domain: math info shs

text:
Patrice Loisel
email: patrice.loisel@inra.fr

Marielle Brunette
email: marielle.brunette@inra.fr

Stéphane Couture
email: stephane.couture@inra.fr

Insurance and forest rotation decisions under storm risk

Keywords: Faustmann Value, insurance, risk-sharing, storm risk, rotation length, subsidy JEL: D81, G22, Q23

Introduction

Natural hazards represent serious threat for forest ecosystems. In particular, storms are the more damageable event for European forest since they are responsible for more than 50% of the European forest damage over the period 1950-2000 [START_REF] Schelhaas | Natural disturbances in the european forests in the 19th and 20th centuries[END_REF]. In the same vein, [START_REF] Gardiner | Living with storm damage to forests[END_REF] reported 130 storms over the last sixty years in Europe, i.e., an average of two storms per year. Since 1999, three main storms generate important damages in European forests, and in particular in France.

Lothar and Martin in 1999 induced 6 billions Euros of damage for the whole France, while Klaus in 2009 generated an average cost of 1.5 billion Euros in the South-western part of France [START_REF] Rakotoarison | The faustmann model under storm risk and price uncertainty: A case study of european beech in northwestern france[END_REF].

The occurrence of storm generates various type of losses both for forest owners and society: loss of marketability, costs of storage and restoration, losses in other income such as hunting leases or losses of carbon sequestration and amenities [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF]. Climate change is expected to increase the frequency and intensity of disturbances [START_REF] Flannigan | Climate change and forest fires[END_REF][START_REF] Fuhrer | Climate risks and their impact on agriculture and forests in switzerland[END_REF] and in particular those of storm [START_REF] Haarsma | More hurricanes to hit western europe due to global warming[END_REF].

To face such events, forest owners may adopt classical risk-sharing strategy like insurance contract.

Various European countries propose insurance contract against storm and/or fire like for example France, Germany, Slovakia, Spain, etc [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF]. Such a contract allows the forest owner to transfer the risk to the insurer. In exchange of the payment of a premium, the owner perceives an indemnity in case of storm occurrence. Another option available for the forest owner to face the increasing risk of storm is to adapt the silvicultural management. A classical risk-reducing strategy in forestry is to reduce the rotation length [START_REF] Lindner | Alternative forest management strategies under climate change: prospects for gap model applications in risk analyses[END_REF][START_REF] Spittelhouse | Adaptation to climate change in forest management[END_REF], i.e., the time from the establishment of the stand to its final felling. Such a strategy allows diminishing both the time of exposure to natural event and the vulnerability of trees due to ageing [START_REF] Bréda | Vulnerability to forest decline in a context of climate changes: new prospects about an old question in forest ecology[END_REF].

Traditionally, in the literature, decisions related to insurance and rotation length have been analyzed separately. The forest owner's insurance decision has been studied in relation to the disincentive role played by public help implemented by government after the occurrence of a disaster [START_REF] Brunette | Public compensation for windstorm damage reduces incentives for risk management investments[END_REF][START_REF] Brunette | The impact of governmental assistance on insurance demand under ambiguity: A theoretical model and an experimental test[END_REF]. A part of the literature also deals with the determination of the optimal insurance premium through an actuarial approach (Holecy and Hanewinkel, 2006;[START_REF] Pinheiro | Forest property insurance : an application to portuguese woodlands[END_REF][START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF]. One of the main conclusion is that, in order to provide relevant forest insurance premium, the insured forest area should increase, leading [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF] to suggest an European forest insurance scheme. In this forest insurance literature, the analysis of insurance decision under risk is generally realized from a static point of view whereas the problem of forest management under risk is typically a dynamic sequential decision problem under uncertainty. It is fundamental to consider this temporal and sequential aspect in the definition of insurance contract because when a risk occurs, the loss covered by insurance strongly depends on the optimal rotation decision of the forest owner. To our knowledge, in this literature, the insurer's behavior is never modeled and integrated into the analysis.

The optimal rotation length is the common research question in forest economics tackled through the classical Faustmann approach [START_REF] Faustmann | Berechnung des wertes welchen waldboden sowie noch nicht haubare holzbestunde fur die weldwirtschaft besitzen[END_REF]. This approach allows the evaluation of the Land Expectation Value (LEV) over an infinite sequence of rotation. Consequently, the optimal harvesting age in a Faustmann framework has been addressed a lot in the literature [START_REF] Naslund | Optimal rotation and thinning[END_REF][START_REF] Schreuder | The simultaneous determination of optimal thinning schedule and rotation for an evenaged forest[END_REF][START_REF] Kao | Simultaneous optimization of thinnings and rotation with continuous stocking and entry intervals[END_REF][START_REF] Roise | A nonlinear programming approach to stand optimization[END_REF][START_REF] Haight | Optimal harvesting with stand density targets: Managing rocky mountain conifer stands for multiple forest outputs[END_REF][START_REF] Jacobsen | Forest value and optimal rotations in continuous cover forestry[END_REF]. However, originally, this approach is deterministic. [START_REF] Reed | The effects of the risk of fire on the optimal rotation of a forest[END_REF] was the first to introduce risk in the Faustmann framework.

He showed that a potential total destruction due to a fire risk acts as an increase in the discount rate.

Such a framework has then been applied to different risks, and recently to disease [START_REF] Macpherson | The effects of disease on optimal forest rotation: a generalisable analytical framework[END_REF]. To our knowledge, few studies focused on the impact of storm risk on optimal rotation length in a Faustmann framework. [START_REF] Haight | Hurricanes and the economics of loblolly pine plantations[END_REF] studied the impact of storm on the expected present value. They showed that age-dependent damage risk and stocking reduction caused by tree mortality have the greatest impact on the relationship between the expected present value and rotation age. [START_REF] Loisel | Impact of storm risk on faustmann rotation[END_REF] showed that considering storm risk leads to earlier optimal thinning than without risk. Recently, [START_REF] Rakotoarison | The faustmann model under storm risk and price uncertainty: A case study of european beech in northwestern france[END_REF] jointly analyzed storm risk and price risk in a Faustmann model. They propose simulations of their theoretical model for European beech stand in Northwestern France. The simulations indicate that the storm risk reduces the optimal rotation length, limits the number of thinning and increases economic loss (but less than price variation).

In this context, it seems that insurance decision and decision related to optimal rotation length have never been studied jointly. Thus, we propose to tackle the question of the impact of insurance decision on the optimal rotation length. For that purpose, we extend the Faustmann optimal rotation model under risk by considering the decision of insurance. Our paper has two additional contributions. First, linked to the rotation and insurance model of the forest owner, we propose to model the microeconomic behavior of the insurer in order to define the components of the insurance contract. Second, in order to analyze insurance decision, we integrate into the model the forest owner's preferences towards risk associated to an expected utility maximization criteria. Such a criteria allows to explicitly consider the forest owner's preferences towards risk, i.e., neutrality, risk aversion or risk loving. Indeed, previous works have proved that forest owners are risk averse on average [START_REF] Musshof | An experimental analysis of the behavior of forestry decision-makers -the example of timing in sales decisions[END_REF][START_REF] Sauter | Faustmann vs. real options theory -an experimental investigation of foresters' harvesting decisions[END_REF][START_REF] Brunette | Attitude towards risk and production decision : An empirical analysis on french private forest owners. Environmental Modeling and Assessment[END_REF]. In addition, the impact of risk aversion on optimal rotation length has been proved in various modeling approaches, like mean-variance approach [START_REF] Ollikainen | A mean-variance approach to short-term timber selling and forest taxation under multiple sources of uncertainty[END_REF][START_REF] Gong | Impact of risk aversion on optimal rotation age[END_REF]Caulfield, 1988;[START_REF] Taylor | Optimum plantation planting density and rotation age based on financial risk and return[END_REF]Valsta, 1992), Wicksellian single rotation framework [START_REF] Alvarez | Does risk aversion accelerate optimal forest rotation under uncertainty[END_REF]), heuristic method [START_REF] Kangas | Incorporating risk attitude into comparison of reforestation alternatives[END_REF][START_REF] Pukkala | A method for integrating risk and attitude toward risk into forest planning[END_REF] or expected utility maximization model [START_REF] Gong | Risk preferences and adaptive harvest policies for even-aged stand management[END_REF]. A majority of papers obtained that risk aversion tends to shorten the optimal rotation length (Caulfield, 1988;[START_REF] Taylor | Optimum plantation planting density and rotation age based on financial risk and return[END_REF][START_REF] Alvarez | Does risk aversion accelerate optimal forest rotation under uncertainty[END_REF][START_REF] Gong | Risk preferences and adaptive harvest policies for even-aged stand management[END_REF] while (Valsta, 1992) found the opposite result. Moreover, one of the main factors explaining the individual's insurance decision is risk aversion. A risk-averse individual is willing to transfer a part of risk to a third party even if it is costly. It is fundamental to consider the risk preferences of the forest owner when analyzing the insurance decision. However, to our knowledge, the forest owner's preferences towards risk have never been considered in a Faustmann modeling.

The rest of the paper is structured as follows. Section 2 presents the model. Section 3 develops the maximization program of the forest owner and the insurer. Finally, Section 4 discusses the results while Section 5 provides the conclusion.

2 Modeling the risk and the insurance contract 2.1 Modeling storm risk and its impact on the stand

We suppose that the occurrence of storm follows a Poisson process, i.e., that storms occur independently of one another, and randomly in time. This is a classical assumption in the literature to model both storm and fire [START_REF] Reed | The effects of the risk of fire on the optimal rotation of a forest[END_REF]Yin and Newman, 1996;[START_REF] Fina | Uncertainty, debt, and forest harvesting: Faustmann revisited[END_REF][START_REF] Ohlson | Multi-attribute evaluation of landscape-level fuel management to reduce wildfire risk[END_REF][START_REF] Armstrong | Using adaptive management to determine requirements of re-introduced populations: the case of the new zealand hihi[END_REF][START_REF] Jacobsen | The regeneration decision: a sequential two option approach[END_REF][START_REF] Loisel | Faustmann rotation and population dynamics in the presence of a risk of destructive events[END_REF]. Thus the distribution of the times between successive storms is an exponential with mean 1/λ: F (x) = 1 -e -λx where λ is the expected number of storms per unit of time. The severity of the storm is given by the random variable A. Let τ be the period of time between the beginning of the stand and, either the storm occurrence or the final harvesting. The storm risk is then described by the couple of random variables (τ, A).

Unlike other types of risk such as fire, the impacts of storm are age-dependent and are low on young stands. Following [START_REF] Schmidt | An inventory-based approach for modeling single-tree storm damage -experiences with the winter storm of 1999 in southwestern germany[END_REF], we assume that, for a tree-height H less equal than H L , there is no damage. The height H L is reached at a time t L , the time t L explicitly depends on the height limit H L and can be deduced using the time dependent function H: H(t L ) = H L . Therefore, in case of storm occurrence before t L , the storm has no impact. At the opposite, in case of storm occurrence after t L , the proportion of damaged trees θ t depends on the severity of the storm A and the characteristics of the stand when the storm occurs (tree height, tree diameter which are age-dependent), hence the proportion of damaged trees is age-dependent. Let α(t) represents the expectation of the proportion of survival trees 1 -θ t .

The sequence of events

Without risk, the forest stand grows from t = 0 (plantation or regeneration) to T , with T the harvesting age. The storm risk modifies the sequence of events as follows:

• If a storm occurs before t L , there is no impact and the stand growth continues.

• If a storm occurs after t L but before T , the proportion of damaged trees is θ , a clear-cutting and a regeneration (or plantation) of the stand is realized.

• If no storm occurs before T , a clear-cutting and a regeneration (or plantation) of the stand is realized at time T . Let τ be the spending time between the beginning of the stand and the first event after t L , i.e., storm occurrence or harvesting at time T . The distribution of the random variable τ is defined for t

L < t ≤ T as F τ (t) = F (t -t L ) = 1 -e -λ(t-t L ) .
Let L(θ, τ ) the loss following a storm occurring at time τ for a proportion of damages trees θ. This loss represents the unexploitability of damaged trees: L(θ, τ ) = θV (τ ) where V (τ ) is the potential final (without risk) income at time τ .

Modeling the insurance contract

An insurance contract is defined as a couple (premium, indemnity), so that we described these two variables in our model.

The contract is characterized by a upper premium P by unit of time (directly linked to the actualization of future losses, see next section) for the full coverage of the damage L(θ, τ ), i.e., damage associated to a storm occurring at time τ . The forest owner chooses a premium P which is a percentage ξ of P, so that P = ξP. If the forest owner chooses ξ = 1 then, s/he is fully insured in case of storm occurrence, in the sense that the the damage is fully covered by insurer. At the opposite, ξ = 0 means no insurance. All the other cases, for 0 < ξ < 1, correspond to partial insurance, where only a part of the damage is covered by insurer, the other part being borne by the forest owner. Then, ξ represents the forest owner's degree of insurance coverage.

The indemnity perceives by the forest owner, in case of a storm occurring at time τ , is then I(θ τ , τ ) = ξL(θ τ , τ ). The indemnity corresponds to the percentage ξ of the loss with, as defined above, 0 ≤ ξ ≤ 1.

The programs of the involved stakeholders

The forest owner's decision variables are the harvesting age T and the percentage ξ as regard to insurance decision, while the insurer chooses the upper premium P. Let's turn to the analysis of each individual's maximization program.

The forest owner's program

In this section, we determine the forest owner's decision related to the rotation length, assuming a given level of insurance. The forest owner is characterized by an utility function u with u ′ (.) > 0 and u ′′ (.) < 0 to represent risk aversion. The forest owner is assumed to maximize the utility of the net economic return from silvicultural activity. This utility of the net economic return Y, actualized at storm occurrence time τ or logging time T , writes as follows:

Y =      P 0 (t, τ ) + H δ (t, τ ) + u(V 1 (θ τ , τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ )) if t L < τ < T P 0 (t, T ) + H δ (t, T ) + u(V (T ) -c 1 )
if τ = T with: -s) ds, the insurance premium paid by the forest owner between the initial time and time t, actualized at time t. -s) ds, the thinning incomes between time t and time t actualized at time τ .

• P 0 (t, t) = ∫ t 0 u(-P)e δ(t
• H δ (t, t) = ∫ t t u(h -P)e δ(t
• V 1 (θ τ , τ ) = (1 -θ τ )V (τ )
, the final income in case of storm occurring at time τ .

• c 1 , the regeneration (or plantation) cost.

• C n (θ τ , τ ) the clearing costs for a storm occurring at time τ .

We deduce the following expression for the Faustmann value J F (see Appendix A):

J F = E[e -δτ Y] 1 -E[e -δτ ] = 1 b(T ) [ a 0 (T )u(-P) + a 1 (T )u(h -P) δ + W F (0, T ) ] where a 0 (T ) = e (δ+λ)T -λt L (1-e -δt ), a 1 (T ) = e (δ+λ)T -λt L -δt - λe (δ+λ)(T -t L ) + δ δ + λ , b(T ) = a 0 (T )+
a 1 (T ) and W F (0, T ) is a modified income such that:

W F (0, T ) =λ ∫ T t L E[u(V 1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))]e (δ+λ)(T -τ ) dτ + u(V (0, T ) -c 1 )
This modified income is composed of two parts. The first part is the integral, with respect to τ , of a reduced final income minus the clearing costs in case of a storm at time τ < T . The second part represents the final income in case of no storm before cutting age T .

This Faustmann value, obtained with storm risk and insurance, differs from the Faustmann value without risk, in several ways. First, the discount rate δ is replaced by the addition of the discount rate and expected number of storms δ + λ (as previously noticed by [START_REF] Reed | The effects of the risk of fire on the optimal rotation of a forest[END_REF] without thinning and Price (2011) with thinning). Second, the utility of the final income W 0 (0, T ) = u(V (T ) -c 1 ) is replaced by W F (0, T ). More precisely, an integral relative to the final cutting minus clearing costs in case of a storm before the cutting age T is added.

The maximum of the Faustmann value is obtained by solving:

max T J F = 1 b(T ) [ a 0 (T )u(-P) + a 1 (T )u(h -P) δ + W F (0, T ) ] (1)
We consider comparative statics relative to the maximization with respect to the rotation period T . We obtain the following results (proof in Appendix B):

Proposition 1 For a risk averse forest owner with differentiated loss and gain risk aversion function

(u(x) = u + x + -u -x -with u + ≤ 1 ≤ u -)
, the optimal cutting age T satisfies: T P > 0 and T increases as the degree of insurance coverage ξ increases.

This result indicates that the risk-sharing strategy and the risk-reducing one are substitutes for the forest owner. Indeed, as the insurance coverage increases, then the forest owner reduces the riskreducing strategy, leading to an increase in the rotation length. This result is in line with the standard result of [START_REF] Ehrlich | Market insurance, self-insurance, and self-protection[END_REF] indicating that insurance and self-insurance are substitutes. Here, the reduction of rotation length may be associated to a self-insurance activity, in the sense that it reduces the damage in case of storm occurrence.

The insurer's program

We assume that the insurer's discount rate is different from the owner's one and is noted δ A . b calculated with discount rate δ is replaced by the corresponding b A with δ A . The Faustmann value J A is:

J A = ξ P δ A -(1 + l f )(1 + m) E[e -δτ I(θ τ , τ )] 1 -E[e -δτ ]
(expectation with respect to θ and τ )

= ξ P δ A -λ(1 + l f )(1 + m) ∫ T t L E[I(θ τ , τ )]e (λ+δ A )(T -τ ) dτ b A (T ) (expectation with respect to θ)
where

E[I(θ τ , τ )] = E[ξL(θ τ , τ )] = ξE[L(θ τ , τ )]
is the expected insurance income, l f is the loading factor and m the security coefficient.

The objective of the insurer is to generate benefits from the insurance activity. In order to have benefit, the insurer must choose P > (1 + l f )(1 + m)P 0 (T ) where P 0 (T ) is the Faustmann value associated to the future losses multiplied by the discount rate δ A for the cutting age T :

P 0 (T ) = δ A E[e -δτ L(θ τ , τ )] 1 -E[e -δτ ] = λ δ A b A (T ) ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ
This constraint depends on the cutting age T chosen by the forest owners. P may not depend on the cutting age T , while in the forest owner's program, T depends on P. In order to ensure a level m for its benefit, the insurer has to evaluate the optimal harvesting age from the forest owner's point of view. P 0 is decreasing with respect to cutting age T . Assuming that the forest owner decides of a cutting age lower than in the case without risk T 0 , then the condition becomes:

P > (1 + l f )(1 + m)P 0 (T 0 ).
The insurer thus has all the information necessary to determine the insurance contract, and to assess the insurance premium and the indemnity.

Analysis of the optimal insurance decision

In this section, we determine the optimal insurance decision considering that the optimal rotation length is given. We assume P = (1 + l f )(1 + m)P 0 (T ) where T is assumed known (and then, not necessarily optimal). The Faustmann value becomes:

J F = a 0 (T )u(-P) + a 1 (T )u(h -P) δb(T ) + P (1 + l f )(1 + m)δ A b A (T ) b(T ) ∫ T t L E[u(V 1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))]e (λ+δ)(T -τ ) dτ ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ (2) + u(V (T ) -c 1 ) b(T )
We consider the derivative of the Faustmann Value J F with respect to insurance premium P. We obtain the following results (proof in Appendix C):

Proposition 2 (i) If the discount rates of the forest owner and the insurer are equal (δ A = δ) then for harvesting age T , premium P * (T ) = 0 is optimal.

(ii) If the forest owner's discount rate and the insurer's discount rate are sufficiently unequal then for This proposition indicates that, in some cases, it may be optimal for a risk averse forest owner to not adopt insurance. This result is not surprising since the premium proposed by the insurer, P, corresponds to a loaded (or unfair) premium. Consequently, the forest owner may consider that the insurance premium is too high and may prefer to not insure despite risk aversion. A way to reduce the insurance premium may be that the public authority subsidizes the premium.

Insurance subsidy

In the literature, the subsidy for the forest insurance premium has already been mentioned in [START_REF] Brunette | The impact of governmental assistance on insurance demand under ambiguity: A theoretical model and an experimental test[END_REF]. They obtained theoretically that, when the government provides a subsidy to the forest owner for her/his insurance premium, then the forest owner should increase the insurance demand.

However, they do not validate this result with the empirical part of the paper. In this article, we adopt another idea which is that, rather than to subsidy the forest owner directly, the government subsidies the insurer.

Indeed, we consider that the government gives a percentage γ of the indemnity to insurer. Consequently, it is as a coinsurance contract where a part γ of the damage is borne by the government while 1 -γ is borne by the insurer (and/or the forest owner in case of partial insurance demand). The risk supported by the insurer is lower, the indemnity paid by the insurer in case of storm occurrence also and the insurance premium too, hence:

P 0 (T ) = (1 -γ) λδ A b A (T ) ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ
We consider the derivative of the Faustmann Value J F with respect to insurance premium P when δ A = δ. We obtain the following results (proof in Appendix D):

Proposition 3 If the discount rates of the forest owner and the insurer are equal (δ A = δ), and if the government gives a percentage γ such that 0 < γ(δ) < γ < γ(δ) < 1 of the indemnity to the insurer then an insurance premium P * (T ) > 0 may be optimal.

By subsidizing the insurer, the government increases the attractiveness of insurance for the forest As underlined by [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF], the adoption of a forest insurance contract is still an exception in Europe. For example, in France, only approximately 4% of the French private forests are insured representing 400,000 ha of private forests. Similar observations are obtained for Spain, Germany or Slovakia [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF]. The literature explores different potential explanations for this low level of insurance, among which the existence of public program to financially help the forest owners after the occurrence of storms. In this context, [START_REF] Brunette | Public compensation for windstorm damage reduces incentives for risk management investments[END_REF] and [START_REF] Brunette | The impact of governmental assistance on insurance demand under ambiguity: A theoretical model and an experimental test[END_REF] showed that a public help reduces the forest owners' insurance demand, but that this negative effect is lower when the help is contingent to insurance adoption rather than fixed (i.e., independent from any insurance contract adoption). In this paper, we provide another potential explanation to the nonadoption of insurance contract by private forest owners, the fact that, in some cases, insurance may be considered as too expensive by the forest owner (see Proposition 2). In addition, as regard to the existing literature, we explore another channel from which the government may participate to the insurance scheme. Indeed, we show that, by subsidizing the insurer, the government encourages the forest owner to adopt insurance contract. Consequently, an horizontal risk sharing between the insurer, the forest owner and the government may be an efficient way to insure storm risk in forest.

Conclusion

In this paper, we propose to extend the classical Faustmann rotation model under risk by considering the insurance decision. This means that, in addition to the decision related to the rotation length, the forest owner also decides on a degree of insurance coverage. Such a challenge requires to model the insurer's behavior (in order to determine the contract) and to consider the forest owner's preferences towards risk (in order to analyze the insurance decision), two supplementary originalities for this paper. In this framework, our results indicate that the forest owner considers that the risk-sharing strategy (insurance) and the risk-reducing one (reduction of the rotation length) are substitutes. In addition, we show that, in some cases, it may be optimal for the forest owner to not adopt insurance. Finally, we conclude that, a way to incite forest owners to adopt insurance contract is to consider that the government subsidizes the insurer.

A The Faustmann Value

As we assume that the storm occurs independently of one another and randomly in time, we deduce the following expression for the land value:

J F = E[ ∞ ∑ i=1 e -δ(τ 1 +τ 2 +...+τ i ) Y i ] = ∞ ∑ i=1 i-1 ∏ j=1 E[e -δτ j ].E[e -δτ i Y i ] = [e -δτ Y] 1 -E[e -δτ ] (3)
From the definition of τ we deduce the expectation E[e -δτ ]:

E[e -δτ ] = ∫ T t L e -δτ dF (τ -t L ) + e -δT (1 -F (T -t L )) = λ + δe -(δ+λ)(T -t L ) δ + λ e -δt L
We deduce the expected net economic return E[e -δτ Y] actualized at initial time:

1 -e -δt δ u(-P) + e -δt -e -δt L δ u(h -P) + e -δt L -e (δ+λ)T -λt L δ + λ u(h -P) + ∫ T t L (E[u(V 1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))])e -δτ dF (τ -t L ) + u(V (T ) -c 1 )e -δT (1 -F (T -t L ))
and hence the result.

B Proof of Proposition 1

To lighten the presentation, in some calculus, we omit the λ dependency in functions a 1 , b and denote a T (resp. b T ) the partial derivative of a 1 (resp. b) with respect to T .

From (1), differentiating J F with respect to the rotation age T gives the following first-order condition:

J T = [ W F (0, T ) + a 1 (T ) δ ((hu -)I h<P + (hu + + P(u --u + ))I h>P ) b(T ) ] T = 0
We consider the case h > P (the other case is simpler). From the first-order condition J T = 0 we deduce:

W F (0, T ) T b(T ) -W F (0, T )b T (T ) + (a T (T )b(T ) -a 1 (T )b T (T )) hu + + P(u --u + ) δ = 0 (4)
Differentiating J T = 0 with respect to P yields: J T T T P + J T P = 0. From J T T < 0, T P and J T P have the same sign. J T P is proportional (with the same sign) to the sum of: γ(τ )e (λ+δ)(t-τ ) dτ , then:

A P = W F (
A ′ P (t) =γ ′ (t)b(t) + γ(t)b T (t) -δγ(t) -(λ + δ)(γ(t)b(t) -A P (t))
=γ ′ (t)b(t) + (λ + δ)A P (t)

From α ′ (t) < 0 we deduce γ ′ (t) > 0, moreover from A P (0) = 0 we deduce that A P (T ) > 0.

C Proof of Proposition 2

We consider the derivative of the Faustmann Value J F with respect to insurance premium P:

∂J F ∂P = -a 0 (T )u ′ (-P) + a 1 (T )u ′ (h -P) δb(T )

+ 1 (1 + l f )(1 + m)δ A b A (T ) b(T ) ∫ T t L E[u ′ (V 1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))L(θ, τ )]e (λ+δ)(T -τ ) dτ ∫ T t L E[L(θ τ , τ
)]e (λ+δ A )(T -τ ) dτ (i) From the concavity of the utility u, and using -P < h -P < V 1 (θ, τ ) + I(θ τ , τ ) -c 1 then:

∂J F ∂P < 1 δ (-1 + 1 (1 + l f )(1 + m) δ δ A b A (T ) b(T ) ∫ T t L E[L(θ τ , τ )]e (λ+δ)(T -τ ) dτ ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ
)u ′ (h -P)

From δ A = δ then b A = b and ∂J F ∂P < 1 δ (-1 + 1 (1 + l f )(1 + m)
)u ′ (h -P) < 0, hence the result.

(ii) ∂J F ∂P is proportional to:

-(a 0 (T )u ′ (-P) + a 1 (T )u ′ (h -P))

δ A b A (T ) ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ + 1 (1 + l f )(1 + m) δ ∫ T t L E[u ′ (V

  0, T ) T P b(T ) -W F (0, T ) P b T (T ) and (a T (T )b(T ) -a 1 (T )b T (T ))(u --u + ) The difference a T (T )b(T ) -a 1 (T )b T (T ) is positive. Concerning A P : W F (0, T ) α(τ ))V (τ )e (λ+δ)(T -τ ) dτ W F (0, T ) T P =λ u + P (1 -α(T ))V (T ) + (λ + δ)W F (0, T ) P From b T (T ) = δ + (δ + λ)b(T )we deduce:A P =λ u + P (1 -α(T ))V (T )b(T ) -δW F (0, T ) α(τ ))V (τ )e (λ+δ)(T -τ ) dτ ] Let γ(t) = λ u + P (1 -α(t))V (t)b(t) then: A P = γ(T )b(T ) -δ ∫ T t L γ(τ )e (λ+δ)(T -τ ) dτMoreover, studiing the behavior of A P (t) = γ(t)b(t) -δ∫ t t L

  1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))L(θ, τ )]e (λ+δ)(T -τ ) dτIf the forest owner's discount rate and the insurer's discount rate are sufficiently unequal then ∂J F ∂P > 0for sufficiently large value of δ, hence ∂J F ∂P is equal to zero for smaller δ and we deduce the existence of a non trivial solution P * (T ) > 0.

Acknowledgments

The UMR BETA is supported by a grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (ANR-11-LABX-0002-01, Lab of Excellence ARBRE).

 Science, 38:430-447, 1992.R. Yin and D.H. Newman. The effect of catastrophic risk on forest investment decisions. Journal of Environmental Economics and Management, 31:186-197, 1996.

D Proof of Proposition 3

If δ A = δ then b A = b and:

∂J F ∂P is positive for sufficiently large value γ and is negative for γ = 0. Hence we deduce that for intermediate γ, ∂J F ∂P may be equal to 0, hence a non null premium P will be optimal.


domain: sdv info

text:
Key concept in pharmacology

˛Biased agonism is a key concept to be distinguish from § Partial or full agonist.

§ Antagonist, inverse agonist.

§ Affinity (K d ), potency pEC 50 q, efficacy (E max ).

˛A bias might be context-dependent (cell type, physiological state, etc.)

˛Biased agonism is becoming a major tool in drug discovery.

ñ Candidate screening requires to accurately quantify bias. § Quipazine is biaised towards PI accumulation with respect to AA production, compared to the reference agonist DOI.

A receptor may adopt several spatial conformations, each of which has different activation pathway profiles. § Conformational selectivity = Ligand-specific modification of the energetic landscape, changing affinities and efficacies of signaling patways.

Kenakin, J Pharmacol Exp Ther (2011) § Quipazine is biaised towards PI accumulation with respect to AA production, compared to the reference agonist DOI. § LSD is not biased. § Bias due to an E max difference.

pRq ´2C ´B ´CB is biaised towards PI accumulation with respect to AA production, compared to the reference agonist DOB.

Outline

What is biased signaling ?

Some examples

Bias calculus -standard method : operational model

Some extension to bias calculus

Biased signaling using dynamical model

Operational model

Dose-response data are fitted with the function

y " E tot τ n rLs n prLs `KA q n `τ n rLs n . § Response at equilibrium of a Michaelis-Menten type model. § K A " Dissociation constant of the couple Ligand/Receptor § τ " R tot {K E , Efficacy coefficient (K E is dissociation constant of the ternary complex)
Black and Leff, Proc. R. Soc. Lond. B (1983) y " E tot τ n rLs n prLs `KA q n `τ n rLs n .

For n " 1, § EC 50 " K A τ `1 § Efficacy y 8 {E tot " τ τ `1 Black and Leff, Proc. R. Soc. Lond. B (1983) y " E tot τ n rLs n prLs `KA q n `τ n rLs n .

For n " 1, § EC 50 " K A τ `1 § Efficacy y 8 {E tot " τ τ `1 Then, we define § Transduction coefficient : R :" log ˆτ K A ˙Black and Leff, Proc. R. Soc. Lond. B (1983) Two ligands (j " 1, 2) and two measured responses (i " 1, 2) : Each dose-response data is fitted with the operational model :

y ij " E i τ n i ij rLs n i prLs `Ka ij q n i `τ n i ij rLs n i .
Two ligands (j " 1, 2) and two measured responses (i " 1, 2) : Each dose-response data is fitted with the operational model :

y ij " E i τ n i ij rLs n i prLs `Ka ij q n i `τ n i ij rLs n i .
For a given response i, we calculate ∆ i logpτ {K a q " logpτ i2 {Ka i2 q ´logpτ i1 {Ka i1 q.

Two ligands (j " 1, 2) and two measured responses (i " 1, 2) : Each dose-response data is fitted with the operational model :

y ij " E i τ n i ij rLs n i prLs `Ka ij q n i `τ n i ij rLs n i .
For a given response i, we calculate Instead of focusing on dose-response curves, we deal with several doses kinetic experiments (here : induced BRET data) ("trick" to minimize variance...)

Original "raw" data II) We fit all data at once, using some common parameters (initial concentration of molecules, measurement parameters...) and some different ones (kinetic parameters...)

II)

We fit all data at once, using some common parameters (initial concentration of molecules, measurement parameters...) and some different ones (kinetic parameters...) IV) After re-optimization, the set of distinct (ligand-specific) kinetic parameters gives us an accurate description of ligand specificity.

Principle

V) Significant differences between parameters is assessed by PLE C1 is biased towards β-arr, compared to cAMP, in comparison to FSH. With a "simpler" model

But consistent results

With a "simpler" model

And "better" parameter identifiability

-3 -2 -1 0 1 2 3 4 0 σ 2σ FSH param. k1 k2 koff Kd k1off k2off -2 -1 0 1 2 3 4 5
Parameter values

  receptor by hCG. § Selective (biased) allosteric modulation Ayoub et al., Mol. Cell. Endocrinol (2016) Many GPCR's are known to have biased ligands ( G / β-arrestin)

  Instead of focusing on dose-response curves, we deal with several doses kinetic experiments (here : induced BRET data)

+

  adjusting the number of data points ... I)We start with a sufficiently detailed chemical reaction network Principle I)We start with a sufficiently detailed chemical reaction network to accurately fit the data (one separate model for each Ligand)

  use L 1 -penalization to find the needed ligand specific parameters Steiert, Timmer and Kreutz, Bioinformatics (2016) III) We use L 1 -penalization to find the needed ligand specific parameters, keeping the fit 'as good as before'

  value several pathways by a Ligand at a given receptor. § Standard quantification has several drawbacks (no time, limited to sigmoid scenario,..). § We gave a kinetic interpretation of Ligand biased, which rely on dynamic (ODE) modeling and parameter estimation with L 1 penalization.

					10 4 parameter #5	parameter #6	parameter #7	parameter #8
	2	PL	-6.133		95% (point-wise)			likelihood likelihood
			-6.1332
			10 6 -6.1334
			10 4	-0.5 log 10 (k1) -0.4 10 4 parameter #9	-1.4 -1.3 -1.2 log 10 (k1off) parameter #10	-2.8 parameter #17 -2.7 log 10 (k2)	6.5 parameter #18 7 7.5 log 10 (kd)
	2	PL	10 2 -6.1332 10 4 -6.133
			-6.1334
					0.5	1	1.5	-0.4	converged fits -0.3 -0.2 -0.06	-0.04	0.1 0.14 0.18
	σ 2σ 2 PL	10 2 -6.1334 -6.1332 -6.133 10 0	C1 rel. param 400 initial objective function value k1 k2 koff Kd k1off k2off converged fits run index (sorted by likelihood) initial objective function 0 100 200 300 log 10 (koff) log 10 (krep) log 10 (relto_239_k1) log 10 (relto_239_k1off) 500 10 4 parameter #19 parameter #20 parameter #21 parameter #22
		0			0.8	0.9	1	0.1	0.2	0.3	3.85	3.9
			10 0	0 log 10 (relto_239_k2) 100	200 log 10 (relto_239_k2off)	300 log 10 (relto_239_kd)	400 0 log 10 (relto_239_koff) 500 5 10
								run index (sorted by likelihood)
	C1 is biased towards β-arr, compared to cAMP, in comparison to
										FSH.

Urban et al., J Pharmacol Exp Ther (2007)

§ Notion of signaling bias to quantify differential activation of several pathways by a Ligand at a given receptor. § Standard quantification has several drawbacks (no time, limited to sigmoid scenario,..). § We gave a kinetic interpretation of Ligand biased, which rely on dynamic (ODE) modeling and parameter estimation with L 1 penalization. § Any other ideas how to define bias ? § How to deal with "fuzzy/noisy" PLE ? § How to deal with non uniqueness of the penalized solution ? § How to perform a model reduction that would lead to both a satisfactory fit and identifiable parameters ?


est destinée au dépôt

  Update in the geographical distribution of the invasive tick Hyalomma marginatum in south of France; First attempts to identify factors favoring its establishment Laurence Vial, Karine Huber, Anaïs Appelgren, Clement Calloix, Claude Andary, Vladimir Grosbois, Renaud Lancelot, Frederic Stachurski To cite this version: Laurence Vial, Karine Huber, Anaïs Appelgren, Clement Calloix, Claude Andary, et al.. Update in the geographical distribution of the invasive tick Hyalomma marginatum in south of France; First attempts to identify factors favoring its establishment. 21. European Society for Vector Ecology Conference (E-SOVE), Oct 2018, Palerme, Italy. hal-02785256


domain: math info

text:
corpus with more than 6,000 transactions, 3 centuries, all related to Castelnau Montratier Individual Transaction q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Ratier Ratier (II) Castelnau Jean Laperarede

Bertrande Audoy

Formally, relational data are: Different relational data types are related to each others a kernel is equivalent to an Euclidean distance:

D(x, x ) := K(x, x) + K(x , x ) -2K(x, x )
from a dissimilarity, similarities can be computed: Different relational data types are related to each others a kernel is equivalent to an Euclidean distance:

S(x, x) := a(x) (arbitrary), S(x, x ) = 1 2 a(x) + a(x ) -D 2 (x,
D(x, x ) := K(x, x) + K(x , x ) -2K(x, x )
from a dissimilarity, similarities can be computed: ∃ two Euclidean spaces E + and E - and two mappings φ + and φ -st: Iterative learning (representation step): all prototypes in neighboring units are updated with a gradient descent like step:

S(x, x) := a(x) (arbitrary), S(x, x ) = 1 2 a(x) + a(x ) -D 2 (x,
D(x, x ) = φ + (x) -φ + (x ) 2 E + - φ -(x) -φ -(x ) 2
p t+1 u ←-p t u + µ(t)H t (d(f (x i ), u))(x i -p t u )
Nathalie Villa-Vialaneix | Learning from (dis)similarity data 16/35

Extension of SOM to data described by a kernel or a dissimilarity [Olteanu and Villa-Vialaneix, 2015a] Data: (x i ) i=1,...,n ∈ R d 1: Initialization: randomly set p 0 1 , ..., p 0 for all u = 1 → U do Representation 6:

U in R d 2: for t = 1 → T do 3: pick at random i ∈ {1, . . . ,
p t+1 u = p t u + µ(t)H t (d(f t (x i ), u)) x i -p t u 7:
end for 8: end for Extension of SOM to data described by a kernel or a dissimilarity [Olteanu and Villa-Vialaneix, 2015a] Data: (x i ) i=1,...,n ∈ X 1: Initialization: randomly set p 0 1 , ..., p 0 for all u = 1 → U do Representation 6:

U in R d 2: for t = 1 → T do 3: pick at random i ∈ {1, . . . ,
p t+1 u = p t u + µ(t)H t (d(f t (x i ), u)) x i -p t u 7:
end for 8: end for Extension of SOM to data described by a kernel or a dissimilarity [Olteanu and Villa-Vialaneix, 2015a] Data: (x i ) i=1,...,n ∈ X 1: Initialization: for all u = 1 → U do Representation 6:

p 0 u = n i=1 β 0 ui φ(x i ) (convex combination) 2: for t = 1 → T do 3: pick at random i ∈ {1, . . . ,
p t+1 u = p t u + µ(t)H t (d(f t (x i ), u)) x i -p t u 7:
end for 8: end for Extension of SOM to data described by a kernel or a dissimilarity [Olteanu and Villa-Vialaneix, 2015a] Data: (x i ) i=1,...,n ∈ X 1: Initialization:

p 0 u = n i=1 β 0 ui φ(x i ) (convex combination) 2: for t = 1 → T do 3: pick at random i ∈ {1, . . . , n} 4: Assignment f t (x i ) = arg min u=1,...,U φ(x i ) -p t u 2 H 5: for all u = 1 → U do Representation 6: p t+1 u = p t u + µ(t)H t (d(f t (x i ), u)) x i -p t u 7:
end for 8: end for Extension of SOM to data described by a kernel or a dissimilarity [Olteanu and Villa-Vialaneix, 2015a] Data: (x i ) i=1,...,n ∈ X 1: Initialization:

p 0 u = n i=1 β 0 ui φ(x i ) (convex combination) 2: for t = 1 → T do 3: pick at random i ∈ {1, . . . , n} 4: Assignment f t (x i ) = arg min u=1,...,U φ(x i ) -p t u 2 H 5: for all u = 1 → U do Representation 6: p t+1 u = p t u + µ(t)H t (d(f t (x i ), u)) φ(x i ) -p t u 7:
end for 8: end for Extension of SOM to data described by a kernel or a dissimilarity [Olteanu and Villa-Vialaneix, 2015a] Data: (x i ) i=1,...,n ∈ X 1: Initialization:

p 0 u = n i=1 β 0 ui φ(x i ) (convex combination) 2: for t = 1 → T do 3: pick at random i ∈ {1, . . . , n} 4: Assignment f t (x i ) = arg min u=1,...,U (β t u ) Kβ t u -2(β t u ) K(., x i ) 5:
for all u = 1 → U do Representation 6:

β t+1 u = β t u + µ(t)H t (d(f t (x i ), u)) 1 i -β t u 7:
end for 8: end for Extension of SOM to data described by a kernel or a dissimilarity [Olteanu and Villa-Vialaneix, 2015a] Data: (x i ) i=1,...,n ∈ X 1: Initialization:

p 0 u ∼ n i=1 β 0 ui x i (convex combination) 2: for t = 1 → T do 3: pick at random i ∈ {1, . . . , n} 4: Assignment f t (x i ) = arg min u=1,...,U D(p t u , x i ) 5:
for all u = 1 → U do Representation 6:

p t+1 u = p t u + µ(t)H t (d(f t (x i ), u)) ∼ x i -p t u 7:
end for 8: end for Extension of SOM to data described by a kernel or a dissimilarity [Olteanu and Villa-Vialaneix, 2015a] Data: for all u = 1 → U do Representation 6:

(x i ) i=1,...,n ∈ X 1: Initialization: p 0 u ∼ n i=1 β 0 ui x i (convex
β t+1 u = β t u + µ(t)H t (d(f t (x i ), u)) 1 i -β t u 7:
end for 8: end for SOMbrero [START_REF] Villa-Vialaneix | Stochastic self-organizing map variants with the R package SOMbrero[END_REF], https://cran.r-project.org/package=SOMbrero stochastic variants of SOM (standard, KORRESP and relational) with a large number of diagnostic plots specific functions to use with graphs and obtain simplified representations Similarities between kernels:

C mm = K m , K m F K m F K m F = Trace(K m K m ) Trace((K m ) 2 )Trace((K m ) 2 )
. Similarities between kernels:

(
C mm = K m , K m F K m F K m F = Trace(K m K m ) Trace((K m ) 2 )Trace((K m ) 2 )
.

(C mm is an extension of the RV-coefficient [START_REF] Robert | A unifying tool for linear multivariate statistical methods: the rv-coefficient[END_REF] to the kernel framework)

∆ i (β) = φ * β (x i ),              φ * β (x 1 ) . . . φ * β (x n )              =              K * β (x i , x 1 ) . . . K * β (x i , x n )             

  Euclidean distances or (non Euclidean) dissimilarities between n entities: symmetric (n × n)-matrix D with positive entries and null diagonal kernels: a symmetric and positive definite (n × n)-matrix K that measures a "relation" between n entities in X (arbitrary space) K(x, x ) = φ(x), φ(x ) networks/graphs: groups of n entities (nodes/vertices) linked by a (potentially weighted) relation (edges) ⇒ symmetric (n × n)-matrix with positive entries and null diagonal W Nathalie Villa-Vialaneix | Learning from (dis)similarity data 8/35 Formally, relational data are: Euclidean distances or (non Euclidean) dissimilarities between n entities: symmetric (n × n)-matrix D with positive entries and null diagonal kernels: a symmetric and positive definite (n × n)-matrix K that measures a "relation" between n entities in X (arbitrary space) K(x, x ) = φ(x), φ(x ) networks/graphs: groups of n entities (nodes/vertices) linked by a (potentially weighted) relation (edges) ⇒ symmetric (n × n)-matrix with positive entries and null diagonal W Similarities between n entities: symmetric (n × n)-matrix S (with usually positive entries) but not necessarily definite positive Nathalie Villa-Vialaneix | Learning from (dis)similarity data 8/35

  x ) various kernels have been proposed for graphs (e.g., based on the graph Laplacian): [Kondor and Lafferty, 2002] Nathalie Villa-Vialaneix | Learning from (dis)similarity data 9/35

[

  ) (time) and O(n 2 ) (space) adjclust: O(nh + n log n) (time) and O(nh) (space) with h the non sparse band around the diagonal Icing on the cake: wrappers for Hi-C datasets and LD datasets model selection methods (broken stick and slope heuristic) corrected dendrogram to avoid reversals [Grimm, 1987] ... and other nice plots to compare data with clustering Nathalie Villa-Vialaneix | Learning from (dis)similarity data 13/35 Application to Hi-C data with data from [Dixon et al., 2012] ) i=1,...,n ⊂ R d are affected to a unit f (x i ) ∈ {1, . . . , U} the grid is equipped with a "distance" between units: d(u, u ) and observations affected to close units are close in R d every unit u corresponds to a prototype, p u (x) in R d Iterative learning (assignment step): x i is picked at random within (x k ) k and affected to best matching unit: f t (x i ) = arg min

  n}

  n}

[

  Olteanu and Villa-Vialaneix, 2015b] Nathalie Villa-Vialaneix | Learning from (dis)similarity data SOMbrero [Villa-Vialaneix, 2017], https://cran.r-project.org/package=SOMbrero stochastic variants of SOM (standard, KORRESP and relational) with a large number of diagnostic plots specific functions to use with graphs and obtain simplified representations [Olteanu and Villa-Vialaneix, 2015b] contains comprehensive vignettes illustrated on 3 datasets corresponding to the three algorithms (iris, presidentielles2002 and lesmis, a graph from "Les Misérables") Nathalie Villa-Vialaneix | Learning from (dis)similarity data 18/35 SOMbrero [Villa-Vialaneix, 2017], https://cran.r-project.org/package=SOMbrero stochastic variants of SOM (standard, KORRESP and relational) with a large number of diagnostic plots specific functions to use with graphs and obtain simplified representations [Olteanu and Villa-Vialaneix, 2015b] contains comprehensive vignettes illustrated on 3 datasets corresponding to the three algorithms (iris, presidentielles2002 and lesmis, a graph from "Les Misérables") Web User Interface (made with shiny) with sombreroGUI() Tested on and approved by an historian! STATIS like framework [L'Hermier des Plantes, 1976, Lavit et al., 1994]

  m and v ∈ R M such that v 2 = 1. feature space. m I {A m k >0} or W = m A m k Nathalie Villa-Vialaneix | Learning from (dis)similarity data 27/35 A kernel preserving the original topology of the data I Similarly to [Lin et al., 2010], preserve the local geometry of the data in the feature spacem I {A m k >0} or W = m A m k Feature space geometry measured by

[Kondor and Lafferty, 2002]

	Principles for learning from relational data Principles for learning from relational data
	Euclidean case (kernel K) Euclidean case (kernel K)	non Euclidean case (non Euclidean
	rewrite all quantities using: rewrite all quantities using: rewrite all quantities using:	dissimilarity D): do almost the same
	K to compute distances and dot K to compute distances and dot K to compute distances and dot	using a pseudo-Euclidean framework
	products products products
	linear or convex combinations of linear or convex combinations of linear or convex combinations of
	(φ(x i )) i to describe all (φ(x i )) i to describe all (φ(x i )) i to describe all
	unobserved elements (centers unobserved elements (centers unobserved elements (centers
	of gravity and so on...) of gravity and so on...) of gravity and so on...)
	Works for: PCA, k -means, linear Works for: PCA, k -means, linear in summary regression, ... regression, ...
	useful simplification: "is the framework Euclidean or not?" (e.g., kernel vs
	non Euclidean dissimilarity)
	Nathalie Villa-Vialaneix | Learning from (dis)similarity data	10/35

x ) various kernels have been proposed for graphs (e.g., based on the graph Laplacian):

[START_REF] Goldfarb | A unified approach to pattern recognition[END_REF]

Hermier des Plantes, 1976, Lavit et al., 1994]

C mm is an extension of the RV-coefficient [Robert and Escoufier, 1976] to the kernel framework) Nathalie Villa-Vialaneix | Learning from (dis)similarity data 26/35 STATIS like framework [L'

Nathalie Villa-Vialaneix | Learning from (dis)similarity data

Principles for learning from relational data Euclidean case (kernel K)

Nathalie Villa-Vialaneix | Learning from (dis)similarity data 10/35

Nathalie Villa-Vialaneix | Learning from (dis)similarity data 14/35

Nathalie Villa-Vialaneix | Learning from (dis)similarity data 16/35

Nathalie Villa-Vialaneix | Learning from (dis)similarity data 20/35

Nathalie Villa-Vialaneix | Learning from (dis)similarity data 25/35

Nathalie Villa-Vialaneix | Learning from (dis)similarity data 26/35

[L'Hermier des Plantes, 1976,[START_REF] Lavit | The ACT (STATIS method)[END_REF]

Similarly to[START_REF] Lin | Multiple kernel learning for dimensionality reduction[END_REF], preserve the local geometry of the data in the feature space.Nathalie Villa-Vialaneix | Learning from (dis)similarity data

Similarly to[START_REF] Lin | Multiple kernel learning for dimensionality reduction[END_REF], preserve the local geometry of the data in the

Nathalie Villa-Vialaneix | Learning from (dis)similarity data 35/35

Note on drawbacks of RSOM Two main drawbacks:

For T ∼ γn iterations, complexity of RSOM is O(γn 3 U) (compared to O(γUdn) for numeric) [START_REF] Rossi | How many dissimilarity/kernel self organizing map variants do we need?[END_REF] Exact solution proposed in [START_REF] Mariette | Accelerating stochastic kernel som[END_REF] RSOM for mining a medieval social network with the heat kernel Individual Transaction q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Ratier Ratier (II) Castelnau Multi-kernel/distances integration How to "optimally" combine several relational datasets in an unsupervised setting?

for kernels K 1 , . . . , K M obtained on the same n objects, search:

.

(C mm is an extension of the RV-coefficient [START_REF] Robert | A unifying tool for linear multivariate statistical methods: the rv-coefficient[END_REF] to the kernel framework)

A kernel preserving the original topology of the data II


Dinh Sébastien Ribeiro

Minh Tran

Marine Deon

André Clément-Demange

Dominique Garcia

Mouman Soumahoro

Aurélien Masson

Valérie Pujade-Renaud

Dinh Minh Tran

Marine Déon

Deletion of the cassiicolin-encoding gene Cas1 from Corynespora cassiicola causes a loss of virulence on rubber tree

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Cassiicolin Cas1 is the major virulence factor involved in the compatible interaction between CCP and susceptible rubber clones. It could be a good candidate for effector-based selection.

Rubber tree (Hevea brasiliensis) is an economically important tropical tree that provides natural rubber. In Asia and Africa, it is affected by the Corynespora Leaf Fall (CLF) disease caused by the necrotrophic fungus Corynespora cassiicola. On the most susceptible rubber clones, C. cassiicola can cause massive defoliation leading to yield losses. Some pathogenic strains secrete a small protein toxin, the cassiicolin 1,2,3 , which is transcriptionally up-regulated during early stages of the disease [START_REF] Déon | Plant science : an international journal of experimental plant biology[END_REF] . Seven isoforms were identified and used to classify the various strains into toxin classes [START_REF] Déon | [END_REF] . Strains belonging to toxin class Cas1 were the most aggressive on tested rubber clones [START_REF] Déon | [END_REF] .

However, strains without cassiicolin gene (toxin class Cas0) may also produce toxic exudates 6 , suggesting the existence of effectors other than cassiicolin. The growth rate, conidia production and percentage of germination were compared in vitro between CCP and ccpΔcas1 strains.

Background

Strain

Objective 3 :Result 2 :

 32 Construction of the Cas1 gene deletion mutantPhysiological analyses of CCP and ccpΔcas1 susceptible clone tolerant cloneVirulence of the ccpΔcas1 strainResult At least two QTL (in yellow) were detected with the purified cassiicolin and with Cas1 culture filtrates, but none with ccpΔcas1. Without cassiicolin Cas1, the CCP strain is avirulent on susceptible clones, except for a few pinpoint symptoms.Deletion mutant was obtained by replacing the Cas1 gene from the highly aggressive strain CCP with a hygromycin-resistance cassette by homologous recombination.The deletion mutant was named ccpΔcas1.

CCP

  and ccpΔcas1 strains were compared for their virulence on four rubber clones (still attached to the plant), by analyzing the extent of symptoms as the mean surface of necrotic tissue (mm²), nine days after inoculation (200 conidia/drop, 6 drops per leaf).

  of necrosed tissue (mm²)

QTL associated with sensitivity of rubber tree to cassiicolin Cas1

12 th Congress of the International Plant Molecular Biology (IPMB)

QTL detection was conducted on the PB260 x RRIM600 family for the response to purified cassiicoline Cas1, to culture of isolates carrying or not a Cas1 gene (including CCP and ccpΔcas1) and to blank treatments (culture medium and water). The response was quantified by conductivity measurement of the induced electrolyte leakage 6 .

Conclusion

Toxin class (Cas1 or Cas0) is indicated for each isolate. QTL are named by the number of the linkage group (g) on which they are located, and their position. Numbers represent the percentage of explained phenotypic variance.


Allan Debelle

Simon Fellous

Alexandre Courtiol

Ted Morrow

Exploring the environmental dependence of sexual conflict using Drosophila

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Fitnesso

  of each hemiclonal line as male or as female in the field (number of offspring using phenotypic markers) Fitness measured as males and as females o Presence of sexually antagonistic genetic variation for fitness o 3 types of lines • High male/low female fitness (HM/LF) • Low male/high male fitness (LM/HF) • concordant male--female fitness (CONC) per line and per sex : 150 cages • 2 blocks = Field 1 & 2 (Aug./Sep. in Montpellier) • ♂ assay : 5 hemiclonal ♂ + 10 ♂competitors + 15 ♀ OR • ♀ assay : 5 hemiclonal ♀+ 10 ♀competitors + 15 ♂ • Type of lines changed position on the fitness space between all environments • Variance decreased in field environments (LRT: Chi--square = 45.645, p<0.001) • No correlation between lab and field environments • No correlation between the 2 field environments either • Negative r MF signal is blurred in both field environments • Absolute fitness decreased in the field • Unfavorable conditions: Lab > Field 1 Blurred signal of sexual antagonism between environments + Absence of sex--specific correlation between environments Ø Environmental dependency of sexual conflict Reduction of variance and mean fitness in the field environments Ø Unfavorable environment masks the expression of genetic variation ? Allan Debelle 1 , Simon Fellous 1 , Alexandre Courtiol 2 , Ted Morrow 3 1 INRA, Centre de Biologie et de Gestion des Populations (UMR 1062 INRA/IRD/Cirad/Montpellier SupAgro), Montferrier--Sur--Lez, France ; 2 Department of Evolutionary Genetics, Leibniz Institute for Zoo and Wildlife Research, Berlin, Germany; 3 School of Life Sciences, University of Sussex, Brighton, U.K. allandebelle@gmail.com @LordFruitFly Environmental variation can maintain genetic variation for fitness (either concordant or sexually antagonistic) Depending on the structure of environmental variation (e.g. autocorrelation), selection will have the opportunity to act consistently on this variation37


domain: math info shs.eco

text:
Patrice Loisel
email: patrice.loisel@inra.fr

Marielle Brunette
email: marielle.brunette@inra.fr

Stéphane Couture
email: stephane.couture@inra.fr

Insurance and forest rotation decisions under storm risk

Keywords: Faustmann Value, insurance, risk-sharing, storm risk, rotation length, subsidy JEL: D81, G22, Q23

Forests are often threatened by storms; and such a threat is likely to increase due to climate change. Indeed, climate change is expected to increase the frequency and intensity of extreme events such as intensive storms. For forest owners, adaptation to climate change will require major adjustments in forest management practices. Then, forest owners will have to take these increasing risks into account through risk-sharing (insurance) and risk-reducing strategies (reduction of rotation length). In this paper, we propose to jointly analyze the forest owner's insurance decision and the rotation age under storm risk. We extend the Faustmann optimal rotation model under risk, first, considering the forest owner's preferences towards risk, and second, integrating the decision of insurance. With this analytical model, we show that unless the forest owner's discount rate and the insurer's discount rate are sufficiently unequal, it is not optimal for the forest owner to adopt insurance. We prove that the rotation length increases as the insurance premium increases, i.e.

substitution between the two coverage mechanisms. Finally, we discuss the potential implications of government policy on the insurance scheme.

Introduction

Natural hazards represent serious threat for forest ecosystems. In particular, storms are the more damageable event for European forest since they are responsible for more than 50% of the European forest damage over the period 1950-2000 [START_REF] Schelhaas | Natural disturbances in the european forests in the 19th and 20th centuries[END_REF]. In the same vein, [START_REF] Gardiner | Living with storm damage to forests[END_REF] reported 130 storms over the last sixty years in Europe, i.e., an average of two storms per year. Since 1999, three main storms generate important damages in European forests, and in particular in France.

Lothar and Martin in 1999 induced 6 billions Euros of damage for the whole France, while Klaus in 2009 generated an average cost of 1.5 billion Euros in the South-western part of France [START_REF] Rakotoarison | The faustmann under storm risk and price uncertainty: A case study of european beech in northwestern france[END_REF].

The occurrence of storm generates various type of losses both for forest owners and society: loss of marketability, costs of storage and restoration, losses in other income such as hunting leases or losses of carbon sequestration and amenities [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF]. Climate change is expected to increase the frequency and intensity of disturbances [START_REF] Flannigan | Climate change and forest fires[END_REF][START_REF] Fuhrer | Climate risks and their impact on agriculture and forests in switzerland[END_REF] and in particular those of storm [START_REF] Haarsma | More hurricanes to hit western europe due to global warming[END_REF].

To face such events, forest owners may adopt classical risk-sharing strategy like insurance contract.

Various European countries propose insurance contract against storm and/or fire like for example France, Germany, Slovakia, Spain, etc [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF]. Such a contract allows the forest owner to transfer the risk to the insurer. In exchange of the payment of a premium, the owner perceives an indemnity in case of storm occurrence. Another option available for the forest owner to face the increasing risk of storm is to adapt the silvicultural management. A classical risk-reducing strategy in forestry is to reduce the rotation length [START_REF] Lindner | Alternative forest management strategies under climate change: prospects for gap model applications in risk analyses[END_REF][START_REF] Spittelhouse | Adaptation to climate change in forest management[END_REF], i.e., the time from the establishment of the stand to its final felling. Such a strategy allows diminishing both the time of exposure to natural event and the vulnerability of trees due to ageing [START_REF] Bréda | Vulnerability to forest decline in a context of climate changes: new prospects about an old question in forest ecology[END_REF].

Traditionally, in the literature, decisions related to insurance and rotation length have been analyzed separately. The forest owner's insurance decision has been studied in relation to the disincentive role played by public help implemented by government after the occurrence of a disaster [START_REF] Brunette | Public compensation for windstorm damage reduces incentives for risk management investments[END_REF][START_REF] Brunette | The impact of governmental assistance on insurance demand under ambiguity: A theoretical model and an experimental test[END_REF]. A part of the literature also deals with the determination of the optimal insurance premium through an actuarial approach (Holecy and Hanewinkel, 2006;[START_REF] Pinheiro | Forest property insurance : an application to portuguese woodlands[END_REF][START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF]. One of the main conclusion is that, in order to provide relevant forest insurance premium, the insured forest area should increase, leading [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF] to suggest an European forest insurance scheme. In this forest insurance literature, the analysis of insurance decision under risk is generally realized from a static point of view whereas the problem of forest management under risk is typically a dynamic sequential decision problem under uncertainty. It is fundamental to consider this temporal and sequential aspect in the definition of insurance contract because when a risk occurs, the loss covered by insurance strongly depends on the optimal rotation decision of the forest owner. To our knowledge, in this literature, the insurer's behavior is never modeled and integrated into the analysis.

The optimal rotation length is the common research question in forest economics tackled through the classical Faustmann approach [START_REF] Faustmann | Berechnung des wertes welchen waldboden sowie noch nicht haubare holzbestunde fur die weldwirtschaft besitzen[END_REF]. This approach allows the evaluation of the Land Expectation Value (LEV) over an infinite sequence of rotation. Consequently, the optimal harvesting age in a Faustmann framework has been addressed a lot in the literature [START_REF] Naslund | Optimal rotation and thinning[END_REF][START_REF] Schreuder | The simultaneous determination of optimal thinning schedule and rotation for an evenaged forest[END_REF][START_REF] Kao | Simultaneous optimization of thinnings and rotation with continuous stocking and entry intervals[END_REF][START_REF] Roise | A nonlinear programming approach to stand optimization[END_REF][START_REF] Haight | Optimal harvesting with stand density targets: Managing rocky mountain conifer stands for multiple forest outputs[END_REF][START_REF] Jacobsen | Forest value and optimal rotations in continuous cover forestry[END_REF]. However, originally, this approach is deterministic. [START_REF] Reed | The effects of the risk of fire on the optimal rotation of a forest[END_REF] was the first to introduce risk in the Faustmann framework.

He showed that a potential total destruction due to a fire risk acts as an increase in the discount rate.

Such a framework has then been applied to different risks, and recently to disease [START_REF] Macpherson | The effects of disease on optimal forest rotation: a generalisable analytical framework[END_REF]. To our knowledge, few studies focused on the impact of storm risk on optimal rotation length in a Faustmann framework. [START_REF] Haight | Hurricanes and the economics of loblolly pine plantations[END_REF] studied the impact of storm on the expected present value. They showed that age-dependent damage risk and stocking reduction caused by tree mortality have the greatest impact on the relationship between the expected present value and rotation age. [START_REF] Loisel | Impact of storm risk on faustmann rotation[END_REF] showed that considering storm risk leads to earlier optimal thinning than without risk. Recently, [START_REF] Rakotoarison | The faustmann under storm risk and price uncertainty: A case study of european beech in northwestern france[END_REF] jointly analyzed storm risk and price risk in a Faustmann model. They propose simulations of their theoretical model for European beech stand in Northwestern France. The simulations indicate that the storm risk reduces the optimal rotation length, limits the number of thinning and increases economic loss (but less than price variation).

In this context, it seems that insurance decision and decision related to optimal rotation length have never been studied jointly. Thus, we propose to tackle the question of the impact of insurance decision on the optimal rotation length. For that purpose, we extend the Faustmann optimal rotation model under risk by considering the decision of insurance. Our paper has two additional contributions. First, linked to the rotation and insurance model of the forest owner, we propose to model the microeconomic behavior of the insurer in order to define the components of the insurance contract. Second, in order to analyze insurance decision, we integrate into the model the forest owner's preferences towards risk associated to an expected utility maximization criteria. Such a criteria allows to explicitly consider the forest owner's preferences towards risk, i.e., neutrality, risk aversion or risk loving. Indeed, previous works have proved that forest owners are risk averse on average [START_REF] Musshof | An experimental analysis of the behavior of forestry decision-makers -the example of timing in sales decisions[END_REF][START_REF] Sauter | Faustmann vs. real options theory -an experimental investigation of foresters' harvesting decisions[END_REF][START_REF] Brunette | Attitude towards risk and production decision : An empirical analysis on french private forest owners. Environmental Modeling and Assessment[END_REF]. In addition, the impact of risk aversion on optimal rotation length has been proved in various modeling approaches, like mean-variance approach [START_REF] Ollikainen | A mean-variance approach to short-term timber selling and forest taxation under multiple sources of uncertainty[END_REF][START_REF] Gong | Impact of risk aversion on optimal rotation age[END_REF]Caulfield, 1988;[START_REF] Taylor | Optimum plantation planting density and rotation age based on financial risk and return[END_REF]Valsta, 1992), Wicksellian single rotation framework [START_REF] Alvarez | Does risk aversion accelerate optimal forest rotation under uncertainty[END_REF]), heuristic method [START_REF] Kangas | Incorporating risk attitude into comparison of reforestation alternatives[END_REF][START_REF] Pukkala | A method for integrating risk and attitude toward risk into forest planning[END_REF] or expected utility maximization model [START_REF] Gong | Risk preferences and adaptive harvest policies for even-aged stand management[END_REF]. A majority of papers obtained that risk aversion tends to shorten the optimal rotation length (Caulfield, 1988;[START_REF] Taylor | Optimum plantation planting density and rotation age based on financial risk and return[END_REF][START_REF] Alvarez | Does risk aversion accelerate optimal forest rotation under uncertainty[END_REF][START_REF] Gong | Risk preferences and adaptive harvest policies for even-aged stand management[END_REF] while (Valsta, 1992) found the opposite result. Moreover, one of the main factors explaining the individual's insurance decision is risk aversion. A risk-averse individual is willing to transfer a part of risk to a third party even if it is costly. It is fundamental to consider the risk preferences of the forest owner when analyzing the insurance decision. However, to our knowledge, the forest owner's preferences towards risk have never been considered in a Faustmann modeling.

The rest of the paper is structured as follows. Section 2 presents the model. Section 3 develops the maximization program of the forest owner and the insurer. Finally, Section 4 discusses the results while Section 5 provides the conclusion.

2 Modeling the risk and the insurance contract 2.1 Modeling storm risk and its impact on the stand

We suppose that the occurrence of storm follows a Poisson process, i.e., that storms occur independently of one another, and randomly in time. This is a classical assumption in the literature to model both storm and fire [START_REF] Reed | The effects of the risk of fire on the optimal rotation of a forest[END_REF]Yin and Newman, 1996;[START_REF] Fina | Uncertainty, debt, and forest harvesting: Faustmann revisited[END_REF][START_REF] Ohlson | Multi-attribute evaluation of landscape-level fuel management to reduce wildfire risk[END_REF][START_REF] Armstrong | Using adaptive management to determine requirements of re-introduced populations: the case of the new zealand hihi[END_REF][START_REF] Jacobsen | The regeneration decision: a sequential two option approach[END_REF][START_REF] Loisel | Faustmann rotation and population dynamics in the presence of a risk of destructive events[END_REF]. Thus the distribution of the times between successive storms is an exponential with mean 1/λ: F (x) = 1 -e -λx where λ is the expected number of storms per unit of time. The severity of the storm is given by the random variable A. Let τ be the period of time between the beginning of the stand and, either the storm occurrence or the final harvesting. The storm risk is then described by the couple of random variables (τ, A).

Unlike other types of risk such as fire, the impacts of storm are age-dependent and are low on young stands. Following [START_REF] Schmidt | An inventory-based approach for modeling single-tree storm damage -experiences with the winter storm of 1999 in southwestern germany[END_REF], we assume that, for a tree-height H less equal than H L , there is no damage. The height H L is reached at a time t L , the time t L explicitly depends on the height limit H L and can be deduced using the time dependent function H: H(t L ) = H L . Therefore, in case of storm occurrence before t L , the storm has no impact. At the opposite, in case of storm occurrence after t L , the proportion of damaged trees θ t depends on the severity of the storm A and the characteristics of the stand when the storm occurs (tree height, tree diameter which are age-dependent), hence the proportion of damaged trees is age-dependent. Let α(t) represents the expectation of the proportion of survival trees 1 -θ t .

The sequence of events

Without risk, the forest stand grows from t = 0 (plantation or regeneration) to T , with T the harvesting age. The storm risk modifies the sequence of events as follows:

• If a storm occurs before t L , there is no impact and the stand growth continues.

• If a storm occurs after t L but before T , the proportion of damaged trees is θ , a clear-cutting and a regeneration (or plantation) of the stand is realized.

• If no storm occurs before T , a clear-cutting and a regeneration (or plantation) of the stand is realized at time T . Let τ be the spending time between the beginning of the stand and the first event after t L , i.e., storm occurrence or harvesting at time T . The distribution of the random variable τ is defined for t

L < t ≤ T as F τ (t) = F (t -t L ) = 1 -e -λ(t-t L ) .
Let L(θ, τ ) the loss following a storm occurring at time τ for a proportion of damages trees θ. This loss represents the unexploitability of damaged trees: L(θ, τ ) = θV (τ ) where V (τ ) is the potential final (without risk) income at time τ .

Modeling the insurance contract

An insurance contract is defined as a couple (premium, indemnity), so that we described these two variables in our model.

The contract is characterized by a upper premium P by unit of time (directly linked to the actualization of future losses, see next section) for the full coverage of the damage L(θ, τ ), i.e., damage associated to a storm occurring at time τ . The forest owner chooses a premium P which is a percentage ξ of P, so that P = ξP. If the forest owner chooses ξ = 1 then, s/he is fully insured in case of storm occurrence, in the sense that the the damage is fully covered by insurer. At the opposite, ξ = 0 means no insurance. All the other cases, for 0 < ξ < 1, correspond to partial insurance, where only a part of the damage is covered by insurer, the other part being borne by the forest owner. Then, ξ represents the forest owner's degree of insurance coverage.

The indemnity perceives by the forest owner, in case of a storm occurring at time τ , is then I(θ τ , τ ) = ξL(θ τ , τ ). The indemnity corresponds to the percentage ξ of the loss with, as defined above, 0 ≤ ξ ≤ 1.

The programs of the involved stakeholders

The forest owner's decision variables are the harvesting age T and the percentage ξ as regard to insurance decision, while the insurer chooses the upper premium P. Let's turn to the analysis of each individual's maximization program.

The forest owner's program

In this section, we determine the forest owner's decision related to the rotation length, assuming a given level of insurance. The forest owner is characterized by an utility function u with u ′ (.) > 0 and u ′′ (.) < 0 to represent risk aversion. The forest owner is assumed to maximize the utility of the net economic return from silvicultural activity. This utility of the net economic return Y, actualized at storm occurrence time τ or logging time T , writes as follows:

Y =      P 0 (t, τ ) + H δ (t, τ ) + u(V 1 (θ τ , τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ )) if t L < τ < T P 0 (t, T ) + H δ (t, T ) + u(V (T ) -c 1 )
if τ = T with: -s) ds, the insurance premium paid by the forest owner between the initial time and time t, actualized at time t. -s) ds, the thinning incomes between time t and time t actualized at time τ .

• P 0 (t, t) = ∫ t 0 u(-P)e δ(t
• H δ (t, t) = ∫ t t u(h -P)e δ(t
• V 1 (θ τ , τ ) = (1 -θ τ )V (τ )
, the final income in case of storm occurring at time τ .

• c 1 , the regeneration (or plantation) cost.

• C n (θ τ , τ ) the clearing costs for a storm occurring at time τ .

We deduce the following expression for the Faustmann value J F (see Appendix A):

J F = E[e -δτ Y] 1 -E[e -δτ ] = 1 b(T ) [ a 0 (T )u(-P) + a 1 (T )u(h -P) δ + W F (0, T ) ] where a 0 (T ) = e (δ+λ)T -λt L (1-e -δt ), a 1 (T ) = e (δ+λ)T -λt L -δt - λe (δ+λ)(T -t L ) + δ δ + λ , b(T ) = a 0 (T )+
a 1 (T ) and W F (0, T ) is a modified income such that:

W F (0, T ) =λ ∫ T t L E[u(V 1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))]e (δ+λ)(T -τ ) dτ + u(V (0, T ) -c 1 )
This modified income is composed of two parts. The first part is the integral, with respect to τ , of a reduced final income minus the clearing costs in case of a storm at time τ < T . The second part represents the final income in case of no storm before cutting age T .

This Faustmann value, obtained with storm risk and insurance, differs from the Faustmann value without risk, in several ways. First, the discount rate δ is replaced by the addition of the discount rate and expected number of storms δ + λ (as previously noticed by [START_REF] Reed | The effects of the risk of fire on the optimal rotation of a forest[END_REF] without thinning and Price (2011) with thinning). Second, the utility of the final income W 0 (0, T ) = u(V (T ) -c 1 ) is replaced by W F (0, T ). More precisely, an integral relative to the final cutting minus clearing costs in case of a storm before the cutting age T is added.

The maximum of the Faustmann value is obtained by solving:

max T J F = 1 b(T ) [ a 0 (T )u(-P) + a 1 (T )u(h -P) δ + W F (0, T ) ] (1)
We consider comparative statics relative to the maximization with respect to the rotation period T . We obtain the following results (proof in Appendix B):

Proposition 1 For a risk averse forest owner with differentiated loss and gain risk aversion function

(u(x) = u + x + -u -x -with u + ≤ 1 ≤ u -)
, the optimal cutting age T satisfies: T P > 0 and T increases as the degree of insurance coverage ξ increases.

This result indicates that the risk-sharing strategy and the risk-reducing one are substitutes for the forest owner. Indeed, as the insurance coverage increases, then the forest owner reduces the riskreducing strategy, leading to an increase in the rotation length. This result is in line with the standard result of [START_REF] Ehrlich | Market insurance, self-insurance, and self-protection[END_REF] indicating that insurance and self-insurance are substitutes. Here, the reduction of rotation length may be associated to a self-insurance activity, in the sense that it reduces the damage in case of storm occurrence.

The insurer's program

We assume that the insurer's discount rate is different from the owner's one and is noted δ A . b calculated with discount rate δ is replaced by the corresponding b A with δ A . The Faustmann value J A is:

J A = ξ P δ A -(1 + l f )(1 + m) E[e -δτ I(θ τ , τ )] 1 -E[e -δτ ]
(expectation with respect to θ and τ )

= ξ P δ A -λ(1 + l f )(1 + m) ∫ T t L E[I(θ τ , τ )]e (λ+δ A )(T -τ ) dτ b A (T ) (expectation with respect to θ)
where

E[I(θ τ , τ )] = E[ξL(θ τ , τ )] = ξE[L(θ τ , τ )]
is the expected insurance income, l f is the loading factor and m the security coefficient.

The objective of the insurer is to generate benefits from the insurance activity. In order to have benefit, the insurer must choose P > (1 + l f )(1 + m)P 0 (T ) where P 0 (T ) is the Faustmann value associated to the future losses multiplied by the discount rate δ A for the cutting age T :

P 0 (T ) = δ A E[e -δτ L(θ τ , τ )] 1 -E[e -δτ ] = λ δ A b A (T ) ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ
This constraint depends on the cutting age T chosen by the forest owners. P may not depend on the cutting age T , while in the forest owner's program, T depends on P. In order to ensure a level m for its benefit, the insurer has to evaluate the optimal harvesting age from the forest owner's point of view. P 0 is decreasing with respect to cutting age T . Assuming that the forest owner decides of a cutting age lower than in the case without risk T 0 , then the condition becomes:

P > (1 + l f )(1 + m)P 0 (T 0 ).
The insurer thus has all the information necessary to determine the insurance contract, and to assess the insurance premium and the indemnity.

Analysis of the optimal insurance decision

In this section, we determine the optimal insurance decision considering that the optimal rotation length is given. We assume P = (1 + l f )(1 + m)P 0 (T ) where T is assumed known (and then, not necessarily optimal). The Faustmann value becomes:

J F = a 0 (T )u(-P) + a 1 (T )u(h -P) δb(T ) + P (1 + l f )(1 + m)δ A b A (T ) b(T ) ∫ T t L E[u(V 1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))]e (λ+δ)(T -τ ) dτ ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ (2) + u(V (T ) -c 1 ) b(T )
We consider the derivative of the Faustmann Value J F with respect to insurance premium P. We obtain the following results (proof in Appendix C):

Proposition 2 (i) If the discount rates of the forest owner and the insurer are equal (δ A = δ) then for harvesting age T , premium P * (T ) = 0 is optimal.

(ii) If the forest owner's discount rate and the insurer's discount rate are sufficiently unequal then for This proposition indicates that, in some cases, it may be optimal for a risk averse forest owner to not adopt insurance. This result is not surprising since the premium proposed by the insurer, P, corresponds to a loaded (or unfair) premium. Consequently, the forest owner may consider that the insurance premium is too high and may prefer to not insure despite risk aversion. A way to reduce the insurance premium may be that the public authority subsidizes the premium.

Insurance subsidy

In the literature, the subsidy for the forest insurance premium has already been mentioned in [START_REF] Brunette | The impact of governmental assistance on insurance demand under ambiguity: A theoretical model and an experimental test[END_REF]. They obtained theoretically that, when the government provides a subsidy to the forest owner for her/his insurance premium, then the forest owner should increase the insurance demand.

However, they do not validate this result with the empirical part of the paper. In this article, we adopt another idea which is that, rather than to subsidy the forest owner directly, the government subsidies the insurer.

Indeed, we consider that the government gives a percentage γ of the indemnity to insurer. Consequently, it is as a coinsurance contract where a part γ of the damage is borne by the government while 1 -γ is borne by the insurer (and/or the forest owner in case of partial insurance demand). The risk supported by the insurer is lower, the indemnity paid by the insurer in case of storm occurrence also and the insurance premium too, hence:

P 0 (T ) = (1 -γ) λδ A b A (T ) ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ
We consider the derivative of the Faustmann Value J F with respect to insurance premium P when δ A = δ. We obtain the following results (proof in Appendix D):

Proposition 3 If the discount rates of the forest owner and the insurer are equal (δ A = δ), and if the government gives a percentage γ such that 0 < γ(δ) < γ < γ(δ) < 1 of the indemnity to the insurer then an insurance premium P * (T ) > 0 may be optimal.

By subsidizing the insurer, the government increases the attractiveness of insurance for the forest As underlined by [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF], the adoption of a forest insurance contract is still an exception in Europe. For example, in France, only approximately 4% of the French private forests are insured representing 400,000 ha of private forests. Similar observations are obtained for Spain, Germany or Slovakia [START_REF] Brunette | An actuarial model of forest insurance against multiple natural hazards in fir (abies alba mill.) stands in slovakia[END_REF]. The literature explores different potential explanations for this low level of insurance, among which the existence of public program to financially help the forest owners after the occurrence of storms. In this context, [START_REF] Brunette | Public compensation for windstorm damage reduces incentives for risk management investments[END_REF] and [START_REF] Brunette | The impact of governmental assistance on insurance demand under ambiguity: A theoretical model and an experimental test[END_REF] showed that a public help reduces the forest owners' insurance demand, but that this negative effect is lower when the help is contingent to insurance adoption rather than fixed (i.e., independent from any insurance contract adoption). In this paper, we provide another potential explanation to the nonadoption of insurance contract by private forest owners, the fact that, in some cases, insurance may be considered as too expensive by the forest owner (see Proposition 2). In addition, as regard to the existing literature, we explore another channel from which the government may participate to the insurance scheme. Indeed, we show that, by subsidizing the insurer, the government encourages the forest owner to adopt insurance contract. Consequently, an horizontal risk sharing between the insurer, the forest owner and the government may be an efficient way to insure storm risk in forest.

Conclusion

In this paper, we propose to extend the classical Faustmann rotation model under risk by considering the insurance decision. This means that, in addition to the decision related to the rotation length, the forest owner also decides on a degree of insurance coverage. Such a challenge requires to model the insurer's behavior (in order to determine the contract) and to consider the forest owner's preferences towards risk (in order to analyze the insurance decision), two supplementary originalities for this paper. In this framework, our results indicate that the forest owner considers that the risk-sharing strategy (insurance) and the risk-reducing one (reduction of the rotation length) are substitutes. In addition, we show that, in some cases, it may be optimal for the forest owner to not adopt insurance. Finally, we conclude that, a way to incite forest owners to adopt insurance contract is to consider that the government subsidizes the insurer.

A The Faustmann Value

As we assume that the storm occurs independently of one another and randomly in time, we deduce the following expression for the land value:

J F = E[ ∞ ∑ i=1 e -δ(τ 1 +τ 2 +...+τ i ) Y i ] = ∞ ∑ i=1 i-1 ∏ j=1 E[e -δτ j ].E[e -δτ i Y i ] = [e -δτ Y] 1 -E[e -δτ ] (3)
From the definition of τ we deduce the expectation E[e -δτ ]:

E[e -δτ ] = ∫ T t L e -δτ dF (τ -t L ) + e -δT (1 -F (T -t L )) = λ + δe -(δ+λ)(T -t L ) δ + λ e -δt L
We deduce the expected net economic return E[e -δτ Y] actualized at initial time:

1 -e -δt δ u(-P) + e -δt -e -δt L δ u(h -P) + e -δt L -e (δ+λ)T -λt L δ + λ u(h -P) + ∫ T t L (E[u(V 1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))])e -δτ dF (τ -t L ) + u(V (T ) -c 1 )e -δT (1 -F (T -t L ))
and hence the result.

B Proof of Proposition 1

To lighten the presentation, in some calculus, we omit the λ dependency in functions a 1 , b and denote a T (resp. b T ) the partial derivative of a 1 (resp. b) with respect to T .

From (1), differentiating J F with respect to the rotation age T gives the following first-order condition:

J T = [ W F (0, T ) + a 1 (T ) δ ((hu -)I h<P + (hu + + P(u --u + ))I h>P ) b(T ) ] T = 0
We consider the case h > P (the other case is simpler). From the first-order condition J T = 0 we deduce:

W F (0, T ) T b(T ) -W F (0, T )b T (T ) + (a T (T )b(T ) -a 1 (T )b T (T )) hu + + P(u --u + ) δ = 0 (4)
Differentiating J T = 0 with respect to P yields: J T T T P + J T P = 0. From J T T < 0, T P and J T P have the same sign. J T P is proportional (with the same sign) to the sum of: γ(τ )e (λ+δ)(t-τ ) dτ , then:

A P = W F (
A ′ P (t) =γ ′ (t)b(t) + γ(t)b T (t) -δγ(t) -(λ + δ)(γ(t)b(t) -A P (t))
=γ ′ (t)b(t) + (λ + δ)A P (t)

From α ′ (t) < 0 we deduce γ ′ (t) > 0, moreover from A P (0) = 0 we deduce that A P (T ) > 0.

C Proof of Proposition 2

We consider the derivative of the Faustmann Value J F with respect to insurance premium P:

∂J F ∂P = -a 0 (T )u ′ (-P) + a 1 (T )u ′ (h -P) δb(T )

+ 1 (1 + l f )(1 + m)δ A b A (T ) b(T ) ∫ T t L E[u ′ (V 1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))L(θ, τ )]e (λ+δ)(T -τ ) dτ ∫ T t L E[L(θ τ , τ
)]e (λ+δ A )(T -τ ) dτ (i) From the concavity of the utility u, and using -P < h -P < V 1 (θ, τ ) + I(θ τ , τ ) -c 1 then:

∂J F ∂P < 1 δ (-1 + 1 (1 + l f )(1 + m) δ δ A b A (T ) b(T ) ∫ T t L E[L(θ τ , τ )]e (λ+δ)(T -τ ) dτ ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ
)u ′ (h -P)

From δ A = δ then b A = b and ∂J F ∂P < 1 δ (-1 + 1 (1 + l f )(1 + m)
)u ′ (h -P) < 0, hence the result.

(ii) ∂J F ∂P is proportional to:

-(a 0 (T )u ′ (-P) + a 1 (T )u ′ (h -P))

δ A b A (T ) ∫ T t L E[L(θ τ , τ )]e (λ+δ A )(T -τ ) dτ + 1 (1 + l f )(1 + m) δ ∫ T t L E[u ′ (V

  0, T ) T P b(T ) -W F (0, T ) P b T (T ) and (a T (T )b(T ) -a 1 (T )b T (T ))(u --u + ) The difference a T (T )b(T ) -a 1 (T )b T (T ) is positive. Concerning A P : W F (0, T ) α(τ ))V (τ )e (λ+δ)(T -τ ) dτ W F (0, T ) T P =λ u + P (1 -α(T ))V (T ) + (λ + δ)W F (0, T ) P From b T (T ) = δ + (δ + λ)b(T )we deduce:A P =λ u + P (1 -α(T ))V (T )b(T ) -δW F (0, T ) α(τ ))V (τ )e (λ+δ)(T -τ ) dτ ] Let γ(t) = λ u + P (1 -α(t))V (t)b(t) then: A P = γ(T )b(T ) -δ ∫ T t L γ(τ )e (λ+δ)(T -τ ) dτMoreover, studiing the behavior of A P (t) = γ(t)b(t) -δ∫ t t L

  1 (θ, τ ) + I(θ τ , τ ) -c 1 -C n (θ τ , τ ))L(θ, τ )]e (λ+δ)(T -τ ) dτIf the forest owner's discount rate and the insurer's discount rate are sufficiently unequal then ∂J F ∂P > 0for sufficiently large value of δ, hence ∂J F ∂P is equal to zero for smaller δ and we deduce the existence of a non trivial solution P * (T ) > 0.

Acknowledgments

The UMR BETA is supported by a grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (ANR-11-LABX-0002-01, Lab of Excellence ARBRE).

 Science, 38:430-447, 1992.R. Yin and D.H. Newman. The effect of catastrophic risk on forest investment decisions. Journal of Environmental Economics and Management, 31:186-197, 1996.

D Proof of Proposition 3

If δ A = δ then b A = b and:

∂J F ∂P is positive for sufficiently large value γ and is negative for γ = 0. Hence we deduce that for intermediate γ, ∂J F ∂P may be equal to 0, hence a non null premium P will be optimal.


Sylvie Muratorio

Julie Gauzere

Aurore Bontemps

Jean-François Rey

Etienne Klein

Tree, Sex and Size: Ecological determinants of male versus female fecundity in three Fagus sylvatica stands

come   L'archive ouverte pluridisciplinaire


Jeanne Simon

Maida Cardoso

Eric Alibert

Pierre Valsesia

Gilles Vercambre

Jean Luc Verdeil

Christophe Coillot

Christophe Goze-Bac

Nadia Bertin

Combining flow-MRI method and modeling approach to assess water fluxes in tomato plant architecture

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

CONCLUSION AND PERSPECTIVES

The flow-MRI method allows us to measure water fluxes in slow flow conditions. The measurements are in agreement with the predictions of the plant model in the same environmental conditions. These results confirm that this method will be a useful tool to analyse plant responses to abiotic stresses and will help to improve the plant model. Sens. Sens. Syst., 5, pp. 137-145, (2016).

[3] : Baldazzi V., Pinet A., Vercambre G., Bénard C, Biais B., Génard M. 2013. In-silico analysis of water and carbon relations under stress conditions. A multi-scale perspective centered on fruit. Front. Plant Sci., doi: 10.3389

Model predictions

Model predictions of xylem fluxes in the MRI conditions, on a virtual plant with the same aerial architecture as the measured plant, and at the same position on the stem : under the third truss.

The assumption made by the model at night is that the fluxes are only due to fruit activity and that the leaves do not transpire.

Flow-MRI measurements

Plant model

We ran a bio physiological functionalstructural plant model [3] until the computed plant achieved the same architecture as the i n v e s t i g a t e d p l a n t . T h e n , t h e M R I environmental conditions were reproduced in order to analyze the fluxes predicted under the third truss.

Flow-MRI method

We used a novel flow-MRI method involving the slice selectivity in the context of a multi slice spin echo sequence

INTRODUCTION MATERIALS AND METHODS

RESULTS

Environmental conditions

The plant is placed horizontally inside the MRI scanner.

The experiments were performed during three consecutive days with no light (extended night conditions). The temperature in the MRI scanner is 20°C.

Perspectives

• Add some echoes to increase the SNR and have better chances to see phloem fluxes.

• A new 1 H MRI probe as been designed to fit the fruit pedicel plants which will allow us to measure the fluxes going to the fruit.

• The acquisition of a new vertical MRI scanner 3T will allow us to make measurements in better environmental conditions and during longer time periods.

  Combining flow-MRI method and modeling approach to assess water fluxes in tomato plant architecture Jeanne Simon 1,2 , Maïda Cardoso 2 , Eric Alibert 2 , Pierre Valsesia 1 , Gilles Vercambre 1 , Jean-Luc Verdeil 3 , Christophe Coillot 2 , Christophe Goze-Bac 2 and Nadia Bertin 1 Water and carbon status throughout growth and development are tightly controlled by the plants and are key components of their response to environmental stresses. Measuring and predicting resource availability and transport within intact plants is a challenge. The objective of this study is to compare the flow-MRI measurements to the prediction of a functional-structural plant model in order to explore the relevance of this MRI method to analyze the plant responses to abiotic stresses and map the ressource availability.

  [1] : Buy S., Le Floch S., Tang N., Sidiboulenouar R., Zanca M., Canadas P., Nativel E., Cardoso M., Alibert E., Dupont G., Ambard D., Maurel C., Verdeil J.-L., Bertin N., Goze-Bac C., Coillot C. 2018. Flip-flop method: A new T1-weighted flow-MRI for plants studies. PLoS ONE 13(3): e0194845. doi: 10.1371 [2] : Coillot C., Sidiboulenouar R., Nativel E., Zanca M., Alibert E., Cardoso M., Saintmartin G., Noristani H., Lonjon N., Lecorre M., Perrin F. and Goze-Bac C., Signal modeling of an MRI ribbon solenoid coil dedicated to spinal cord injury investigations, J.

  [1]. Two pulse sequences measuring a slice inflow and outflow weighted are performed, allowing slow flow sensitive imaging. The experiments have been performed on a Agilent MRI scanner working at 9.4T. A 1 H MRI openable saddle-coil tuned at 400MHz has been designed to fit the stem of plants[2]. We applied this method to the main stem of a tomato plant (cv : WVA106), under the third truss. Histological measurements were performed after the MRI experiment.This work has been carried out thanks to the support of the « Investissements d'avenir » programme (Labex Agro:ANR-10-LABX-0001-01), under the frame of I-SITE MUSE (ANR-16-IDEX-0006).

ACKNOWLEDGMENTS

  Agilent MRI scanner 9.4T Openable saddle-coil Openable saddle-coil around the stem of a tomato plant Schematic representation of how the model describes the network of source and sink organs as well the transport of resources within this network [3]

1

  Institut National de la Recherche Agronomique (INRA), PSH (UR 1115), Avignon, France ; 2 Charles Coulomb Laboratory (L2C-BioNanoNMRI team), UMR 5221 CNRS -University, Montpellier, France ; 3 Centre de Coopération Internationale en Recherche Agronomique pour le Développement (CIRAD), UMR AGAP, Montpellier, France jeanne.simon@inra.fr

  mm 3 /s giving 0.13 g/h Xylem fluxes predicted after three days of extended night: 0.048 mm 3 /s giving 0.17 g/h Active surface = 0.23 mm 2 and v = 0.33 mm/s Xylem fluxes

3D representation of the whole computed plant (A) and 3D mapping of relative flow (B) in the whole plant at night at the beginning of the experiment. (C) Zoom on the first truss.