Loup Rimbaud

Gael Thébaud

Samuel S Soubeyrand

Emmanuel Jacquot

Model-based design and assessment of management strategies for epidemics in a heterogeneous landscape

come

Model-based design and assessment of management strategies for epidemics in a heterogeneous landscape

Loup Rimbaud 1 , Gaël Thébaud 2 , Samuel Soubeyrand 3 and Emmanuel Jacquot 2 1 Montpellier SupAgro, UMR 385 BGPI, F-34398 Montpellier cedex 5, France. Epidemics are often managed by strategies mainly based on expert opinions, although the spread of a disease results from complex interactions between biological processes, to which may be added human interventions. Thus, it is no easy task to optimize disease management, especially when alternative strategies cannot be tested experimentally.

Therefore, this project aims to develop a comprehensive spatiotemporal model which simultaneously simulates the propagation of an epidemic in a heterogeneous landscape and the impact of several innovative management strategies. The key parameters of the model will first be identified by a sensitivity analysis and, when possible, estimated through experiments or statistical analyses of epidemiological data. Then, a wide range of potential management strategies will be tested through the model by Monte Carlo simulations.

The model developed in this project will be applied to sharka, caused by Plum pox virus (PPV, genus Potyvirus), a quarantine pathogen in the European Union. Indeed, this model is interesting to circumvent the difficulty in carrying out experiments due to the sanitary status of this pathogen. Besides, the improvement of management strategies for this disease is vital because it is the most devastating viral disease of stone fruits (e.g. peach, apricot and plum trees), with a cost of more than 10 billion Euros between 1975 and 2005 in the world. In France, the management of the epidemics is based i) on regular visual inspections of the orchards and ii) on the removal of symptomatic trees. The first step of the project is to make the model flexible enough to include this current management strategy, by taking into account the mandatory observation, sampling and control processes.

Finally, the general approach of this Ph.D project may be used to help managing other sanitary crises due to epidemics.


Romain Guy

Catherine Laredo

come

Inference for epidemic data using diffusion processes with small diffusion coefficient

Inference for epidemic data using diffusion processes with small diffusion coefficient

Romain Guy 1,2 , Catherine Laredo 1,2 , Elisabeta Vergu Abstract One of the simplest and most appropriate models for the study of epidemic spread is the SIR (Susceptible-Infectious-Removed) model, in which the successive transitions of individuals between states are described in various mathematical frameworks. Epidemic data being often partially observed and also temporally aggregated, parametric inference through likelihood-based approaches is rarely straightforward, whatever the mathematical representation used. Although these last years methods based on data augmentation, able to deal with different patterns of missingness were developed, they do not provide systematic solutions since they are limited by the amount of augmented data and hence by computation times. In this context, diffusion processes provide a good approximation of epidemic dynamics and, due to their analytical power, allow shedding new light on inference problems of epidemic data. Indeed, the normalization by the population size N of the continuous time Markov jump process leads to an ODE system. Before passing to the limit, the SIR dynamics is described by a bidimensional diffusion with small diffusion coefficient proportional to = 1/ √ N . We consider here a multidimensional diffusion (X t ) t≥0 with drift coefficient b(α, X t ) and diffusion coefficient σ(β, X t ). The diffusion is discretely observed at times t k = k∆ on a fixed time interval [0, T ] with T = n∆. We study Minimum Contrast Estimators (MCE) derived from a Gaussian process approximating (X t ) t≥0 for small . We obtain consistent and asymptotically Gaussian estimators of (α, β) as → 0 and ∆ = ∆n → 0 under the condition " 2 n" bounded , and for fixed ∆ consistent and asymptotically Gaussian estimator of α. Our results extend [START_REF] Genon-Catalot | Maximum contrast estimation for diffusion processes from discrete observations[END_REF] to multidimensional diffusions with arbitrary diffusion coefficient and [START_REF] Gloter | Estimation for stochastic differential equations with a small diffusion coefficient[END_REF] to low frequency data. Finally, we simulate various epidemics dynamics with Markov jump process, we compare our MCE to other commonly used estimators: least squares for noisy observed ODE, maximum likelihood for Markov jump processes. Our findings are very promising: on low and high frequency data, our estimator is very close to the estimator based on the complete observation of the Markov jump process . We are currently extended these results to the more realistic case of partially observed data.


Jérôme Ogée

Lisa Wingate

A process-based interpretation of the stable C and O isotope signals in the soil-plant-atmosphere continuum using the ecosystem model MuSICA

may come

A process-based interpretation of the stable C and O isotope signals in the soil-plant-atmosphere continuum using the ecosystem model MuSICA Jerome Ogee, Lisa Wingate


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Conclusion-

  Daily gas emissions from the dairy sheds in Mirecourt Unit, for grassland and mixed crop livestock systems during two winter periods(2009-10 and 2010-11) MS deep litter system has emitted 3 to 4 times more gases than the GS cubicles systemOur measurements appear consistent with Brachet (2007), except for NH 3 Simplified method rather easy to operate and non invasive -Further calculations and analyses to do in order to check sampling method and analyses through H 2 0, P and K balances -Impossible to conclude if the considered organic dairy systems are really less emitting than conventional ones:-from manure production to spreading, there are many ways… -even if GS seems very promising !


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Sophie Schbath

Analyse statistique de réseaux biologiques

Keywords: Many scientific fields : sociology, physics, "internet", biology. Biological networks : protein-protein interaction networks, regulatory networks, metabolic networks sociology, physics, "internet", biology. Biological networks : protein-protein interaction networks, regulatory networks, metabolic networks. Statistical aspects : -network inference, -statistical properties of given networks (degrees, diameter, clustering coefficient, modules, motifs etc.), -random graph models

Nature of the data :

-n individuals (n large), -but also n 2 couples.

Focus on exceptional motifs = motifs appearing more frequently than expected. [Milo et al. (02),Zhang et al. (05),Lacroix et al. (06), Lee et al. (07),Taylor et al. (07)] Topological motif : connected pattern of interconnection → an occurrence in the network is an isomorphic subgraph Ex : particular regulatory units like feed-forward loop or bi-fan motifs. Interpretation of over-represented topological motifs : they are thought to reflect functional units which combine to regulate the cellular behavior as a whole.

Coloured motif : multiset of node colours, e.g. {•, •, •} → an occurrence in the network is a connected subgraph with the appropriate node colours Interpretation of over-represented coloured motifs : they are thought to reflect groups of cooperative enzymes (reaction networks).

How to assess the exceptionality of a motif ?

Step 1 To count the observed number of occurrences N obs (m) of a given motif m (out of my scope)

Its significance is assessed with the p-value P{N(m) ≥ N obs (m)} (the probability to get as much occurrences at random)

Step 2 To choose an appropriate random graph model

Step 3 To get the distribution of the count N(m) under this model Analytical approaches :

The most popular random graph model is the Erdös-Rényi model (nodes are connected independently with proba p) Some theoretical works exist on Poisson and Gaussian approximations of topological motif count distribution (see [Janson et al. (00)] for an overview) BUT

• only for particular motifs ("balanced" property),

• the Erdös-Rényi model is not a good model for biological networks (e.g. it does not fit the degrees).

Simulated approaches :

Random networks are generated by edge swapping, (degrees are preserved)

Empirical distributions for motif counts are obtained leading either to p-values or to z-scores

BUT

• huge number of simulations required to estimate tiny p-values,

• z-scores are compared to N (0, 1) which is not always appropriate,

• edge swapping does not define a probabilistic random graph model.

To propose probabilistic random graph models

• adapted for biological networks,

• allowing probabilistic calculations,

• with efficient estimation procedures.

[Daudin, Picard, Robin ( 08

Random graphs

A random graph G is defined by :

• a set V of fixed vertices with |V| = n, • a set of random edges X = {X ij , (i, j) ∈ V 2 } such that X ij = 1 if i and j are connected, 0 otherwise
• and a distribution on X ij .

Examples :

• the Erdös-Rényi model,

• the Stochastic Block Model (=mixture of ER models),

• the Expected Degree Distribution model.

P(X ij = 1) = p Degrees are Poisson distributed K i := j =i X ij ∼ B(n -1, p) ≈ P((n -1)p)
Bad fit of Erdös-Rényi model on biological networks due to heterogeneous connection probabilities along the network.

Stochastic Block Model (or "Mixnet")

Vertices are spread into Q groups.

Conditionally to the group of vertices, edges are independent and

X ij | {i ∈ q, j ∈ ℓ} ∼ B(π q,ℓ )
π q,ℓ is the connection probability between groups q and ℓ.

Degrees are distributed according to a Poisson mixture

K i ∼ q α q B(n -1, π q ) with π q = ℓ α ℓ π q,ℓ
Introduced by [Nowicki and Snijers (2001)] Classical maximum likelihood procedures fail log-likelihood L(X) not calculable because of hidden groups (Z, Z i is the group of node i).

EM algorithm, classical to fit mixture models, cannot be used because P(Z | X) is not computable (all vertices are potentially connected, no local dependence).

Variational approach (iterative procedure)

maximization of L(X) -KL(P(Z | X), Q R (Z)) where Q R is the best approximation of P(Z | X) within a class of 'nice' distributions. ⇒ estimator of P(Z i = q|X).
analytical expressions for α q and π q,ℓ Choice of Q : heuristic penalized likelihood criterion inspired from BIC (ICL)

Topological motifs

Let m be a motif of size k (connected graph with k vertices, k << n).

• m is defined by its adjacency matrix (also denoted by m) :

m uv = 1 iff nodes u ↔ v (m uv = 0 otherwise).
• Let R(m) be the set of non redundant permutations of m (so-called "versions").

• Ex : 3 versions of the V motif at a fixed position (i, j, k). The total count N(m) of motif m is then :

m m ′ m ′′ 2 4 0 1 1 1 0 0 1 0 0 3 5 2 4 0 1 0 1 0 1 0 1 0 3
N(m) = α∈I k m ′ ∈R(m) Y α (m ′ )
Warning : N(m) = number of induced subgraphs ("m = G α ").

(X i 1 ,j 1 , . . . , X i ℓ ,j ℓ ) = D(X i ′ 1 ,j ′ 1 , . . . , X i ′ ℓ ,j ′ ℓ )
(H2) Independence of disjoint occurrences we have [Picard,Daudin,Koskas,Schbath,Robin (08)]

EN(m) = n k |R(m)|µ(m).
where µ(m) := EY α (m) = P(m occurs at α) and

VarN(m) = k s=0 C(n, k, s) m ′ Ω s m ′′ µ(m ′ Ω s m ′′ ) -[EN(m)] 2 .
where m ′ Ω s m ′′ is a super-motif composed of the union of two overlapping occurrences of m ′ and m ′′ sharing s common vertices.

(m) = EN 2 (m) -[EN(m)] 2 . We then calculate EN 2 (m) = E   α,β∈I k m ′ ,m ′′ ∈R(m) Y α (m ′ )Y β (m ′′ )   , = E   k s=0 |α∩β|=s m ′ ,m ′′ ∈R(m) Y α∪β (m ′ Ω s m ′′ )   = k s=0 C(n, k, s) m ′ Ω s m ′′ µ(m ′ Ω s m ′′ ),
where m ′ Ω s m ′′ is a super-motif composed of the union of two overlapping occurrences of m ′ and m ′′ sharing s common vertices.

µ(m) = p e(m)
Mixture of ER model (Mixnet/SBM) with Q groups, proportions α 1 , . . . , α Q and connection probabilities π q,ℓ

µ(m) = Q c 1 =1 . . . Q c k =1 α c 1 . . . α c k 1≤u<v ≤k π muv cu,cv .
Exact distribution unknown. Several approximations exist in the literature under specific conditions (motif and model) :

• Poisson distribution [Bollobas (81), Barbour (82), Karónski and Ruci ński ( 83 • PPI network : 706 proteins and 1420 interactions (edges).

• Mixnet was fitted to the network → 4 groups of connectivity. Coloured motifs : Expected count

Motif

EN(m) = α∈I k EY α (m) = n k P(m occurs at α) = n k g(k, p) × k! c∈C s(c)! k i=1 f (m i ) :=γ(m)
where g(k, p) is the probability for an ER(p) graph of size k to be connected [Gilbert,59] :

g(k, p) = 1 - k -1 i=1 k -1 i -1 g(i, p)(1 -p) i(k -i) .
(g(1, p) = 1). Let us just compute EN 2 (m).

EN 2 (m) =

α∈I k β∈I k E[Y α (m)Y β (m)].
Q m (α, β) = P(C(α) = C(β) = {m 1 , . . . , m k }). = m * ⊂m γ(m * )[γ(m -)] 2 s(m * )
color term :

Q m (α, β) = P(C(α) = C(β) = {m 1 , . . . , m k }). = m * ⊂m γ(m * )[γ(m -)] 2 s(m * )
connectedness term :

K (α, β) = P(G(α) and G(β) are connected) =            g(k, p), if ℓ = k g 2 (k, p), if ℓ = 0 or 1.
Q m (α, β) = P(C(α) = C(β) = {m 1 , . . . , m k }). = m * ⊂m γ(m * )[γ(m -)] 2 s(m * )
connectedness term : Both parameters of the PA distribution can be derived from the first 2 moments of the count.

K (α, β) = P(G(α) and G(β) are connected) =            g(k, p), if ℓ = k g 2 (k, p), if ℓ = 0
N gm = n g k -1 N gm , g = 1, 2
Box-Cox Transformation to make the counts "more" Gaussian :

N * gm = 2( N gm -1), g = 1, 2
Euclidian distance on z-scores :

d 2 (N * 1 , N * 2 ) = ||(Σ * 1 ) -1/2 (N * 1 -EN * 1 ) -(Σ * 2 ) -1/2 (N * 2 -EN * 2 )|| 2 2
where EN * g and the covariance matrix Σ * g = (Cov(N * g (m i ), N * g (m j ))) i,j can be calculated from EN g and the covariance matrix (Cov(N g (m i ), N g (m j ))) i,j (previous part).

Another approach

To model the vector N = (N 1 , N 2 , . . . , N M ) Need for a "multidimensional (compound) Poisson distribution with given covariance matrix" Our choice = the multivariate Poisson-log normal distribution from [Aitchison and Ho,89] :

  edges : consecutiveness in the metabolic network Main characteristics : several thousands of nodes (n) sparsity (nb of edges = O(n) ) heterogeneous connexions nodes may be coloured (biological function, class of reaction, cellular localization etc.) Looking for local structures Breaking-down complex networks into functional modules or basic building blocks : [Shen-Orr et al. (02)] → network motifs : topological motifs and/or coloured motifs.

  Let α = (i 1 , . . . , i k ) ∈ I k be a possible position of m in G. G α denotes the subgraph (V i 1 , . . . , V i k ). Non strict occurrences :m occurs at position α ⇔ m ⊆ G α Random indicator of occurrence : Y α (m) Y α (m) = 1 I{m occurs at position α} = 1≤u,v ≤k X muv iuiv .

  )]• Gaussian distribution[Barbour et al. (87)] • Compound Poisson distribution [Stark (01)] Simulation study Model = mixnet with 2 groups, n = 200, etc. For expectedly frequent motifs : ), Poisson (-) and Geometric-Poisson (-) ), Poisson (-) and Geometric-Poisson (-) Sophie Schbath (INRA -MIG) Statistics for biological networks Orsay, 23 juin 2011

  occurs at α and β) =K (α,β)×Qm(α,β).whereK (α, β) = P(G(α) and G(β) are connected) Q m (α, β) = P(C(α) = C(β) = {m 1 , . . . , m k }).

  with S. Robin and L. Benaroya) Motif-based distance Normalization by the size of the graphs : Since EN(m) = n k P(m occurs at α), we define :

  N m ∼ P(e λm ) Λ ∼ N (µ, Σ) (µ, Σ) can be explicitly derived from the expectation and covariance matrix of N. Distance = euclidian distance between Λ 1 and Λ 2 Λ is estimated by E(Λ | N) Limitations : Σ may be not positive No analytical expression for E(Λ | N) (Monte Carlo)

  )]. A mixture model for random graphs. Statis. Comput.

	Part 2
	Mixture model for random graphs
	(Stochastic Block model)

[Birmelé (07)]. A scale-free graph model based on bipartite graphs. Disc. Appl. Math. [Mariadassou, Robin, Vacher (10)]. Uncovering structure in valued graphs : a variational approach. Ann. Appl. Statist. [Latouche, Birmele, Ambroise (10)] Overlapping Stochastic Block Models with Application to the French Political Blogosphere. Annals of Applied Statistics [Daudin, Pierre, Vacher (10).] Model for Heterogeneous Random Networks Using Continuous Latent Variables and an Application to a Tree-Fungus Network. Biometrics [Latouche, Birmele, Ambroise (11)] Variational Bayesian Inference and Complexity Control for Stochastic Block Models. Statistical Modelling [Gazal, Daudin, Robin (11)]. Accuracy of variational estimates for random graph mixture models. J. Comput. Comput. Simul. Sophie Schbath (INRA -MIG) Statistics for biological networks Orsay, 23 juin 2011

  N obs E mixnet N σ mixnet (N) P(GP ≤ N obs ) P(GP ≥ N obs )

	14113 75 98697 112490 1058 3535 79 0 Sophie Schbath (INRA -MIG) Assessing the significance of 13602 2659 66.9 20.4 94578 27039 93741 27257 516.6 208.7 2897 1120 34.8 20.0 0.17 0.45 8.5 10 -1 Statistics for biological networks Orsay, 23 juin 2011 Part 3 4.06 -1 3.31 -1 4.12 -1 2.34 -1 1.33 -2 2.63 -1 3.11 -2 coloured motif frequencies

Sophie Schbath (INRA -MIG)

Statistics for biological networks Orsay, 23 juin 2011

Sophie Schbath (INRA -MIG)Statistics for biological networks Orsay, 23 juin 2011

• mean and variance of the count in a wide class of random graph models, • relevant distribution to approximate the count distribution.[Matias, Schbath, Birmelé, Daudin and Robin (06)] Network motifs : mean and variance for the count, REVSTAT. 4 31-51.[Picard,Daudin, Schbath and Robin (08)] Assessing the exceptionality of network motifs, J. Comput. Biol.[Schbath, Lacroix and Sagot (09)] Assessing the exceptionality of coloured motifs in networks, EURASIP Sophie Schbath (INRA -MIG) Statistics for biological networks Orsay, 23 juin 2011

Acknowledgement net subgroup Sophie Schbath (INRA -MIG) Statistics for biological networks Orsay, 23 juin 2011

Compound Poisson distribution

Distribution of Z i=1 T i when Z ∼ P(λ) and T i 's iid.

Particularly adapted for the count of clumping events : Z is the number of clumps and T i is the size of the i-th clump.

All network motifs are overlapping : they occur in clumps. We proposed to use a Geometric-Poisson(λ, a) distribution, i.e. when T i ≈ G(1a)

• analogy with sequence motifs [S. (95)],

• (λ, a) can be calculated according to EN(m) and VarN(m) :

Indicator of occurrence at position α : Y α (m) The motif count distribution is then approximated by a Geometric-Poisson distribution.

-→ approximate p-value P(N(m) ≥ N obs (m)).

Sophie Schbath (INRA -MIG) Statistics for biological networks Orsay, 23 juin 2011 color term :

Aim

Let G 1 and G 2 be two coloured graphs of size n 1 and n 2 (typically reaction networks from 2 different species).

Each graph is caracterized by the count vector of M given motifs of size k : N g = (N g1 , N g2 , . . . , N gM ), for g = 1, 2.

Questions :

Do they share common exceptional motifs ?

Have both graphs similar k-motif compositions ?

Whose motifs are the most discriminant ?

Idea : to define a motif-based distance taking care of the deviations from the models, the dependence between motif counts.

Sequential distance

Consider all single motif sets (dim(N * 1 )=dim(N * 2 )=1), and take

Consider all motif pairs ( m 1 , m j ) with m j = m 1 , and take

and so on


Background

• Growth Models -Assume straight stems -Regular branching pattern • But trees are not "perfect", especially in windy environments environments

-For example: trees can lose their top in a storm -Results in:

Poor stem form Large branches

• Descriptive codes are recorded in New Zealand Permanent Sample Plots

Descriptive codes -related to branching

  To display the risk zone for worst tree form • Perception from New Zealand industry -Too difficult to integrate branch response to damage into modelling systems -They already have "fudge-factors" that account for this type of loss

	• BW: Basket Whorl • DL: Double leader • FK: forked, two leaders above 1.4 m • ML: multileader Descriptive codes -related to stem form • SW: gentle sweep • BS: extreme butt sweep • CK: crooked • LN: growing with excessive lean • Proportion of different defect codes varies with site conditions Radiata pine trials • Proportion of trees with descriptive codes tends to increase with increasing mean annual windspeed • Treatment significant: 5 of 8 sites • Seedlot significant: 4 of 8 sites • Long-internode seedlot (selected for few branch Douglas-fir trial in North Island Douglas fir trial • Site: mean annual windspeed 8.34 km/hr • Planted 1982 • Final stocking either 250, 500 or 750 stems/ha At each step depending of age/ tree density/genetic treatment At a thinning step in order to constraint the thinning • Thinned once at age 10.93 years Douglas fir Trial Douglas-fir trial How to integrate descriptive codes into forest How to integrate descriptive codes into Douglas-fir Trial management tools? forest management tools • Time trend in descriptive codes for unpruned plots • Fork, most recorded defect. Crook also common difference with -Add a descriptive code to the individual tree • No significant with different final crop stockings • Tree growth model • GIS system
	• DT: dead top • DT: dead top • TO: top out • LR: leader replacement • RC: ramicorn • MF: undefined malformation • SO: socketing • SO: socketing • TP: toppled, roots damaged but tree alive Mean annual Mean annual windspeed 8.57 km/hr Mean annual windspeed -Stems pruned 250 or 500 sph years 13.56 km/hr cluster) generally more trees with "defects" than cluster) generally more trees with "defects" than other seedlots • Seedlots selected for high and low wood density -• Unpruned or pruned regimes In connection with a wind model to model the wind In connection with a wind model to model the wind • Unpruned or pruned regimes • On average 9% of breakage probability • Pruning regimes combination of: trees have a • Crown remaining after pruning lift: 4, 6 or 8 m descriptive code at • Wood quality model similar amounts of defects -Nominal prune height 6 or 12 m (did not reach 12 age 20 years, and -Add a descriptive code to the individual tree m) 18% by age 29 To integrate a decrease in wood quality due to tree form

-

Acknowledgements

• Future Forest Research for use of data • Carolyn Andersen for extracting the tree data • Duncan Harrison for extracting wind data • TRANZFOR for funding Celine's visit to New Zealand Zealand

Descriptive codes -forked trees • Trees forked below 1.4 m counted as separate trees • For this study, such trees assigned descriptive code LF (low fork)

This study: Analysis of stem descriptive codes • Radiata Pine trials -Genetics and stocking -Eight radiata pine trials containing a range of improved seedlots including ones developed for wood density wood density • Douglas-fir trial -Stocking and Pruning -One trial with a range of silvicultural treatments Radiata pine trials 5 trials planted in 1992, 3 planted in 1994 Each trial contains: -Up to 7 seedlots -Two silvicultural treatments 1. Plant 500 stems/ha and leave 1. Plant 500 stems/ha and leave 2. Plant 1000 stems/ha, thin to 400 at mean crop ht of 7 to 8 m and prune to 2 m


domain: shs sdv

text:
Sophie Legras
email: sophie.legras@dijon.inra.fr

Jean Cavailhès
email: jean.cavailhes@dijon.inra.fr

Urban form and sustainable development $

Keywords: polycentrism, urban costs, communication costs, commuting costs, environmental policy, greenhouse gas emissions, land conversion Q54, Q58, R12, R21

In this paper we study how the urban structure affects two environmental outcomes : greenhouse gas emissions from commuting and the conversion of agricultural/natural land ressources. The urban structure is characterized by its degree of polycentrism and by the share of urban and peri-urban dwellers. Polycentrism may lead to opposite effects on these indicators as lower commuting costs in secondary centers lead to lower land rents that may allow households to consume more residential land; furthermore commuting lenght may decrease, if workers commute to the closest business district, inducing less GHG emissions. The environmental impact of the urban structure is very dependant on residential lot sizes, fixed within zones but different between them. Then we address the implementation of two types of instruments to comply with an exogenous GHG target, a commuting tax/subsidy and a communication tax/subsidy for firms established in the secondary center.

Introduction

Cities are home to an increasing share of the world population. It is therefore important to understand how urban development affects their environmental performance. Increasingly, jobs, hence households, are decentralized within cities : polycentrism is becoming a major feature of the urban landscape worldwide [START_REF] Anas | Urban Spatial Structure[END_REF]. Despite a good understanding of the economic forces favoring the ermergence of polycentric structures, there is no real consensus on their environmental performance. This paper analyses how polycentrism affects two important environmental issues associated with urban development : the emission of greenhouse gases (GHG) by commuters and the conversion of surrounding agricultural/natural land.

The emergence of polycentric urban structures can be explained by the combination of three main factors: the increase in population, the increase in commuting costs and the fall in firms' communication costs between decentralized locations and the central business district (CBD). With increasing city size, urban costs (housing and commuting) tend to rise. This has to be compensated for by firms through higher wages. This creates an incentive for firms to locate outside the main center, in secondary centers where urban costs are lower. Indeed, Timothy and Weaton [START_REF] Timothy | Intra-Urban Wage Variation, Employment Location and Commuting Times[END_REF] report large variations in wages according to intra-urban location (15% higher in central Boston than in outlying work zones, 18% between central Minneapolis and the fringe counties). As they enjoy living on large plots and/or move along with firms, workers may also want to live in suburbia [START_REF] Glaeser | Sprawl and urban growth[END_REF]. The relocation of firms is further facilitated by a fall in communication costs, costs related to the fact that the primary centers retain some specific services, such as high finance, so that firms established in secondary centers incur an access cost to these services. The development of new information and communication technologies in the last decades has greatly reduced these costs, without totally suppressing the need to access these services. Cavailhès et al. [START_REF] Cavailhès | Trade and the structure of cities[END_REF] analyze how the interplay between these various costs and the trade costs of the inter-regional shipment of commodities affects both the inter-regional and the intra-urban location of firms in a two-regions model. They show that under some conditions, polycentric agglomerations outperform monocentric cities on economic grounds : "the emergence of subcenters within cities is a powerful strategy for large cities to maintain their attractiveness" [8, p.384]. Absent from this analysis, though, are environmental concerns.

Urban development is associated with numerous damage to the environment : the transport of people and goods contributes to local air pollution and GHG emissions [START_REF] Kahn | Green cities : urban growth and the environment[END_REF] [START_REF] Glaeser | The greenness of cities : carbon dioxide emissions and urban development[END_REF]; the increase in impervious surfaces alters the functioning of water ecosystems [START_REF] Lohse | Forecasting relative impacts of land use on anadromous fish habitat to guide conservation planning[END_REF]; land fragmentation is detrimental to biodiversity [START_REF] Merenlender | Exurban development influences woodland bird composition[END_REF]; etc. In this paper, we focus on two of the most pressing issues that are directly affected by the urban structure : the conversion of surrounding agricultural and natural land and the emission of GHG from the transport of people.

Indeed, urban spread, whether continuous or leagfrogging, implies high rates of conversion of agro-forested or natural land into developed land. Ahearn et al. [START_REF] Ahearn | A discussion of recent land-use trends[END_REF] note that "in large part, this new use of previously undeveloped land is for rural residences, often times on the fringe of urban areas, and often times with large lot sizes". These large peripherical residential lots are explained by the fundamental trade-off of urban economics, between cost of land and cost of commuting to the central business district, or Clark's Law on density gradients [START_REF] Clark | Urban population densities[END_REF]. These mechanisms operate in an actual world that is heterogenous, with natural or administrative boundaries and zoning, and positive or negative amenities. Urban growth boundaries that exist in many U.S. metropolises or greenbelts of U.K. cities constitute spatial discontinuities : before and beyond, the lot sizes are different, due to different supplies of urban public goods, zoning, etc. Instead of the negative exponential function [START_REF] Clark | Urban population densities[END_REF], we prefer to use a non-smooth lot size gradient between urban and periurban areas. In France, for instance, the lot size of single-detached houses increases from 400 m2 downtown to 600 m 2 in the suburbs and 900-1000 m 2 in the peri-urban belt 1 . The administrative limits create breaks, which may be due to a discontinuity in local public goods availability: in the city, inhabitants enjoy a lot of urban public goods (urban transport system, high-standard schools, etc.) that are not -or less-available in peri-urban areas.

The pressing importance of the second environmental variable we study is illustrated by the following quote from the former Executive Secretary of the United Nation Framework Convention on Climate Change : "given the role that transport plays in causing greenhouse gas emissions, any serious action on climate change will zoom in on the transport sector" 2 . Commuters are the main contributors to GHG emissions by the transport of people [START_REF] Gaigné | Are conpact cities environmentally friendly?[END_REF], and even if substantial reductions in polluting emissions might be warranted by technological innovation [START_REF] Kahn | Green cities : urban growth and the environment[END_REF], these reductions will not suffice to stabilize the contribution of the transport sector to GHG emissions [START_REF]Greenhouse gas emission trends and projections in Europe[END_REF]. A large empirical literature is devoted to analyzing the impact of city size and structure on GHG emissions, via commuting [START_REF] Bento | The efficiency and distributional impacts of alternative anti-sprawl policies[END_REF][START_REF] Kahn | Green cities : urban growth and the environment[END_REF][START_REF] Brownstone | The impact of residential density on vehicule usage and energy consumption[END_REF][START_REF] Glaeser | The greenness of cities : carbon dioxide emissions and urban development[END_REF]. Among the theoretical contributions, Gaigné et al. [START_REF] Gaigné | Are conpact cities environmentally friendly?[END_REF] investigate the relevance of city compactness -understood as high-density urban areas -as a way to mitigate GHG emissions from the inter-regional shipping of commodities and the intra-regional commuting of workers. On the one hand, agglomeration in one region reduces the level of pollution from commodity shipping between regions; on the other hand, it increases the amount of commuting within the only metropolis, hence the associated polluting emissions. The authors advocate the combination of a density-increasing policy and of a policy encouraging polycentrism in order to achieve the best environmental outcome. Given our focus on a single metropolis, we analyse the GHG emissions from commuting only.

In this paper, we build on [START_REF] Cavailhès | Trade and the structure of cities[END_REF] and [START_REF] Gaigné | Are conpact cities environmentally friendly?[END_REF] to propose an investigation of the environmental performance of a metropolis made of a primary compact city surrounded by a residential peri-urban belt (where density is lower) and secondary cities and their peri-urban belts. More precisely, we look at how the degree of polycentrism, that is the share of inhabitants settled in the secondary centers, impact on both environmental variables of interest. We focus on this urban structure, leaving inter-regional trade issues aside, and consider two environmental impacts of urban development. We consider different lot sizes in the primary and secondary cities and in the peri-urban belts, to capture the stylized fact that lot sizes vary according to the location. In a second step, we study the efficient management of GHG emissions and undeveloped land preservation at the agglomeration level, by introducing environmental policies in this urban economics framework.

We describe our modeling strategy in Section 2 and derive the decentralized equilibrium in Section 3. Section 4 analyses the environmental outcomes of the equilibrium urban structure. In section 5 we discuss various policy instruments to ensure compliance with an exogenous GHG target. Section 6 concludes.

The Model

(i) The spatial structure. We consider a one-dimensional metropolis with mobile workers, one sector and two primary goods, labor and land. This metropolis is made of three urban areas. Its centre is a punctiform central business district (CBD); two punctiform secondary business districts (SBDs) -if any -are located towards the right and the left of the CBD. Jobs are located in the CBD and the SBDs. Workers are housed in cities that spread around the CBD and SBDs, and in peri-urban belts -if any -beyond. The settlement is dense in cities, and scattered in the peri-urban belts. Without loss of generality, we focus on the right-hand side (RHS) of the metropolis (see Figure 1). Hereafter, subscript c refers to the CBD and subscript s to the SBD, while u and p denote, respectively, city and peri-urban belts. x = 0 is the center of the metropolis, x uc the eastern limit of the city of the primary urban area and x pc the eastern limit of the peri-urban belt of this primary urban area. Around the SBD (x os ) lies a city between x ps and x us surrounded by peri-urban belts. The eastern limit of the metropolis is x pse . We denote X uc = [0, x uc ], X pc = [x uc , x pc ], X L ps = [x pc , x ps ], X us = [x ps , x us ] and X R ps = [x us , x pse ] the various segments of the metropolis.

(ii) Firms. Let M be the total number of workers and N the total number of identical firms. Technology in manufacturing is such that producing Q(k) units of the composite good requires a given number φ of labor units. Firms do not consume land and are free to locate in either the CBD or the SBD. They pay a wage w j , j ∈ {c, s} depending on which business district they are settled in. To simplify notations, without loss of generality we assume M = 1. The profit of the CBD firms is Π c , and Π s the profit of the SBD firms. The latters bear a fixed communication cost K > 0 reflecting the fact that some high standard services or facilities (such as banking, airports, etc.) are only available in the CBD : firms located in the SBD have to visit those services periodically, at a cost. Profit functions for a firm k are set as follows, assuming that, within the metropolis, the output price does not vary and the cost of shipping the good is nul :

Π c (k) = P f Q(k) -φw c , (1)
Π s (k) = P f Q(k) -φw s -K, (2)
where P f denotes the output price and Q(k) the quantity of output. Since both types of firms face the same demand function, setting the profits to zero, we obtain the following equilibrium condition :

w c -w s = K φ (3)
Hence, the equilibrium wage wedge increases with the communication costs faced by firms in the SBD (Appendix 4 illustrates this difference in the case of the metropolis of Toulouse).

(iii) Households. A proportion α ( 1 3 ≤ α ≤ 1) of households live in the primary urban area, among which αβ 1 live in the city and α(1 -β 1 ) live in the peri-urban belt, with 0 < β 1 ≤ 1. The secondary urban area also comprises a city, where (1 -α)β 2 households live, and a peri-urban belt, where (1 -α)(1 -β 2 ) households reside, with 0 < β 2 ≤ 1.

Households are assumed to comprise a single worker. They enjoy a utility U(q i j , h i j ) = q i j +v ln h i j , i ∈ {u, p} and j ∈ {c, s}, through the consumption of two goods : a quantity q i j > 0 of an aspatial composite good at unitary price P and a residential good, with lot size h i j (x) and rent R j (x). The residential lot size depends on which segment of the city households reside in; typically we assume that peri-urban lot size h pc is the same in all peri-urban zones, and that it is higher than urban lot sizes h uc and h us . We do not make a priori assumption on how urban lot sizes compare : in the standard urban economics case, h us > h uc since the rent functions are lower in the secondary centers than in the primary ones (as illustrated in Figure 2). However, the case of h us < h uc may arise in a heterogenous space with externalities (crime rate, pollution, congestion in the primary city, amenities in the secondary city, etc.), or when there exist limits to urban growth of the secondary cities that do not apply to the primary one. Appendix 4 illustrates a secondary urban lot size lower than that of Toulouse, a large and spread metropolis, and its satellites, attractive medium-sized cities.

To sum up, we assume the following : h pc = h ps > h uc , h us and without loss of generality, we set h uc = 1. With three different lot sizes, we only imperfectly account for the increase in lot size with distance from the city center and the decrease with population observed in the real world. However, this formalisation provides a richer depiction of this reality than the assumption of a fixed lot size over the entire urban system (see Appendix 4).

Households incur a cost of commuting between their residential location and the closest business district where they work, either CBD or SBD. Hence we assume no cross-commuting. However, we consider that commuting costs differ between urban and peri-urban areas because of network congestion in the former. Hence, distances travelled in a city have a unitary cost t = 1 while those travelled in a peri-urban area have a cost of 0 < θ < 1.

Absentee landowners allocate land to the highest bidder. Noting Ψ j (x) the bid rent of urban households and Φ j (x) the bid rent of peri-urban households, j ∈ {c, s}, the rent function is :

R j (x) = max{Ψ j (x), Φ j (x), R A } (4)
The opportunity rent, which is the agricultural land rent, R A is set equal to 0 for simplicity.

Consequently, households' budget constraint depend on the location of their residence :

x ∈ X uc , w c -x = Pq uc (x) + h uc R c (x) (5.uc) x ∈ X pc , w c -θ (x -x uc ) -x uc = Pq pc (x) + h pc R c (x) (5.pc) x ∈ X us , w s -|x 0s -x| = Pq us (x) + h us R s (x) (5.us) x ∈ X L ps , w s -θ(x ps -x) -(x 0s -x ps ) = Pq us (x) + h ps R s (x) (5.psL) x ∈ X R ps , w s -θ(x -x ps ) -(x ps -x 0s ) = Pq ps (x) + h ps R s (x) (5.psR)
The spatial equilibrium urban structure is analysed in the following section. To do so, we derive the equilibrium share of population in the primary urban area, α * , and of urban/peri-urban population β * 1 and β * 2 , and assess how the parameters of the model impact on these equilibrium levels.

Equilibrium urban structure

Inside a zone i j the lot size h i j is constant. Consequently, the equilibrium consumption of the aspatial good is the same within each zone. The bid rents are derived by equating the urban costs (housing plus commuting) within each area. Refer to Appendix 1 for the expressions of the bid rents and to Figure 2 for an illustration. The worker located in x pc is indifferent about her workplace. She pays a land rent R c (x pc ) = R s (x pc ) = R A = 0 and she consumes a residential lot of size h pc = h ps . Consequently, she consumes the same level of composite good q(x pc ) whatever her workplace. Equating the budget equations (5.pc) and (5.psL) evaluated at x = x pc , we obtain a condition on the wage difference between the CBD and the SBD :

w c -w s = θ 2x pc -x uc -x ps + x uc -x 0s + x ps (5)
The wage differential compensates for the gap in commuting costs. Combining Equations ( 5) and ( 3), we obtain a condition on the various segment limits for given θ, K and φ :

K φ = θ 2x pc -x uc -x ps + x uc -x 0s + x ps (U1)
The above expression ensures the equality of utilities between peri-urban households. To complete the spatial equilibrium, we also look for the following conditions :

U(q uc , h uc ) = U(q pc , h pc ) (U2) U(q us , h us ) = U(q ps , h pc ) (U3)
From (U1)-(U2)-(U3), we can express the composite good consumption levels as functions of the residential lot sizes, or vice versa : either the residential lot size is exogenous, resulting from cadastral division or urban zoning policy, or the available quantities of the composite good are exogenous, for instance if they are local public goods. We adopt the former vision in the paper.

Solving for α, β 1 and β 2 in (U1)-(U2)-(U3) and replacing the zonal limits by their values (see Appendix 2) we obtain the equilibrium shares of population when h pc > max{1, h us }:

α * = 1 h us + 2 [h us + 4 K φ -4Pv V] , with V = ln(h pc )h pc (1 -θ)(h us -1) + ln(h us )(h pc -1)(θh pc -h us ) θ(h pc -1)(h pc -h us ) (6)
β * 1 (α) = 1 - 2Pv ln h pc α * θ(h pc -1) (7)
β * 2 (α) = 1 - Pv ln h pc h us (1 -α * )θ(h pc -h us ) (8)
The following condition must apply to ensure that the primary center attracts at least one third of the population and not more that the total population :

V - 1 6 θ(h us -1) < K φ < 1 2 + V (C1)
Given the importance of the communication costs in shaping the polycentric nature of the metropolis in our framework, we express the conditions necessary for the validity of α * ∈ [0, 1] with respect to these costs. Not surprisingly, they need not be too high nor too low to ensure that the level of polycentrism, expressed as a share of the total population, is consistent. Note that setting

θ = h us = h pc = 1 in (U1)-(U2)-(U3), we obtain α * = 1 3 + 4 3 K φ .
Consistent with previous studies [START_REF] Cavailhès | Trade and the structure of cities[END_REF][13], under the assumption of a constant lot size and of a constant transport rate, the primary city is home to at least a third of the total population, since two subcenters are considered. Furthermore, the higher the communication costs incurred by the secondary center firms, the larger the primary city.

Since h pc > max{1, h us }, it can be easily shown that

β * 1 (α), β * 2 (α) ≤ 1.
To ensure that both urban/peri-urban population shares are positive, the following conditions on the equilibrium polycentrism level must hold:

0 < 2Pv ln(h pc ) θ(h pc -1) < α * < 1 - Pv ln(h pc /h us ) θ(h pc -h us ) (C2)
To ensure that the equilibrium shares of urban/peri-urban populations are consistent, the polycentrism level should not be extreme, nor too low nor too high.

Proposition 1. The impact of firm-related parameters on the urban form is non ambiguous : the system is more polycentric, the primary urban area is more compact and the secondary ones are more peri-urbanized when the communication costs between the CBD and the SBD are low. The impact of household-related parameters are not so clear-cut : an increase in commuting costs or in the preference for lot size may induce either a higher or a lower polycentrism level, and larger or smaller peri-urban belts, depending on the relative urban and peri-urban lot sizes.

Proof. The impact of communication costs is straightforward :

∂α * (β * 1 , β * 2 ) ∂K = 4/φ 2 + h us > 0 (9)
A high wage difference, that is a high communication cost between the primary and secondary centers, tends to be in defavor of polycentrism (remember that a high α * denotes a low level of polycentrism). Communication costs impact first on the location of firms; but they also have an impact on the equilibrium urban/peri-urban repartition of population : rising communication costs increase the share of primary urban agents, and decreases that of secondary urban agents. Figure 2 provides the intuition behind this mechanism : more households need to accomodated in the primary city and/or the primary peri-urban blet. The bid rent function of the primary city and/or peri-urban belts rises, so that in all cases x uc moves to the right. A similar reasoning applies to explain the decrease in β * 2 .

To assess the sign of the impact of the other parameters of the model on the equilibrium polycentrism level, (C3)-(C5) establish contidions on the ratio of lot size differences.

Commuting costs favor polycentrism if the following applies (see Appendix 3 for full derivation) :

h us -1 h pc -1 < h us ln h us h pc ln h pc (C3)
They have both a positive direct impact on β * 1 (α) and β * 2 (α) and an indirect impact through α * which sign depends on compliance or not with (C3). When (C3) holds, rising commuting costs increase the share of secondary urban agents and have an impact which sign cannot be derived on the share of primary urban dwellers; if (C3) does not hold they increase the share of primary urban agents, and their aggregate impact on secondary urban agents cannot be signed.

Preference for residential lot size and the commodity price have an impact of the same direction on the level of polycentrism. When θ > h us h pc , they always always favor polycentrism; otherwise they do so if :

h us -1 h pc -1 > (h us -θh pc ) ln h us h pc (1 -θ) ln h pc (C4)
As for transport costs, they have both a negative direct impact on β * 1 (α) and β * 2 (α) and an indirect impact through α * which sign depends on compliance with (C4) : increasing preference for lot size decreases the share of primary urban dwellers if (C4) does no't hold and decreases that of secondary urban dwellers if it holds. As for above, in the other cases, the aggregate sign cannot be determined.

Finally, we look at the impact of residential lot size on the various components of the urban structure. We can establish that increasing h pc favors polycentrism when the following applies :

h us -1 h pc -1 < h us ln h us (h pc -1) + (h us -1)(h pc -h us ) (h 2 pc -h us ) ln h pc (C5)
The direct impact of the peri-urban lot size on the shares of urban dwellers can be shown to be positive. Then, as above, the aggregate impact depends on how lot size affects the level of polycentrism and is difficult to sign in some cases.

The case of the secondary urban lot size is more difficult to tract analytically -it can only easily be shown that it has no impact on β 1 .

The environmental impact of polycentrism

This section is devoted to analysing under which conditions the polycentric structure may be conducive to less land conversion and aggregate distances travelled. We focus on these two environmental variables as they capture important debates related to urban development and the same features of the urban structures impact on them : residential lot size, peri-urban belt width and polycentrism level. However, as we will show, these environmental features may still arise as conflicting objectives.

Land conversion

The first environmental criteria we analyse is the level of land conversion, noted S , which is the total width of the (half) city we study:

S = x pse = α[β 1 + h pc (1 -β 1 )] + (1 -α)[h us β 2 + h pc (1 -β 2 )] (10)
The higher S , the higher the detrimental impact of urban development on surrounding natural and agricultural land.

Proposition 2. Polycentrism decreases the pressure of the metropolis on land ressources, if the secondary urban lot size is lower than unity.

Proof. To sign the impact of the level of polycentrism on the total city width we proceed as follows :

dS dα = ∂S ∂α + ∂S ∂β 1 ∂β * 1 ∂α + ∂L ∂β 2 ∂β * 2 ∂α (11)
that can be rearranged into :

dS dα = (h pc -h us ) β * 2 -(1 -α) ∂β * 2 ∂α -(h pc -1) β * 1 + α ∂β * 2 ∂α (12)
The above expression is positive, meaning that polycentricy reduces land conversion, if the following applies :

h pc -h us h pc -1 > 1 ⇔ h us < 1 (C6) since it is easy to show that β * 2 -(1 -α) ∂β * 2 ∂α = 1 and β * 1 + α ∂β * 1 ∂α = 1.
The impact of polycentrism on land ressources depends on the relative difference in urban/peri-urban lot sizes : due to the assumptions that h ps = h pc and h us = 1, this amounts to comparing h us to 1. Monocentrism entails more land conversion when (C6) holds, that is when the lot size is smaller in the secondary city than in the primary one.

Consequently, any alteration that increases the monocentric structure of the metropolis will be in favor of less land consumed when (C6) does not apply : higher communication costs, higher transport costs (if (C3) doesn't apply), and a lower preference for lot size (if (C4) doesn't apply). Conversely, if (C3)-(C4) and (C6) apply, less land consumed is obtained by favoring a more polycentric structure, through reduced communication costs, lower transport costs and a higher preference for lot size.

Commuting distances

The aggregate length of commuting L is the second environmental variable we study. Through this variable we capture the impact in terms of GHG emissions of the urban form. Hence we assume that there is a fixed linear relation between the distance travelled and the level of emission : we abstain from the consideration of the effect of congestion on the intensity of emissions, for instance.

L = 2 i j x∈X ji 1 h ji x dx = h pc 4 - h pc -h us 4 (1 -α) 2 β 2 - h pc -1 4 αβ 1 (2 -αβ 1 ) (13)
Proposition 3. Increasing the level of polycentrism may induce a higher or a lower aggregate distance travelled by commuters, depending on the relative residential lot sizes.

Proof. To study the impact of city structure on aggregate distance travelled, we consider the following :

dL dα = ∂L ∂α + ∂L ∂β 1 ∂β * 1 ∂α + ∂L ∂β 2 ∂β * 2 ∂α (14)
Rearranging the above terms, accounting for the fact that β * 2 -(1-α)

dL dα = (1 -α)(1 -h us ) -2 pv θ [2 ln h pc -ln h us ] (15)
If h us > 1, monocentrism should increase distances travelled if α > 1 + 2 Pv θ(h us -1) [2 ln h pcln h us ] > 1 : this is not possible by definition of α. Monocentrism always decreases distances travelled.

If h us < 1, monocentrism increases distances travelled if α < ᾱ = 1 -2 Pv θ(1-h us ) [2 ln h pc -ln h us ].
In other words, if the monocentrism level is low enough, reducing distances travelled is achieved by tending towards a more polycentric urban structure. Further note that ᾱ > 0 if 1h us > 2Pv θ [2 ln h pcln h us ] : h us has to be sufficiently smaller than 1 to ensure that the condition 0 < α < ᾱ is achievable.

As for land conversion, the relative residential lot size difference plays a major role in assessing whether monocentrism or polycentrism encourages less kilometers travelled. Then again, any alteration of the exogenous parameters that impact on the urban structure can play in favor or in defavor of an environmental improvement.

The possibility of conflicting environmental objectives

Proposition 4. Depending on the relative values of the lot sizes in the various segments of the metropolis, the impact of the urban structure on GHG emissions and land conversion can be characterized by three cases only : (i) polycentrism as the win-win solution, (ii) monocentrism as the win-win solution and (iii) polycentrism as a land preserving solution, at the expense of increased GHG emissions.

Proof. Drawing from Propositions 2 and 3, the impact of the metropolis on the two environmental variables of interest can be assessed through the difference in urban residential lot sizes.

• for 1h us < 0 we have shown that ∂S /∂α < 0 and ∂L/∂α < 0 : monocentrism is the win win solution (case A);

• for 0 < 1h us < ᾱ we have shown that ∂S /∂α > 0 and ∂L/∂α < 0 : polycentrism reduces land conversion but increases GHG emissions (case B);

• for 1h us > ᾱ we have shown that ∂S /∂α > 0 and that ∂L/∂α > 0 if α < ᾱ : polycentrism may appear arise as the win win solution (case C).

Both cases A and C are exemples of win-win situations where decreasing the distances travelled entails less land consumed. However the implication in terms of urban structure is different : in case A a more monocentric metropolis is the environmentally-friendly structure, while in case C, the polycentric metropolis is the sustainable solution. The mixed case B implies conflicting policy prescriptions in terms of urban structure to manage either S or L : while polycentrism reduces land conversion, it increases GHG emissions. The implications in terms of public policy design are developped in the following section.

Public policies in a polycentric metropolis

In this section we address the design of public policies to induce compliance with an exogenously determined environmental target. We assume that the metropolis planner is constrained by an exogenous target on the aggregate level of GHG emissions, decided at a higher level of government. This constraint on GHG emissions is translated in our framework into a constraint on the aggregate commuting distance travelled, implying the same assumption of fixed emission rates as in the previous section. We analyse efficient tax/subsidy schemes that would allow the metropolis planner to comply with this target. Given the interplay between distances travelled, commuting costs and communication costs, we want to assess the effectiveness of two policy bases : commuting and communication costs. Due to the cumbersome nature of the analytical derivation of the tax schemes, we have recourse to numerical simulations. After deriving the efficient schemes, we assess how they impact the metropolis structure and hence the other environmental variable of interest in the model, land conversion. Remember that we look at an isolated metropolis, hence we do not account for how the urban form affects its competitiveness with respect to the rest of the world. Consequently, we are only interested in how different environmental objectives interplay, while Gaigné et al. [START_REF] Gaigné | Are conpact cities environmentally friendly?[END_REF] analyse the links between environmental and economic objectives.

Result 1. A decrease of a given level of GHG emissions L(K ini ) -T is achievable by applying a communication tax/subsidy : t com = K * -K ini . The resulting decrease in distances travelled may necessitate a more (cases B and C) or less (case A) polycentric urban structure and entail more (case C) or less (cases A and B ) land conversion.

Proof. We are looking for a value of K * such that an exogenous GHG target T is respected. The simulations are undertaken as follows : for K = K ini , we calculate the resulting L(K ini ). Then we randomly draw values of T < L(K ini ) and derive the values of K * that would lead the economy to that particular GHG emission target, respecting the definition domains of α * , β * 1 and β * 2 .

A tax/subsidy on communication costs proves an efficient GHG reduction strategy. Decreasing communication costs induce firms to settle in the SBD, attracting households and conducing to a more polycentric setting. In case C, this translates into lower commuting distances hence lower GHG emissions, but also into lower land conversion. In cases A and B, in order to comply with the GHG reduction target, it is necessary to go towards more monocentric urban structures, and this is achieved by taxing communication costs. However, the GHG reducing policy has opposite effects on land ressources : while it is also environmentally beneficial in case A, it increases land conversion in case B. This is due to the straightforward relation that exists between urban structure and communication costs : they affect positively the polycentrism level, and in an opposite fashion the urban/peri-urban population shares. As a consequence, relying on a communication costs based strategy to manage GHG emissions is a win-win solution in cases A and C, but in case B environmental objectives are still conflicting. This changes when commuting costs are the basis of policy-design. Result 2. A decrease of a given level of GHG emissions L(θ ini ) -T is obtained by imposing a transport tax/subsidy : t tr = θ * -θ ini . The induced decrease in distance travelled may necessitate either a higher (cases B and C) or a lower (case A) level of polycentrism and it always entails less land conversion.

Proof. We apply the same simulation strategy as for the transport tax analysis to the resolution of L(θ) = T .

A tax/subsidy on peri-urban commuting costs proves an efficient urban planning strategy -within the restricted framework of our analysis, where the efficiency of a policy instrument is assessed by its capacity to induce a given level of GHG emissions. The way it impacts on the urban structure and, consequently, on land conversion, is different, though, from the communication tax/subsidy case. Indeed, due to the complex link between commuting costs and the two types of population shares, α and the β i s, there is no straightforward relation between how increasing or decreasing transport cost will affect aggregate distances and land conversion. As a matter of fact, cases A and C both remain win-win situations, with the same difference as above : polycentrism as a double environmental improvement for case A, monocentrism as the win-win solution in case A. However, case B is interesting since land conversion and aggregate distances travelled are no longer conflicting objectives when commuting is taxed : the urban structure is affected in such a way that both distances and land conversion are reduced by increasing transport costs -this translates into a more polycentric and more urban metropolis. Hence, the two policy strategies we have chosen to study induce different urban structures, hence different impacts on land conversion, to achieve the same GHG target.

Concluding remarks

In this paper we presented an urban economics model that allows for the development of secondary business districts, given a communication cost with the central business district, and of peri-urban belts where the population density is lower than that in compact city centers and where commuting to work is faster (hence its unitary cost is lower). Thus this model relaxes the assumptions that are often relied on of a monocentric city and/or of a unique residential lot size and/of a fixed transport cost. With this model we can analyse two environmental effects of the urban structure : the conversion of agricultural or natural land to urban uses and the emissions of GHG from commuting. The model also allows studying the effects of policy instruments on these two environmental variables : a commuting tax/subsidy and a communication tax/subsidy.

Polycentrism appears and increases when the communication costs decrease, which favors the decentralisation of firms. We show that the urban form has an impact on the two environmental variables we study. We show that there exist three polar cases of the environmental performance of the urban form : monocentrism as the win-win environmental solution, polycentrism as the win-win solution and monocentrism as the way to reduce GHG emissions at the expense of more land consumed. When polycentrism is a competitive economic equilibrium in inter-regional exchanges [START_REF] Cavailhès | Trade and the structure of cities[END_REF], it may also be environmentally beneficial.

These results suggest that a policy that modifies the structure of the metropolis can be used for environmental purposes; in this paper to limit the emissions of GHG. To restrain commuting-related GHG emissions, transport taxes are the usual suspects. We also analyse the efficiency of an instrument directly targeted at impacting on the development of secondary centers, a tax/subsidy on the communication costs that firms established in the secondary business district have to bear. These two instruments consist in reducing GHG by impacting on the urban form, but they have different impacts on the urban structure, hence on land conversion. The consideration of a budget-balanced scheme targeting both commuting and communication costs is under consideration for future developments.

The environmental attractiveness of the polycentric structure relies on the existence of attractive secondary urban areas, both for the firms (by reducing the commuting costs) and for the consumers. This motivates research in progress in two directions : theoretical to analyse how the differentiated availability of local public goods in the various zones may explain the residential lot size discrepancies; empirical, to estimate this relationship.

These developments also lead to public policy recommandations, in particular, in terms of regional planning.

In France, the DATAR seeks to favor an urban network constituted of regional metropolises, with a national and international level of competitiveness, linked to secondary centers structuring rural areas. Such configurations may or not be environmentally sustainable according to the transport and communication costs and the LPG supplies -all of which constitute potential bases for public policies that allow -or not -to conciliate economic and environmental objectives.

Ψ c (x 0 ) = Ψ c (x pc ), Ψ sl (x 0s ) = Ψ sl (x ps ). c stands for central, s for secondary, l for left-hand side of the secondary city and r for its right-hand side.

Ψ c (x) = x uc + θ h pc (x pc -x uc ) -x (16)
Φ c (x) = θ h pc (x pc -x) (17)
Ψ sl (x) = θ h pc (x ps -x pc ) + 1 h us (x -x ps ) (18)
Ψ sr (x) = θ h pc (x ps -x pc ) + 1 h us (2x 0s -x ps -x) (19)
Φ sl (x) = θ h pc (x -x pc ) (20)
Φ sr (x) = θ h pc (2x 0s -x pc -x) (21)
Appendix 2 : Analytical expression of the zonal limits

Given the proportions of households living in each type of zone, and the lenght occupied by their residence, it is easy to provide an analytical definition of the limits of the zones :

x uc = 1 2 αβ 1 x pc = x uc + 1 2 h pc α(1 -β 1 ) x ps = x pc + 1 4 h pc (1 -α)(1 -β 2 ) x 0s = x ps + 1 4 h us (1 -α)β 2 x us = x 0s + 1 4 h us (1 -α)β 2 x pse = x us + 1 4 h pc (1 -α)(1 -β 2 ) 1 ∂α * ∂α * ∂θ ( 23
)
∂β * 2 (α * , β * 1 ) ∂θ = 4pv ln h pc h us (1 -α * )θ 2 (h pc -h us ) + ∂β * 2 ∂α * ∂α * ∂θ (24) ∂α * (β * 1 , β * 2 ) ∂v = -4p[ln h pc (h us -1)h pc (1 -θ) + ln h us (h pc -1)(θh pc -h us )] θ(h us + 2)(h pc -1)(h pc -h us )) (25) ∂β * 1 (α * , β * 2 ) ∂v = -2p ln h pc α * θ(h pc -1) + ∂β * 1 ∂α * ∂α * ∂θ (26)
∂β * 2 (α * , β * 1 ) ∂v = -4p ln h pc h us (1 -α * )θ(h pc -h us ) + ∂β * 2 ∂α * ∂α * ∂θ (27)

Appendix 4. Stylized facts from France

As usual in urban economics, the outcomes of the foregoing model have been obtained at the cost of many simplifying assumptions. We have to assess the relevance of this model vis-à-vis the actual world. We present here, with empirical data from the French context, stylized facts that illustrate some theoretical assumptions (existence of peri-urban belts, absence of cross-commuting, variations in lot size), and some results that illuminate the main contrasting situations regarding the environnemental outcomes of the model.

As in many developed countries, the French urban structure is polycentric. The case of Toulouse and its satellites illustrates this pattern (see Figure 6). Toulouse, with more than 500,000 jobs and 1.2 million inhabitants is the fifth French urban area [START_REF] Brutel | Un maillage du territoire français[END_REF]. The city accommodates an international airport, a well-known university, high-tech industries and businesses (Airbus, banking, etc.). It is surrounded by satellites, medium-sized urban areas 4 that count around 75,000 people: Albi, Montauban, Auch, etc. Another key assumption of the model is no cross-commuting between the CBD and the SBDs. Daily commuting to Toulouse from the satellite urban areas is rare: the round-trip from the town hall of Albi to the town hall of Toulouse is 134 minutes (non-rush periods). Finally, firms of the satellites cities use the upper services located in Toulouse, bearing a communication cost, another key assumption of the model.

Regarding the urban structure, our modeling framework does not assume any geographical scale of analysis: scales of CBD/SBD interactions that differ from the Toulouse case are relevant within the framework of this model, provided that the SBDs are not to far from the CBD (so that SBDs' firms can interact with CBD's services) but far enough to exclude cross-commuting by workers. This situation occurs between medium-sized urban areas (several hundreds ot thousand inhabitants) and smaller satellite towns (several tens of thousands inhabitants). To illutrate this case, we use the example of Caen, a medium-sized metropolis (400,000 inhabitants), surrounded by urban areas that are smaller (40,000 inhabitants or so: Bayeux, St Lo, Vire, Flers, Lisieux, Dives-sur-mer). In urban economics, it is usual to say that the lot size increases in the distance from the CBD, due to the trade off between land cost and commuting cost. We slightly modify this mechanism by assuming that this change in distance occurs in stages. The situation around Toulouse is illustrated by Table 1 5 : given the administrative borders, the mean lot size of single-detached houses in central Toulouse is 327 m 2 ; then in the suburbian communes of the 'pôle urbain' it varies according to the distance (from less than 20 minutes to more than 30 minutes) between 900 and 1,200 m 2 , leading to a large mean lot size for all the communes of the 'pôle urbain' (the city of the model): 1,013 m 2 . In the peri-urban belt, the lot sizes fluctuate around 2,000 m 2 , a much higher level than in the pôle urbain, that is quite steady. Our assumption of an evolution in plateaus of the lot sizes is quite realistic regarding the suburbs and the peri-urban belt. Another break occurs between the inner city (300 m 2 ) and its suburbs (around 1,000 m 2 ); however the model would be untractable with three lot sizes in the primary urban area, and three other sizes in the secondary one. The individual data from the housing surveys allows us to calculate that the mean salary in the urban area of Toulouse is 20,074 euros, and 16,664 euros in the satellite urban areas. A salary equation à la Mincer [START_REF] Mincer | Changes in Wage Inequality[END_REF] allows estimating the K φ . We obtain for Toulouse : w c = 1298(±900) + w s (R 2 = 0.45). The wage wedge is significant at a 15% threshold, and is in favor of the primary urban area. Finally, we assume that the commuting cost is lower in the peri-urban belt than in cities. Using the Housing surveys data for all the French large and medium-sized urban areas, we calculate that the commuting speed is 39 km/h in the central city and its suburbs and 49.3 km/h in the peri-urban belt. 6For the following observed values of the lot sizes (h us , h pc ), the following Corollaries illustrate Propositions 2-4 in the French cases of Caen (h us = 1.2, h pc = 1.76), Toulouse (h us = 0.81, h pc = 2.16) and Besançon (h us = 0.73, h pc = 1.07). Note that in the Toulouse and Besançon cases, the lot size in the secondary 'poles urbains' is much lower than in the 'pole urbain' of the primary city, leading to positive environmental outcomes of polycentrism.

With these values of the parameters, corollaries of Propositions 2, 3 and 4 are:

Corollary 1. In Caen, monocentrism entails less land conversion : given the lot size differences, reducing sprawl would necessitate a more monocentric structure. Besançon and Toulouse are cases where the reverse applies : reducing the hold on land ressources goes with a more polycentric structure.

Corollary 2. Caen and Toulouse is a case where monocentrism entails less kilometers travelled : given the lot size differences, reducing commuting distances would necessitate a more monocentric structure. In Beançon, the reverse applies : polycentrism is compatible with a GHG reduction target.

Corollary 3. Caen and Besançon are examples of win-win situations where altering the urban structure induce a decrease in both GHG emissions and land conversion; however in Caen the structure has to be more monocentric, in Besançon polycentrism should be favored. Toulouse illustrates the mixed case where a more monocentric structure is needed to reduce GHG emissions, at the expense of increased land conversion.

Figure 1 :

 1 Figure 1: The metropolis

Figure 2 :

 2 Figure 2: Urban and peri-urban bid rents

∂β * 2 ∂α = 1 and β * 1 +α ∂β * 1 ∂α = 1

 211 , and replacing the β * i by their value in the remaining expression, we obtain : 1 2

Figure 3

 3 Figure 3 illustrates the three cases with the following numerical values for the parameters 3 : p = 1, v = 0.1, wc = 1, w cw s = 0.1, θ = 0.9 and for A : h us = 1.2 and h pc = 2, for B : h us = 0.7 and h pc = 2 and for C : h us = 0.85 and h pc = 1.05.

Figure 3 :

 3 Figure 3 : Kilometers travelled and land conversion as function of the degree of polycentrism for, from left to right, cases A, B and C.

Figure 4 :

 4 Figure 4 : Kilometers travelled and land conversion as function of the degree of polycentrism for, from left to right, cases A, B and C, with the efficient communication tax/subsidy scheme.

Figure 5 :

 5 Figure 5 : Kilometers travelled and land conversion as function of the degree of polycentrism for, from left to right, cases A, B and C, with the efficient commuting tax/subsidy scheme.

Figure 6 :

 6 Figure 6 : The metropolis of Toulouse

Table 1 :

 1 Residential lot size according to distance to center -Toulouse

		number of obs.	mean lot size (m 2 )
	city of Toulouse	526	327
	suburbs 0-20 mn from center	187	1245
	suburbs 20-25 mn from center	357	921
	suburbs 25-30 mn from center	385	1200
	suburbs > 30 mn from center	312	1227
	peri-urban belt < 30 mn	196	1935
	peri-urban belt 30-35 mn	113	2435
	peri-urban belt 35-40 mn	151	1876
	peri-urban belt > 40 mn	100	1909

Authors' calculation from the Housing surveys by the INSEE.

Yvo de Boer, Speech to the Ministerial Conference on global Environment and Energy in Transport, 15 January 2009.

The value of v of used in the simulations is lower than in standard empirical urban economics. Indeed, the surface and its exponent v in the utility function apply to the raw plot; while the structure is in the aspatial good because its price is footloose.

In the 2010 statistical zoning of aires urbaines, a French large or medium urban area is a densely built-up "pôle urbain" (city-center +suburbs) offering more than

5,000 jobs, and a peri-urban belt made up of communes where the built-up area is not adjacent to the pôle urbain (separated by agriculture, forests, etc.), and from which 40% or more of the working population commutes daily to another commune, generally to the "pôle urbain".

We use individual economic data from housing surveys conducted by the INSEE (Institut National de la Statistique et des Etudes Economiques)in 1984, 1988, 1992, 1996, 2002, and 2006. We thank the INSEE for providing us computing and technical equipments required for econometric treatments on its secured distant center.

Distances come from the software Odomatrix[START_REF] Hilal | ODOMATRIX. Calcul de distances routières intercommunales[END_REF] which computes distances by the road network, expressed in kilometers and minutes during peak and off-peak traffic periods.

Appendix 1 : Bid rent functions

In each segment of the city, the fixed lot size assumption applies. Then the urban costs (housing plus commuting) are equated among the households in each segment. Consequently, in each segment the slope of the bid rent function is the opposite of the ratio between the commuting cost and the lot size prevaling in the segment.

The bid rent functions for urban and peri-urban households, respectively Ψ and Φ, are derived under the above assumption and accounting for the following conditions : Ψ c (x uc ) = Φ c (x uc ) , Ψ sl (x ps ) = Φ sl (x ps ), Appendix 3 : impact of parameters on the population shares

∂β * 1 (α * , β * 2 ) ∂θ = 2pv ln h pc α * θ 2 (h pc -1) + ∂β *


Catherine Bastien
email: catherine.bastien@orleans.inra.fr

Collaborative Research in forest genomics and tree breeding : Mutual benefits and future contribution to adaptation of forests

Breeders provided :

• Pedigrees :

• Phenotypic information for a wide range of traits Molecular Geneticists provided :

•

•A

  Why accelerated artificial selection from Forest Tree Breeding could assist adaptation of forests to climate change ? • Dissection of genetic variation a common interest in tree breeding and genomics • Results from 'Phenotype-Genotype' collaborative research • New perspectives from high throughput tree genome sequencing Outline estimate that landowners can realize net present values of $125 to over $750/ha across a range of productivity and silvicultural management regimes simply by planting the best genotypes that are currently available from commercial nurseries…" 2006. Journal of Forestry 104:352-358 tree breeding and in genomics Up to now mainly through Phenotype in tree breeding • Resemblance of phenotypes of related individuals is a proxy of Breeding Value • Variation of phenotype of a given individual across environments describes phenotypic plasticity • Breeding values allow prediction of adaptive and economical gains from selection E N V I R O N N E M E N T • DNA sequences of genes and individual genomes • Nucleotide variation and Structural variants • Nb of genes involved in traits of interest, their effects and their position on genome • Gene expression and gene function need : • to increase precision of genetic evaluation • to reduce selection age • to increase selection intensity • to develop more multi-trait selection • to better monitor genetic diversity Molecular markers (h 2 =1) tightly linked to genes involved in control of target traits can help

New perspectives from high throughput tree genome sequencing

Large genomes for forest trees

Reference Genome Sequence


Dr Matieu Henry

Dr Nicolas Picard

Building tree allometric equation to assess volume and biomass : from the field to the prediction

1. Forest stratification (theorical concept); 2. Three approaches for land use change assessment; 3. Assessing national land use & carbon stock changes; 4. Different methods to assess forest carbon stocks and carbon stock changes; 5. Harmonization of tree allometric equations to maximize the use of existing tree allometric equations; 6. Definitions; 7. Content of the course.

CONTENT

• adequate, i.e., capable of representing land-use categories, and conversions between land-use categories, as needed to estimate carbon stock changes and greenhouse gas emissions and removals;

• consistent, i.e., capable of representing land-use categories consistently over time, without being unduly affected by artificial discontinuities in time-series data;

• complete, which means that all land within a country should be included, with increases in some areas balanced by decreases in others, recognizing the biophysical stratification of land if needed (and as can be supported by data) for estimating and reporting emissions and removals of greenhouse gases; and

• transparent, i.e., data sources, definitions, methodologies and assumptions should be clearly described.

1.Forest stratification (theorical concept) Under the IPCC 2006:

Generally, all data should be:

1.Forest stratification (theorical concept)

The land-use categories for greenhouse gas inventory reporting are:

(i)Forest Land
This category includes all land with woody vegetation consistent with thresholds used to define Forest Land in the national greenhouse gas inventory. It also includes systems with a vegetation structure that currently fall below, but in situ could potentially reach the threshold values used by a country to define the Forest Land category.

(ii) Cropland (iii) Grassland (iv) Wetlands (v) Settlements (vi) Other Land

Under the IPCC 2006:

Land-use categories and sub-categories may be further stratified on the basis of land-use practices and biophysical characteristics in order to create more homogeneous spatial units as may be used for emissions estimation 1.Forest stratification (theorical concept) Strata are areas assumed to contain trees from a single population (in terms of statistical prediction).

Within stratum vegetation characteristics will be similar in terms of structure, species composition, size class, growth form, age distribution, and canopy characteristics.

The strata used for sampling biomass will be limited by the available vegetation classification and mapping schemes and the scale of estimation required, which in this case is national.

Stratifying an forest area by its carbon stocks rather than by forest vegetation can reduce uncertainty and costs

Table 1. Land use change matrix. This matrix describes the change trajectories of the six categories defined under the IPCC GHG Guidelines ( 2006): FL = Forest Land, GL = Grass Land, CL = Crop Land, SL = Settlement Land, OL = Other Land. These categories are firstly divided into managed and unmanaged land, then further subdivided into homogeneous strata (i.e. forest types, grassland types, etc.). Columns and row totals show the total land area for each sub-category at times T0 (Initial Area) and T1 (Final Area), the Net Change (last row) is estimated as the difference between T0 and T1.

Land use change matrix

Different methods to assess forest carbon stocks and carbon stock changes

This study also demonstrates that the biomass estimates are primarily driven by the biomass reference data while the type of spatial maps used for their stratification has a smaller, but not negligible, impact. Avitabile et al. ( 2011) Stem volume is function of a tree's height, basal area, shape, and depending on definition, bark thickness. Volume is often measured for specific purposes, and the interpretation of the volume estimate will depend on the units of measurement, standards of use, and other specifications… For example:

5.Harmonization of tree allometric equations to maximize the use of existing tree allometric equations

Biological volume is the volume of stem with branches trimmed at the junction with the stem, but usually excluding irregularities not part of the natural growth habit (e.g. malformation due to insects, fungi, fire, and mechanical damage).

Merchantable volume: excludes some volume within irregularities of the bole shape caused by normal growth in addition to those irregularities not part of natural growth.

For example, the volume contained in the swelling around a branch node may be excluded because this volume could not be utilized (by a nominated user).

Gross volume estimates would include defective and decayed wood.

Net volume estimates would exclude defective and decayed wood.

  national land use & carbon stock changes

  Kenzo et al. (2009) Henry et al. (2010)

  1. Basis for biomass assessment: from the biology to the AE (Complexity of tree growth and biomass allocation) 3. Field work: example from case studies 2. Sampling strategy and stratification 4. Data entering, data management and QC 5. Graphical analysis and first data interpretation

Table 1 .1. Use change matrix within a land use subcategory.

 1 This table subcategorizes the change trajectories on forest land into their management activities. As an example, this matrix focuses on degradation, Sustainable Management of Forests (SMF) and Enhanced Carbon Stocks (ECS), affecting evergreen lowland rainforests.

	3. Assessing national land use & carbon stock changes
	Degraded Burning Degraded Selective logging Degraded Insect Infestation SMF ECS (restoration) ECS (afforestation/ reforestation) Conservation Final Initial (B)	Degraded 4 Burning	Degraded 5 Selective	Degraded Insect 0 Infestation	SMF 4 1	ECS 12 (restoration)	ECS 15 (afforestation/	Conservation 0	...	4 5 4 1 12 15 0 Final Area	Forest activity data matrix
	...
	Initial Area	4	5	0	5	12	15	0	0	41
	Net change (Δ = T0-T1)	0	0	4	-4	0	0	0	0	0

...

Different methods to assess forest carbon stocks and carbon stock changes

	Method 1: Average carbon stocks / stratums / land uses Method 2: modelling
	AGC	ST1 FL Managed lowland forest FL Managed > 1000m ST2 ST2 ST1	Etc…
	variables ST3 NDVI/ NDFI, AVI, GVI etc…	ST3	ST1
	Remote sensing/ optical sensors		Remote sensing/ radar sensors	Remote sensing/ laser sensors
	Ecological zones identified >>	ecology, climate, topography
	Land use identified Stratum Landsat, MODIS, Ikonos, Quickbird, DEMOS	>> >>	boundary mapping, remote sensing field measurements/ remote sensing Envisat, PALSAR ERS, JERS, Lidar

References


Context: When there can be a lot of different trees in a given forest….

I guess we are lost…., and is it necessary to fell all these beautiful trees to measure their biomass?

What is Allometry ?

Broad definition : within a given population, there is a statistical relationship between the size of an organism and the size of any part of it (Gould, 1966) H D

CD

For example: between height and diameter; diameter and crown size; biomass and diameter; etc..

What is Allometry ?

The literature on biomass equations dangles between two opposites sides: A-The group of West, Brown and Enquist have been developing an appealing theory of biological allometry relying on the fractal properties of branching networks, referred as allometric biomass partitioning theory (APT) by McCarthy and Enquist (2007) s1 accounting for the constraint of biomechanical stability s2 accounting for the minimization of hydrodynamic resistance through the vascular network.

Two main parameters:

From West et al. 1997: When taking s1=1 to fit the hypothesis of volume fillin and uniform biomechanical constraints, the tree mass is predicted to be related to the tree diameter raised to a power a=8/3≈2.67 But rather stands for an average tree model to explore biomass variations among plant size orders than being predictive for single species

What is Allometry ?

The literature on biomass equations dangles between two opposites sides:

B-The very large group of purely statistical equations, with little regard to the understanding of the biological processes involved. Such models are only reliable within their domain of validity Often calibrated on small number of trees, covering a little part of tree size variations for a given species

Equations of various forms

Catalogues and databases start to be available for all continents (ex: Zianis for Europe; Navar for south America; Henry for Africa)

Leading to too specialized and restricted models

What is Allometry ?

A good candidate set of volume or biomass equations should be simultaneously:

(i)consistent: standardized biomass partition and additivity of tree compartments, (ii) generic: common form of the models whatever the tree species or the forest structure. Meaningful parameters (ie related to the biology).

(iii) robust: system operating correctly across a wide range of operational conditions with a low sensitivity to the sampling design and the working hypotheses (iv) accurate.

  Nutrient content prediction  Nutrient content equationsWhat is Allometry ?More restrictive definition : proportionality between the relative increments of two metrics measured on an organism (Huxley, 1924)Where a gives the proportionality between the relative increments, b gives the proportionality between biomass and diameter (given a) and c is the biomass of the tree when D=0 (if D was measured at a height different from zero)

-From

  There are some range of variations of wood production at a given top height (variations related to the stockability issue) of Stand density (global pressure on the tree) (stand density in itself (Sd), stand basal area (G), hartbecking spacing factor based on tree growth without competition, Reinecke density index (Rdi) and stand density Index (Sdi) based on the self-thinning law, etc…) -an Index of Social status of the tree (between tree competition) (h/Ho, d/Do, the relationship between radial increment and tree diameter, etc…..)Biological conceptsTree and stand growth: case of even-aged and monospecific forests -C-Biomass partitioning in the tree -Ring area increases linearly from the top of the tree to the crown basis and From Adu-Bredu et al. 2008, Gomat et al. 2011 From Saint-André et al. 2005, Genet et al. 2010

Tree growth encompasses primary growth (height) and secondary growth (cambium activity) : a highly complex process

1-both internal (changes in wood properties with ontogeny APT ) and external factors (e.g., growing conditions, social status of the tree OPT) are of importance concerning tree biomass relationship -The main consist in identifying the proportion between APT and OPT 2-The observed pattern may vary strongly from one species to another, but our hypothesis is that species of similar traits (e.g., crown architecture, wood structure) would exhibit similar to identical From biology to allometric equations