = x, π ′ x = 1 -π x , π ′ τ = 0, ν ′ x = -ν x , ν ′ τ = 0, U ′ = x, V ′ = 1 or x = x 0 e τ , π x = 1 -e -τ , ν x = 0, π τ = x 0 , ν τ = 1 U = x -x 0 , V = τ
There is a switching curve S 1 that appears in the solution. The switching condition for the first player is the following

A u = (1-e -τ )(xz+1)-x(1-e -τ )z-x = (1-e -τ )-x = 0 Then S 1 : x = 1 -e -τ
for x ≤ 1 2 ; indeed, the curve x = 1 -e -τ can only be crossed by solutions if x(T ) ≤ 1 4 , which corresponds to an intersection at x = 1 2 . First step of the construction of the characteristics is shown on Fig. 1. The blue line S 1 is the switching curve for the first player. Moving from the left side of S 1 to the right one the values π x and π τ are changing smoothly. But the values of ν x and ν τ have a jump, because the corresponding HJB-equation is changing. On the right side of it, the HJBequation has the form

-ν + τ + ν + x (1 -x + xz) -V z + 1 = 0
where the + superscript denotes the values "after" the jump in reverse time.

A normal vector to the switching surface S 1 equals to

ν = ∇S 1 = (∂S 1 /∂x, ∂S 1 /∂τ ) = (-1, 1 -x)
Due to the continuity of the value function The argument in the Heaviside function of the second control v becomes equal to

∇V + = ∇V -+ kν or ν + x ν + τ = ν - x ν - τ + k -1 1 -x = -k 1 + k(1 -x) with (ν - x , ν - t ) = (0, 1). Substitution into the HJB-equation gives k = - τ z 2(1 -x) + xz Then ν + x = τ z 2(1 -x) + xz , ν + τ = 1 - (1 -x)τ z 2(1 -x) + xz
A v + = (2(1 -x)µ -(1 -2x)z)τ 2(1 -x) + xz
The expression in the denominator

xz + 2(1 -x) is always positive since x < 1/2 on S 1 . Then A v + > 0 if 2(1 -x)µ - (1-2x)z > 0 or x > x * = (z -2µ)/2(z -µ).
This holds for all possible x if z < 2µ. Otherwise there is an internal point

x = x * such that A v + < 0 for 0 ≤ x < x * and A v + > 0 for x * < x ≤ 1/2.
The following situation corresponding to the values z = 8 and µ = 2 is shown on Fig. 2.

B. Segment AB

Suppose that the coordinate x * exists (0 < x * < 1/2) and consider the characteristic field emitted from the segment AB. In addition let's denote the coordinate of the switch on S 1 as (x 1 , τ 1 ). Both controls equals to one (predators are feeding, preys continue to eat), u(τ and the characteristic system is written in the form

1 + 0) = v(τ 1 + 0) = 1,
x ′ = -(z -1)x -1, x(τ 1 ) = x 1 , π ′ x = -π x , π x (τ 1 ) = 1 -e -τ1 = x 1 , π ′ τ = -π τ z, π τ (τ 1 ) = x 1 e -τ1 = x 1 (1 -x 1 ), ν ′ x = -ν x , ν x (τ 1 ) = τ 1 z 2(1 -x 1 ) + x 1 z , ν ′ τ = -ν τ z, ν τ (τ 1 ) = 1 - (1 -x 1 )τ 1 z 2(1 -x 1 ) + x 1 z , U ′ = -U z, U (τ 1 ) = x 1 (1 -e -τ1 ) = x 2 1 , V ′ = -V z + 1, V (τ 1 ) = τ 1 Then x = x 1 e -(z-1)(τ -τ1) - 1 -e -(z-1)(τ -τ1) z -1 π x = x 1 e -(τ -τ1) , π τ = x 1 (1 -x 1 )e -z(τ -τ1) ν x = ν x (τ 1 )e -(τ -τ1) , ν τ = ν τ (τ 1 )e -z(τ -τ1) U = x 2 1 e -z(τ -τ1) , V = τ 1 e -z(τ -τ1) + 1 -e -z(τ -τ1) z
The switching surface for the second player can be obtained from the equality

A v = ν x (1 + x(z -µ)) -V (z -µ) = 0 or S 2 : τ 1 ze -(τ -τ1) 2(1 -x 1 ) + x 1 z (1 + x(z -µ)) -τ 1 e -z(τ -τ1) + 1 -e -z(τ -τ1) z (z -µ) = 0 where x = x 1 e -(z-1)(τ -τ1) - 1 -e -(z-1)(τ -τ1) z -1 , x 1 = 1 -e -τ1
Finally, we have the situation shown on Fig. 3.

C. A singular arc

Since the unchanged continuation of the terminal solution that is tangent to S 1 and solutions that lead to S 2 leave an empty space between them, there must exist a singular arc S σ that is emitted from this point along which the first player applies the intermediate control u = u σ while the control of the second player remains the same v = 1. To define such a curve S σ let one write the necessary conditions of its existence

h u = -π τ + x(1 -π x ) = 0 A u = π x (xz + 1) -U z -x = 0, A u ′ = 0
We have

A u ′ = {A u h u } = ∂h u ∂π x ∂A u ∂x + π x ∂A u ∂U + π τ ∂A u ∂τ + ∂h u ∂π τ ∂A u ∂U - ∂A u ∂π x ∂h u ∂x + π x ∂h u ∂U - ∂A u ∂π τ ∂h u ∂τ + π τ ∂h u ∂U = -1 + x(1 + z(π x -1)) + π x + zπ τ = 0
where the braces {• •} denote the Jacobi (Poisson) brackets.

Then the conjugate variables are expressed as follows

π x = 1 -x, π τ = x(1 -x)
The value function of the first player

U = x(1 -x) + 1 -2x z
The expression for the singular control u σ can be obtained using the equality of the second derivative of A u to zero

A u ′′ = {{A u h u }h u } = {{(π x -1 + x)h u }h u } = -2x + u σ (xz + 2) = 0 Then u σ = 2x xz + 2
We have also

x ′ = - x 2 z xz + 2 , x(τ 1 ) = 1 2 , τ1 = ln 2
Finally we can integrate this differential equation with the given initial conditions and obtain the following expression for the singular arc

S σ : -ln x + 2 xz - 4 z = τ, τ ≥ τ1
Consider the moment of switching from u = u σ to u = 1 at some point of the singular arc (x σ , τ σ ). The HJB-equation is changed to the following equation

-ν σ τ + ν σ x (1 -x + xz) -V z + 1 = 0
and there is a jump in the costate variables of the second player.

Along the singular arc S σ :

V ′ = 1 -V zu σ = 1 -V 2xz xz + 2 , x ′ = - x 2 z xz + 2 , V (ln 2) = ln 2, x(ln 2) = 1 2
This can be rewritten in the form

dV dx = 2V xz -xz -2 x 2 z , V (1/2) = ln 2 Therefore V = 4x 2 ln 2+ (1 -2x)(4 + 8x + 16x 2 + 3xz(1 + 2x)) 6xz V σ
Choose the variable s = x as a parameter of the singular arc. In this case

∂τ ∂s = - 1 x - 2 x 2 z
and

∂V ∂s = - 2 + 4x 3 (8 -3z(2 ln 2 -1)) 3x 2 z = ν σ x -ν σ τ 1 x + 4 x 2 z
After the substitution to the changed HJB-equation we have the following values of conjugate variables

ν σ τ = 2 + 3xz + 4x 3 (8 -3z(2 ln 2 -1)) 6(xz + 1) , ν σ x = 4 + 3xz -4x 3 (8 -3z(2 ln 2 -1)) 6x(xz + 1)
These values are the initial conditions of characteristic trajectories emitted from the singular curve S σ in the domain 11. For this domain we have again

x ′ = -(z -1)x -1, x(τ σ ) = x σ , π ′ x = -π x , π x (τ σ ) = 1 -x σ , π ′ τ = -π τ z, π τ (τ σ ) = x σ (1 -x σ ), ν ′ x = -ν x , ν x (τ σ ) = ν σ x , ν ′ τ = -ν τ z, ν τ (τ σ ) = ν σ τ , U ′ = -U z, U (τ σ ) = x σ (1 -x σ ) + 1 -2x σ z , V ′ = -V z + 1, V (τ σ ) = V σ
The switching condition

A v = ν x (1 + x(z -µ)) -V (z -µ) = 0 or S σ 2 : ν σ x e -(τ -τ σ ) (1 + x(z -µ))- V σ e -z(τ -τ σ ) + 1 -e -z(τ -τ σ ) z (z -µ) = 0 where x = x σ e -(z-1)(τ2-τ σ ) - 1 -e -(z-1)(τ2-τ σ ) z -1
This curve is indicated on Fig. 4 by the segment CD.

IV. BI-SINGULAR SOLUTION

A. Derivation of the Hamilton-Jacobi (HJ) equations

A simple Nash argument shows that it is not possible to have the region 00 inside the domain 0ACD (see Fig. 4). Indeed, let us consider the segment 0A; in forward time, a solution initiated on the right of 0A with the controls (0, 0) would be such that the predators only reproduce until the final time, while the preys first hide until reaching 0A and then eat until the final time. Obviously, the preys can improve their situation from there by instead choosing to eat all the time, since they are not threatened by the preadtors, which are reproducing. The controls (0, 0) can therefore not be part of a Nash solution on the right of 0A. Further analysis of A + u and A + v shows that no solution with bang-bang control values can be emitted from 0A.

The phenomenon of the appearance of the bi-singular solution for the class of nonzero-sum differential games has been recently investigated [START_REF] Hamelin | Uncoupling Isaacs equations in twoplayer nonzero-sum differential games. Parental conflict over care as an example[END_REF]. To define necessary equations for the value function in the bi-singular domain let one eliminate the controls v and u from the pairs of the equations

h u = -π τ -π x x(1 -µ(1 -v)) -µU (1 -v) + x = 0, A u = π x (xz + 1)v -U vz -x = 0 h v = -ν τ -ν x x(1 -µ) -µV + 1 = 0, A v = ν x (u(xz + 1) -µx) + V (µ -uz) = 0
This leads to the following PDEs which are denoted HJequations (instead of the previous HJB-equations):

ĥu -π τ -π x x(1 -µ) -µU + x - µx(π x x -U ) π x (xz + 1) -U z = 0 ĥv -ν τ -ν x x(1 -µ) -µV + 1 = 0
The bi-singular controls can be obtained from the solution of these PDEs

u σ = µ(ν x x -V ) ν x (xz + 1) -V z , v σ = x π x (xz + 1) -U z

B. Solution of the second HJ-equation

Consider the HJ-equation regarding the second player ĥv = -ν τ + ν x x(µ -1) -µV + 1 = 0 with singular control

u σ = µ(ν x x -V ) ν x (xz + 1) -V z (0 ≤ u σ ≤ 1)
The Characteristic system has the form

x ′ = - ∂ ĥv ∂ν x = -x(µ -1), ν ′ x = ∂ ĥv ∂x + ν x ∂ ĥv ∂V = -ν x , ν ′ τ = ∂ ĥv ∂τ + ν τ ∂ ĥv ∂V = -µν τ , V ′ = -ν x ∂ ĥv ∂ν x -ν τ ∂ ĥv ∂ν τ = -µV + 1, x(τ 1 ) = x 1 , V (τ 1 ) = τ 1
If we take the segment 0A as the boundary we need to recalculate the values of the costate variables ν x (τ 1 ) and ν τ (τ 1 ):

ν x (τ 1 ) ν τ (τ 1 ) = -k 1 + k(1 -x)
regarding the HJ-equation

-ν τ (τ 1 ) + ν x (τ 1 )(µ -1)x 1 -µτ 1 + 1 = 0 Then k = - µτ 1 µx 1 + 1 -2x 1 and ν x (τ 1 ) = µτ 1 µx 1 + 1 -2x 1 , ν τ (τ 1 ) = 1 - (1 -x 1 )µτ 1 µx 1 + 1 -2x 1 Then x = x 1 e -(µ-1)(τ -τ1) , ν x = ν x (τ 1 )e -(τ -τ1) , ν τ = ν τ (τ 1 )e -µ(τ -τ1) , V = τ 1 e -µ(τ -τ1) + 1 -e -µ(τ -τ1) µ
By substitution of this solution we can write the expression for the singular control u σ . On the segment 0A it equals to

u σ (τ 1 ) = µν x (τ 1 )x 1 -µτ 1 ν x (τ 1 )(x 1 z + 1) -τ 1 z = µ(1 -2x 1 ) (1 -2x 1 )z -µ
which belongs to the interval between zero and one. The singular control equals exactly one at the boundary point A:

u σ (τ 1 ) x1=x * = 1
The result is shown on Fig. 5. The horizontal line x = 0 is at level u σ = µ/z. Now let's take the segment AC into account. If we denote the coordinates of the point on AC as (x 2 , τ 2 ) ∈ S 2 and corresponding to it the point on AB as (x 1 , τ 1 ) we can write

x 2 = x 1 e -(z-1)(τ2-τ1) - 1 -e -(z-1)(τ2-τ1) z -1 ν x (τ 2 ) = τ 1 ze -(τ2-τ1) 2(1 -x 1 ) + x 1 z ,
ν τ (τ 2 ) = 1 - (1 -x 1 )τ 1 z 2(1 -x 1 ) + x 1 z e -z(τ2-τ1) V (τ 2 ) = τ 1 e -z(τ2-τ1) + 1 -e -z(τ2-τ1) z with u σ (τ 2 ) = 1. Then x = x 2 e -(µ-1)(τ -τ2) ν x = ν x (τ 2 )e -(τ -τ2) , ν τ = ν τ (τ 2 )e -µ(τ -τ2) V = V (τ 2 )e -µ(τ -τ2) + 1 -e -µ(τ -τ2) µ
Using these expressions we can obtain the value of the singular control u σ . The solution is shown on Fig. 6 by the set of blue curves started from the segment AC. The surface closer to us indicates the previous solution.

The characteristics of the bi-singular field (which are not the game trajectories) intersect the field (11) coming, in backward time, from the singular arc. Therefore we can construct the singular line of the dispersal type such that V σ = V 11 on it, where the value V σ denotes the solution

Construction of the dispersal line CE

For one part of CE we should compare characteristics emitted from 0A and characteristics emitted from the singular arc. They will give a lower part of CE. For another part we should consider the field (11) and bi-singular characteristics emitted from segment AC.

For the characteristics from 0A we have

x 11 = x 1 e -(µ-1)(τ -τ1) , V 11 = τ 1 e -µ(τ -τ1) + 1 -e -µ(τ -τ1) µ
for the characteristics from AC

x 11 = x 2 e -(µ-1)(τ -τ2)

V 11 = τ 1 e -z(τ2-τ1) + 1 -e -z(τ2-τ1) z e -µ(τ -τ2) +

1 -e -µ(τ -τ2) µ where (τ 1 , x 1 ) ∈ 0B and (τ 2 , x 2 ) ∈ AC.

If we consider a primary we can write

x σ = x σ e -(z-1)(τ -τ σ ) - 1 -e -(z-1)(τ -τ σ ) z -1 , V σ = 1 -e -z(τ -τ σ ) z + e -z(τ -τ σ ) 4(x σ ) 2 ln 2+ (1 -2x σ )(4 + 8x σ + 16(x σ ) 2 + 3x σ z(1 + 2x σ )) 6x σ z
where the coordinates (x σ , τ σ ) belong to the singular arc S σ . The coordinates (τ 3 , x 3 ) of the point on CF can be obtained through equations

V 11 τ =τ3 = V σ τ =τ3
= -π τ + x(1 -π x )+ µ(π x x -U ) 1 - x π x (xz + 1) -U z = 0
with respect to singular control in the form

v σ = x π x (xz + 1) -U z
The corresponding characteristic system can be written analogously to the previous case

x ′ = -(µ -1)x + µxU (π x (xz + 1) -U z) 2 , π ′ x = 1 -π x - µ(π x x -U ) π x (xz + 1) -U z , π ′ τ = -µπ τ 1 - π x x (π x (xz + 1) -U z) 2 , U ′ = x -µU - µx(U 2 z + π 2 x x(xz + 1) -2U π x (xz + 1)) (π x (xz + 1) -U z) 2
with initial condition on the segment 0A

x(τ 1 ) = x 1 , π τ (τ 1 ) = x 1 (1-x 1 ), π x (τ 1 ) = x 1 , U (τ 1 ) = x 2 1 Notice that ∂ Ũ ∂n ′ = (π x x -U ) ′ = -µ(π x x -U ) But in our case (π x x -U ) τ =τ1
= 0. Therefore π x x -U = 0 all the time. This leads to the characteristic equation for π x of the form π ′ x = 1 -π x with initial condition on 0A: α(τ 1 ) = 1 -e -τ1 = x 1 . Then π x = 1 -e -τ and the singular control can be written in the form directly depending on x and τ :

v σ = x 1 -e -τ
For the characteristic field emitted for the segment AC the situation is more complicated since the conjugate variables have a jump on it. Suppose that the segment S 2 is defined through the following conditions

S 2 : τ 1 ze -(τ -τ1) 2(1 -x 1 ) + x 1 z (1 + x(z -µ)) -τ 1 e -z(τ -τ1) + 1 -e -z(τ -τ1) z (z -µ) = 0 x = x 1 e -(z-1)(τ -τ1) - 1 -e -(z-1)(τ -τ1) z -1 (x 1 = 1 -e -τ1 )
Suppose that τ 1 = τ 1 (x, τ ). If we differentiate the last equation for x we obtain Then we can derive the expressions for ∂S 2 /∂x and ∂S 2 /∂τ but we will omit this part since they have too complicated a form. Taking into account that

∂τ 1 ∂x = 1 -x 1 + x 1 z (1 -x + xz)(2(1 -x 1 ) + x 1 z) , ∂τ 1 ∂τ = 1 -x 1 + x 1 z 2(1 -x 1 ) + x 1 z 0.2 A C 0.6 Τ 0.5 1 2 v Σ
π + x = x 1 e -(τ -τ1) +k ∂S 2 ∂x , π + τ = x 1 (1-x 1 )e -z(τ -τ1) +k
∂S 2 ∂τ we can substitute these values into the HJ-equation

-π + τ +x(1-π + x )+µ(π + x x-U ) 1 - x π + x (xz + 1) -U z = 0 with U = x 2 1 e -z(τ -τ1
) and obtain the equation w.r.t. the scalar k. After that, we can write the new values of the conjugate variables π +

x and π + τ . We should notice that there are two possible solutions k 1 and k 2 . The value of k 1 is negative on AC and equals to zero at point A. But this branch corresponds to the situation when v σ 1. Another branch of the solution with k 2 > 0 corresponds to the values v σ ∈ (0, 1), see Fig. 8.

The resulting characteristic field for the segment 0A and AC is shown on Fig. 9.

D. Construction of the real game trajectories

Using the constructed solution of the decomposed system of two HJ-equations, we can compute the pair of singular controls u σ and v σ in the domain 0ACE. Thus, we can construct the real game trajectories in the bi-singular domain. They are shown on Fig. 10. We can identify our main difficulty on this figure. Even though the whole space was filled with the characteristics for the decomposed system of two HJ equations, there is an empty subdomain in the final portrait of optimal trajectories (shaded area of Figure 10). This is a strong indication that some important things may have been missed in the process of deriving the solution.

V. CONCLUSIONS

We did not finalize the solution and did not cover the whole phase space by the field of characteristics. But our work is still in progress and we will continue our research. Nevertheless, we would like to point out that in the paper we investigated the transition from bang-bang controls to bisingular ones and apply the method of characteristics to a system of two HJB-equations. Although the fact that we did not find the solution in a subdomain of the phase space can be crucial and a new idea of the solution can change the whole picture drastically we think that the proposed solution will remain fulfilled in a sufficiently large area of the phase space after. The theory of nonzero-sum differential games is not welldeveloped yet. There are no theorems of existence and uniqueness of their solution [START_REF] Hamelin | Uncoupling Isaacs equations in twoplayer nonzero-sum differential games. Parental conflict over care as an example[END_REF], [START_REF] Cardaliaguet | Existence and uniqueness of a Nash equilibrium feedback for a simple non-zero-sum differential game[END_REF], [START_REF] Bressan | Semi-cooperative strategies for differential games[END_REF] so that it might be possible that no Nash equilibrium exist from the initial states in the "void" zone. There might also be several, may be an infinity, of Nash equilibria. There is no strict classification of possible singular surfaces as one has been done for zero-sum differential games, see for example [9, p. 121]. Therefore this work is only an attempt to study a particular nonzero-sum differential games with an application in behavioral ecology.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Appearance of the switching curve S 1

2 Fig. 3 .

 23 Fig. 3. The switching curve S 2

2 σFig. 4 .

 24 Fig. 4. Construction of the switching curve S σ 2

Fig. 5 .Fig. 6 .

 56 Fig. 5. Solution of the 2nd HJ-equation with a boundary at 0A

Fig. 7 .

 7 Fig. 7. Final construction of the characteristic field for the 2nd HJ-equation

  of the first HJ-equation Consider the following PDE ĥu

Fig. 8 .

 8 Fig. 8. Singular control v σ on the boundary of the bi-singular domain

Fig. 9 .

 9 Fig. 9. Characteristic field for the first HJ-equation

Fig. 10 .

 10 Fig. 10. Real game trajectories recovered from the bi-singular solution

VI. ACKNOWLEDGMENTS This research has been supported by the 'Lutins&Co' COLOR project of INRIA Sophia Antipolis-Méditerranée and the 'IA2L' project of the SPE department of INRA. The work of the first author has been partially supported by the Russian Foundation for Basic Research (grant 07-04-00418).


Dario Corradini

Dario Marrocchelli

Paul A Madden

Mathieu Salanne

Effect of dispersion interactions on the properties of LiF in condensed phases

Classical molecular dynamics simulations are performed on LiF in the framework of the polarizable ion model. The overlap-repulsion and polarization terms of the interaction potential are derived on a purely non empirical, first-principles basis. For the dispersion, three cases are considered: a first one in which the dispersion parameters are set to zero and two others in which they are included, with different parameterizations. Various thermodynamic, structural and dynamic properties are calculated for the solid and liquid phases. The melting temperature is also obtained by direct coexistence simulations of the liquid and solid phases. Dispersion interactions appear to have an important effect on the density of both phases and on the melting point, although the liquid properties are not affected when simulations are performed in the N V T ensemble at the experimental density.

INTRODUCTION

Dispersion interactions between two chemical entities arise from the instantaneous correlation of the fluctuations of their electron densities [START_REF] London | The general theory of molecular forces T[END_REF]. They are the only attractive interactions in noble gases and are therefore at the origin of the existence of their pure condensed phases [START_REF] Van Der Waals | The equation of state for gases and liquids Nobel[END_REF]. But dispersion interactions are generally weaker than the other intermolecular interactions in molecular systems and their role in systems for which stronger attractive forces are present is hard to establish.

In density functional theory (DFT) calculations, dispersion interactions are difficult to capture. Several methods have been developed, in which they are treated directly by devising new functionals [START_REF] Vydrov | Dispersion interactions from a local polarizability model[END_REF][START_REF] Vydrov | Nonlocal van der Waals density functional: The simpler the better[END_REF] or effectively by adding an analytical term to existing functionals [START_REF] Grimme | Accurate description of van der Waals complexes by density functional theory including empirical corrections[END_REF]. Following these developments, it was shown that dispersion interactions impact structure and dynamics in a variety of systems; for example in water at ambient [START_REF] Lin | Importance of van der Waals interactions in liquid water[END_REF] or supercritical conditions [START_REF] Jonchiere | Van der Waals effects in ab initio water at ambient and supercritical conditions[END_REF]. But the most affected quantity is usually the density, which is understimated by as much as 20 % in water when dispersion interactions are not accounted for [START_REF] Schmidt | Isobaricisothermal molecular dynamics simulations utilizing density functional theory: an assessment of the structure and density of water at near-ambient conditions[END_REF]. Similar deviations are found in crystalline systems [START_REF] Allen | Tin monoxide: Structural prediction from first principles calculations with Van der waals corrections[END_REF]. In ionic materials, the effect of dispersion is expected to be smaller due to the presence of a strong attractive Coulombic interaction between species with different charges. This leads to strong charge ordering effects, and the overlap-repulsion interaction sets the distance of closest approach between two ions. The polarization (induction) interaction also plays a well-identified role and it is at the origin of the stabilization of non-trivial crystal structures and of the crosslinking of the coordination shells of multivalent cations in ionic melts [START_REF] Salanne | Polarization effects in ionic solids and melts[END_REF]. Nevertheless, it was shown by Kirchner et al. that in "room-temperature" ionic liquids, the inclusion of dispersion interactions in DFT calculations modifies the vibrational and dynamic properties of the system [START_REF] Grimme | Performance of dispersion-corrected density functional theory for the interactions in ionic liquids[END_REF][START_REF] Malberg | The bulk and the gas phase of 1-ethyl-3-methylimidazolium ethylsulfate: dispersion interaction makes the difference[END_REF][START_REF] Pensado | Effect of dispersion on the structure and dynamics of the ionic liquid 1-ethyl-3-methylimidazolium thiocyanate[END_REF].

DFT-based molecular dynamics is however limited to short simulation times and small system sizes, so that it is hard to establish clearly the role of dispersion on a wide range of properties. Classical molecular dynamics is an alternative option, provided that the force field accurately separates the various contributions from Coulombic, repulsion, polarization and dispersion interactions. In recent years, we have developed a series of methods which make possible the derivation of such force fields for inorganic ionic materials [START_REF] Aguado | Multipoles and interaction potentials in ionic materials from planewave-DFT calculations[END_REF][START_REF] Madden | From first-principles to material properties[END_REF][START_REF] Rotenberg | From localized orbitals to material properties: building classical force fields for nonmetallic condensed matter systems[END_REF][START_REF] Salanne | Including many-body effects in models for ionic liquids[END_REF]. In short, we use a polarizable ion model, and all the parameters except the ones concerning dispersion are fitted to standard condensed phase DFT calculations. The emphasis on "condensed phase" here is because the physical properties of ions are strongly affected by their coordination environment and the use of ab initio data on isolated ions or on the interaction energies of small clusters leads to hopelessly inadequate force-fields. This applies particularly to the dispersion coefficients (and polarizabilities). For our procedure the reference data includes both the dipole moments on each ion and the total force which is exerted on them, allowing the polarization and the repulsion terms to be fitted independently (formal charges are used for the ions, which automatically sets the Coulombic interaction). In addition, some parameters such as the condensed phase polarizability of the ions can directly be calculated [START_REF] Heaton | Condensed phase ionic polarizabilities from plane wave density functional theory calculations[END_REF][START_REF] Salanne | Polarizabilities of individual molecules and ions in liquids from first principles[END_REF][START_REF] Molina | Ions in solutions: Determining their polarizabilities from first-principles[END_REF]. The dispersion coefficients have to be treated separately, because on one hand the dispersion effects are not yet reliably represented in the DFT calculations and on the other they contribute relatively weakly to the forces on the ions, which means that they do not strongly affect the quality of the fit to the DFT data with the classical force-field.

In the present work, we perform simulations of LiF in the crystal and liquid phases. We compare the thermodynamic, structural and dynamic properties obtained with two different sets of condensed phase dispersion coefficients, obtained either from the Coupled Hartree-Fock (CHF) theory [START_REF] Fowler | Calculations of 2-body and 3-body dispersion coefficients for ions in crystals[END_REF] or from a recent method [START_REF] Rotenberg | From localized orbitals to material properties: building classical force fields for nonmetallic condensed matter systems[END_REF][START_REF] Silvestrelli | Van der Waals interactions in DFT made easy by Wannier functions[END_REF] involving the use of the Maximally Localized Wannier Functions (MLWFs) [START_REF] Marzari | Maximally localized generalized Wannier functions for composite energy bands[END_REF][START_REF] Marzari | Maximally localized Wannier functions: Theory and applications[END_REF] in DFT, as well as with the case where the dispersion interactions are set to zero. LiF was chosen because it is a material for which we expect that the effect of the dispersion interactions is among the lowest. Dispersion is negligible for the Li + cation due to its small radius. Also, F -is the halide anion with the smallest polarizability.

NUMERICAL METHODS

Polarizable Ion Model

The polarizable ion model includes Coulombic, dispersion, overlap repulsion and polarization components [START_REF] Madden | Covalent" effects in "ionic" systems[END_REF]. First the Coulombic term is:

V Coul = i<j q i q j r ij (1)
where q i is the charge on ion i, and formal charges are used throughout. The dispersion component includes dipole-dipole and dipole-quadrupole terms

V disp = - i<j f ij 6 (r ij ) C ij 6 (r ij ) 6 + f ij 8 (r ij ) C ij 8 (r ij ) 8 (2)
where

C ij 6 (C ij 8 )
is the dipole-dipole (dipole-quadrupole) dispersion coefficient, and f ij n are damping functions [START_REF] Tang | An improved simple model for the van der Waals potential based on universal damping functions for the dispersion coefficients[END_REF], describing the short-range penetration correction to the asymptotic multipole expansion of dispersion [START_REF] Stone | Theory of intermolecular forces[END_REF] (f ij n (0) = 0 and f ij n (∞) = 1). They take the form

f ij n (r ij ) = 1 -e -b ij n r ij n k=0 (b ij n r ij ) k k! (3)
and the parameters b ij n represent the distance at which the correction begins to be taken into account. The repulsion overlap component is given by

V rep = i<j B ij e -a ij r ij (4)
Finally the polarization part of the potential includes charge-dipole and dipole-dipole terms:

V pol = i<j q i µ j α g ij D (r ij ) -q j µ i α g ji D (r ij ) T (1) α - i<j µ i α µ j β T (2) αβ + i 1 2α i | µ i | 2 . ( 5
)
Here T

(1) α and T

(2)

αβ are the charge-dipole and dipoledipole interaction tensors and α i is the polarizability of

Ion pair A ij a ij b ij D = b ji D c ij D F --F -282.3 2.444 - - F --Li + 18.8 1.947 1.834 1.335 Li + -Li + 1.0 5.0 - - TABLE I:
Fitted parameters for the repulsion and polarization terms for LiF (atomic units). The fluoride polarizability was set to 7.9 atomic units, and the Li + is not polarizable.

ion i. Again, we include some short-range effects which are due to the high compression of the ions in condensed ionic materials [START_REF] Fowler | Fluctuating dipoles and polarizabilities in ionic materials: calculations on LiF[END_REF][START_REF] Domene | A transferable representation of the induced multipoles in ionic crystals[END_REF][START_REF] Jemmer | Dipole and quadrupole polarization in ionic systems: ab initio studies[END_REF]. These short-range induction effects are straightforwardly included through the use of damping functions similar to the ones used in the dispersion term:

g ij D (r ij ) = 1 -c ij D e -b ij D r ij 4 k=0 (b ij D r ij ) k k! . (6)
Here c ij D is a parameter that reflects the amplitude of this damping at ion j due to the presence of i and b ij D again is a range parameter.

The instantaneous values of the dipole moments { µ i } N are obtained by minimization of V pol with respect to these variables: they will therefore depend on the instantaneous positions of neighboring ions and consequently change at each timestep in an MD run. The interaction potential can therefore be seen to contain three additional degrees of freedom (induced dipoles), which describe the state of the electron charge density of the ions. When calculating the forces on the ions in an MD simulation, these electronic degrees of freedom should have their "Born-Oppenheimer" values, which minimize the total potential energy, for every atomic configuration. We search for the ground state configurations of these degrees of freedom at each time step, using a conjugate gradient routine [START_REF] Press | Numerical Recipes[END_REF]. The dynamics is thus similar to the so-called Born-Oppenheimer first-principles molecular dynamics, as implemented, for example, in the CP2K code [32].

We perform an Ewald summation of all electrostatic interactions and also of dispersion [START_REF] Chen | Fast Ewald sums for general van der Waals potentials[END_REF]. Thus, all those interactions are free from truncation errors. The shortrange repulsion, which is an exponentially decaying function of distance, is however truncated beyond a distance equal to, at maximum, half the shortest dimension of the simulation cell.

Parameterization

Parameters for the repulsion and polarization terms were obtained from a generalized force-fitting procedure described elsewhere [START_REF] Aguado | Multipoles and interaction potentials in ionic materials from planewave-DFT calculations[END_REF][START_REF] Salanne | Including many-body effects in models for ionic liquids[END_REF]. This approach has been TABLE II: F --F -dispersion parameters for the three simulation setups (atomic units). b ij 6 = b ij 8 = 1.9, and the other C6 and C8 coefficients (for Li + -Li + and Li + -F -) are set to zero in all cases. used successfully for a series of oxide and fluoride materials [START_REF] Burbano | A dipole polarizable potential for reduced and doped CeO2 obtained from first principles[END_REF][START_REF] Marrocchelli | Understanding chemical expansion in non-stoichiometric oxides: Ceria and zirconia case studies[END_REF][START_REF] Salanne | Cooperative mechanism for the diffusion of Li + ions in LiMgSO4[END_REF][START_REF] Tangney | An ab initio parametrized interatomic force field for silica[END_REF][START_REF] Han | Polarizable interatomic force field for TiO2 parametrized using density functional theory[END_REF]. Note that the PBE functional was used to provide the reference data to which the parameters were fitted [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]. From our experience, this functional can be considered as "dispersion-free" in the case of ionic materials. The obtained parameters are provided in Table I. As for the dispersion term, three different cases are tested. In the first case considered, we set it to zero. In the second one ("CHF"), we have taken C 6 and C 8 parameters which reproduce the Coupled Hartree-Fock calculation by Fowler et al. [START_REF] Fowler | Calculations of 2-body and 3-body dispersion coefficients for ions in crystals[END_REF]. Finally, in the third case ("Wannier"), these parameters were determined from a calculation of the MLWFs [START_REF] Marzari | Maximally localized generalized Wannier functions for composite energy bands[END_REF][START_REF] Marzari | Maximally localized Wannier functions: Theory and applications[END_REF] using the procedure described in references [START_REF] Rotenberg | From localized orbitals to material properties: building classical force fields for nonmetallic condensed matter systems[END_REF][START_REF] Salanne | Including many-body effects in models for ionic liquids[END_REF]. The three sets of dispersion coefficients are provided in Table II.

The only interaction which differs between the different cases is the F --F -one. The sum of the three pairwise additive terms is plotted for each parameters set in Figure 1, in the region corresponding to first neighbour typical distances. The inset shows the dispersion term only. It appears obvious that the dispersion term only provides a tiny contribution (between 10 and 15 kJ mol -1 for the shortest distance) to the total interaction potential (which takes values ranging between 450 and 750 kJ mol -1 in the same region).

Simulation details

We consider the rocksalt (B1) 8×8×8 (2048 LiF units) crystal structure for the solid phase. We build the crystal using the LiF lattice parameter at ambient conditions, a = 4.026 Å. The solid thus obtained is equilibrated in the N P T ensemble [START_REF] Martyna | Constant pressure molecular dynamics algorithms[END_REF], at T = 300 K and ambient pressure, using a thermostat time constant τ T = 1 ps and an isotropic barostat time constant τ P = 2 ps. Equilibrium is rapidly reached over 10 ps simulations. We then simulate the crystal for accumulating the values of the density over another 20 ps.

For the liquid phase, we do two series of simulations. On the one hand we perform N V T runs at T = 1200 K and at the experimental density, at ambient pressure, ρ exp = 1.77 g cm -3 [START_REF] Janz | Molten Salts: Volume 1, Electrical Conductance, Density, and Viscosity Data[END_REF]. The simulation cell for the liquid phase contains 500 LiF units. The temperature is controlled by a Nosé-Hoover thermostat chain [START_REF] Nosé | A molecular dynamics method for simulations in the canonical ensemble[END_REF][START_REF] Hoover | Canonical dynamics: equilibrium phase-space distributions[END_REF] with a time-constant τ T = 10 ps.

On the other hand we also perform N P T simulations at T = 1200 K and at ambient pressure in order to determine the equilibrium density produced by each of the three sets of parameters. In this case we use 432 LiF units. The pressure is controlled by applying a barostat [START_REF] Martyna | Constant pressure molecular dynamics algorithms[END_REF], with τ P and τ T set to 10 ps.

The melting point calculations are performed by direct simulations of the coexisting crystalline and molten phase in the N P T ensemble [START_REF] Espinosa | On fluid-solid direct coexistence simulations: The pseudohard sphere model[END_REF][START_REF] Conde | Determining the phase diagram of water from direct coexistence simulations: The phase diagram of the TIP4P/2005 model revisited[END_REF][START_REF] Hong | Solid-liquid coexistence in small systems: A statistical method to calculate melting temperatures[END_REF]. The simulation cell contains 896 LiF units and it has a large aspect ratio, initially L x = L y = L z /4. First we perform N V E simulations that allow us to roughly estimate the melting temperature as described in Ref. [START_REF] Lanning | Solidliquid coexistence in ionic systems and the properties of the interface[END_REF]. Starting from the final ionic configurations from the N V E runs, which contain a crystalline and liquid region close to coexistence, we then proceed to perform N P T direct coexistence simulations at ambient pressure and at several temperatures in proximity of the melting temperature estimated by the N V E method. The thermostat time constant is set to 1 ps and the anisotropic barostat time constant is set to 2 ps. The angles between the cell vectors are fixed to π/2, so that the symmetry of the cell remains orthogonal, but the lengths of the sides of the simulation cell are able to fluctuate independently in order that the calculated pressure tensor is, on average, isotropic with the diagonal elements equal to ambient pressure. The time step for the integration of the equations of motion is 1 fs. As the simulation progresses, we monitor the extent of the crystalline and liquid regions over a 500 ps run. For temperature significantly above or below T m , the whole cell tends towards becoming molten or crystalline and this enables us to identify a temperature at which it is possible to conduct a run at which the extent of the liq- uid and solid regions can coexist over the length of the simulation.

RESULTS

Density

Equilibrium densities extracted from the N P T simulations of solid and liquid LiF at respective temperatures of 300 and 1200 K are provided in Table III. The comparison with experiments shows that similar differences are obtained in both cases when dispersion effects are included. The largest difference is observed when they are omitted, with an underestimation of 15.3 % for ρ liq . Such a deviation is of the same order of magnitude as that obtained in liquid water at room-temperature from DFT calculations using PBE or BLYP functionals [START_REF] Schmidt | Isobaricisothermal molecular dynamics simulations utilizing density functional theory: an assessment of the structure and density of water at near-ambient conditions[END_REF]. When the dispersion effects are included, the liquid density remains underestimated, by 4.5 % and 8.5 % in the Wannier and CHF cases, respectively. We note that in order to reproduce more accurately the experimental density, larger values would be necessary for C 6 and C 8 parameters. Such an underestimation of the density was already observed in our previous work on LiF-ThF 4 , while a perfect agreement was obtained for LiF-NaF-KF and NaF-ZrF 4 mixtures [START_REF] Salanne | Heat-transport properties of molten fluorides: determination from first-principles[END_REF][START_REF] Dewan | Molecular dynamics simulation of the thermodynamic and transport properties of the molten salt fast reactor fuel LiF-ThF4[END_REF] (both studies used the CHF parameters). This shows that although the polarizable ion model transferability for multiple physical and chemical conditions is well established, a completely transferable model would require more complex functional form, in which "environmental effects" are taken into account. These environmental effects have mainly been studied for the polarizability [START_REF] Jemmer | Environmental effects on anion polarizability: variation with lattice parameter and coordination number[END_REF][START_REF] Wilson | Models of environmental effects on anion polarizability[END_REF], and since dispersion effects follow a similar dependence on the electronic cloud extension as does the former, equivalent functional forms could be employed.

Liquid properties at fixed density

In liquid water, the dispersion interaction was shown to impact strongly the structure and dynamics at ambient [START_REF] Lin | Importance of van der Waals interactions in liquid water[END_REF] and supercritical [START_REF] Jonchiere | Van der Waals effects in ab initio water at ambient and supercritical conditions[END_REF] conditions, even at fixed density. In particular, the first peaks of the O-O and O-H radial distribution functions and the diffusion coefficients were shown to vary substantially. This is not the case for LiF: as can be seen in Figure 2, no difference is observed for any of the three partial radial distribution functions. This is because in such a simple molten salt, the structure mostly arises from a competition between the Coulombic interaction, which induces strong charge ordering, and the overlap-repulsion which sets the firstneighbour distance. The dispersion term is much weaker than those two and therefore does not change the structure.

The diffusion coefficients are extracted from molecular dynamics simulations using Einstein's relation, i.e. from the long-time slope of mean squared displacement:

D (α) = lim t→∞ 1 6N α t Nα i∈α,i=1 |r i (t) -r i (0)| 2 , (7)
where N α is the total number of atoms of type α, and r i (t) is the position of atom i of type α at time t. We obtain similar values for the three simulation setups, i.e. 7.4 • 10 -5 and 10.0 • 10 -5 cm 2 s -1 for F -and Li + ions, respectively. Note that these values are in excellent agreement with the experimental ones [START_REF] Sarou-Kanian | Diffusion coefficients and local structure in basic molten fluorides: in situ NMR measurements and molecular dynamics simula-tions[END_REF][START_REF] Levesque | Structure and dynamics in yttrium-based molten rare earth alkali fluorides[END_REF]. Again, this situation is very different from that observed in liquid water, in which a diffusion coefficient two to three times larger was obtained when dispersion effects were included in DFT-based molecular dynamics [START_REF] Lin | Importance of van der Waals interactions in liquid water[END_REF] (although these numbers might be mitigated due to the use of short simulation times). In conclusion, due to the predominance of Coulombic and repulsion interactions, in contrast with the case of water, dispersion interaction does not play a major role on the physico-chemical properties of molten LiF at a given density.

Solid-liquid interface

The melting point is usually considered as a very stringent test of a force-field, since it depends on the relative free energies of the crystal and liquid phases. The melting point can be determined by creating a cell containing the solid and liquid at coexistence. The interface has to be constructed by combining two separate bulk simulations of the crystalline and molten phases each equilibrated at the same pressure and the estimated melting temperature [START_REF] Lanning | Solidliquid coexistence in ionic systems and the properties of the interface[END_REF][START_REF] Aguado | New insights into the melting behavior of MgO from molecular dynamics simulations: the importance of premelting effects[END_REF]. The cell parameters for the liquid phase have to match those of the crystalline simulation. The two cells are then placed in a supercell, with a large aspect ratio, as shown on Figure 3. A first estimation of the melting point is obtained via short simulations performed in the N V E ensemble, before longer simulations are performed in the N P T ensemble. In the first stage, the system is run for a short period of time (1 ps) in a constant volume simulation with the thermostat set at the estimated coexistence temperature (or by regular velocity rescaling) to remove excess energy caused by bringing the solid and liquid together. Great care must be taken during these initial stages to ensure that one or other phase is not destroyed. Once the excessive relaxation energy has been removed, the system can be run on for a further 50 ps in a N V E simulation to ensure complete equilibration. We then perform a 50 ps production run, again in the N V E ensemble, during which we determine the average temperature and pressure. Other (T ,P ) coexistence points can then be sampled by performing additional simulations where the initial kinetic energy of the system is rescaled to a value above that of coexistence and then the system is allowed to re-equilibrate in an N V E simulation [START_REF] Lanning | Solidliquid coexistence in ionic systems and the properties of the interface[END_REF].

The variation of P versus T at coexistence is shown on Figure 4 for the three sets of dispersion parameters. At P = 1 atm, we obtain approximate melting temperatures of 980 K, 1115 K and 1170 K for the case with null dispersion and with the CHF and Wannier parameters, respectively. We then perform long simulations in the N P T ensemble, where the target pressure is set to 1 atm while several target temperatures close to the estimated T m are studied. We sample the potential energy and the cell length along the z direction during these N P T simulations. The results are shown on Figure 5. Both quantities remain constant only when the liquid and solid are coexisting [START_REF] Conde | Determining the phase diagram of water from direct coexistence simulations: The phase diagram of the TIP4P/2005 model revisited[END_REF]. We extract refined melting temperatures of 1000 K, 1108 K and 1170 K for the case with null dispersion and with the CHF and Wannier parameters, respectively. This trend is easily understood by the fact that increasing the dispersion effects will result in a stabilization of the most condensed phase, i.e. the solid, with respect to the less condensed phase. Nevertheless, the observed differences are large (up to > 100 K) despite the fact that no important differences were observed for the single phase structural properties. The CHF case provides a very accurate estimate of the melting point since the experimental value is 1118 K. This result is somewhat surprising since the model using the Wannier dispersion parameters provided the best estimate of the densities of both the solid and liquid phases, again showing the difficulty of building a force field of perfect accuracy for all the properties, even if it includes polarization effects.

CONCLUSIONS

In conclusion, we have shown in this paper that, although it brings a tiny contribution to the total energy, the dispersion interaction has important effects on the properties of LiF. On the one hand, these effects are not observed in fixed volume simulations of the liquid, for which the structure and the dynamics are independent of the C 6 and C 8 terms. On the other hand, the predicted equilibrium densities are affected in both the liquid and solid phases: an underestimation of the experimental data by as much as 15 % is observed when dispersion effects are omitted. But the strongest differences are obtained for the melting point, a quantity which reflects the free energy difference between the solid and liquid phase. This means that as soon as free energyrelated quantities have to be evaluated, special attention must be paid to using a correct parameterization of the dispersion coefficients.

To carry out this parameterization on a purely firstprinciples basis, which is what we have attempted here, is not straightforward. Here we have tested two sets of parameters. The first one was extracted from condensedphase Coupled Hartree Fock calculations [START_REF] Fowler | Calculations of 2-body and 3-body dispersion coefficients for ions in crystals[END_REF], while the second one was obtained from the determination of the Maximally Localized Wannier Functions in a simple con-densed phase DFT calculation [START_REF] Rotenberg | From localized orbitals to material properties: building classical force fields for nonmetallic condensed matter systems[END_REF][START_REF] Silvestrelli | Van der Waals interactions in DFT made easy by Wannier functions[END_REF]. Both of them provide reasonably good predictions for the densities and the melting points, but more systematic calculations on a series of materials would be needed for identifiying the better parameterization procedure. It is to be expected that the developments which are currently being made for treating accurately dispersion interactions in DFT calculations [START_REF] Vydrov | Dispersion interactions from a local polarizability model[END_REF][START_REF] Vydrov | Nonlocal van der Waals density functional: The simpler the better[END_REF][START_REF] Grimme | Accurate description of van der Waals complexes by density functional theory including empirical corrections[END_REF] will provide useful routes.

FIG. 1 :

 1 FIG.1: Sum of the three pairwise additive terms for the F --F -interaction (inset: dispersion term only). The distance range corresponds to the extent of the first peak of the F-F partial radial distribution function in liquid LiF (shown in Figure2).

FIG. 2 :

 2 FIG.2: Radial distribution functions in liquid LiF. Simulations were performed at 1200 K, at the experimental density ρexp = 1.77 g cm -3 .

FIG. 3 :

 3 FIG. 3: Illustration of the simulation cell used to determine the LiF melting point (green: F -, blue: Li + ).

FIG. 4 :

 4 FIG.4:(P ,T ) plots for LiF at the solid-liquid coexistence for the three sets of dispersion parameters.

FIG. 5 :

 5 FIG. 5: Variation of the potential energy (left) and cell length along the z axis (right) at several temperatures during N P T simulations performed at several temperatures.

TABLE III :

 III Equilibrium densities of solid LiF at T = 300 K and of liquid LiF at T = 1200 K.

	Setup	ρ sol (g cm -3 ) ρ liq (g cm -3 )
	Experiment	2.64	1.77
	No dispersion	2.32	1.50
	CHF	2.38	1.62
	Wannier	2.42	1.69

Acknowledgement

The work has been performed under the HPC-EUROPA2 project (project number: 228398) with the support of the European Commission -Capacities Area -Research Infrastructures.


Emmanuel Blanchard

French Colonial Police

The question of policing is consubstantial to that of colonization, and was raised at a very early stage, not just in diplomatic or military terms, but also with respect to the various manners of civility, regulation, and discipline that were to be expected from a good policing service. The notion of colonial policing, however, can hardly account for the many policing practices encountered in the various territories, with their diverse judicial and political statuses. It tends to foster a uniform vision of policing organization and practices, even though none of this was ever centralized, or even standardized, in this vast empire. Most importantly, it tends to conceal the fact that policing tasks were handled by multiple organizations and agents, many of whom were not, technically, actual police. Still, the very existence and usage of this phrase-colonial policing-suggests that, regardless of place and time, colonial authorities had their own set of policing practices and violent behaviors, which were partly distinct from what was prevalent in metropolitan France.

From a chronological point of view, the legacy of slavery and plantation power as well as the continued domination over various territories (e.g. some West Indies or Indian Ocean islands) ust be considererd: the hiatus between the first French colonial empire (built under the Ancien régime, during the 17th century in particular) and the second (refounded on the rubble of Napoleon's defeats, starting with the conquest of Algeria in 1830) should not be overstressed.

Only in the late 19th century, however, considering the developments and reforms happening in the home country, did colonial policing issues gradually and partially gain autonomous status, distinct from wider concerns about the frailty of colonial domination. Indeed, colonial domination only exceptionally reached the "pacification" stage, even though this goal was continually re-asserted by civilian authorities, closely associated to their military counterparts.

Colbert complained as early as 1667 that "very little concern has been shown so far in New France for the policing and civil life of Algonquians and Huron". He considered this "longstanding neglect" a mistake that ought to be remedied promptly [START_REF] Havard | Les forcer à devenir Cytoyens". État, Sauvages et citoyenneté en Nouvelle-France (XVIIe-XVIIIe siècle)[END_REF]). The word "police" was at the time highly polysemic in French. Indeed, the indeterminate nature of the term speaks volumes, insofar as police forces, under the Ancien Régime already, were associated to a civilizing mission with a particular focus on extending the reach of law. On its own, however, the "law and order" diptych cannot fully account for 19-20 th century colonial policing in the French empire -the period studied here.

The issue of policing organization and practices in the colonial context had to wait until the 19 th century to be raised in concrete fashion. In the early days, the task of pacifying newlyconquered territories was largely entrusted to the military. Only in areas specifically designed to receive European settlements was the creation of policing bodies seriously considered. A police commissioner (commissaire de police), for instance, was appointed in the immediate aftermath of the capitulation of the Dey of Algiers, on 5 July 1830. As evidenced by available documents, his meager staff spent many years focusing exclusively on managing and disciplining the "populace" and the "unwanted" freshly disembarked from Europe. Until the late 19th century, archives make scant mention of the "natives" ("indigènes"), who seldom appeared as a problem to be tackled by policing agents. The "gendarmerie of Africa", which was also created in the wake of the landing in Alger regency (June 1830) and was primarily conceived as a provost body for the supervision of locally-stationed troops, occasionally took part in combats for the extension of colonized territories. Only under the Third Republic was it endowed with law and order enforcement responsibilities with regard to indigenous populations not governed by military administration.

Whenever it was felt that the action of armed forces should be supplemented by the creation of police units, the dominant method consisted in importing or adapting models or institutions that were prevalent in metropolitan France, two of which -the gendarmerie and the Paris prefecture of police -were frequent sources of inspiration. These flows of policing practices and frameworks from the home country to the colonies should not, however, lead us to underestimate other legacies that also contributed to the shaping of colonial policing: the settlers themselves, but also privately-owned companies, the Army, indigenous troops, country policemen and rangers, colonial officials whose prerogatives could be extended with respect to the remoteness of their post (such as the gardes-cercle) have all been endowed with policing and penal capacities right until the decolonization era. Most importantly, in remote countryside areas, colonial authorities have endeavored to build upon local hierarchies and authorities, thus sparing themselves the task of having to establish specific policing bodies.

While state order reigned supreme in "islands of domination" (in cities and along railways mainly), it was replaced in many places (plantations, concessions) by private authorities or by traditional social control mechanisms, so that the policing and judicial systems of settlers were only called upon as a last resort option, or instrumentalized to pressure the promoters of local order.

Given this proliferation of policing agents and organizations, trying to distinguish between the "civilian" and the "military" makes little sense [START_REF] Blanchard | La professionnalisation policière en situation coloniale : détour conceptuel et explorations historiographiques[END_REF]). The ongoing concern was cost-effective, non-staff-intensive law and order maintenance. This perspective, however, left little room for insights going beyond the hackneyed remarks about natives being receptive only to the language of force. Hence, troubles were systematically interpreted as potential rebellions that might upset the entire edifice of colonial domination: despite many natives having been enrolled and various segments of the population having voluntarily rallied, while others had pursued arrangements and interests that they would be keen to stand for, colonial power remained exogenous. It was never able to take root in true cultural hegemony or in any political legitimacy reaching beyond the small circle of direct beneficiaries of this mode of government. A major indicator of the frailty of colonial power at the time is the way policing agents, administrative elites, and even the community of ordinary settlers tended to construe dreaded but simple attacks on private property as suggesting a general dissent against the balance of power-if not threats on their own existence.

Characteristics such as a lack of specialization of policing functions (which were closely linked to tax levying and statute labor) and a limited interest for crime repression, although not peculiar to the colonial setting, nevertheless do point to the principles of cheap government and under-management. The pattern has been well described by George Orwell with respect to Burma [START_REF] Orwell | The Complete Works of George Orwell[END_REF]), but it also applied to most territories comprising the French empire. In an even more fundamental way, this pertained to some of the inner contradictions of the political project regarding settlements: enticed by separatism and rejecting any increase of tax pressure-which they felt should exclusively burden indigenous populations, settlers tended to impede the development of a specialized, professional policing apparatus. Thus it is that in the years 1898-1900, their representatives in the départements of Algeria kept complaining of growing insecurity, while simultaneously pressuring the authorities to obtain the withdrawal of the gendarmerie, which they considered too costly and too formal in its enforcement style.

The only positive aspects in their eyes were the reinforcement of the action of armed forces and of emergency powers (in particular the 1881 "Code de l'indigénat", a set of specific criminal rules for natives, which in fact legalized long-standing practices). Such tensions with civil authorities and police organizations could be found in many other places until the decolonization era. A case in point is the Kingdom of Cambodia under French protectorate, where the colonial administration, far from deferring to financiers and settlers, actually tended to moderate their excesses, providing a legal and practical framework that prevented extreme prejudice toward the natives, which might otherwise have triggered an uprising [START_REF] Guérin | Paysans de la forêt à l'époque coloniale : la pacification des habitants des Hautes Terres du Cambodge[END_REF]). The buffering role played by police forces in such circumstances should, however, not be overstated. Colonial policing did embrace the whole array of founding violence (land expropriation, physical constraint, attacks against local cultural and religious practices…) traditionally associated to "despotic" colonial administrations that, as was the case, placed themselves under the auspices of the "sword" (Orwell 2009).

The judicial inflation that can be observed in a colonial situation -"the empire of law" described by Emmanuelle Saada [START_REF] Saada | The Empire of Law: Dignity, Prestige, and Domination in the Colonial Situation[END_REF]) -was accompanied by the deployment of "coercive networks" [START_REF] Sherman | Tensions of Colonial Punishment: Perspectives on Recent Developments in the Study of Coercitive Networks in Asia, Africa and the Carribean[END_REF]), of which police forces were a mere segment. This removes a lot of interest from any attempts at reporting on policing institutions in a colonial setting from a historical perspective, since the coercive practices experienced by the colonized could hardly be accounted for. In the case of the French empire, the task is even made almost impossible by the fact that no centralized agency was ever put in charge of designing and evaluating such policies -i.e. there was no equivalent of the British Colonial Police Service.

Nor was any formal, shared employment scheme ever deployed throughout the colonies.

While some officers did circulate across colonies, they made up a tiny minority of the numerous police deployed in the empire. Furthermore, ambitious and thorough historical research focusing on the local variations and characteristics of an empire encompassing a great many different institutional frameworks is made impossible by the sheer scarcity of available studies: no overview, no synthesis, no comparative research have been published so far, which is why the French case cannot be considered on par with the UK [START_REF] Sinclair | At the End of the Line. Colonial Policing and the Imperial Endgame, 1945-1980[END_REF] or the Netherlands models [START_REF] Bloembergen | The Perfect Policeman: Colonial Policing, Modernity, and Conscience on Sumatra's West Coast in the Early1930s[END_REF]. Indeed, the most recent synthetic works published on French police forces have almost nothing to say on the imperial dimension of policing, focusing instead on metropolitan France [START_REF] Anderson | In thrall to political change. Police and gendarmerie in France[END_REF], even though the latter can hardly said to have been estranged from the overseas at the time.

CURRENT RESEARCH

Recent or current research by young scholars, as well as the abundance of studies on the gendarmerie (Luc 2010), do however allow us to outline what the mechanisms of policing have been in certain territories. On a general level, Florence Bernault's insights on colonial prisons, which were often administered by policing bodies, are a good starting point: "the colonial prison did not apprehend delinquents according to a single judicial register (…) it sought to maintain the racial and juridical hierarchies upon which colonial rule was founded, thus reproducing a violent, personalized, and circumscribed power (…) As the vanguard of these confining tactics, colonial prisons thus illuminate how European regimes in Africa survived as enterprises of unending conquest" (Bernault 2003).

Florence Bernault rightly insists that the conquest endeavor was an ongoing one.

"Pacification" was never taken for granted, but remained a violent, uninterrupted process, which however varied in intensity according to place and time. But it is no less true that initial periods of military domination were generally followed by deliberate attempts at establishing civilian authorities, epitomized, among other symbols, by their police forces. In many territories gendarmes remained for a long time (sometimes as late as Word War II) the only actual, however furtive, colonial presence identified by the locals.

Using a-necessarily reductive-process approach, the timeline highlighted by John

McCracken regarding the British empire, and more precisely Malawi (McCracken 1992), can be adapted to the French empire: an "era of coercion" encompassing the first decades of domination, followed by an "era of control", then one of "collaboration" (the interwar period), and finally a "crisis of authority" after World War II. Clearly, this pattern is not set in stone, and periods may overlap. Most importantly, it must be inscribed in singular colonial trajectories, which depend heavily on when the conquest was made as well as on the frequency and punch of violent actions of resistance to colonial order.

The gendarmerie has been the main colonial police force: its military police dimension (both as a provost body and because of the military status of gendarmes) made it a privileged tool in the conquest phase and during the "era of coercion". For instance, on 16 May 1843, about 30 gendarmes were involved in the battle of Taguin, or more precisely in a raid against the smala of Abd El-Kader [START_REF] Jaulin | La gendarmerie dans la guerre d'Algérie[END_REF], which resulted in the taking of more than 3,000 prisoners and major spoils (a "feat" that still adorns the flag of the French national gendarmerie). In Madagascar the gendarmerie's contribution to pax gallica, as imposed by Gen. Gallieni, was rather short-lived (1901-1904, although a gendarmerie detachment was re-established in 1931). But in the whole empire, the colonial gendarmerie did take part in numerous battles, regardless of the precise dates of territorial annexations. In 1918, it supported the French Army after France was granted a mandate on the Levant from the League of Nations.

Incidentally, the Lebanon gendarmerie was on the front line -and targeted -during the 1925 Druze insurgency (Faisant de Champchesnel 2010). From this moment on and until the accession to independence of most French colonies, however, locals gendarmes tended to side with insurgents, and the resulting struggles, pitting gendarmes vs. gendarmes, highlighted the limits of the French policy of fueling ethnic and religious divides. Even where the colonization process was initiated in the diplomatic framework of trans-imperial competition instead of bringing about a war of conquest, the phrase "era of coercion" refers to those moments when French authorities were trying to tighten their grip. The kingdom of Cambodia, where a protectorate treaty had been signed between king Norodom and France in 1863, is a case in point: certainly, the 1885 uprising was stifled by military interventions, but it was used as an opportunity to reinforce the militiamen of the indigenous guard, who also acted as a police force. In particular, they were in charge of supervising tax collection and organizing statute labor days, up to and including the Highlands, an area where dissent toward central authority, whether Khmer of French, was at its highest [START_REF] Guérin | Paysans de la forêt à l'époque coloniale : la pacification des habitants des Hautes Terres du Cambodge[END_REF]).

The "era of control" should be understood both as: a) the moment when military head count decreases and powers are transferred to civilians (for instance in 1871 in main districts of Algeria) and b) the period when new policing methods are being developed to control people, following technological innovations from metropolitan France. In Madagascar at the turn of the century, General Gallieni extensively resorted to photography for the census and racial classification of the population. Generally speaking, although colonial police forces tended to be understaffed and underfunded, they were at the cutting-edge of anthropometric and identification techniques. These intellectual, technological and human investments were markedly associated to specific representations of crime as pertaining to entire populations, which in many areas of the empire translated into collective fines and punishments.

From a technical and organizational point of view, the protectorate of Tunisia was a precursor, equipped, as early as 1890, with a judicial identification service similar to that created only a few years before at the Prefecture of police. Indeed, in the early 1890s, Alphonse Bertillon himself played a key part in the establishment of the Algiers judicial identification service. Similar bureaus were subsequently opened in the AOF (French West Africa), the AEF (French Equatorial Africa), Madagascar and Indochina. But the most accomplished version of this anthropometric form of control -the best-researched at least, thanks to the works of Ilsen About (About 2011) -occurred in Saigon. The anthropometric identification bureau was institutionalized in 1897, as part of the immigration services.

Carding (forcing people to have an ID), fingerprints and anthropometry were made instrumental in a vast census and registration project encompassing all natives and foreign Asians. Even though opposition from multiple sources (congregations, consular representatives, part of indigenous elites…) eventually caused the shelving of this ambitious project, by 1917 the bureau had managed to gather some 700,000 records, complete with fingerprints. Indochina in this domain acted as a laboratory, ahead of the home country.

The emphasis on identification as a trait of colonial policing is further enhanced if one integrates metropolitan France into the global picture of the colonial empire [START_REF] Rosenberg | Policing Paris. The Origins of Modern Immigration Control Between the Wars[END_REF]). In Paris, after an identity card for foreigners had been established in 1917, French nationals of colonial origin were the next target [START_REF] Blanchard | La police parisienne et les Algériens[END_REF]. A North African brigade was thus created at the Paris Prefecture of police in 1925, with multiple prerogatives (administrative and social in particular) aiming to feed a database that would list all colonized people originating from North Africa (90% of whom were Algerian Muslims enjoying French citizenship, meaning that they needed no specific documents whatsoever to settle in metropolitan France). Yet these panoptic aspirations were never entirely fulfilled: given the rules governing the political representation of certain segments of the colonial world, from the 1930s onwards (especially after World War II), the colonized were sometimes able to oppose such designs, which in any case appeared to be limited by the sheer scope of their technical and financial requirements. Still, from the interwar period onwards, in metropolitan France as in the empire, political surveillance of the colonized became a policing priority. In 1922 in Dakar, the creation of a Central Service for the Policing and Safety of French Western Africa (Service central de police et de sûreté de l'AOF), whose structure was replicated in each of the component colonies (Service central de police et de sûreté du Sénégal, Service central de police et de sûreté du Dahomey…) epitomized this concern. The general objective of controlling mobility and preserving disciplined, segregated urban spaces for the benefit of European minorities came on top of a hidden agenda: thwarting the ability of the bettereducated colonized (the "evolved") to travel and act (Brunet-La Ruche 2012). These people were suspected -sometimes quite rightly so -of anti-colonialist or nationalist activism. As a matter of fact, during the interwar period, and especially in the AOF, this channeling of policing means and services into political surveillance contributed to relocating the fears and representations of insecurity from rural areas to the cities, where criminality was on the rise but did not seem to be eliciting any tailored policing response [START_REF] Fourchard | Lagos and the invention of juvenile delinquency in Nigeria, 1920-1960[END_REF]).

The weight of political defiance toward indigenous elites partly explains why the notion of an "era of collaboration" does not apply to the French empire quite as it does to the British empire. No movement of indigenization proper has ever taken place in the higher ranks of the policing hierarchy. To a certain extent, policing management in the French empire was more reminiscent of the Belgian approach than the British method. In Belgian Congo for instance, the Force publique created in 1885 never entrusted any natives with leadership positions [START_REF] Lauro | Maintenir l'ordre dans la colonie-modèle. Notes sur les désordres urbains et la police des frontières raciales au Congo belge (1918-1945)[END_REF]). The same applied to most of the French Empire: with the exception of a handful of commissioners, "Africanization", and more generally indigenous access to managing positions, was only engaged in the very last years of colonization. Moreover, such developments were seldom considered as preparing a transition toward independence. In some cases, the requirements (educational, legal, etc.) were raised to such a level and the will to confront political and violent protests was so strong that any increase of staffing levels was matched by a corresponding growth in the ratio of European-born recruits (such was the case for instance for the gendarmerie in Algeria). None of the French police bodies ever surmounted the quandary of importing mainland operational methods and gradually implementing a shared legal framework without making it increasingly difficult to gather intelligence from the locals [START_REF] Thomas | Empire of intelligence. Security Services and Colonial Disorder after[END_REF]). Indeed, the chasm from local circles was such that intelligence services often had to resort to broad, hackneyed ideological patterns (the Communist threat, pan Arabism, the influence of foreign nations…) which actually stood in the way of taking the full measure of protests or local separatist movements. In defense of these services, it should be said that nationalist, religious, and communist movements and motivations were so enmeshed -during rural uprisings in Vietnam (1930)(1931) for instancethat deciphering the redistribution of anti-colonialism was an extremely difficult task.

Anti-colonialism was in fact eventually recognized from an institutional perspective -the African Democratic Rally (RDA), founded in 1946, is a prime example -and was sometimes able to weigh in on policy and policing developments. Political surveillance, which had been considerably reinforced all along the 1940s, was thus gradually loosened as the recognition of African states made progress: in 1958, both the Intelligence and the Judicial Identity services of the AOF were even officially disbanded, but the gendarmerie managed to grab some of their files and prerogatives (Papa Dramé 2010). Most importantly, in the wake of World War II, recruitment processes became more bureaucratic. The written word as well as legal proceedings took greater importance, both in daily tasks and in terms of career development:

French-speaking, higher-educated colonized people massively took over the intermediate echelons of the hierarchy. Joël Glasman has convincingly shown how, in the case of Togo, such developments were both a) inscribed in how part of the population adjusted to the colonizers standards and b) loaded with contradictions and oppositions, not only within policing bodies but also among the local elites [START_REF] Glasman | Penser les intermédiaires coloniaux : notes sur les dossiers de carrière de la police du Togo[END_REF]. This standardization process, however, should not be overestimated: even in the départments of Algeria, where metropolitan rules and frameworks had supposedly been best integrated, the creation of the first constabulary academy was only decided in 1949, one year prior to that of the AOF in Dakar. Indeed, colonial police forces were never known to put a strong emphasis on disciplining and educating their personnel. Thus, restraining the use of force was never considered a priority, or even a reality. In the mid-1950s, Albert Memmi could write of the colonizer, with good reason, that "a machine-gun burst into a crowd of colonized causes him merely to shrug his shoulders" [START_REF] Memmi | The Colonizer and the Colonized[END_REF]. Even during allegedly quiet periods -in terms of violent unrest against colonial domination -, marked by a relative professionalization of policing by civil bodies instead of military ones, this reliance on shootings and killings was noticeable: in Tunisia in 1937-1938, both in the mining area of Gafsa (March 1937) and in Tunis (9 April 1938), union (CGT) and political (Néo-Destour) rallies and demonstrations triggered bloody repression. Massive use of gunfire by policemen, gendarmes, and colonial troops killed dozens.

During the so-called "crisis of authority" period, these shootings grew in both magnitude and frequency. In May 1945, for instance, in Sétif and Guelma (Algeria), vigilante European settlers sided with the police forces, who in turn had received support from the Army [START_REF] Peyroulou | Guelma 1945 : une subversion européenne dans l'Algérie coloniale[END_REF]. They coordinated not just to suppress Algerian unrests, but also to arrest, intern and quite often kill or dispose of anyone suspected of belonging to the separatist movement. Part of the casualties, which numbered in the thousands, were actually caused by shelling, both from sea and air. Ever since the interwar period, in the Maghreb and Middle East in particular, air policing had been used not only for land surveying, but also to subdue outbreaks by the colonized: firepower was supposed to make up for scant police and military staffing on the field [START_REF] Omissi | Air Power and Colonial Control: The Royal Air Force 1919-1939[END_REF]. Inspired by the Italian conquest of Libya (the 1912 bombings) as well as the action of the Royal Air Force in Afghanistan and Iraq in particular (1919)(1920), the French also resorted to this ploy. After a few experiments in Morroco (the Rif War), they made the most of air domination to suppress the Druze rebellion of 1925-1926.

The city of Damascus was massively shelled, and the residents of Druze and Muslim neighborhoods subjected to what can only be dubbed a terrorization policy. This particularly lethal use of air policing was designed to match the movements of Army troops and police forces, the gendarmerie in particular. The idea was to make up for their notorious and accepted understaffing.

Still, the "crisis of authority" period saw a rapid increase of police staffing: notwithstanding the decolonization wars (Indochina, Cameroon, Nigeria), the growing intensity and attendance of protest demonstrations, as well as the ever-increasing burden of the standards and regulations in metropolitan France left no other option. To develop but one example, the head count of the AOF gendarmerie (including Togo) rose from less than 300 in 1946 to almost 1,200 in 1957 (all of them "white", bar two). Auxiliary gendarmes (black Africans) numbered 700 in 1946 and more than 2,625 in 1957. In Togo alone, from the 1930s to the 1950s, police staffing levels doubled, but -more importantly maybe -remained quite low at less than 1,000 men at the end of the colonial era [START_REF] Blanchard | La professionnalisation policière en situation coloniale : détour conceptuel et explorations historiographiques[END_REF]. Hence, even though the relative weight of uniformed service personnel (vigilantes, guards, gendarmes, constables…) was particularly significant in the colonial State (from 1/3 to 1/2 of the entire personnel in Togo; a lower proportion whenever more Europeans were present), the overseas police-topopulation ratio remained lower than in the home country. While in the 1950s, France had on average one police per 250 citizen (a ratio that was only matched in the most watched cities in Africa, such as Abidjan), the figure could drop down to less than one per 1,500 in certain cities in AOF (such as Kayes in French Sudan). This lack of personnel is sometimes put forward as an explanation to the disproportionate use of force by policing authorities: gunfire in this case is interpreted as a means of compensating the powerlessness of outnumbered police doing their best to keep the crowd at a distance.

Much like its British counterpart (e.g. with the 1919 Amritsar massacre), the French empire was marked by the permanence of this logic consisting of slaughtering protesting crowds of colonized people, as evidenced by the now well-documented case of the December 1952 repression of the Casablanca riot. These casualties, however, which numbered in the hundreds each time, can hardly be attributed to any panic on the part of supposedly outnumbered, and therefore overwhelmed, personnel: not only was the police receiving support from colonial troops and vigilante settlers, but the same logic of massive, lethal gunfire against demonstrating colonized was also at work in metropolitan France at the same period [START_REF] Blanchard | La police parisienne et les Algériens[END_REF]. While the Paris police had stopped using firearms to disperse demonstrations (even unauthorized ones) as early as the 1930s, gatherings of colonized people were treated differently. On 14 July 1953, for instance, six Algerian demonstrators were killed by police fire on the place de la Nation. In 1961, dozens of Algerians succumbed to the blows and bullets of the Paris police, who on [START_REF]Profiling Minorities. A study of Stop-and-Search Practices in Paris[END_REF] October conducted what has got to be considered a pogrom [START_REF] House | Paris 1961: Algerians, State Terror, And Memory[END_REF]. More than the format of policing or the actual behavior of demonstrators (police response hardly varied, irrespective of the peacefulness of demonstrations), it is the racial-political condition of the colonized that determined both the nature and degree of the policing techniques employed.

The last years of the colonization era had very little to do with a transition period. What entire segments of the police forces offered instead was a violent toughening of their stance. In Algeria, the police and gendarmes were largely involved in the massive use of torture during the war of independence. The collusion between a majority of policemen and the most extreme supporters of French Algeria (the OAS, Secret Army Organization, in particular) both contributed to destabilizing legal authorities (in May 1958 and April 1961) and to protracting the fights beyond the cease-fire signed in March 1962. Aside from this well-known example, parts of Africa were also plagued by counterrevolutionary warfare, especially Cameroon, where the struggle against the partisans and maquis of the Union of the Peoples of Cameroon (UPC), sparked amidst extreme violence in the late 1940s, lasted a further decade after the independence had been proclaimed in 1960 [START_REF] Mbembe | La naissance du maquis dans le Sud-Cameroun, 1920-1960: Histoire des usages de la raison en colonie[END_REF].

POSTCOLONIAL LEGACIES

The "racial coloration" of the colonial situation pointed out by Georges Balandier i s undoubtedly one of the major characteristics of colonial policing [START_REF] Balandier | The Sociology of Black Africa: Social Dynamics in Central Africa[END_REF]. "Racial segregation" might actually be more to the point, insofar as local police were never able to impose the legitimacy of their prerogatives on European settlers: Amandine Lauro's remark about Belgian Congo -black police were not even allowed to pick up drunk white people in the gutter -also applied in the French part of that colony [START_REF] Lauro | Maintenir l'ordre dans la colonie-modèle. Notes sur les désordres urbains et la police des frontières raciales au Congo belge (1918-1945)[END_REF]. Conversely, and more tragically, Europeans never gave up their individual right to punish the colonized: whenever police action was considered not to be harsh enough, it would be complemented by punitive raids and the creation of vigilante militias. Beyond such paroxystic circumstances, urban police forces were the main tool for managing populations, and their primary task was to safeguard the feeling of close-knit community, the hygiene, and the tranquility of the "European city". They were, however, never granted the wish to restrict indigenous access to European residential spaces to the sole servants. It is true that -as opposed to what was the case in Ghana or British Nigeria -no official segregationist policy (creating separate neighborhoods for Europeans, migrants or minorities) was ever enforced in the cities of the French empire. Still, policing authorities -through vice squads in particular -did restrict and keep under surveillance the least movement across racial and colonial borders.

The question of prestige was so pressing for Europeans that many of them were actually not welcome in the colonies. Those who were allowed to settle there had to conform to a set of rules that worked as just as many markers of the superiority of French civilization. Close attention was paid to gender relations and roles: civil servants in particular were briefed on the subject, and deprecatory discourses on the sexuality or virility/feminity of the colonized abounded. Police forces, in their surveillance reports for instance, contributed to spreading these stereotypes. More importantly, they played a direct part in the regulation of the sex trade. Studies by Christelle Taraud on North Africa [START_REF] Taraud | La prostitution coloniale. Algérie, Tunisie[END_REF] and Isabelle Tracol-Huynh on Vietnam have shown that prostitution regulationism in the colonies cannot be measured by the yardstick of hygienism, the primacy of the patriarchal family model, and the extension of police prerogatives, as was the case in metropolitan France. There was of course no lack of such dimensions, but the primary issue was the drawing of clear gender and racial lines in the government of the people.

In Vietnam, where marriages between colonizers and colonized were more frequent than in North Africa, and mixed-blood children more numerous, indigenous concubines of "Europeans" were even considered prostitutes (Tracol-Huynh 2010). Prostitutes, in turn, were de facto organized into a regulatory and penal hierarchy based on racial criteria. Other trade categories, loosely defined but highly visible, such as dancers or singers, were also under tight surveillance, which varied, however, depending on the means available to the police, the proportion of settlers in the overall population, and how the majority population responded to such regulatory and policing intrusions. In the Shanghai French concession for instance, eradicating or even regulating the massive prostitution -which predated colonial domination -was never really considered [START_REF] Henriot | Prostitution and Sexuality in Shanghai. A social history 1849-1949[END_REF]. Still, the fact that healthcare protective measures, however scarce, only catered for European clients clearly indicates that, even in this highly specific colony, the seeds of a racialized population policy had been sown. This policy also reached metropolitan France, especially through the police or military institution consisting of organizing prostitution spaces (such as the military brothels a.k.a "bordels militaires de campagne") dedicated to immigrants or North African troops. This was one among many aspects of the distinctive brand of policing applied to the colonized, whose condition was deeply affected -especially during the Algerian war of independence -by recurrent identity checks, frequent roundups, and administrative detention. They were also targeted by positive policies such as welfare (access to healthcare, job search support…) and collective housing (accommodation facilities for migrant workers that were built starting in the 1930s, with a second wave launched in the late 1950s). While sometimes improving their living conditions and facilitating their understanding of mainland rules and bureaucracy, which differed significantly from what they had known overseas, these programs nevertheless followed a logic of control that did not vanish after the independences. To assess the weight of colonial legacies, one only has to look at the careers of former colonial administrators or officers, who started out managing agencies in charge of supervising the colonized (especially during the Algerian revolution) before heading immigrant support services. Such careers also epitomize the hybridization of policing repertoires, which became a blend of population control, political surveillance, and welfare.

Controversies on the colonial matrix governing the police control of immigrant or racialized populations did not, however, cease with the retirement of the last personnel to have started out at the time of the colonies or in the ranks of these particular agencies. Today, the colonial "difference principle" no longer is a founding principle of specific measures and programs. Sociological studies, however, unanimously demonstrate the existence of racialized gradations in terms of the ability to stand up for one's rights against policing authorities, police influence, and the use of force-especially lethal force. In France today, the bestknown aspects of the racialization of policing practices are police "blunders" (often a euphemism for "killings") and the repeated, often illegal identity checks performed on young, non-white males: depending on their clothing habits, black or Arab-looking young men are 2 to 15 times more likely to have their identity checked than other categories of the population (Jobard, Lévy, 2009). While certainly not all targeted persons can be linked to the former French colonial empire (or even its current remains, notably in the West Indies and Indian Ocean -the Mayotte and Réunion islands), it cannot be denied that the control of the colonized was one of the historical operators in the process of policing racialization. The weight of colonial policing's legacy has thus become a controversial issue, fiercely debated both in academic circles and in the media of contemporary France.

Colonial continuities are no monopoly of the former home country: many a policing institution has been passed on, almost "as is", to newly independent states. The few local police have often contributed to the training of younger generations, with the support of the French cooperation services (after the independences, as before, many police commissioners from West African states attended Saint-Cyr au Mont d'Or police academy, in the Lyon area).

The most striking and symbolic legacy has got to be the flock of national gendarmeries in former French colonies: 150 years after Napoleon's conquests, the decolonization era acted as a second age for the spreading of the gendarmic model. From Ivory Coast or Mali to Syria and Lebanon, through Tunisia or Algeria in particular, the forces that stemmed from the French legions of colonial gendarmerie have played their part in several of the episodes of repression and political transition that are currently at the heart of geopolitical news. Beyond