Stéphanie Aulong
email: aulong@ensam.inra.fr

Charles Figuières
email: figuiere@ensam.inra.fr.

Robert Lifran
email: lifran@ensam.inra.fr

Negotiation processes for the protection of biodiversity

Keywords: biodiversité, processus de négociation, mécanisme de vote, révélation des préférences biodiversity, negotiation processes, voting scheme, preferences revelation

On considère un pays en développement détenant le potentiel de conservation de la biodiversité et des pays développés qui bénéficient de cette biodiversité sans pouvoir la produire. Au statu quo, une augmentation de l'effort de conservation serait néfaste pour le pays en développement, alors que les pays développés et le monde dans son ensemble en profiteraient. Autrement dit, la conservation de la biodiversité est un bien public global. Le problème de négociation est le suivant : comment organiser des transferts compensatoires des pays développés vers le pays en développement afin de produire un niveau de biodiversité Pareto optimal, sachant que : 1) chacun des pays développés est susceptible de se comporter en free-rider et laisser les autres pays contribuer, 2) il n'existe pas d'autorité supranationale qui détient l'information nécessaire sur le consentement à payer des pays pour la biodiversité et qui a le pouvoir d'imposer une règle de transfert socialement efficace ? Ce papier explore comment et dans quelle mesure la théorie des processus d'allocation des ressources apporte des réponses à ce problème. Il est centré sur les propriétés incitatives des processus de négociation proposés et leur capacité à respecter la contrainte de souveraineté des Etats.

Introduction

Consider a developing country that has the potential for biodiversity conservation, and developed countries that beneÞt from biodiversity but are not in position to produce it. From the statu quo, some incremental protections of biodiversity would be harmful for the developing country but would ben-eÞt the developed countries and the world as a whole. This global public good dimension, it is argued, stems from "...moral, ethical, cultural, aesthetic and purely scientifc reasons" (Brundtland report, World commission on Environment and Development, 1987, p. 13); the adocates of this public good aspect also emphasize that the wider the biologic diversity, the richer the genetic information, which is useful for the production of chemicals, for biotechnolgies, etc. Such a situation calls for compensation transfers from the developed countries to the developing country to sustain a higher (Pareto optimal) level of biodiversity. The difficulties attached to such a solution are well understood: i) each developed country has an incentive to free-ride on transfers conceded by others, ii) no supranational authority exists that has both the necessary relevant information on countries's concerns about biodiversity, and the power to impose a socially beneÞcial proÞle of transfers. [START_REF] Barrett | The Biodiversity Supergame[END_REF] suggests that countries may Þnd their ways to a Pareto optimal outcome without the intervention of any public authority. His argument rests on the possibility for countries to devise threat strategies, whereby each country contributes a Pareto optimal level of transfers as long as no deviation from any other country is detected; such a deviation would trigger a punishment by a fall back to a non cooperative play, which is harmful for the deviator. Conditions are identiÞed for the above strategies to form a negotiation-proof Nash equilibrium. The possibility also exists of trigger equilibrium strategies that are Pareto improving over the static Nash equilibrium.

The present contribution to this debate departs from the above repeated game argument in two important ways. Firstly, we relax the assumption of fully-rational agents who are able to contemplate highly sophisticated strategies in order to maximize their discounted total payoff; instead we assume that the countries are myopic, in the sense that when choosing an action, they do so only on the basis of its immediate effect, one period ahead. Secondly, an institution (traditionally called a planning bureau in the related literature, and which we prefer to call an international agency) is called upon to organize the exchange of key pieces of information with the hope to implement Pareto improving changes as a function of the collected data. The question we ask is: could a Pareto optimal level of biodiversity emerge from a negotiation process organized by an international agency that is able to respect countries' sovereignty and to neutralize the countries' temptation to manipulate the information? Our approach belongs to the theory of resources allocation processes which, in its current state, can provide some answers to this question, though unresolved conceptual issues would deserve a particular attention, with a view to better contributing to the debate over the protection of biodiversity.

Put differently, our paper stands at the intersection of two strands of literature. It Þrst belongs to the economic literature on Biodiversity, offering theoretical answers for issues in relation with the organization of Þnancial transfers from developed to developing countries. It also, modestly, contributes to the dynamic theory of resources allocation, with the introduction of a majority voting scheme into a discrete-time process. The resulting process builds on an earlier one, proposed by [START_REF] Champsaur | Stability theorems with economic applications[END_REF], and further studied by [START_REF] Schoumaker | Strategic behaviour in a discrete-time procedure[END_REF], who focused on the agents' incentives to manipulate the exchange of information with the institution. Actually, we end up with an explicit formulation of [START_REF] Bowen | The interpretation of voting in the allocation of economics resources[END_REF]'s procedure of iterative voting for increments of a public good. It is worthwhile noting that the proposed institutional framework looks like actual institutions such as the Global Environmental Facility (GEF) created in 1991. The GEF is an independent Þnancial organization that provides grants to developing countries for projects that "beneÞt the global environment and promote sustainable livelihoods in local communities". GEF's projects address six global environmental issues, one of which is biodiversity. The major Þnancial tool of the Convention on Biological Diversity is the GEF fund. GEF's contributors are voluntary donors, most of whom are States. It funds the "incremental" or additional costs associated with transforming a project with national beneÞts into one with global environmental beneÞts.

The discussion is structured as follows. The following section formalizes the particular aspect of the biodiversity issue we want to investigate, and illustrates the free-rider problem. Section 3 explains what answers can be provided by the dynamic theory of resources allocation. The proposed procedure can be given the interpretation of a negotiation process. Section 3 also describes some shortcomings of this approach. Section 4 explores how to get rid of them; to do so we qualify the process by introducing a voting scheme. The last section concludes.

Modelling the financing of biodiversity as a public good

The problem is modelled along the lines of [START_REF] Barrett | The Biodiversity Supergame[END_REF]. Consider one developing country (country 0) that has the potential for biodiversity conservation, and n developed countries (labeled country 1, 2, ..., n) that beneÞt from biodiversity but are not in position to produce it, generally for geographical and climatic reasons. Let S * stands for the units of biodiversity that the developing country is willing to protect unilaterally 1 . By deÞnition of S * , any incremental protection of biodiversity, S, in an admissible interval2 ¤ 0, S ¤ , is detrimental for country 0, and this will be interpreted as the developing country's cost of conserving biodiversity, which we shall assume to be a strictly increasing and convex function: C(S), with C 0 (S) > 0, C 00 (S) ≥ 0, S ∈ ¤ 0, S ¤ .

For a developed country i, the beneÞts of conserving additional units of global biodiversity above S * are given by:

B i (S ), i = 1, ..., n, (1)
a concave function, increasing for all S ∈ ¤ 0, S ¤ . Let M i denotes a money transfer from developed country i to country 0. This transfer is paid out of country i's exogenous income w i . If y i = w i -M i stands for the income left for any other use, the net beneÞts of developed country i read as:

NB i (S, y i ) = B i (S ) + y i . (2)
The aggregate payment made by the developed countries is M = P M i , a Global Biodiversity Conservation Fund in the words of [START_REF] Barrett | The Biodiversity Supergame[END_REF]. This fund is meant to cover country 0's cost, therefore M = C(S); since there is an upper bound on S, there is an upper bound on the amount of money that can be collected for the protection of biodiversity, i.e. M = C ¡ S ¢ . To simplify, it will be assumed that countries' endowments w i are such that P w i ≤ M . Also, for a given M, and since the cost function is invertible, we can express:

S = C -1 (M ), (3)
which is the affordable level of biodiversity conservation when the payments sum-up to M . Country i's net beneÞts can be restated as:

NB i (M i , M -i ) = B i (C -1 (M i + M -i ) ) + w i -M i , (4)
= Φ i (M i + M -i ) + w i -M i (5)
where

M -i = P j6 =i M j , and
Φ i = B i oC -1 .
With the assumptions made on B i and C, the function Φ i is increasing and concave over the admissible range for M. We shall also make the following assumption to ensure the existence of interior Pareto outcomes and Nash equilibria:

lim M →0 B i0 C -10 (M) > 1 and lim M → ! w i B i0 C -10 (M ) < 1.
This is a classical model of voluntary contributions to a public good, M, where the marginal cost to produce this public good is 1.

Contributions are Pareto optimal, or efficient, when they satisÞed the Bowen-Lindhal-Samuelson conditions; here those conditions require that the sum of countries marginal beneÞts from the Conservation Fund is equal to its marginal cost , i.e.

n X i=1 Φ i ´= 1 . (6)
On the other hand, a voluntary (Nash) contributions equilibrium is characterized by the following marginal conditions:

Φ i ´= 1 , i = 1, ..., n. (7)
From expressions (11) and ( 7), one can deduce that voluntary contributions are not efficient.

The following example allows further precision.

Example 1 [START_REF] Barrett | The Biodiversity Supergame[END_REF] dealt only with the following example. Let the associated cost for country 0 be C(S) = cS 2 /2, c > 0.

For a developed country i, the beneÞts of conserving additional units of global biodiversity are given by:

B i (S) = b n µ a i S - S 2 2 ¶ , b, a i > 0, i = 1, ..., n, (8)
a concave function, increasing for all S ∈ [0, a i ] . It should be noted that country i 0 s beneÞts depend on the number of participants, n.

Therefore the net beneÞts of developed country i read as:

NB i (S, y i ) = b n µ a i S - S 2 2 ¶ + y i . (9)
The affordable increment of biodiversity, given the overall payments received by the developing country M = P M i , is:

S = r 2M c . (10)
Plugging ( 10) into ( 9), country i's net beneÞts are:

b n

" a i µ 2 (M i + M -i ) c ¶ 1/2 - M i + M -i c # + w i -M i .
The Bowen-Lindhal-Samuelson conditions are then

n X i=1 b n " a i c µ 2 (M i + M -i ) c ¶ -1/2 - 1 c # = 1 . (11)
And the voluntary (Nash) contributions equilibrium solves:

b n

" a i c µ 2 (M i + M -i ) c ¶ -1/2 - 1 c # = 1. ( 12
)
From expressions (11) and ( 12), voluntary contributions are clearly not efficient. Often, the voluntary contributions equilibrium falls short of the social optimum. This last property holds and is particularly easy to check in the symmetric case, where countries are identical so that a i = a , ∀i; then (11) becomes:

b

" a c µ 2 (M ) c ¶ -1/2 - 1 c # = 1 ,
and gives the efficient levels of transfers and units of biodiversity:

c M = a 2 b 2 c 2(b + c) 2 , b S = ab (b + c) .
Note that there exists an inÞnite number of combinations for national contributions that are consistent with the above levels, i.e. M = P M i is unique but the M i s are not. In other words, there is an inÞnity of Pareto optima. Also the Þrst order conditions for a voluntary Nash equilibrium now read as:

b n " a c µ 2M c ¶ -1/2 - 1 c # = 1 ,
which yields the following transfers and units of biodiversity:

M = a 2 b 2 c 2(b + nc) 2 , S = ab (b + nc) .
When there are at least two developed countries, from the above expressions one readily deduces the following intuitive inequalities:

M < c M , S < b S .
3 Negotiation processes to move to a social optimum

The question we want to challenge now is how to induce the countries to move from the inefficient voluntary contributions equilibrium, or any other inefficient starting point, to an efficient outcome, given a number of constraints, mainly of informational nature? This is a classical implementation problem. The literature on mechanism design has suggested a number of mechanisms that somehow solve the problem; one such mechanism based on a process approach has attracted our attention, for it can be given a negotiation interpretation 3 . It is also appealing on several other counts, which we shall develop below.

The approach rests on a sequence of small size changes. Changes are decided by a international agency, conditionally on the result of an exchange of information with the developed countries: a biodiversity increment is proposed by the international agency and countries are asked whether (and eventually how much) they value this change. After collection of those local data, the proposed change is, or is not, implemented by the international agency, according to a rule which is common knowledge. Desired properties of the sequence of changes are: i) their repetition drives the economy from any arbitrary starting point (for instance the voluntary contributions equilibrium) to a Pareto efficient state that is universally preferred to the starting point, ii) the international agency need not know the national interests in Biodiversity conservation that are parameters of asymmetric information (in Example 1, those parameters are the a i s, and we shall assume that the parameter c is public knowledge 4 ).

In a nutshell, this sequential approach is made of repeated cost/beneÞt analyses with, presumably, seductive incentive properties. It has also strong bearings with the contingent valuation method, which aims to measure the social desirability of a given environmental change, by directly asking people what they are willing to pay (or receive) for the change.

The sequential approach could be contrasted with one-shot approaches characterized by a direct jump from the prevailing situation to the target. The former has a more realistic ßavor, but it rests on the implicit assumption that some unspeciÞed constraints hinder a direct jump. As an effort to justify those constraints, it has been argued that one-shot changes require that agents communicate to the international agency their entire utility functions, that is, larger-size messages than those required to convey local data; if the agents' per-period budget constraints cannot cover the associated costs to collect and transmit the data, this can be seen as an economic justiÞcation for small-size messages. In our biodiversity context, it can also be advocated that, since lands can be switched from agricultural use to biodiversity protection only gradually, such a gradual pattern ought to be taken as an exogenous constraint in our problem.

The procedure given correct reports of private pieces of information

In order to facilitate the explanation of the procedure, this section deals with honest countries, i.e. countries that truthfully report their private information when they are asked to. But we shall investigate the case where countries can strategically report in the next subsections, and by contrast, emphasize the issue of manipulation of information.

Consider an arbitrary state of affairs (S, y 1 , ..., y n ) , and suppose the international agency is contemplating the possibility to change the level of biodiversity fund S by an amount s. The international agency asks each country two pieces of information:

• what is the maximum amount of money country i is willing to give up to enjoy a level of biodiversity S + s? If m + i denotes the amount that will leave country i's net beneÞts unchanged, upon truthful report it is by deÞnition a solution to the equation:

NB i (S, y i ) = NB i ¡ S + s, y i -m + i ¢ Example 2 (Example 1 continued) Using Barrett's framework, the above equation gives b n • a i S - S 2 2 ¸+ y i = b n • a i (S + s) - (S + s)¨2 2 ¸+ y i -m + i , or m + i = b n • a i - 2S + s 2 ¸s .
• what is the minimum amount of money required by country i to compensate a reduction of biodiversity to the level Ss? Honest countries would report a number m - i that solves

NB i (S, y i ) = NB i ¡ S -s, y i + m - i ¢ Example 3 (Example 1 continued) b n • a i S - S 2 2 ¸+ y i = b n • a i (S -s) - (S -s)¨2 2 ¸+ y i + m - i , or m - i = b n • -a i + 2S -s 2 ¸s .
Then the international agency computes:

• γ + , the cost to increase the biodiversity protection up to the level S +s, that is C (S + s) -C(S),

• and γ -, the money saved by reducing the level of biodiversity down to Ss, formally C(S) -C (Ss) .

There are then three mutually exclusive possibilities5 :

1. P m + i -γ + > 0, 2. P m - i -γ -< 0, 3. P m + i -γ + ≤ 0 and P m - i -γ -≥ 0.
If situation 1 occurs, the international agency increases the level of biodiversity to S +s and each country is asked to pay m + i to cover the cost. Besides it receives a share δ i ≥ 0 ¡ with P δ i = 1 ¢ of the social surplus P m + iγ + . Therefore its net beneÞts in the new situation are:

NB i ¡ S + s, y i -m + i ¢ + δ i ³ X m + i -γ + ´= NB i (S, y i ) + δ i ³ X m + i -γ +
The right hand side of this equality follows from the deÞnition of m + i , and clearly shows that no country experiences a decrease in utility after the change.

If situation 2 occurs, and provided that s ≤ S, the international agency decreases the level of biodiversity to Ss. If s > S then S is cut down to zero. Each country is paid m - iδ i ¡P m - iγ -¢ . Country i's net beneÞts in the resulting situation are:

NB i ¡ S -s, y i + m - i ¢ -δ i ³ X m - i -γ -´= NB i (S, y i ) -δ i ³ X m - i -γ -
Ágain the deÞnition of m - i is used to obtain this equality, which shows that every country gains from the change.

In any other situation, either the procedure stops, or the level of biodiversity is not changed but the increment s is divided by two.

More compactly, and expliciting the temporal dimension of the variables, the dynamics describing the process are:

s t+1 =    s t if either P i m + i,t -γ + t > 0 , (i) or P i m - i,t -γ - t < 0, (ii) st 2
otherwise.

S t+1 =    S t + s t if (i) holds, max {0, S t -s t } if (ii) holds, S t
otherwise.

y i t+1 =    y i t -m + i,t + δ i ¡P i m + i,t -γ + t ¢ if (i) holds , y i t + m - i,t -δ i ¡P i m - i,t -γ - t ¢ if (ii) holds , y i t otherwise.
with δ i ≥ 0, P i δ i = 1. It can be shown that this procedure leads to a Pareto optimum (Champsaur, Drèze and Henry (1977)), and countries beneÞts are non decreasing along the procedure, as explained above. At a steady state s t = 0, S t+1 = S t ; also lim t→+∞ y t+1y t = -m i + δ i ( P i m iγ) = 0, which means that all countries end-up contributing the same effort for biodiversity conservation in the long run, whatever their differences in income w i , their interest in the environment a i , and their share of the surplus δ i .

Incentive properties of the procedure

How are we to think that countries will report their private information truthfully? And what if they typically do not so? This issue is viewed as a non-cooperative game with the reports φ + i,t and φ - i,t as countries' strategies and the beneÞts from the resulting allocation as payoffs. Here the assumption is made that countries are myopic, for they ignore all the possible effects of their reports more than one period ahead. Will the reports be equal to the true local data m + i,t and m - i,t ? In the continuous-time version of the procedure, the question has been investigated by [START_REF] Roberts | Incentives in Planning Procedures for the Provision of Public Goods[END_REF], who shown that, at the unique Nash equilibrium, agents misreport their true data, yet the convergence property to a Pareto optimal allocation does not collapse. Schoumaker (1977), in the discrete-time version, also obtained the property that φ + i,t ≤ m + i,t and φ - i,t ≥ m - i,t , a similar result with one difference though.

The Nash equilibrium of the message game is not unique; rather there is a continuum of Nash equilibria. Although the other nice properties of the continuous-time process are maintained in the discrete-time version, this multiplicity of equilibria raises an important coordination problem.

A majority voting procedure

In this Subsection we shall introduce a majority voting scheme in the procedure. There are three reasons to try such a change. Firstly, we shall see that the voting scheme is a way out to the coordination problem alluded to above. A similar majority voting scheme has been analyzed in the continuous-time process by [START_REF] Green | Dynamic approaches[END_REF], who showed that there is still convergence to a Pareto optimum, but the individual rationality property is lost. Our challenge is Þrst to offer an adequate discrete-time version of majority voting in such processes, and to ascertain the robustness (or the lack) of the properties of convergence and individual rationality. The discussion in the previous subsection makes clear that one cannot infer those properties hold in the discrete-time formulation on the mere basis that they hold in the continuous-time formulation.

Secondly, with this change we end-up with an explicit formalization of the procedure described by [START_REF] Bowen | The interpretation of voting in the allocation of economics resources[END_REF] in the section "Individual voting on increments to existing outputs" p. 40-42; given the importance of this paper in the public economic literature, there is an intrisic interest in this formalization exercise, for it opens the possiblity to clarify the assumptions on the agents' preferences and the kind of individual behaviors underwhich Bowen's procedure sucessfully deÞnes a trajectory converging to a Pareto optimum.

Lastly, it can be argued that actual practices are closer to voting procedures than to processes asking people what they are willing to pay / receive for a change. From this point of view, this section can be seen as an attempt to assess what properties can be expected from real procedures. Refereeing back to the contingent valuation method, it is worth noting that asking for people's votes instead of willingnesses to pay is in the spirit of one of the guidelines prescribed by the famous NOOA panel (see [START_REF] Portney | The contingency valuation debate: why economists should care?[END_REF] for more details) 6Assume there is an odd number of developed countries n > 2. Starting from a situation where there is no incremental protection of biodiversity, within each time period, developed countries are offered the possibility to vote for increasing the biodiversity level of an exogenous positive increment s t , against the statu quo; they are also offered the possibility to vote for decreasing the biodiversity level of the same increment s t , against the statu quo7 .

Let v i,t denote country i's Þrst vote, which can take two values:

v i,t = ½ 1 to vote for an increase S t+1 = S t + s t , 0 to vote for S t+1 = S t . (13)
Similarly, w i,t denotes country i's second vote, which can also take two values:

w i,t = ½ 1 to vote for a decrease S t+1 = S t -s t , 0 to vote for S t+1 = S t . (14)
At each point of time, the result of the Þrst vote is not made public before the second vote has taken place. This trick is used to prevent any strategic manipulation of the second votes by the Þrst votes.

The new procedure also speciÞes that the cost of an eventual increment of biodiversity is shared equally among the countries, that is each country bears γ + t n . Identically the gain of cost to decrease the level of biodiversity is shared equally, so each country would beneÞt from γ - t n . When P i v i,t > (n -1)/2 the majority calls for an increase of S. When P i w i,t > (n -1)/2 the majority prefers to decrease S. Otherwise the level of biodiversity remains the same, but the increment is halved.

A v i,t -vote is truthful if

v i,t = ( 1 when m + i,t > γ + t n , 0 otherwise . ( 15
)
and a w i,t -vote is truthful if

w i,t = ( 1 when m - i,t < γ - t n , 0 otherwise . ( 16
)
following question: is the contingent valuation method capable of providing estimates of existence values that are reliable enough to be used in natural resource damage assessments?

Assumption 4 When a country i gets the same utility by voting v i,t = 1 or

v i,t = 0, if m + i,t > γ + t /n it chooses v i,t = 1. If m + i,t ≤ γ + t /n, it chooses v i,t = 0.
Assumption 5 When a country i gets the same utility by voting w i,t = 1

or w i,t = 0, if m - i,t > γ - t /n it chooses v i,t = 0. If m - i,t ≤ γ + t /n, it chooses w i,t = 1.
Those two behavioral assumptions can be given the interpretation of some form of strategic-risk aversion. They both say that, when the strategic uncertainty can entail a reduction of utility, countries prefer to vote for the statu quo.

Proposition 6 Under Assumptions 4 and 5, truthful voting is a dominant strategy for every country.

Proof. Consider the situation from the point of view of country i when m + i,t > γ + t /n. Given the votes of the other countries, if i,tγ + t /n < 0, whereas v i,t = 0 produces no loss; Þnally if P j6 =i v j,t ≤ (n -1)/2 and P h v h,t ≤ (n -1)/2 then v i,t = 1 and v i,t = 0 both produce no loss. This time, whatever the v-votes of the other countries, the best plan for country i is to vote v i,t = 0.

P j6 =i v j,t > (n -1)/2, whatever v i,t the net increase of utility is m + i,t -γ + t /n > 0; if P j6 =i v j,t ≤ (n - 1)
The proof for w-votes follows the same logic.

Assumption 7 At each point of time, the distribution of the (true) willingness to pay m + i,t , and the distribution of (true) compensations to accept m - i,t

are such that the average equals the median.

In Example 1, this assumption takes a simple form: it is satisÞed when the distribution of the parameters a i s is such that the average equals the mean. From Assumption 7 it follows that, at each instant, the distribution of the willingness to pay for increasing biodiversity net of the imputed cost, m + i,t -(1/n)γ + t , is such that the median equals the mean. This property also holds, at each instant, for the distribution of the compensations for biodiversity reduction net of the imputed gain, m - i,t -(1/n)γ - t . Since reporting the truth is a dominant strategy for all countries, one has Proposition 8 At a Nash equilibrium of the voting game, under Assumption 7

X i v i,t > n -1 2 ⇔ X i m + i,t > γ + t ,
and

X i w i,t > n -1 2 ⇔ X i m - i,t < γ - t .
As a corollary of the above proposition, it is important to realize that, at each moment, there cannot be a majority for both an increase and a decrease. Indeed, remember that only one of the following three situations is possible:

1. P m + i,t -γ + t > 0, 2. P m - i,t -γ - t < 0, 3. P m + i,t -γ + t ≤ 0 and P m - i,t -γ - t ≥ 0.
What can be the effect of the voting scheme in each situation? If situation 1 prevails, from Proposition 8 the outcome of the v-vote is a majority for the increase; and the w-vote will result in the statu quo. Symmetrically, in situation 2 the v-vote outcome is the statu quo, whereas the w-vote outcome is an decrease of S. Finally in situation 3, both votes produce the statu quo.

After collection of v and w-votes, the procedure increases (decreases) the biodiversity level by s t if there is a majority which favors either option; otherwise the step size is halved. Symbolically, the dynamics are:

s t+1 =    s t if either P i m + i,t -γ + t > 0 , (i) or P i m - i,t -γ - t < 0, (ii) st 2 otherwise. S t+1 =    S t + s t if (i) holds, max {0, S t -s t } if (ii) holds, S t
otherwise.

y i t+1 =    y i t -γ + t /n if (i) holds , y i t + γ - t /n if (ii) holds , y i t otherwise.
Starting from an arbitrary state (s 0 , S 0 , y i 0 ), the above system deÞnes a unique trajectory for the vector (s t , S t , y i t ) .

Proposition 9 Under Assumptions 4, 5 and 7, any trajectory solution to the process with the majority voting scheme asymptotically converges to a biodiversity steady state which is Pareto optimal.

Proof. The property that the steady state of the process is Pareto optimal is obvious. As for the stability property, to establish the proof it is sufficient to Þnd a Lyapounov function (see LaSalle (1986)). Let us denote W (S, y 1 , ..., y n ) = P i NB i (S, y i ) the utilitarian measure of welfare. Clearly the function f (S, y 1 , ..., y n ) = W ( Ŝ, ŷ1 , ...., ŷn )-W (S, y 1 , ..., y n ), where ( Ŝ, ŷ1 , ..., ŷn ) is the utilitarian Pareto optimum, qualiÞes as a Lyapounov function. Indeed:

1. it is a continuous function with f ( Ŝ, ŷ1 , ..., ŷn ) = 0, 2. along any outcome of the process, f (S, y 1 , ..., y n ) ≥ 0, 3. it is strictly decreasing along the process, at any point but the steady state.

This second process has two comparative advantages over the previous one. First, truthful behavior is a dominant strategy, which is the strongest incentive property one can obtain from a mechanism. Since agents are free from any strategic uncertainty when they devise their optimal behavior, the procedure could be also applied to a situation where countries do not know each other preferences. The second advantage is that there is no coordination problem any more. But those strengths come at a cost. First Assumption 7 is very demanding. If it does not hold, there is no guaranty that the process will converge any more: it may stop below the Pareto optimal level of biodiversity (a second best result). Furthermore, even when all the required assumptions hold, there is no guaranty that every country will see its utility increased along and at the steady state of the procedure. To see why in some cases, the voting scheme could lead to changes which are harmful for some countries, note that at each step where the increment is not halved, the process creates a surplus ( P i m + i,tγ + t or γ - t -

P i m - i,t
) that is not redistributed, contrary to the process outlined in Section 3.1. As a result a voted increase is harmful for countries such that m + i,t -¡ 1 n ¢ γ + t < 0, and a voted decrease is harmful for countries characterized by m - i,t -

¡ 1 n ¢ γ - t > 0.
In practice, to make assumption 7 less stringent, the sharing rule could be modiÞed to take into account of further information on countries, when available, for instance a measure of their wealth like GDP, with the hope to increase the probability to make the average and median parameters coincide. And, in order to reduce the probability of hurting some countries along the way, the process could require a stronger majority to enact a change, such that a 66.66 percent rule instead of 50 percent rule.The higher the threshold, the lower the probability to hurt some countries, but the higher the difficulty to implement socially desirable changes, therefore the slower the convergence to a Pareto optimal state.

Conclusion

How does the conclusions drawn from the two proposed processes compare with the conclusions from Barrett's repeated game argument? Barrett's main result is that when the gain from cooperation is large enough, cooperation cannot be sustained as a non cooperative outcome. Here on the contrary, such large gains can be implemented via both processes. The main weakness of the Þrst process is the potential coordination problem raised by the multiplicity of Nash equilibria that support a given Pareto optimum. The assumption of myopic behaviors might also be viewed as another weakness (of both processes), though some may argue that the assumption of super rational countries is an even worse alternative way to conceptualize behaviors. The coordination problem can be overcome when the process is qualiÞed with a majority voting scheme (Section 3.3). Besides the resulting equilibrium is made of dominant strategies. Unfortunately, it has been shown that at the same time the property of individual rationality is lost, i.e. the second process may be harmful for some countries. Besides, only under a traditional (and demanding) assumption of equality between average and median values of the marginal net beneÞts, does the second process converge to a Pareto optimum. In pratice, the second device can be qualiÞed to improve its welfare properties, but cannot completely evacuate the veto of some countries. A challenging further research would be to Þnd a way to discard the coordination issue of the Þrst process, while preserving its other nice properties. This paper is a Þrst step in that direction.

  /2 and P h v h,t > (n-1)/2 then v i,t = 1 produces a net increase of utility of m + i,t -γ + t /n > 0, and v i,t = 0 produces no gain; Þnally if P j6 =i v j,t ≤ (n-1)/2 and P h v h,t ≤ (n -1)/2 then v i,t = 1 and v i,t = 0 both produce no gain. Therefore, whatever the v-votes of the other countries, under Assumption 4 the best plan for country i is to vote v i,t = 1.

	If on the contrary m + i,t ≤ γ + t /n, when v i,t the net loss of utility is m + i,t -γ + t /n ≤ 0; if P j6 =i v j,t > (n -1)/2, whatever P j6 =i v j,t ≤ (n -1)/2 and P h v h,t > (n -1)/2, then v i,t = 1 produces a net loss of utility of m +

Units of biodiversity are treated here as positive real numbers. As stressed by Barret, measuring the biodiversity is by no means a simple problem. The analysis of this paper makes sense only to the extent that this problem has been sorted out.

The upper bound S is exogenously given. If S is to be interpreted as the number of acres devoted to wilderness, clearly this number cannot go to inÞnitum.

For the negotiation interpretation, see[START_REF] Tulkens | Dynamic processes for public goods. An institutionoriented survey[END_REF]' survey, in particular Section

4, and Chander and Tulkens (1992).

This assumption could easily be relaxed by treating the developing country as an additionnal agent involved in the negotiation process. It would introduce no new conceptual difficulties.

Explanations are given in Champsaur, Drèze and Henry (1977) on page 285; see also[START_REF] Schoumaker | Strategic behaviour in a discrete-time procedure[END_REF], page 368.

The panel, chaired by K. Arrow and R. Solow, was asked to provide advice to the

Note that the procedure is very similar to that of the referendum in the case of an indivisible project.


Sylvie Benzoni-Gavage

Local well-posedness of nonlocal Burgers equations

Keywords: Nonlinear surface wave, amplitude equation, smooth solutions, blow-up criterion. 2000 Mathematics Subject Classification: 34K07, 35L60

This paper is concerned with nonlocal generalizations of the inviscid Burgers equation arising as amplitude equations for weakly nonlinear surface waves. Under homogeneity and stability assumptions on the involved kernel it is shown that the Cauchy problem is locally well-posed in H 2 (R), and a blow-up criterion is derived. The proof is based on a priori estimates without loss of derivatives, and on a regularization of both the equation and the initial data.

Introduction

We consider a nonlocal generalization of the inviscid Burgers equation (1.1)

∂ t u + ∂ x Q[u] = 0 ,
where Q is a quadratic nonlocal operator given in Fourier variables by

(1.2) FQ[u](k) = Λ(k -, ) u(k -) u( )d ,
when u is Schwartz. (Throughout the paper F denotes the Fourier transform in one space dimension.) Equations of this type arise in particular as amplitude equations for weakly nonlinear waves [START_REF] Hunter | Nonlinear surface waves[END_REF][START_REF] Alì | Hamiltonian equations for scale-invariant waves[END_REF][START_REF] Benzoni-Gavage | Weakly nonlinear surface waves and subsonic phase boundaries[END_REF]. The kernel Λ is piecewise smooth and satisfies the following conditions (When Λ ≡ 1/2, (1.1) is nothing but the classical inviscid Burgers equation.) The symmetry condition is not actually a restriction: any kernel Λ defining an operator Q as in (1.2) can be changed into a symmetric one, by change of variables → k -in the integral. The reality condition ensures that Q[u] is real valued if u is so (which is equivalent to u(-k) = u(k)). When (1.1) is an amplitude equation, the homogeneity condition is linked to the scale invariance of surface waves. Homogeneity of degree zero ensures in particular that Λ is bounded, since it is piecewise smooth, and that its singularities occur along rays. We shall assume these singularities are located exclusively on the axes k = 0, = 0 and on {(k, ) ; k + = 0}, which is clearly compatible with all other assumptions. The one that we call structure condition may look mysterious at first glance. The weaker condition (v) was pointed out by Hunter [START_REF] Hunter | Nonlinear surface waves[END_REF] as formally ensuring the linearized stability of constant states. Alì, Hunter and Parker [START_REF] Alì | Hamiltonian equations for scale-invariant waves[END_REF] have observed that (iv) yields a Hamiltonian structure for (1.1). More precisely, if (iv) holds true, (1.1) equivalently reads

(1.3) ∂ t u + ∂ x δH[u] = 0 ,
where the Hamiltonian is defined by

(1.4) FH[u](k) = 1 3 Λ(k, ξ) u(k) u(ξ) u(-k -ξ) dk dξ
if u is Schwartz. In this framework -that is, assuming (iv) -Hunter has shown the local existence of periodic solutions of (1.3) [START_REF] Hunter | Short-time existence for scale-invariant Hamiltonian waves[END_REF]. The purpose of this paper is to show the local existence of smooth solutions to (1.1) on the real line, and more precisely the local well-posedness of (1.1) in H 2 (R), assuming the stability condition (v) but not (iv). Indeed, (v) turns out to be sufficient to get a priori estimates for (1.1) without loss of derivatives. To prove well-posedness we shall then use a regularization method as proposed by Taylor [7, p.360].

A priori estimates

Let us rewrite (1.1) as

(2.5) ∂ t u + 2B[u, ∂ x u] = 0 , where (2.6) FB[u, w](k) = Λ(k -, ) w(k -) u( )d .
The operator B is well-defined and bilinear on S (R)×S (R), and it is clearly symmetric because of the symmetry property of Λ. Furthermore, by straightforward inspection, (2.7)

∂ x B[u, w] = B[∂ x u, w] + B[u, ∂ x w] ,
so that, because also of the symmetry of B, (1.1) and (2.5) are indeed equivalent for smooth solutions.

In addition, the operator B extends to a continuous operator from H 1 (R) × L 2 (R) to L 2 (R). Indeed, by its definition (2.6), Plancherel's theorem and L 1 -L 2 convolution estimates we have

(2.8) B(u, w) L 2 = FB(u, w) L 2 ≤ Λ L ∞ u L 1 w L 2 Λ L ∞ u H 1 w L 2 .
Here above and throughout the paper, the symbol means 'less or equal to a harmless constant times'. In a similar but more symmetric way -distributing derivatives equally on u and w -, we also find that B is a continuous bilinear operator in H 1 (R) and H 2 (R), with

(2.9) B(u, w) H 1 Λ L ∞ u H 1 w H 1 , (2.10) B(u, w) H 2 Λ L ∞ ( u H 1 w H 2 + u H 2 w H 1 ) .
Now we consider, for a given (smooth enough) u, the linear equation

(2.11) ∂ t v + 2B[u, ∂ x v] = 0 ,
and we look for a priori estimates without loss of derivatives. All functions of the space variable x will be taken with values in R, and we shall use repeatedly the corresponding property ( u(-k) = u(k)) in the Fourier variable.

Lemma 2.1 We assume that Λ is C 1 outside the lines k = 0, = 0, and k + = 0, and has C 1 continuations to the sectors delimited by these lines. If in addition it satisfies (ii), (iii), (v), then the solutions of (2.11) satisfy the following a priori estimates

(2.12) d dt v 2 L 2 ≤ C(Λ) F(∂ x u) L 1 v 2 L 2 , (2.13) d dt v 2 H 1 ≤ C(Λ) F(∂ x u) L 1 v 2 H 1 , (2.14) d dt v 2 H 2 ≤ C(Λ) u H 2 v 2 H 2 ,
where C(Λ) depends only on Λ L ∞ , ∇Λ L ∞ (D) , and ∇Λ L ∞ (I) with

D = {(-1 + θ, -θ) ; 0 < θ < 1 } , I = {(1, -θ) ; 0 < θ < 1 } .
Proof. If v is a solution of (2.11), we have

d dt v(t) 2 L 2 = d dt v(t) 2 L 2 = -2 Re i(k -)Λ(k -, ) v(k -, t) u( , t) v(-k, t) d dk .
The integral here above can be split into

- i Λ(k -, ) v(k -, t) u( , t) v(-k, t) d dk + ikΛ(k -, ) v(k -, t) u( , t) v(-k, t) d dk .
By Fubini and the Cauchy-Schwarz inequality the modulus of the first integral is bounded by

Λ L ∞ F(∂ x u) L 1 v(t) 2 L 2 .
We now concentrate on the real part of the second one, equal to

ikΛ(k -, ) v(k -, t) u( , t) v(-k, t) d dk - ikΛ( -k, -) v( -k, t) u(-, t) v(k, t) d dk .
By change of variable (k, ) → (k -, -) in the first integral here above, we obtain

Re ikΛ(k -, ) v(k -, t) u( , t) v(-k, t) d dk = - i Λ(k, -) v(k, t) u(-, t) v( -k, t) d dk + ik (Λ(k, -) -Λ( -k, -)) v( -k, t) u(-, t) v(k, t) d dk ,
where the first integral is bounded again by Λ

L ∞ F(∂ x u) L 1 v(t) 2 L 2 .
As to the second integral, the factor

Λ(k, -) -Λ( -k, -) = Λ(k, -) -Λ( -k, -)
(by the reality assumption (ii)) is zero if the structure assumption (iv) is satisfied. It turns out that we can also deal with the integral under the (weaker) stability assumption (v). Indeed, this integral can be split into the sum of

|k|≤| | ik (Λ(k, -) -Λ( -k, -)) v( -k, t) u(-, t) v(k, t) d dk , whose modulus is bounded by 2 Λ L ∞ F(∂ x u) L 1 v(t) 2
L 2 , and of

|k|>| | ik (Λ(k, -) -Λ( -k, -)) v( -k, t) u(-, t) v(k, t) d dk .
This integral can be decomposed again as the sum of four integrals, each taken on a sector on which (k, ) → Λ(k, -) -Λ( -k, -) is smooth. These four sectors are {0 < < k} and {-k < < 0}, and their images by the center symmetry (k, ) → (-k, -). By the reality assumption (ii) it is sufficient to estimate the integrals on the first two sectors. Now for 0

< < k, Λ(k, -) -Λ( -k, -) = Λ(1, -/k) -Λ(-1 + /k, -/k) by (iii). Since Λ(1, 0-) = Λ(-1, 0-)
by (ii) and (v), we may rewrite the above equality as

Λ(k, -) -Λ( -k, -) = Λ(1, -/k) -Λ(1, 0-) + Λ(-1, 0-) -Λ(-1 + /k, -/k) ,
and thus obtain the bound, for 0 < < k,

|Λ(k, -) -Λ( -k, -)| ≤ (max D |∂ 1 Λ| + max I |∂ 2 Λ|) k ,
where D, I are the line segments joining respectively the points (-1, 0)-(0, -1) and (1, 0)-(1, -1).

The very same bound is obtained for -k < < 0, by using (ii). So finally we get

|k|>| | ik (Λ(k, -) -Λ( -k, -)) v( -k, t) u(-, t) v(k, t) d dk ≤ 4 (max D |∂ 1 Λ| + max I |∂ 2 Λ|) F(∂ x u) L 1 v(t) 2 L 2 .
This proves the L 2 estimate (2.12). The derivation of higher order estimates is a little bit trickier but follows the same lines. (It is to be noted that no commutator estimate is required.) We have

d dt ∂ n x v(t) 2 L 2 = d dt k 2n | v(k, t)| 2 dk = -2 Re ik 2n (k -)Λ(k -, ) v(k -, t) u( , t) v(-k, t) d dk .
As in the case n = 0, we make a change of variables (k, ) → (k -, -) in the integral, and leave its conjugate unchanged. This yields

Re ik 2n (k -)Λ(k -, ) v(k -, t) u( , t) v(-k, t) d dk = i k (k -) 2n Λ(k, -) -(k -)k 2n Λ( -k, -)) v(k, t) u(-, t) v( -k, t) d dk
To estimate this integral, the idea is to distribute in a suitable way the powers of k, , and k -among v(-k, t), u( , t), and v(k -, t) respectively. Let us begin with n = 1.

As in the case n = 0, the contribution of the domain |k| ≤ | | to the integral is harmless. Indeed, we have

|k|≤| | ik 2 (k -)Λ(k -, ) v(k -) u( ) v(-k) d dk ≤ Λ L ∞ |(k -) v(k -) u( ) k v(-k)| d dk ≤ Λ L ∞ F(∂ x u) L 1 ∂ x v 2 L 2
For the other part we observe that

|k|>| | k (k -) 2 Λ(k, -) -(k -)k 2 Λ( -k, -)) v(k, t) u(-, t) v( -k, t) d dk ≤ |k|>| | |Λ(k, -) k v(k, t) u(-, t) ( -k) v( -k, t)| d dk + |k|>| | |k (Λ(k, -) -Λ( -k, -))) k v(k, t) u(-, t)( -k) v( -k, t)| d dk ,
where the former admits again the bound

Λ L ∞ F(∂ x u) L 1 ∂ x v 2 L 2
, and the latter can be estimated exactly as before (in the case n = 0). This proves in turn that

d dt ∂ x v 2 L 2 ≤ 4 ( Λ L ∞ + 2 max D |∂ 1 Λ| + max I |∂ 2 Λ|) F(∂ x u) L 1 ∂ x v 2 L 2 ,
which together with (2.12) implies (2.13). We observe that, as for a (local) transport equation, we have an H 1 estimate which does not require more derivatives on the coefficient (u) than the L 2 estimate. A difference with local transport equations though, is that F(∂ x u) L 1 plays the role of the smaller norm ∂ x u L ∞ . Nevertheless, observing that F(∂ x u) L 1 will be bounded provided that u belongs to H s , s > 3/2, we can hope to deal with the well-posedness of the nonlinear Cauchy problem in these spaces, as for the classical Burgers equation. However, to avoid fractional derivatives, we shall deal with well-posedness in H 2 only.

This means we need an H 2 a priori estimate, which are going to derive now. A first, easy way consists in deducing it from the former estimates, which may written as

(2.15) | v, B[u, ∂ x v] | ≤ C(Λ) F(∂ x u) L 1 v 2 L 2 , (2.16) | ∂ x v, ∂ x B[u, ∂ x v] | ≤ C(Λ) F(∂ x u) L 1 ∂ x v 2 L 2 ,
Then by (2.7) we have

∂ 2 x v, ∂ 2 x B[u, ∂ x v] ≤ ∂ 2 x v, ∂ x B[∂ x u, ∂ x v] + ∂ 2 x v, ∂ x B[u, ∂ 2 x v] ,
where the latter term is bounded by

C(Λ) F(∂ x u) L 1 ∂ 2 x v 2 L 2 thanks to (2.16
), and, by Cauchy-Schwarz and (2.9), the former is bounded by Λ

L ∞ ∂ 2 x v L 2 ∂ x u H 1 ∂ x v H 1 up
to a (harmless) multiplicative constant. Therefore, up to substituting a larger positive constant (depending only on Λ) for C(Λ),

(2.17) ∂ 2 xx v, ∂ 2 xx B(u, ∂ x v) ≤ C(Λ) u H 2 v 2 H 2 ,
which together with (2.15) and (2.16) proves (2.14). We now prove (2.14) in a more direct (and more technical) way, which yields an additional estimate for the nonlinear equation (2.5). Proceeding as described before for higher order estimates, we have

d dt ∂ 2 x v(t) 2 L 2 ≤ i k (k -) 4 Λ(k, -) -(k -)k 4 Λ( -k, -)) v(k, t) u(-, t) v( -k, t) d dk ≤ ((k -) 3 -k 3 )Λ(k, -) k v(k, t) u(-, t) (k -) v( -k, t) d dk + k 3 (Λ(k, -) -Λ( -k, -)) k v(k, t) u(-, t) (k -) v( -k, t) d dk .
We can estimate the first integral by decomposing

(k -) 3 -k 3 = (k -) 2 -k 2 -3 k (k -) .
This yields the bound

Λ L ∞ 2 F(∂ x v) L 1 ∂ 2 x u L 2 ∂ 2 x v L 2 + 3 F(∂ x u) L 1 ∂ 2 x v 2 L 2 .
Concerning the second integral, we split it again. We have

|k|≤| | k 3 (Λ(k, -) -Λ( -k, -)) k v(k, t) u(-, t) (k -) v( -k, t) d dk ≤ 2 ( -k + k) (Λ(k, -) -Λ( -k, -)) k v(k, t) u(-, t) (k -) v( -k, t) d dk ≤ 4 Λ L ∞ F(∂ x v) L 1 ∂ 2 x u L 2 ∂ 2 x v L 2 , while |k|>| | k 3 (Λ(k, -) -Λ( -k, -)) k v(k, t) u(-, t) (k -) v( -k, t) d dk ≤ 4 (max D |∂ 1 Λ| + max I |∂ 2 Λ|) |k|>| | (k -+ ) k 2 v(k, t) u(-, t) (k -) v( -k, t) d dk ≤ 4 (max D |∂ 1 Λ| + max I |∂ 2 Λ|) F(∂ x u) L 1 ∂ 2 x v 2 L 2 + F(∂ x v) L 1 ∂ 2 x v L 2 ∂ 2 x u L 2 .
So we recover the H 2 estimate (2.14) by using the Cauchy-Schwarz inequality to bound F(∂ x w) L 1 by w H 2 for w = v and w = u. In the special case u = v we obtain a more precise estimate for the nonlinear equation (2.5), namely

(2.18) d dt u 2 H 2 ≤ C(Λ) F(∂ x u) L 1 u 2 H 2 .
In fact, by the very same procedure, we can obtain also the

H 3 estimate (2.19) d dt u 2 H 3 ≤ C(Λ) F(∂ x u) L 1 u 2 H 3 .
Indeed, for all smooth enough solutions u of (2.5) we have

d dt ∂ 3 x u(t) 2 L 2 ≤ i k (k -) 6 Λ(k, -) -(k -)k 6 Λ( -k, -)) u(k, t) u(-, t) u( -k, t) d dk ≤ ((k -) 5 -k 5 )Λ(k, -) k u(k, t) u(-, t) (k -) u( -k, t) d dk + k 5 (Λ(k, -) -Λ( -k, -)) k u(k, t) u(-, t) (k -) u( -k, t) d dk
To bound the first integral we use the identity

(k -) 5 -k 5 = -(k -) 2 3 -k 2 3 -5 k 2 (k -) 2 -3 k (k -) 3 ,
in which only the last term seems to be a problem (because when multiplied by k(k -) it yields a distribution of derivatives as 2 + 2 + 3 instead of 1 + 3 + 3). But of course we can bound

|k 2 (k -) 2 3 | by |k||k -| 3 | | 3 + |k -||k| 3 | | 3
. Therefore, we find that

((k -) 5 -k 5 )Λ(k, -) k u(k, t) u(-, t) (k -) u( -k, t) d dk Λ L ∞ F(∂ x u) L 1 ∂ 3 x u 2 L 2 .
As regards the other integral, we observe that

|k|≤| | k 5 (Λ(k, -) -Λ( -k, -)) k u(k, t) u(-, t) (k -) u( -k, t) d dk ≤ 3 ( -k + k) 2 (Λ(k, -) -Λ( -k, -)) k u(k, t) u(-, t) (k -) u( -k, t) d dk ≤ 8 Λ L ∞ F(∂ x u) L 1 ∂ 3 x u 2 L 2 , while |k|>| | k 5 (Λ(k, -) -Λ( -k, -)) k u(k, t) u(-, t) (k -) u( -k, t) d dk ≤ 4 (max D |∂ 1 Λ| + max I |∂ 2 Λ|) |k|>| | (k -+ ) 2 k 3 u(k, t) u(-, t) (k -) u( -k, t) d dk ≤ 8 (max D |∂ 1 Λ| + max I |∂ 2 Λ|) F(∂ x u) L 1 ∂ 3 x u 2 L 2 . 2 3
S ε (ξ) = S 1 (εξ)
with S 1 real valued, C ∞ with compact support, taking the value 1 at zero and of absolute value not greater than 1. Clearly, for all ε ≥ 0, S ε is a bounded operator on H s (R) for each s ∈ R, with

(3.20) S ε H s →H s ≤ 1 .
Furthermore, for all ε > 0, S ε is a regularizing operator, with

(3.21) S ε H s →H s+σ ε -σ
for all s ∈ R and σ ≥ 0, and we have the error estimate

(3.22) S ε u -u H s ε σ u H s+1
for all ε ≥ 0. Here above, the multiplicative constants hidden in the symbol depend on (s, σ) but not of ε of course. Let us consider the following regularization of (1.1)

(3.23) ∂ t u ε + S ε B (u ε , ∂ x S ε u ε ) = 0 .
For ε > 0, the mapping u → S ε B (u, ∂ x S ε u) is locally Lipschitz in H 2 (R), and more precisely, for all ε ∈ (0, 1], if u and v belong to H 2 (R)then

S ε B (u, ∂ x S ε u) -S ε B (v, ∂ x S ε v) H 2 ≤ S ε B (u -v, ∂ x S ε u) H 2 + S ε B (v, ∂ x S ε (u -v)) H 2 ≤ C ε Λ L ∞ ( u H 2 + v H 2 ) u -v H 2 .
Therefore, the Picard iteration scheme

u ε 0 := u 0 , u ε k+1 : t ∈ [0, T ε ] → u ε k+1 (t) := - t 0 S ε B (u ε k (τ ), ∂ x S ε u ε k (τ )) dτ , k ∈ N is well defined and convergent in B R := {u ; u H 2 ≤ R} provided that 2CR Λ L ∞ T ε ≤ ε.
This shows the existence of a solution

u ε ∈ C 1 (0, T ε ; H 2 (R)) of (3.23) such that u ε (0) = u 0 .
This solution is unique and depends continuously on u 0 by Gronwall's lemma. Indeed, if v ε ∈ C 1 (0, T ε ; B R ) is another solution of (3.23) we have

u ε (t) -v ε (t) H 2 ≤ u ε (0) -v ε (0) H 2 + 2CR ε Λ L ∞ t 0 u ε (τ ) -v ε (τ ) H 2 dτ , hence u ε (t) -v ε (t) H 2 ≤ (1 + e 2CRt ε Λ L ∞ ) u ε (0) -v ε (0) H 2 ≤ (1 + e) u ε (0) -v ε (0) H 2
for t ≤ T ε (by assumption on T ε ). This shows continuous dependence on initial data for (3.23), and uniqueness within the ball B R . Unconditional uniqueness follows from a classical connectedness argument.

Let us now redefine T ε as the maximal time of existence of the solution of (3.23) with initial data u ε (0) = u 0 ∈ B R . By the construction hereabove we have T ε ≥ ε/(2CR Λ L ∞ ). It remains to show that T ε is positively bounded by below when ε → 0, and that u ε converges to a solution of (1.1) in H 2 .

As a first step, we show that u ε (t) H 2 is bounded independently of ε > 0 and t ∈ [0, T ] for some positive T . This relies on the a priori estimates (2.15) (2.16) (2.17). Indeed, since S ε is a self-adjoint operator and commutes with ∂ x ,

d dt u ε 2 H 2 = -2 S ε u ε , B(u ε , ∂ x S ε u ε ) -2 ∂ x S ε u ε , ∂ x B(u ε , S ε ∂ x u ε ) -2 ∂ 2 xx S ε u ε , ∂ 2 xx B(u ε , ∂ x S ε u ε ) , hence d dt u ε 2 H 2 ≤ 6C(λ) u ε 3 H 2 ,
and after integration

u ε (t) H 2 ≤ u 0 H 2 1 -3C(λ)t u 0 H 2 for all t ∈ [0, T ε ) such that t < 1/(3C(λ)R). As a consequence, T ε cannot be lower than 1/(3C(λ)R) (otherwise, denoting R ε := R/(1 -3C(Λ)T ε R), we could extend u ε behond T ε , restarting from u ε (t ε 0 ) with t ε 0 = T ε -ε/(4CR ε Λ L ∞ )
as initial data, which would contradict the fact that T ε is maximal). From now on, we choose T < 1/(3C(λ)R). By the argument above, T ε > T for all ε > 0, and (u ε ) ε>0 is bounded in C (0, T ; H 2 (R)).

The heart of the matter then consists in showing that (u ε ) ε∈(0,ε 0 ] satisfies the Cauchy criterion in C (0, T ; L 2 (R)). For 0 < ν ≤ ε, we have

d dt u ε -u ν 2 L 2 = -2 u ε -u ν , S ε B(u ε -u ν , ∂ x S ε u ε ) -2 u ε -u ν , S ε B(u ν , ∂ x (S ε u ε -S ν u ν )) -2 u ε -u ν , (S ε -S ν )B(u ν , ∂ x S ν u ν ) .
We are going estimate these three terms separately. By Cauchy-Schwarz and (2.8) (2.9) we can estimate the first and last terms

|2 u ε -u ν , S ε B(u ε -u ν , ∂ x S ε u ε ) | Λ L ∞ u ε H 2 u ε -u ν 2 L 2 , |2 u ε -u ν , (S ε -S ν )B(u ν , ∂ x S ν u ν ) | Λ L ∞ u ε -u ν L 2 ε u ν H 1 u ν H 2 .
As to the middle term, using again that S ε is self-adjoint, we can split it as

-2 S ε u ε -S ν u ν , B(u ν , ∂ x (S ε u ε -S ν u ν )) + 2 (S ε -S ν )u ν , B(u ν , ∂ x (S ε u ε -S ν u ν )) .
By Cauchy-Schwarz and (2.8) again we have

|2 (S ε -S ν )u ν , B(u ν , ∂ x (S ε u ε -S ν u ν )) | Λ L ∞ ε u ν 2 H 1 ( u ε H 1 + u ν H 1 ) ,
and by (2.15),

|2 S ε u ε -S ν u ν , B(u ν , ∂ x (S ε u ε -S ν u ν )) | C(Λ) ∂ x u ν H 1 S ε u ε -S ν u ν 2 L 2 C(Λ) ∂ x u ν H 1 ( u ε -u ν L 2 + ε u ν H 1 ) 2 .
Adding these estimates altogether, and using that u ε H 2 , u ν H 2 , are uniformly bounded by R we obtain the (rather crude) estimate (for some modified constant C(Λ))

d dt u ε -u ν 2 L 2 ≤ C(Λ) R u ε -u ν 2 L 2 + C(Λ) R 3 ε ,
which yields by integration, using that u ε (0) = u ν (0),

u ε (t) -u ν (t) 2 L 2 ≤ C(Λ) R 3 ε e e C(Λ) R t .
Therefore, u ε is convergent in C (0, T ; L 2 (R)) as ε goes to zero. It remains to show some additional regularity for its limit u. In fact, since (u ε (t)) is uniformly bounded in H 2 (R), we have

u(t) ∈ H 2 (R) for all t ∈ [0, T ]. By L 2 -H 2 interpolation, this implies that u ε converges to u in C (0, T ; H s (R)) for all s ∈ [0, 2)
. Now, by interpolation between (2.8) and (2.9), we have

(3.24) B(v, w) H s Λ L ∞ v H 1 w H s ,
for all s ∈ [0, 1] and v ∈ H 1 , w ∈ H s . This will enable us to show that ). Indeed, we have the pointwise time estimate

S ε B (u ε , ∂ x S ε u ε ) converges to B(u, ∂ x u) in C (0, T ; H s (R)) for s ∈ [0, 1
S ε B (u ε , ∂ x S ε u ε ) -B(u, ∂ x u) H s ≤ ε (1-s)/2 B (u ε , ∂ x S ε u ε ) H 1 + B (u ε , ∂ x S ε u ε ) -B(u, ∂ x u) H s ε (1-s)/2 u ε H 1 u ε H 2 + B (u ε , ∂ x (S ε u ε -u)) H s + B (u ε -u, ∂ x u) H s ε (1-s)/2 u ε H 1 u ε H 2 + u ε -u H 1 u H s+1 .
(For simplicity we have included Λ L ∞ in the symbol.) Therefore, being the limit of ∂ t u ε in the sense of distributions, ∂ t u = -B(u, ∂ x u) belongs to C (0, T ; H s (R)). This shows that u is in C 1 (0, T ; H s (R)) for all s ∈ [0, 1). Then we can prove that u is the unique solution of (1.1) in C (0, T ; H s+1 (R)) ∩ C 1 (0, T ; H s (R)) ∩ L ∞ (0, T ; H 2 (R)) with initial data u 0 . For, if v were another one, we would have

d dt u -v 2 L 2 = -2 u -v, B(u -v, u) -2 u -v, B(v, ∂ x (u -v)) K u -v 2 L 2 .
Here above we have used Cauchy-Schwarz to estimate the first term, (2.15) for the second, and K is a uniform bound for ( u H 2 + v H 2 ) on the time interval [0, T ]. So we have by integration

u(t) -v(t) 2 L 2 ≤ e Kt u(0) -v(0) 2 L 2 .
We already know that u belongs to L ∞ (0, T ; H 2 (R)). To conclude that u is actually in C (0, T ; H 2 (R)) we invoke weak topology arguments. Since u ε converges to u in C (0, T ; H s (R)) for all s ∈ [0, 2), by density of the dual H -s (R) of H s (R) in H -2 (R), we see that u ε (t) converges uniformly on [0, T ] to u in H 2 w (R), the Sobolev space H 2 (R) equipped with the weak topology. By a similar argument, for all t 0 ∈ [0, T ], u(t) tends to u(t 0 ) in H 2 w (R) when t goes to t 0 , which implies in particular lim inf

t→t 0 u(t) H 2 ≥ u(t 0 ) H 2 .
Therefore, to prove the strong limit

lim t→t 0 u(t) -u(t 0 ) H 2 = 0 , it suffices to prove lim sup t→t 0 u(t) H 2 ≤ u(t 0 ) H 2 .
Proof. By a slight modification of the argument used above for the Cauchy problem with non-regularized initial data, we easily see that for all ε > 0, the Cauchy problem with regularized initial u ε (0) = S ε β u 0 admits a unique local solution in H 3 , the maximal time of existence T ε being at least of order of ε 1+β / u 0 H 2 (like ε/ S ε β u 0 H 3 ). In addition, by integration of

d dt u ε 2 H 2 ≤ 6C(Λ) u ε 3 H 2 ,
we obtain

u ε (t) H 2 ≤ u ε (0) H 2 1 -3C(Λ)t u ε (0) H 2 ≤ u 0 H 2 1 -3C(Λ)t u 0 H 2 for all t ∈ [0, T ε ), hence T ε ≥ 1/(3C(Λ)R if u 0 H 2 ≤ R.
From now on we take T < 1/(3C(Λ)R), in such a way that u ε (t) H 2 is uniformly bounded for t ∈ [0, T ] and ε > 0, which also implies a uniform bound for

F(∂ x u ε ) L ∞ (0,T ;L 1 (R)) .
Now, revisiting the proof of (2.12) (2.13) (2.18) (2.19) (just using that | S ε | ≤ 1) we find that

d dt u ε 2 H 3 ≤ C(Λ) F(∂ x u ε ) L 1 u ε 2 H 3 .
By integration, this yields

u ε (t) 2 H 3 ≤ u ε (0) 2 H 3 e C(Λ) F (∂xu ε ) L 1 ((0,T )×R) , hence (3.27) u ε C (0,T ;H 3 (R)) S ε β u 0 H 3 ε -β u 0 H 2 .
Thanks to this estimate and the uniform bound of u ε C (0,T ;H 2 (R)) , say R , we can now show that (u ε ) ε>0 satifies the Cauchy criterion not only in C (0, T ; L 2 (R)) (as done before) but also in C (0, T ; H 2 (R)). For m ∈ N, 0 < ν ≤ ε, we have

1 2 d dt ∂ m x (u ε -u ν ) 2 L 2 = -∂ m x (u ε -u ν ), S ε ∂ m x B(u ε -u ν , ∂ x S ε u ε ) -∂ m x (S ε u ε -S ν u ν ), ∂ m x (B(u ν , ∂ x (S ε u ε -S ν u ν ))) + (S ε -S ν )∂ m x u ν , ∂ m x (B(u ν , ∂ x (S ε u ε -S ν u ν )) -∂ m x (u ε -u ν ), (S ε -S ν )∂ m x B(u ν , ∂ x S ν u ν ) .
For convenience we call I m i , i = 1, 2, 3, 4, the terms above. We first concentrate on the case m = 1. By Cauchy-Schwarz and (2.9),

|I 1 1 | ≤ ∂ x (u ε -u ν ) L 2 B(u ε -u ν , ∂ x S ε u ε ) H 1 ∂ x u ε H 1 u ε -u ν 2 H 1 .
By the energy estimate (2.16) and the error estimate (3.22),

|I 1 2 | F(∂ x u ν ) L 1 ∂ x (S ε u ε -S ν u ν ) 2 L 2 F(∂ x u ν ) L 1 ( ∂ x (u ε -u ν ) 2 L 2 + ε 2 ∂ x u ν 2 H 1 )
.

By Cauchy-Schwarz, (2.9), (3.22), and (3.27),

|I 1 3 | (S ε -S ν )∂ x u ν L 2 B(u ν , ∂ x (S ε u ε -S ν u ν )) H 1 ε 2 ∂ x u ν H 1 u ν H 1 ( u ν H 3 + u ε H 3 ) ε 2-β ∂ x u ν H 1 u ν H 1 u 0 H 2 .
By Cauchy-Schwarz, (2.10), and (3.27),

|I 1 4 | ε ∂ x (u ε -u ν ) L 2 B(u ν , ∂ x S ν u ν ) H 2 ε 1-β ∂ x (u ε -u ν ) L 2 u ν H 2 u 0 H 2 .
By adding all four estimates we get (recalling also the L 2 estimate)

1 2 d dt u ε -u ν 2 H 1 R u ε -u ν 2 H 1 + (R ) 3 ε 2 + (ε 2(1-β) + ε 2-β ) R R 2 .
Therefore, by integration we obtain

u ε (t) -u ν (t) 2 H 1 ≤ e C t ( u ε (0) -u ν (0) 2 H 1 + (R ) 2 ε 2 + (ε 2(1-β) + ε 2-β ) R 2 ) ,
where C is proportional to R . Since u ε (0) -u ν (0) H 1 ε u 0 H 2 by (3.22), we receive a uniform estimate (3.28) u ε -u ν C (0,T ;H 1 (R)) = O(ε 1-β ) .

Let us now turn to the estimate of u ε (t) -u ν (t) H 2 . By Cauchy-Schwarz, (2.10), and (3.28)

|I 2 1 | ≤ ∂ 2 x (u ε -u ν ) L 2 B(u ε -u ν , ∂ x S ε u ε ) H 2 u ε -u ν H 2 ( ∂ x u ε H 2 u ε -u ν H 1 + ∂ x u ε H 1 u ε -u ν H 2 ) ∂ x u ε H 1 u ε -u ν 2 H 2 + O(ε 1-2β ) u ε -u ν H 2 .
By the energy estimate (2.17), the error estimate (3.22), and the uniform bound (3.27)

|I 2 2 | F(∂ x u ν ) L 1 ∂ 2 x (S ε u ε -S ν u ν ) 2 L 2 F(∂ x u ν ) L 1 ( ∂ x (u ε -u ν ) 2 L 2 + ε 2 ∂ x u ν 2 H 2 ) ∂ x u ν H 1 ( ∂ x (u ε -u ν ) 2 L 2 + ε 2(1-β) u 0 2 H 2 )
. By Cauchy-Schwarz, (2.9), (3.22), and (3.27),

|I 2 3 | (S ε -S ν )∂ 2 x u ν L 2 B(u ν , ∂ x (S ε u ε -S ν u ν )) H 2 ε ∂ 2 x u ν H 1 ( u ν H 1 (S ε u ε -S ν u ν ) H 3 + u ν H 2 (S ε u ε -S ν u ν ) H 2 ) ε u ν 2 H 2 ( u ε -u ν H 2 + ε -β u 0 L 2 ) .
The most 'dangerous' term is in I 2 4 . It can dealt with by first integrating by part, which leads to

I 2 4 = -∂ 3 x (u ε -u ν ), (S ε -S ν )∂ x B(u ν , ∂ x S ν u ν ) ,
hence by Cauchy-Schwarz, (2.10), and (3.27),

|I 2 4 | ε ∂ 3 x (u ε -u ν ) L 2 B(u ν , ∂ x S ν u ν ) H 2 ε 1-2β u 0 2 H 2 u ν H 2 .
By summation of these four estimates with the estimates obtained for first order derivatives, we finally arrive at an inequality of the form

1 2 d dt u ε -u ν 2 H 2 u ε -u ν 2 H 2 + O(ε 1-2β ) .
As a consequence, we get

  (i) symmetry: Λ(k, ) = Λ( , k) ∀k, ∈ R , (ii) reality: Λ(-k, -) = Λ(k, ) ∀k, ∈ R , (iii) homogeneity: Λ(αk, α ) = Λ(k, ) ∀k, ∈ R, and α > 0 , (iv) structure: Λ(k + ξ, -ξ) = Λ(k, ξ) ∀k, ξ ∈ R ,the latter being possibly replaced by (v) stability: Λ(1, 0-) = Λ(1, 0+) .

  u ε -u ν C (0,T ;H 2 ) u ε (0) -u ν (0) H 2 + O(ε 1/2-β ) = o(1) .by(3.25) applied to u 0 , s = 2, and σ = 0.2

  Well-posednessOnce we have a priori estimates in H 2 (R), a fairly general method (see for instance[7, p. 360]) to actually prove well-posedness in H 2 (R) consists in regularizing (1.1) in such a way that the regularized problem is merely solvable by the Cauchy-Lipschitz theorem and that its solutions converge in a suitable manner to solutions of the original problem.A simple, and natural way to regularize (1.1) is by means of Fourier multipliers. In what follows we shall use a Fourier multiplier S ε of symbol

Acknowledgement. The author thanks Jean-François Coulombel for fruitful discussions.

Now, as shown before (substituting t -t 0 for t),

for all ε > 0 and t ∈ [t 0 , T ], hence

and finally lim sup

The inequality for t t 0 can be obtained in a similar way by reversing time. We have thus proved the following. Theorem 3.1 Assuming the kernel Λ is C 1 outside the lines k = 0, = 0, and k + = 0, has C 1 continuations to the sectors delimited by these lines, and satisfies (ii), (iii), (v), for all u 0 ∈ H 2 (R) there exists T > 0 and a unique solution u ∈ C (0, T ;

Continuous dependence with respect to initial data in H 2 demands a little more work. Theorem 3.2 Under the assumptions of Theorem 3.1 the mapping

is continuous.

To prove this result, we shall use a trick originally introduced by Bona and Smith for KdV [START_REF] Bona | The initial-value problem for the Korteweg-de Vries equation[END_REF] (also see [START_REF] Benzoni-Gavage | On the well-posedness for the eulerkorteweg model in several space dimensions[END_REF]), and regularize initial data by means of S ε β for a suitable β > 0. In this respect we shall make the further assumption that S ε equals one on the interval [-1/ε, 1/ε]. This implies that for all s ∈ R, σ ≥ 0, for all u ∈ H s (R),

and moreover, for any sequence (u n ) n∈N converging to u in H s (R),

, uniformly with respect to n . We take a β ∈ (0, 1/2). Then, under the assumptions of Theorem 3.1, for all R > 0 there exists T > 0 such that for all u 0 ∈ H 2 (R) of norm not greater than R, for all ε > 0, the Cauchy problem for (3.23) and initial data u ε (0) = S ε β u 0 admits a unique solution u ε ∈ C (0, T ; H 3 (R)). Furthermore, we have

and u ε converges in C (0, T ; H 2 (R)) to a solution u of (1.1) such that u(0) = u 0 , with the following rate in H 1 :

. Proof of Theorem 3.2. It amounts to proving that for any sequence (u n 0 ) n∈N tending to u 0 in H 2 (R), the solutions u n of the Cauchy problems

Let us take (u n 0 ) n∈N tending to u 0 in H 2 (R). We first observe that since (u n 0 ) n∈N is bounded in H 2 , by Lemma 3.1 the solution u n of (3.29) in H 2 is well-defined on some interval [0, T ] independent of n, as well as the solutions u ε n and u ε of the regularized Cauchy problems

Furthermore, by Lemma 3.1 we also have that u ε -u C (0,T ;H 2 ) goes to zero, and revisiting its proof with the help of (3.26), we also find that u ε n -u n C (0,T ;H 2 ) goes to zero uniformly in n. We can now conclude by an ε/3-(or more appropriately here an η/3)-argument. Indeed, for all t ∈ [0, T ], for all n ∈ N, for all ε > 0,

For η > 0, there exists ε 0 such that for all ε ∈ (0, ε 0 ), for all t ∈ [0, T ], for all n ∈ N,

If we choose an ε ∈ (0, ε 0 ), as we have seen in the proof of Theorem 3.1,

by (3.20). So u n (t) -u(t) H 2 can be made less than η for n large enough. 2

Another result that comes out from the proof of Lemma 3.1 is the following blow-up criterion, which generalizes the well-known blow-up criterion for the classical inviscid Burgers equation (lim t T ∂ x u L 1 (0,T ;L ∞ (R)) = +∞ if T is a finite, maximal time of existence).

Corollary 3.1 Under the assumptions of Theorem 3.1, if u ∈ C (0, T ; H 2 (R) is a solution of (1.1) such that F(∂ x u) belongs to L 1 ((0, T ) × R) then u can be extended beyond T .


Anne-Marie Chèvre
email: chevre@rennes.inra.fr

Anne Ponce De Leon

Eric Jenczewski

Frédérique Eber

Régine Delourme

Michel Renard

Hortense Brun²
email: brun@rennes.inra.fr

Introduction of blackleg resistance from Brassica rapa into Brassica napus

Keywords: Brassica napus, Brassica rapa, blackled resistance, specific resistance genes

For diversification of specific resistance genes to Leptosphaeria maculans, Brassica rapa (AA, 2n=20) populations were screened. One resistant plant was selected within a forage population. A BC 1 progeny was produced after two crosses of this plant to a susceptible B. rapa doubled haploid line and screened with six isolates. Resistance to different isolates proves to be under mono or oligogenic control. Resistance genes are organized in a cluster. The same B. rapa plant was crossed twice to 'Darmor' oilseed rape variety, susceptible at the cotyledon stage but partially resistant at the adult stage. The BC 1 oilseed rape progeny was analyzed with two isolates and showed the same results as in B. rapa BC 1 progeny. From 2n=38 resistant BC 1 plants, two additional backcrosses were performed in which the different resistant genes were separated through recombination. The results will be analyzed according to breeding for a durable resistance.

INTRODUCTION

Blackleg caused by Leptosphaeria maculans is one of the most damaging diseases of oilseed rape (Brassica napus L., AACC, 2n=38). Several sources of resistance have already been described: they are either total and specific, efficient from the cotyledon stage and under mono or oligogenic control [START_REF] Chèvre | Characterization of Brassica nigra chromosomes and of blackleg resistance from B. nigra-B. napus addition lines[END_REF][START_REF] Chèvre | Selection of stable Brassica napus-B. juncea recombinant lines resistant to blackleg (Leptosphaeria maculans) 1 : Identification of molecular markers, chromosomal and genomic origin of the introgression[END_REF][START_REF] Balesdent | New avirulence genes in the phytopathogenic fungus Leptosphaeria maculans[END_REF] or partial, efficient at the adult stage, and under polygenic control [START_REF] Pilet | Identification of loci contributing to quantitative field resistance to blackleg disease, causal agent Leptosphaeria maculans (Desm.) Ces. et de Not., in winter rapeseed (Brassica napus L.)[END_REF]. Combination of different resistances is likely to be the only issue to produce material durably resistant. Nevertheless, a diversification of resistance genes is needed since various single ascospore isolates were found to be virulent at the cotyledon stage on almost all major resistance genes even those introduced from related species, Rlm10 from B. nigra [START_REF] Brun | Presence in Leptosphaeria maculans populations of isolates virulent on resistance introgressed into Brassica napus from the B. nigra B genome[END_REF] or Rlm6 from B. juncea [START_REF] Somda | Pathogenicity of Leptosphaeria maculans isolates on a Brassica napus-B. juncea recombinant line[END_REF]. We found different specific resistance genes into a B. rapa variety. The genetic analysis within B.rapa and after introduction into an oilseed rape variety is presented.

MATERIALS AND METHODS

Plant materials: One resistant plant, C1.3 was selected within an old French forage B.rapa variety. It was crossed twice to a susceptible doubled haploid line, Z1 (kindly provided by AAFC, Canada) to produce F1 and from one F1 plant the BC1 progeny. The same C1.3 plant was crossed twice to a winter oilseed rape variety, 'Darmor'. From the five BC1 resistant plants selected, crosses were performed with two oilseed rape variety either 'Darmor' or 'Yudal', a Korean variety. Fungus materials: Six isolates were used. Some of the avirulent genes (AvrLm) they carried, were identified on a differential set with the known resistance genes (table 1). Resistance tests at the seedling stage. After inoculations performed on cotyledons as described by Williams and Delwiche 1979, symptoms were scored at 14 and 21 days using a 1 (resistant)-11 (susceptible) scale according to lesion size, the occurrence of necrosis or chlorosis and the presence of pycnidia. Each plant was tested with two isolates, one on each cotyledon. Cytogenetic studies. Flow cytometry and meiotic behavior analyses were performed as described by [START_REF] Chèvre | Characterization of Brassica nigra chromosomes and of blackleg resistance from B. nigra-B. napus addition lines[END_REF] and by [START_REF] Eber | Chromosome number of oilseed rape (Brassica napus)wild radish (Raphanus raphanistrum) spontaneous hybrids and of their progeny estimated by flow cytometry[END_REF].

RESULTS

Genetic analyses in the BC1 progeny within B.rapa

Screening with S7 and R2 isolates revealed that resistance to S7 isolate is conferred by two dominant independent genes (121R:46S) whereas resistance to R2 isolate is controlled by one dominant gene (88R:72S). The low number of recombinant plants indicated that the resistance genes are linked (table 1). Additionally, at least one S7 resistance gene is closely linked to the R2 resistant gene as no recombinant susceptible to S7 isolate and resistant to R2 isolate was observed.

Resistance to each of the other isolates tested (MX4.3 and 290, S14 and p27d) is controlled either by one gene in distortion or by two partially linked genes. Whatever the test, the low frequency of recombinant plants indicated that resistance genes are linked to one another (table 1). Genetic analyses in the BC1 progeny obtained from crosses to B.napus The three F1 resistant plants showed the expected genomic structure (AAC, 2n=29). Their average meiotic behavior (8.83 univalents + 9.99 bivalents + 0.04 quadrivalents) indicated that the two A genomes paired regularly. Flow cytometry analyses showed that the number of chromosomes segregated in the BC1 progeny obtained after backcrossing the F1 AAC plants to B.napus. A higher frequency of plants close to 38 chromosomes as oilseed rape was produced when oilseed rape was used as female (Figure 1)

Pathological test was performed on the whole BC1 progeny with S7 and R2 isolates, each on one cotyledon. As in the B.rapa progeny, resistance to S7 segregated as two dominant independent genes (61R:17S) whereas resistance to the R2 isolate was conferred by one dominant gene (30R:28S). Among the 45 plants phenotyped with S7 and R2 isolates, 11 recombinant plants, resistant to one isolate and susceptible to the other, were observed but only one plant was susceptible to S7 isolate and resistant to R2 isolate. This result indicates at least three genes are involved and two of them linked.

Five resistant BC1 plants showing 37 or 38 chromosomes were selected and backcrossed to oilseed rape. Whatever the mother plant, resistance to the R2 isolate still segregated as one single gene. Only one BC2 progeny gave the same result as in the previous generation i.e. two independent dominant genes for the resistance to S7 isolate. In the other BC2 plants, the resistance to S7 isolate segregated as one dominant gene either closely linked to the resistance to the R2 isolate in three progenies (0 or 1cM) or more distant from R2 gene (23.6cM) in the fourth progeny.

Three 2n=38 BC2 plants resistant to both isolates were selected. In all cases, the resistance to each isolate, S7 and R2, was conferred by one dominant gene. The R2 and S7 resistance Figure 1: Segregation of the chromosome number in the BC1 progeny from AAC F1 hybrids genes linked in the BC2 progeny were totally linked in the BC3 generation whereas a distance of 17cM was found between the two genes in the other progeny.

DISCUSSION

The C1.3 B.rapa plant contained different resistance genes. According to the avirulent genes currently identified in the six isolates used, some resistance could correspond to Rlm1, Rlm6 Rlm7 and Rlm10. Some of them have only been described in related species i.e. B.nigra (Rlm74, [START_REF] Chèvre | Characterization of Brassica nigra chromosomes and of blackleg resistance from B. nigra-B. napus addition lines[END_REF]B.juncea (Rlm6, Chèvre et al. 1997). Molecular mapping (data not shown) indicated that all these genes are linked.

Crossing the same B.rapa plant to an oilseed rape variety allowed the production of F1 resistant plants (AAC) with a meiotic behavior close to the expected one, 10 bivalents between the A genomes and 9 univalents from the C genome. The segregation of resistance genes carried by the B.rapa A genome can be therefore analyzed in the progeny. The distribution of the chromosome number at the BC1 generation is the same as the one reported by [START_REF] Namai | Inducing cytogenetical alterations by means of interspecific and intergeneric hybridizations in Brassica crops[END_REF]. A higher frequency of plants with 2n=38 was obtained when the female parent was the oilseed rape plant. However, no difference was observed for the segregation of resistance genes which was the same as the one observed in the B. rapa progeny. Only the genetic distances between resistance genes and molecular markers were modified in the BC1 hybrids (data not shown). In the following BC generations we showed that it is possible to separate the two genes involved in the resistance to the S7 isolate; one of them is closely linked to the gene conferring resistance to the R2 isolate. From the BC4 resistant plants, doubled haploid line will be produced and tested with different isolates carrying different AvrLm genes in order to characterize all the resistance genes. Additionally, the size and location of the cluster as well as the variation in genetic distances among the generations will be defined by molecular mapping.

We showed that it is possible to introduce in oilseed rape resistance genes from one of its progenitor, B.rapa. Different varieties will be produced with different specific resistance genes alone or combined in varieties carrying a polygenic partial resistance. The impacts of these different genetic constructions on the efficiency and durability of blackleg resistance will be studied under field conditions.

Table 1 :

 1 Number of plants in the BC1 progeny from B. rapa crosses showing a resistantresistant (RR), a resistant-susceptible (RS), a susceptible-resistant (SR), or a susceptiblesusceptible (SS) phenotype after testing with both isolates, one on each cotyledon

	Phenotypes

ACKNOWLEDGEMENTS

We thank our colleagues, Magali ERMEL and Jean-Claude LETANNEUR for their technical assistance. Agriculture and Agri-Food Canada are acknowledged for providing the susceptible doubled haploid line of B.rapa, Z1.