domain: math sde

text:
O Gimenez

C Crainiceanu

C Barbraud

S Jenouvrier

B J T Morgan
email: b.j.t.morgan@kent.ac.uk

Semiparametric Regression in Capture-Recapture Modeling

Keywords: auxiliary variables, Bayesian inference, demographic rates, environmental covariates, penalized splines, WinBUGS

Capture-recapture models were developed to estimate survival using data arising from marking and monitoring wild animals over time.

Variation in survival may be explained by incorporating relevant covariates.

We propose nonparametric and semiparametric regression methods for es-

Introduction

Understanding population structure and changes in that structure for wild animals is essential for both species conservation and management. Because of human activities, it appears crucial to explain and forecast the effects of climatic and environmental perturbations on population dynamics. The analysis of data arising from observations of marked animals is an important tool for estimating demographic parameters that govern population change.

In the last forty years, a challenging research topic has been the estimation of wild animal survival, and when possible, to explain variations in survival using auxiliary variables such as time, age of animal or relevant covariates like temperature or rainfall. Many traditional models exhibit a productmultinomial likelihood structure, allowing inference in a unified context by classical maximum likelihood [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: A unified approach with case studies[END_REF] through user-friendly software like MARK [START_REF] White | Program MARK: survival estimation from populations of marked animals[END_REF] or M-SURGE [START_REF] Choquet | M-SURGE : new software specially designed for multistate capture-recapture models[END_REF]. The Bayesian approach has been proposed as an alternativesee [START_REF] Brooks | Bayesian animal survival estimation[END_REF] for a review.

To estimate survival probability, the modeling is usually embedded in the Generalized Linear Model (GLM) framework [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: A unified approach with case studies[END_REF]. A logit link for survival probabilities is frequently used but other functions are possible [START_REF] Williams | Analysis and Management of Animal Populations[END_REF]; covariates may be readily incorporated, and here we will focus on environmental covariates that vary over sampling occasions but remain constant over individuals, as defined by [START_REF] Pollock | The use of auxiliary variables in capture-recapture modelling: an overview[END_REF]. Most frequently, covariates are related to survival by a linear or a quadratic function, on the logit scale. However, in general this may be unrealistic and we give three examples to motivate a nonlinear alternative. First, it has been shown that global indices such as the North Atlantic Oscillation (NAO) could relate to population dynamics in complex nonlinear ways [START_REF] Mysterud | Nonlinear effects of large-scale climatic variability on wild and domestic herbivores[END_REF] see also [START_REF] Stenseth | Climate, changing phenology, and other life history traits: nonlinearity and match-mismatch to the environment[END_REF] for a general discussion). Secondly, survival can be nonlinearly related to population density via a threshold effect [START_REF] Lima | Numerical fluctuations in the northern short-tailed shrew: evidence of non-linear feedback signatures on population dynamics and demography[END_REF]. Thirdly, survival as a function of age may exhibit non-linear patterns, through senescence defined as a reduction in survival among old individuals [START_REF] Loison | Age-specific survival in five populations of ungulates: Evidence of senescence[END_REF][START_REF] Catchpole | Sexual dimorphism, survival and dispersal in red deer[END_REF]. In these examples and in many others, a nonparametric alternative avoids strong parametric assumptions and is of interest in itself. It might also suggest a new, scientifically relevant, parametric model if one is needed.

In this paper we applied Generalized Additive Models (GAMs) ideas popularized by [START_REF] Hastie | Generalized Additive Models[END_REF] that extend the traditional GLM framework. Rather than specifying a fixed link between survival and covariates in the model, the shape of the relationship is determined by the data, using penalized splines [START_REF] Ruppert | Semiparametric Regression[END_REF]. Our choice has been guided by the equivalence between a penalized spline formulation of the nonparametric problem with Generalized Linear Mixed Models (GLMMs) that simplifies further extensions.

The paper is organized as follows. In the next section, we give the likelihood for classical survival models, and the nonparametric regression of survival probabilities on covariates is established. In Section 3, we consider a natural extension to the nonparametric model, when a semiparametric regression model for survival is introduced. As well as including the nonparametric component, this allows us to model a parametric component at the same time. Section 4 gives the details of the Bayesian inference and its implementation through MCMC simulations. Section 5 gives the results of a simulation study which validate the ability of our approach to capture various nonlinearities in survival. Section 6 illustrates our method using data from a 40-year study of individually marked Snow petrels (Pagodroma nivea), in trying to relate their survival to a climate covariate. The last section gives general conclusions and discusses the potential of our approach.

Theory

CJS likelihood

We assume here that our capture-recapture study includes I + 1 sampling occasions at which animals are caught or observed, so that I recaptures or re-observations may actually be made. On each occasion, new unmarked animals are given unique marks and then released. Previously marked animals can also be sampled, and after their identity is recorded they are also released back into the studied population. This protocol gives rise to a set of animal encounter histories, made up of 1 and 0 depending respectively on whether an animal is detected or not. [START_REF] Cormack | Estimates of survival from the sighting of marked animals[END_REF], [START_REF] Jolly | Explicit estimates from capture-recapture data with both death and immigration-stochastic model[END_REF] and [START_REF] Seber | A note on the multiple-recapture census[END_REF] independently derived the likelihood for such capture-recapture data, and this model is referred to as the CJS model. [START_REF] Schwarz | Estimating animal abundance: review III[END_REF] and [START_REF] Williams | Analysis and Management of Animal Populations[END_REF] give reviews of the CJS model and its applications. Note that the model includes time-variation in parameters, but no age-variation. It may therefore be appropriate for describing the survival of adult animals. Data are frequently summarized in an upper triangular array, m, called the m-array, where m ij , i = 1, . . . , I, j = i + 1, . . . , I + 1, is the number of animals released at time t i and subsequently recaptured for the first time at time t j . Also the column vector R contains the R i , i = 1, . . . , I, which are the numbers of marked animals released into the population at times t i ; these comprise newly marked animals and those previously marked animals that are recaptured at time t i . Under the assumption that animals are independent (see e.g. [START_REF] Williams | Analysis and Management of Animal Populations[END_REF] for a description of CJS model assumptions and consequences of possible violation), the likelihood is product-multinomial

[m|φ, p, R] ∝ I i=1 χ R i -r i i I+1 j=i+1 φ i p j j-1 k=i+1 φ k (1 -p k ) m ij (1)
where [X] denotes the distribution of X, φ i , i = 1, . . . , I, is the probability that an animal survives to time t i+1 given that it is alive at time t i and p j , j = 2, . . . , I +1 denotes the encounter probability of being detected at time t j (see e.g. [START_REF] Brooks | Bayesian animal survival estimation[END_REF]. We adopt the convention that a null sequence has product 1 so that for example j-1 k=i+1 φ k (1 -p k ) = 1 for j = i + 1. Other terms involve r i = I j=i+1 m ij , the number of animals subsequently recaptured after release at time t i and χ i , the probability that an animal, alive at time t i , is not subsequently encountered. This can be calculated recursively as [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: A unified approach with case studies[END_REF].

χ i = 1 -φ i {1 -(1 -p i+1 )χ i+1 }, with χ I+1 = 1 (e.g.

Nonparametric regression of survival

We consider a nonparametric regression model for the probability that an animal survives from time t i to time t i+1 of the form

logit(φ i ) = f (x i ) + ε i , i = 1, . . . , I (2)
where x i is the value of the covariate for the ith sampling occasion, ε i are i.i.d N (0, σ ε ), ε i is independent of x i and f is a smooth function. Here, the random effects {ε i } allow us to model the residual sampling-occasionto-sampling-occasion variation not described by the covariates alone [START_REF] Barry | The analysis of ring-recovery data using random effects[END_REF]. Variations on the model of Equation ( 2) include:

• Semiparametric regression models in which some of the predictors enter linearly in the model, as illustrated in Section 3, and

• Models including interactions between covariates which are discussed in the last section.

Penalized splines using the truncated polynomial basis [START_REF] Ruppert | Selecting the number of knots for penalized splines[END_REF] were used to model the smooth function

f (x|η) = β 0 + β 1 x + . . . + β P x P + K k=1 b k (x -κ k ) P + (3)
where

P ≥ 1 is an integer, η = (β 1 , . . . , β P , b 1 , . . . , b K ) T is a vector of regres- sion coefficients, (u) p + = u p I(u ≥ 0) and κ 1 < κ 2 < . . . < κ K are fixed knots.
The crucial problem in using relation (3) is the choice of the number and the position of the knots. A small number of knots may result in a smoothing function that is not flexible enough to capture variability in the data, whereas a large number of knots may lead to overfitting. Similarly, the position of the knots will influence estimation. We used a penalized splines approach inspired by the smoothing splines of [START_REF] Green | Nonparametric regression and Generalized Linear Models[END_REF]. First, the number of knots is chosen to ensure enough flexibility. Following Ruppert

T b ≤ λ (4)
where λ is called the smoothing parameter. Equations ( 3) and ( 4) lead to the so-called P-splines approach (see e.g. [START_REF] Lang | Bayesian P-splines[END_REF]. Because roughness is controlled by the penalty term (4), once a minimum number of knots is reached, the fit given by a P-spline is almost independent of the knot number and location [START_REF] Ruppert | Selecting the number of knots for penalized splines[END_REF].

P-spline models can be fruitfully expressed as GLMMs, which facilitates their implementation in standard software [START_REF] Ngo | Smoothing with mixed model software[END_REF][START_REF] Crainiceanu | Bayesian Analysis for Penalized Spline Regression Using WinBUGS[END_REF], and above all provides a unified framework for generalizations of the nonparametric model. Indeed, we first note that the P-splines approach is equivalent to minimizing

I i=1 {logit(φ i ) -f (x i |η)} 2 + 1 λ η t Dη , ( 5
)
where D is a known positive semi-definite penalty matrix. The truncated spline penalty matrix is

D = 0 P ×P 0 P ×K 0 K×P Ω K ,
where a standard choice for Ω K is I K . To avoid overfitting, the matrix D penalizes only the coefficients of the spline basis functions (x -κ k ) P + . Let φ = (φ 1 , . . . , φ I ) T , X be the matrix with the ith row X i = (1, x i , . . . , x P i ) T , and Z be the matrix with ith row

Z i = {(x i -κ 1 ) P + , . . . , (x i -κ K ) P + } T . If we divide Equation (5) by the error variance σ 2 ε we obtain 1 σ 2 logit(φ) -Xβ -Zb 2 + 1 λσ 2 b T b , where β = (β 0 , . . . , β P ) T and b = (b 1 , . . . , b K ) T . Define σ 2 b = λσ 2 ε ,
(φ) = Xβ + Zb + ε, cov b ε = σ 2 b I K 0 0 σ 2 ε I I (6)
for which E(logit(φ)) = Xβ and cov(logit(φ)) = σ 2 ε V where V = I I + λ 2 ZZ T [START_REF] Brumback | Comment on Variable selection and function estimation in additive nonparametric regression using data-based prior, by Shively, Kohn, and Wood[END_REF].

Note that the connection between the P-spline model and the mixed model of Equation ( 6) allows us to extend the nonparametric model to incorporate other nonparametric components as well [START_REF] Ruppert | Semiparametric Regression[END_REF].

Semiparametric regression of survival

In the preceding section, a regression model for survival over a continuous predictor modeled as a smooth function was considered. In this section, we extend this model by including qualitative predictors assumed to enter the model linearly. Without loss of generality, we considered only one parametric categorical component s with one non-parametric component smoothing a continuous predictor x by linear P-splines. We want to let the relationship between logit(φ i ) and x i vary differently but in parallel according to the variable s i taking discrete values, i.e.

logit(φ

i ) = β 0 + γs i + β 1 x i + K k=1 b k (x i -κ k ) + + ε i , i = 1, . . . , I. ( 7
)
The GLMM representation can also be used to handle the semiparametric model. Let us adjust the matrix X so that its ith row is

X i = (1, s i , x i ) T and β = (β 0 , γ, β 1 ) T , while the ith row of matrix Z is Z i = {(x i -κ 1 ) + , . . . , (x i - κ K ) + } T .
Then the mixed model defined by Equation ( 6) can still be used to describe the semiparametric regression defined in Equation ( 7) [START_REF] Ruppert | Semiparametric Regression[END_REF].

Bayesian inference

In this section, we focus on the Bayesian analysis of the nonparametric model defined in Section 2.2. However, within the GLMM framework introduced before, the extension to additive and semiparametric models is straightforward (see Section 6).

The frequentist approach would require maximising the likelihood, which is obtained by integrating the distribution [m|φ, p, R] over the random effects ε i and b k . This is therefore a problem involving a high dimensional integral that could be handled by using approximations like Laplace's method [START_REF] Chavez-Demoulin | Bayesian inference for small-sample capturerecapture data[END_REF][START_REF] Wintrebert | Joint modeling of breeding and survival of Rissa tridactyla using frailty models[END_REF] or asymptotic arguments [START_REF] Burnham | Evaluation of some random effects methodology applicable to bird ringing data[END_REF]. For fitting our models, we opted for a Bayesian approach through Gibbs sampling. Invoking conditional independence properties, a first step is achieved by recursively factorizing the posterior distribution to give:

[β, b, ε, σ 2 b , σ 2 ε , p, R|m] ∝ [m|φ, p, R][φ|β, b, ε][β][b|σ 2 b ][ε|σ 2 ε ][σ 2 b ][σ 2 ε ][p]. ( 8
)
Even if one is only interested in the marginal posterior distribution of a subset of parameters, high-dimensional integrations have to be carried out. In general, such complex integrals are intractable analytically and we made use of MCMC methods which provide powerful computer-intensive methods for making approximations (e.g. [START_REF] Brooks | Markov chain Monte Carlo method and its application[END_REF]. We employed Gibbs sampling (e.g. [START_REF] Casella | Explaining the Gibbs sampler[END_REF], however, in the context of capture-recapture model parameter estimation, generally full conditional distributions are nonstandard [START_REF] Brooks | Bayesian animal survival estimation[END_REF][START_REF] Barry | The analysis of ring-recovery data using random effects[END_REF][START_REF] Johnson | Autoregressive models for capturerecapture data: A Bayesian approach[END_REF], so that usual random variate generation algorithms cannot be used.

Instead, more elaborate algorithms are needed such as adaptive rejection sampling or Metropolis-within-Gibbs sampling (see [START_REF] Gilks | Full conditional distributions[END_REF] for a review).

We therefore used software WinBUGS [START_REF] Spiegelhalter | WinBUGS User Manual[END_REF], which performs the latter.

Simulation study

Before turning to the real example, we conducted a simulation study to provide empirical support for our approach. We considered two scenarios with different forms for the underlying nonlinear regression function f of Equation ( 2). Study 1 used the regression function f (x) = 2.2 if x ≤ -0.06 and f (x) = 2.08-2 x otherwise. This function is a broken line which mimics a threshold effect, for instance the covariate might represent an environmental constraint on resources, which negatively affects survival only above a given level. The xs were equally spaced on [-1.5, 1.5], and the error variance σ 2 ε was equal to 0.1. Study 2 used the regression function f (x) = 1.5 g((x -0.35)/0.15) -g((x -0.6)/0.1) where g(x) = exp(-x 2 /2)/ √ 2π. This function exhibits non-trivial non-linear patterns, which could correspond to complex relationships between climatic conditions and survival. The xs were equally spaced on [0, 1], and the error variance σ 2 ε was equal to 0.02. For both studies, we simulated 50 capture-recapture data sets covering 26 sampling occasions, so that 25 survival probabilities had to be estimated, with 100 newly marked individuals per occasion. The capture probability was set constant and equal to 0.7.

For 5 randomly chosen data sets, we first ran two overdispersed parallel MCMC chains to check if convergence was reached. As a result, we decided to use 100,000 iterations with 50,000 burned iterations for posterior summarization. Details on the priors used and the convergence assessment can be found in Section 6. We then applied our non-parametric approach on each data set, using linear P-splines with 6 knots. For each x value, we computed the median along with a 95% confidence interval for the posterior medians of f and then back-transformed in order to compare the estimated survival curve to its true counterpart.

The results are shown in Figure 1. For each of the two examples, our approach was successful in capturing the nonlinearities in the survival function.

Note that in Study 1, a relatively simple regression function was specified, resulting, for the same number of knots and sample size, in better precision than for Study 2.

[Figure 1 about here.]

Application to Snow petrels data

We illustrate the approach of the paper with data from a 40-year study on individually marked Snow petrels, nesting at Petrels Island, Terre Adélie, from 1963-2002. Two previous studies have showed that a large part of the variation in annual survival was explained by climatic covariates such as the extent of sea-ice and air temperature [START_REF] Barbraud | Effect of sea-ice extent on adult survival of an antarctic top predator: the snow petrel pagodroma nivea[END_REF]Jenouvrier, Barbraud and Weimerskirch, unpublished results). Here, for illustration, we used only a subset of the whole dataset, from 1973-2002 (I = 29, 630 males and 640 females), and considered the Southern Oscillation Index (a covariate denoted by SOI) as a summary of the overall climate condition, with positive (respectively negative) values of the SOI corresponding to cold (respectively warmer) climatic conditions. While the NAO is a useful synthesis of climatic variables that might affect ecology in the Northern hemisphere (see Section 1), the SOI provides its counterpart for the Southern hemisphere (see [START_REF] Stenseth | Studying climate effects on ecology through the use of climate indices: the North Atlantic Oscillation, El Niño Southern Oscillation and beyond[END_REF] for a general discussion). The SOI is available from the Climatic Research Unit (http://www.cru.uea.ac.uk/cru/data/soi.htm).

Preliminary analysis of goodness-of-fit of the CJS model identified lack of fit due to the presence of transients (146 males and 169 females were seen only once) [START_REF] Pradel | Capturerecapture survival models taking account of transients[END_REF] and trap-dependence [START_REF] Pradel | Flexibility in survival analysis from recapture data: Handling trap-dependence[END_REF]. The transients were removed, and trap-dependence was handled by considering different capture probabilities depending on whether a capture occurred or not at the previous sampling occasion.

We modeled the survival probability nonparametrically as a function of the SOI using P-splines. The effect of this covariate was additively differentiated according to the sex of individuals. We used linear splines (P = 1) but quadratic or even cubic splines could have been used instead, resulting mainly in a smoother estimated survival curve [START_REF] Ruppert | Semiparametric Regression[END_REF]. We used K = 6 knots chosen so that the kth knot is the sample quantile corresponding to probability k/(K + 1). Note that the covariate SOI was first standardized in order to avoid numerical instabilities and to improve MCMC mixing [START_REF] Gilks | Strategies for improving mcmc[END_REF]. We therefore considered the following

model logit(φ l i ) = β 0 + γSEX + β 1 SOI i + 6 k=1 b k (SOI i -κ k ) + + ε i (9)
where φ l i is the survival probability over the interval [t i , t i+1 ] for l = male (SEX = 0) or l = female (SEX = 1) and SOI i denotes the SOI in year i, i = 1, . . . , I. The random effects {b k } are independent as well as the {ε i }.

Let us denote φ = (φ f emale 1 , . . . , φ f emale 28 , φ male 1 , . . . , φ male 28 ) T . Then, in matrix notation, Equation ( 9) can be expressed in the form of Equation ( 6)

using β = β 0 γ β 1 T X =          1 1 SOI 1 . . . . . . . . . 1 1 SOI 28 1 0 SOI 1 . . . . . . . . . 1 0 SOI 28         
for the fixed effects and

b = b 1 . . . b 6 T Z =    (SOI 1 -κ 1 ) + . . . (SOI 1 -κ 6 ) + . . . . . . . . . (SOI 28 -κ 1 ) + . . . (SOI 28 -κ 6 ) +   
for the random effects.

The model proposed here differs from the semiparametric approach presented before in that the sex parametric component acts at the individual level rather than on sampling occasions. The likelihood is therefore slightly modified consisting of the product of two sub-components, one for each sex, based on the product-multinomial structure of the m-array (e.g. [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: A unified approach with case studies[END_REF].

To completely specify the Bayesian nonparametric model, we need to provide prior distributions for all parameters. Specifically, we chose

[p i+1 ] = Beta(A p , B p ), [ε i ] = N (0, σ 2 ε ), i = 1, . . . , I [β 0 ] , [β 1 ] , [γ] = N (0, σ 2 β ), [b k ] = N (0, σ 2 b ), k = 1, . . . , K,
where the parameter σ b controls the degree of smoothing for the covariate.

Following [START_REF] Brooks | Bayesian animal survival estimation[END_REF], we chose A p = B p = 1 which leads to a uniform distribution, while following [START_REF] Ruppert | Semiparametric Regression[END_REF], σ 2 β was set to 10 6 , and priors for hyperparameters were chosen as

σ 2 b , σ 2 ε = Γ -1 (0.001, 0.001).
All priors were selected as sufficiently vague in order to induce little prior knowledge, but can be easily refined if required. We generated two chains of length 100,000, discarding the first 50,000 as burn-in. These simulations took approximatively 25 hours on a PC (512Mo RAM, 2.6GHz CPU). Convergence was assessed using the Gelman and Rubin statistic, also called the potential scale reduction, which compares the within to the between variability of chains started at different and dispersed initial values [START_REF] Gelman | Inference and monitoring convergence[END_REF]. We found that the Markov chains exhibit moderate autocorrelation but poor mixing regarding the parameters b k s and βs. We thus tried lowrank thin-plate splines because in that case the posterior correlation of the parameters is generally smaller than for other bases. However, in our example, this only improved the mixing slightly, so that we decided to retain the truncated polynomial basis throughout, coupled with chains of adequate length to achieve convergence. According to our experience, inference based on P-splines within the Bayes framework may be sensitive to the choice of priors, especially regarding σ b (see Crainiceanu et al., 2004a for a discussion of prior distributions for nonparametric P-spline regression). In order to check for the robustness of our results, we ran our model using different priors and in all cases there were only minimal changes.

We Posterior medians, standard deviations, and 95% credible intervals are given in Table 1.

[Table 1 about here.]

Because it does not contain 0, the posterior credible interval for parameter γ suggests that the sex of individuals affects the survival probability.

As demonstrated by other studies (Jenouvrier, Barbraud and Weimerskirch, unpublished results), male petrels survive better than females, whatever the climatic conditions (see Figure 2).

Of particular interest, it appears that survival is nonlinearly related to the SOI covariate (Figure 2). When the SOI increases, survival first decreases and then stabilizes. From a biological point of view, lower values of the SOI may favor access to prey, whereas higher values may improve prey abundance [START_REF] Loeb | Effects of sea-ice extent and krill or salp dominance on the Antarctic food web[END_REF], resulting in the non-linearity found.

[Figure 2 about here.]

In order to know if the nonparametric part our model was needed, we compared the nonparametric model with the simple standard approach in which the SOI is just entered linearly on the logistic scale. From Figure 2, the linear curve (dotted line) differs clearly from the nonparametric curve (solid line), but the 95% credible interval (dashed lines) for the latter partly contains the former, which means that this difference is only marginal. This conclusion was supported by the DIC values [START_REF] Spiegelhalter | Bayesian measures of complexity and fit[END_REF] and the credible intervals for the b k s, which include zero. However the nonlinearity has a biological explanation, and as we can see from Figure 2, in this example we require more years of data corresponding to large values of SOI in order to discriminate better between the two models.

Note that the mean encounter probabilities were 67% for males and 61%

for females if a capture occurred at the previous occasion, and 62% for males and 58% for females if not. This sex-dependent positive trap-effect is in agreement with a recent study on Snow petrels (Jenouvrier, Barbraud and Weimerskirch, unpublished results).

Discussion

This paper presents a Bayesian approach for nonparametric modeling of survival estimated using capture-recapture data, where smooth functions were modeled as penalized splines. Extensions such as additive and semiparametric models are straightforward within the unified framework based on the mixed model representation. In addition, due to the hierarchical structure of our Bayesian approach, the degree of smoothness is data-driven and controlled by the smoothing parameter estimated jointly with the unknown regression parameters.

The modelling of this paper does not include interactions between covariates. For example, an interaction between sex and a climatic covariate would involve considering different smooth functions for males and females [START_REF] Coull | Simple incorporation of interactions into additive models[END_REF]. Following the suggestion of a referee, we considered this interaction for the real data but it did not appear to improve the fit. An interaction between two continuous covariates can be achieved by using bivariate smoothing [START_REF] Ruppert | Semiparametric Regression[END_REF]. For example, it would be interesting to include an interaction between population density and climate in a model [START_REF] Coulson | Age, sex, density, winter weather, and population crashes in Soay sheep[END_REF], requiring an extension of the power truncated function basis to a tensor product basis [START_REF] Green | Nonparametric regression and Generalized Linear Models[END_REF].

In this paper we dealt with goodness-of-fit by first of all applying standard procedures to the CJS model, which identified transients, which were excluded, and the presence of trap-dependence, which was included in the semiparametric model. Any further lack of fit was accommodated in part through the inclusion of the random effect terms in Equation ( 2), which are seen to be needed from the estimate of σ ε in Table 1.

The work of this paper has wider application than just to the CJS model, e.g. in models with age-dependence of survival, including modelling senescence (e.g. [START_REF] Catchpole | Sexual dimorphism, survival and dispersal in red deer[END_REF].

en utilisant des données d'un suivi de 40 années concernant des individus nichant sur l'ile des Pétrels, en Terre Adélie.

  and let κ k be "equally-spaced sample quantiles" i.e. the sample quantiles of the x i 's corresponding to probabilities k/(K + 1). Other choices are possible, such as equally spaced knots within the domain of x, and Crainiceanu et al. (2004a) provide a simulation study comparing these two alternatives with a discussion. Then, following Ruppert et al. (2003) a quadratic penalty is placed on b, which is here the set of jumps in the P th derivative of f (•|η) so that with Equation (3) we associate the constraint b

  consider the vector β as fixed parameters and the vector b as a set of random parameters with E(b) = 0 and cov(b) = σ 2 b I K . If (b T , ε T ) T is a normal random vector and b and ε are independent, then an equivalent model representation of the P-spline model in the form of a GLMM is logit

Figure 1 .Figure 2 .

 12 Figure1. Performance of the non-parametric approach for estimating nonlinearities in the survival probability (top: Study 1, and bottom: Study 2; see text for details). For both scenarios, 50 simulated capture-recapture data sets were used. The solid line is the true regression function, the dashed line is the median of the 50 estimated posterior medians and the dotted lines indicate the associated 95% confidence interval.

Les modèles de capture-recapture servent à estimer la survie d'une population sauvage, grâce à des données issues du marquage et du suivi dans le temps d'individus. Il est d'une importance toute particulière de pouvoir expliquer les variations de survie en fonction de variables judicieuses. Nous développons des modèles de régression nonparamétriques et semiparamétriques pour la probabilité de survie des modèles de capture-recapture. Nous nous plaçons dans un cadre Bayésien, et l'estimation des paramètres s'effectue grâce à des méthodes MCMC. Nous illustrons notre travail par l'étude de la survie de Pétrels des neiges comme une fonction non-linéaire d'une variable climatique,

Acknowledgements

The authors would like to thank V. Grosbois and R. Pradel for very stimulating and helpful discussions. O. Gimenez's research was supported by a Marie-Curie Intra-European Fellowship within the Sixth European Community Framework Programme. B.J.T. Morgan was supported by a Leverhulme Fellowship.

Résumé


G Pinay

M M Hefting

Patterns of denitrification rates in European alluvial soils under various hydrological regimes

Keywords: riparian zone, river floodplain, microbial process, Europe, Climate change

1) (2)* ; B. Gumiero (3) ; E. Tabacchi (4

2. The aim of this study was to measure denitrification rates under a wide range of current climatic conditions and hydrological regimes within Europe (from latitude 64 o N to latitude 42 o Nand from longitude 2° W to longitude 25° E), i) to determine the response patterns of this microbial process under different climatic and hydrological conditions, ii) to identify denitrification proxies robust enough to be used at the European scale.

3. Denitrification activity was significant in all the floodplain soils studied whatever the latitude. However, we found an increase in rates of an order of magnitude from high to mid latitudes. Maximum rates (above 30 g N m -2 month -1 ) were measured in the maritime conditions of the Trent floodplain.

These rates are similar to mineralization rates measured in alluvial soils and of the same order of magnitude as the amount of N stored in herbaceous plants in alluvial soils.

Introduction

World wide predicted temperature increases for the 21 st century range between 1.4°C and 4.7°C [START_REF] Richards | Change in global temperature: a statistical analysis[END_REF]IPCC, 2001). Although the importance of temperature in regulating physiological processes is unquestionable, predicting the relationship between ecological process and temperature is complicated. Indeed, as the Earth warms, a general intensification of the hydrological cycle is also expected to occur [START_REF] Miller | The impact of global warming on river runoff[END_REF].

Precipitation, evapotranspiration and runoff are all expected to increase together with extremes events, i.e. floods and droughts [START_REF] Loaiciga | Global warming and the hydrological cycle[END_REF]. For instance, drastic changes are expected in Bangladesh with a shift in the timing of flood distribution and an increase in their magnitude and duration [START_REF] Mirza | Global warming and changes in the probability of occurence of floods in Bangladesh and implications[END_REF][START_REF] Mirza | The implications of climate chnage on floods of the Ganges, Brahmaputra and Meghna Rivers in Bangladesh[END_REF]. Consequently, a lot of effort is being placed into modelling change in hydrological regimes [START_REF] Bailey | Effects of temperature and root on denitrification in soil[END_REF][START_REF] Najjar | The potential impacts of climate change on the mid-atlantic coastal region[END_REF][START_REF] Chang | The effects of climate change on stream flow and nutrient loading[END_REF][START_REF] Pabich | Relationship between DOC concentration and vadose zone thickness and depth below water table in groundwater of Cape Cod, USA[END_REF][START_REF] Roy | The impact of climate change on seasonal floods of a southern Quebec River basin[END_REF][START_REF] Stone | Impacts of climate change on Missouri River Basin water yield[END_REF][START_REF] Burlando | Effects of transient climate change on basin hydrology. 1. Precipitation scenarios for the Arno River, Central Italy[END_REF][START_REF] Benito | Magnitude and frequency of flooding in the Tagus Basin (Central Spain) over the last Millennium[END_REF]. However, global-scale generalizations of relationships between climate change and extreme floods is difficult because climate [START_REF] Nakagawa | Asynchronous climate change in the North Atlantic and Japan during the last termination[END_REF] and hydrological responses vary regionally [START_REF] Knox | Large increases in flood magnitude in response to modest changes in climate[END_REF]. For instance, [START_REF] Knox | Large increases in flood magnitude in response to modest changes in climate[END_REF] reported in a 7,000-year geological study on the Upper Mississippi River that small changes in the mean annual temperature (1-2°C) and mean annual precipitation (10-20%) can change dramatically the magnitude of floods.

Several papers have already addressed the water resources implications of global warming, among others [START_REF] Falkenmark | Water scarcity as a key factor behind global food insecurity: Round table discussion[END_REF][START_REF] Lettenmaier | Water resources implications of global warming: a US Regional perspective[END_REF][START_REF] Jackson | Water in a changing world[END_REF][START_REF] Middelkoop | Impact of climate change on hydrological regimes and water resources management in the Rhine Basin[END_REF] Yet very few authors have considered the ecological impact of climate change on river ecosystems [START_REF] Poff | Ecological response to and management of increased flooding caused by climate change[END_REF]. However the functioning of rivers and their floodplains depends to a large extend on periodic flooding; the intensity, timing, frequency and duration of these events are essential to maintaining a mosaic of habitats [START_REF] Benke | Flood pulse dynamics of an unregulated river floodplain in the Southeastern U.S. coastal plain[END_REF] and a high productivity both in rivers and their floodplains [START_REF] Junk | The flood pulse concept in river-floodplain systems[END_REF][START_REF] Naiman | The ecology of interfaces: riparian zones[END_REF]. The high productivity of river ecosystems is largely due to riparian zones which control the energy, nutrients and organic matter fluxes both in longitudinal [START_REF] Schlosser | Riparian vegetation and channel morphology impact on spatial patterns of water quality in agricultural watersheds[END_REF][START_REF] Pinay | Geomorphic control of denitrification in large river floodplain soils[END_REF] and lateral directions [START_REF] Peterjohn | Nutrient dynamics in an agricultural watershed: Observations on the role of a riparian forest[END_REF], Haycock, Pinay & Walker, 1997). However, riparian zone's existence is also largely controlled by the timing and duration of flood events [START_REF] Salo | River dynamics and the diversity of Amazon lowland forest[END_REF][START_REF] Gregory | An ecosystem perspective of riparian zone[END_REF][START_REF] Conner | Water level variability and litterfall productivity of forested freshwater wetlands in Louisiana[END_REF][START_REF] Clawson | Changes in production and nutrient cycling across a wetness gradient within a floodplain forest[END_REF]. More specifically, water regime changes, through alteration of the frequency, duration, and period of occurrence of water levels, directly affect nutrient cycling in alluvial soils by controlling the duration of oxic and anoxic phases [START_REF] Patrick | Nitrogen loss from flooded soils[END_REF][START_REF] Ponnamperuma | The chemistry of submerged soils[END_REF][START_REF] Keeney | The nitrogen cycle in sediment-water systems[END_REF] and, in turn, affect nitrogen endproducts and availability [START_REF] Hefting | Water table elevation controls on soil nitrogen cycling in riparian wetlands along a European climatic gradient[END_REF].

In Europe, scenarios of change of the hydrological regime forecast an overall increase in the inter-annual variability of runoff, an increase of the average annual runoff in northern Europe and a decrease in the south [START_REF] Arnell | The effect of climate change on hydrological regimes in Europe: a continental perspective[END_REF]. Moreover, the timing and duration of high and low flow events may shift, especially in the eastern part of the continent. Ultimately, these changes will affect the rates of nitrogen cycling in riparian wetlands and their plant productivity. In this context, we focused on denitrification in floodplain soils, one of the microbial processes involved in the nitrogen cycling. This process is interesting for several reasons: It is very sensitive to the redox status of the soils and sediments, which is controlled by the hydrological regime of the rivers [START_REF] Knowles | Denitrification[END_REF]. High rates have been measured in floodplain soils [START_REF] Johnston | Sediment and nutrient retention by freshwater wetlands: effects on surface water quality[END_REF][START_REF] Groffman | Nitrate dynamics in riparian forests: microbial studies[END_REF][START_REF] Pinay | Geomorphic control of denitrification in large river floodplain soils[END_REF]. It is one of the main biological process emitting and reducing nitrous oxide, a greenhouse gas (Groffman et al., 1998;[START_REF] Blitcher-Mathiesen | Denitrification as a sink for dissolved nitrous oxide in freshwater riparian fen[END_REF][START_REF] Hefting | Nitrous oxide emission and denitrification in chronically nitrate-loaded riparian buffer zones[END_REF]. It has been proved to be the main process responsible for the buffering capacities of riparian zones against nitrate diffuse pollution [START_REF] Peterjohn | Nutrient dynamics in an agricultural watershed: Observations on the role of a riparian forest[END_REF][START_REF] Pinay | Spatial and Temporal Patterns of Denitrification in a Riparian Forest[END_REF][START_REF] Haycock | Buffer Zones: Their Processes and Potential in Water Protection[END_REF][START_REF] Sabater | Nitrogen removal by riparian buffers along a European climatic gradient: Patterns and factors of variation[END_REF].

The aim of this study was: i) to measure denitrification rates in floodplain soils under a wide range of current climatic conditions and hydrological regimes within Europe, ii) to determine the response patterns of this microbial process under these different climatic conditions, iii) to identify denitrification proxies robust enough to be used at the European scale, and iv) to discuss the possible change of denitrification activity in floodplain soils under forecast climate change scenarios.

Study sites

We selected seven rivers in Europe to encompass a large range of climatic and hydrological regimes from maritime to continental regions (Fig. 1; Table 1): The Vindel River in the North of Sweden (64°N) and the Helge River in the South of Sweden (56°N), the River Trent in the United Kingdom (52°N near Nottingham), the Aar River in Switzerland (47°N near Meienried), the Garonne River in Southern France (near Toulouse 44°N), the Po River in Northern Italy (near Pontelagoscuro 43°N) and the Danube in Romania (Near Braïla 42°N). In maritime areas, e.g. Trent, the seasonal regime reflects the seasonal variability in rainfall and evapotranspiration. In continental areas, e.g. the Danube, the peak runoff season corresponds to snowmelt. In between these two extremes, there exist many mixed regimes representing local climatic conditions (Krasovskaia et al., 1994;[START_REF] Arnell | The effect of climate change on hydrological regimes in Europe: a continental perspective[END_REF]. For instance, northern (Vindel River, Helge River) and Alpine regions of Europe (Aar River) have a snowdominated hydrology, while the Garonne and the Po Rivers are influenced both by snowmelt, respectively from the Pyrenees and the Alps mountains, and by the balance of a rainfall and evapotranspiration.

The riparian zones studied were located in the annually flooded alluvial reaches of each river. The main source of nutrient input to the riparian floodplain originated from flood deposits. In most cases, the study sites were forested and included similar species or genera at each site. The most common genera were willow (Salix), alder (Alnus), oak (Quercus), elm (Ulmus), and poplar (Populus). The main species of the understorey varied between regions:

Carex and Calamagrostis in the Vindel River, Phragmites and Deschampsia in the Helge River, Urtica and Phragmites in the Aar River, stinging nettle (Urtica) and Impatiens in the Trent and the Garonne Rivers, Solidago and Urtica in the Po River, Agrostis and Bidens in the Danube River. At the Swedish sites, the soils were typical podzols, which were regularly flooded. At the other sites, the soils belonged to the Fluvents series of alluvial entisoils.

Materials & Methods

River discharge and rainfall were monitored by hydrometric stations near the study sites in each country. Within each river valley, five to ten forested riparian locations were selected in different geomorphological contexts, i.e. meandering and straight channels, different soil grain size, but at the same relative altitude to ensure similar flooding duration and frequency. This sampling effort represents altogether about 2500 different samples collected between 1996 and 1999. At each location, three different plots, each of one square meter, were chosen monthly. At each site above-ground herbaceous vegetation was harvested monthly (except when soil was frozen) and oven-dried at 60°C to determine the dry mass. Soil temperature was measured hourly using a datalogger (Tinytalk II; Gemini Dataloggers, U.K., range -10 + 40°C).

At each quadrat, a composite soil sample was collected from the upper 10 cm of soil, the most active zone in a biological sense, after the litter was discarded. A sub-sample was oven-dried for 24 h at 105°C to determine fresh and dry mass and percent moisture. Percentage of sand was determined by sieving (50 µm mesh) after pre-treating the samples with hydrogen peroxide and dispersal with sodium hexametaphosphate solution [START_REF] Day | Particle fractionation and particle-size analysis[END_REF]. Ten grams (fresh mass) of each soil sample were extracted with 150 ml of 2 M KCl. The extract was filtered and analysed for N-NO 3 -N with a Technicon Autoanalyzer [START_REF] Technicon | Technicon instrument system. Technicon method guide[END_REF]. Nitrate is expressed in g N-NO 3 m -2 of soil.

Denitrification (DNT) was assayed by the static core acetylene inhibition method [START_REF] Yoshinari | Acetylene inhibition of nitrous oxide reduction by denitrifying bacteria[END_REF]. Nine intact cores (length 10 cm, diameter 5 cm) collected monthly or every two months from each of the 15 sites were capped with rubber serum stoppers and then amended with acetone-free acetylene to bring core atmosphere concentration to 10 KPa (10 % V/V) acetylene and 90 kpa air. These cores were incubated at field temperature for periods of between 1 and 4 h, and the rate of denitrification calculated as the rate of nitrous oxide (N 2 O) accumulation in the head space in 4 hours. Head-space gas samples were removed from all cores, stored in evacuated collection tubes (Venoject, Terumo Scientific N.J), sent to Toulouse, and subsequently analysed within a few days using a gas chromatograph (GC Varian 3300) equipped with an electron capture detector (ECD 63 Ni) and Porapak Q columns. Results are expressed in g N m -2 month -1 with the assumption that the value measured is somehow representative of the 2 weeks before and 2 weeks after the sampling time. Even though it is well known that the acetylene block method has some problems; it is a reasonably robust method, especially with systems with moderate or high nitrate fluxes such as riparian soils. See [START_REF] Groffman | Bad solutions to a difficult problem: Methods for measuring denitrification[END_REF] for a review. Therefore it provides a fair estimate of denitrification rates when applied on short term essays and allows inter sites comparisons.

Statistical analyses

We used Multivariate Adaptative Regression Splines (MARS) to relate the response variable denitrification with five relevant predictors, namely soil silt plus clay (%), herbaceous plant biomass (g dry mass/ m 2 ), soil moisture (%), g N-NO3 (m -2 ) and temperature (°C). Introduced by [START_REF] Friedman | Multivariate adaptive regression splines[END_REF], MARS approximates the underlying relationship between the response variable and the predictors using a set of adaptative piecewise linear regressions called basis functions (BFs). A typical BF is max(0,x-c) (resp. max(0,c-x)) taking values x (resp. -x) for all values of x greater (resp. lower) than some threshold c and 0 otherwise. The constant c is usually referred to as knot where a change in the slope of the BF occurs. MARS models were fitted using the MARS software (Salford Systems, http://www.salford-systems.com/mars.php). Because results produced by MARS can be affected by multicolinearity in the set of explanatory variables, we checked that the predictors were uncorrelated using standard Pearson correlations and a principal component analysis (results not shown).

Results

European scale

We combined all the data from the different study sites, sampling locations and sampling dates to analyse the patterns of denitrification rates at the European scale. Several local studies have already shown that soil temperature, moisture and grain size influence denitrification rates. The objective here was to determine whether these simple parameters could be used as proxy of denitrification activity at the large scale; in other words, if they would be robust enough to encompass local, regional and seasonal variability. Our expectation was that denitrification activity would be positively correlated to alluvial soil moisture and temperature. We found a high variability in denitrification rates ranging from zero to 30 g N m -2 month -1 (Fig. 2). Maximum denitrification activity was not found at maximum soil moisture but between 50 and 80% weight/weight (Fig. 2A). The relationship between denitrification activity and soil grain size was not linear either (Fig. 2B). The highest denitrification activity was measured in soil with ca 80% silt & clay. A negative exponential relationship was found between soil nitrate content and denitrification activity (Fig. 2C). High denitrification rates were measured in soils with low to medium nitrate concentration. Moreover, at this large scale of analysis there was also a negative exponential relationship between denitrification rates in alluvial soils and herbaceous biomass production (Fig. 2D).

Highest denitrification activity was measured in soils with low herbaceous biomass production. At this European scale denitrification presented a bimodal pattern with high rates at both low (ca 5-10°C) and mild (ca 15-20°C) soil temperature (Fig. 2E).

We used Multivariate Adaptative Regression Splines (MARS) to relate the response variable denitrification with five relevant predictors, namely soil silt & clay (SC %), herbaceous plant biomass (HPB g dry mass/ m 2 ), soil moisture (SM %), nitrate (N g N-NO 3 m -2 ) and temperature (TEMP °C). The MARS algorithm searches for the optimal combination of BFs therefore allowing maximum flexibility in describing any functionalincluding nonlinearform of the response variable.

The basis functions selected by MARS were: BF1 max(0,SM 77.390); BF2 max(0, 77.390 SM); BF3 max(0, N 1.660); BF4 max(0,1.660 N); BF6 max(0,17.000 TEMP); BF7 max(0,SM 56.310) BF6; BF9 max(0,SM 75.120) BF3; BF11 max(0,SM 58.310); BF13 max(0, TEMP 15.

                  
From this equation, it is clear that a model with interaction effects outperforms a pure additive model, which is confirmed by the difference in Generalized Cross-Validation (GCV) of the two models, 493.562 vs. 529.599 respectively. Interestingly, the explanatory variable Soil Moisture is involved in all interactions through basis functions BF7, BF9 and BF11, which points out its importance in predicting denitrification. In Table 2 we presented the ranking of predictors for denitrification by order of importance (%), which is the relative importance of predictors as compared to the best one (soil moisture here). It confirms that soil moisture was the most important variable determining denitrification, and to a lesser extent by order of importance temperature, nitrate, then marginally herbaceous plant biomass. It is interesting to note that soil texture (silt & clay content, i.e. SC) did not play any role in the final model.

River scale

The relationship between denitrification activity and alluvial soil temperature was reanalysed at the local scale, considering the denitrification patterns river by river (Fig 3). The lowest denitrification rates were measured in the Vindel and the Helge alluvial soils but they were of the same order of magnitude as the values obtained at the other sites. Different seasonal patterns were found across Europe: in the Vindel, Helge, Aar and Danube Rivers, which correspond respectively to cold high latitude, high altitude and continental sites, denitrification rates were higher at highest temperatures characteristic of the period from mid spring to late summer. In these sites, no significant denitrification rates were measured at times of low temperature. In the Trent, Garonne and Po rivers, corresponding respectively to oceanic and warm temperate climates, denitrification rates showed a bimodal pattern being higher at both high and low (i.e. 3-5°C) soil temperatures.

Detailed analysis of the temporal pattern of denitrification in the Garonne alluvial soils, for instance, revealed that both rainfall and flood events can trigger microbial activity (Fig. 4). High denitrification rates were measured with high soil temperature and moderate rainfall (Fig. 4A). However, during high-water periods and various rainfall events, high denitrification rates were also measured even at times of low soil temperature (Fig. 4B).

Similarly, rainfall and flood events can cause high denitrification rates in the alluvial soils of the Po River (Fig 5) under a wide range of soil temperatures (from 5 to 20°C). Even a relatively brief rainfall event (100 mm in two days in June Fig. 5D2) in an otherwise dry period can trigger high denitrification rates.

Discussion

Patterns of denitrification activity and controlling factors at the European scale

Denitrification activity was significant in all the floodplain soils regardless the latitude (Fig. 2E). Yet, we found a difference in rates of an order of magnitude from the high to the mid latitudes. Maximum rates (above 30 g N m -2 month -1 ) were measured in the maritime conditions of the Trent floodplain (Fig. 3). These rates are in accordance with those measured in other wetland sites [START_REF] Merrill | Factors controlling denitrification rates in upland and swamp forests[END_REF][START_REF] Pinay | Spatial and Temporal Patterns of Denitrification in a Riparian Forest[END_REF][START_REF] Naiman | Beaver Influences on the Long-Term Biogeochemical Characteristics of Boreal Forest Drainage Networks[END_REF][START_REF] Johnston | Nutrient dynamics in relation to geomorphology of riverine wetlands[END_REF][START_REF] Pinay | Potential denitrification activity in the landscape of a western Alaska drainage basin[END_REF].

Soil moisture

The most important control variable was soil moisture, which was controlled by hydrological events. This was confirmed by the MARS analysis. Soil moisture was involved in all interactions through basis functions BF7, BF9 and BF11, which points out its importance in predicting denitrification. Flooding frequency and duration are controlled by local topography; low areas being flooded more frequently and for longer periods than higher ones, producing large variations in biogeochemical patterns at the metre scale [START_REF] Pinay | Topographic Influence on Carbon and Nitrogen Dynamics in Riverine Woods[END_REF][START_REF] Pinay | Short term hydrologic variations and nitrogen dynamics in beaver created meadows[END_REF]. Biogeochemical processes, especially for nitrogen, are sensitive to the oxidation status of the soil. For instance, ammonification of organic nitrogen can be realized both under aerobic and anaerobic conditions. Nitrification process, which is strictly aerobic, can occur only in aerated soils or sediments. As a consequence, in permanently anaerobic conditions, the organic nitrogen mineralization processes result in the accumulation of ammonia.

Other processes such as nitrogen dissimilation or denitrification are strictly anaerobic, requiring saturated soils to operate. Therefore the end products of nitrogen cycling in riparian soils are directly controlled by the groundwater table with important implications for floodplain productivity. Even so, maximum denitrification activity was not found at soil saturation but between 60 and 80% of maximum soil moisture (Fig. 2A). This can be explained by the fact that partial waterlogging conditions favour the co-existence of aerobic and anaerobic sites at the soil micro-scale. This, in turn, allows the occurrence in nearby microsites of both strictly aerobic and anaerobic biogeochemical processes involved in the nitrogen cycle and enhances organic matter decomposition and nitrogen loss through denitrification in flooded soils [START_REF] Reddy | Effect of alternate aerobic and anaerobic conditions on redox potential, organic matter decomposition and nitrogen loss in a flooded soil[END_REF][START_REF] Groffman | Denitrification hysteresis during wetting and drying cycles in soil[END_REF]. Similarly, alternate aerobic and anaerobic conditions triggered by short-term periodicity of flooddrainage cycles or short rainfall events (Fig. 5) will provide similar results and therefore enhance denitrification process, providing that soil temperature is above 0°C.

Soil Temperature

The second most important control variable which was confirmed by the MARS analysis was soil temperature (Table 2). At the European scale denitrification presented a bimodal pattern with high rates at both low and mild soil temperature (Fig. 2E). It was found that in cold winter regions, i.e. high latitudes (Vindel and Helge Rivers), high altitudes (Aar River) and continental regions (Danube River), very low temperatures during the winter period limits microbial activity in general and denitrification in particular (Fig. 3). Therefore, whatever the hydrologic event, flood or rainfall, it did not trigger any significant denitrification activity during these low-temperature periods. Under milder winter conditions, i.e. Trent, Garonne and Po Rivers, denitrification rates could be significant throughout the year (Fig. 3). Flooding or rainfall could cause high denitrification rates (Figs. 4 & 5) by controlling waterlogging conditions in the alluvial soils and the duration of oxic and anoxic phases [START_REF] Ponnamperuma | The chemistry of submerged soils[END_REF][START_REF] Keeney | The nitrogen cycle in sediment-water systems[END_REF][START_REF] Patrick | Nitrogen transformations in submerged soils[END_REF].

Nitrate

The third control variable on denitrification after soil temperature and soil moisture, confirmed by the MARS analysis, was soil nitrate concentration (Table 2). At the microbial scale, nitrate is a limiting factor for denitrification. Yet, the global negative relationship between denitrification activity and soil nitrate content (Fig. 2C) reveals that nitrate can be used as proxy to determine whether denitrification occurs in alluvial soils. Yet, low or medium soil nitrate concentrations do not provide much information on the rate of nitrogen turn over in alluvial soils because we measured a wide range of denitrification rates under these soil nitrate concentration. High denitrification rates measured in soils with low nitrate concentration suggest that denitrification rates are of the same order of magnitude as N mineralization rates. See Johnston, (1991) for a review.

Herbaceous plant biomass

Competition between plant and denitrification for N uptake in floodplain soils entails negative exponential relationship between herbaceous plant biomass and denitrification (Fig. 2D). It was a significant variable in the denitrification model developed by the MARS procedure (Table 2). Considering that herbaceous plant biomass contains an average of 2% N; then the N denitrified in alluvial soil is in the same order of magnitude as the N stored in herbaceous plants (Fig. 2D). Yet, denitrification constitutes a true removal of N since the end products are in gaseous form [START_REF] Knowles | Denitrification[END_REF]. Consequently denitrification represents a significant loss of nitrogen from floodplain systems; its increase could limit plant biomass production.

Soil texture

Denitrification activity was low in alluvial soils with a low percentage of silt and clay (Fig. 2B). Above a threshold of ca. 60% of silt and clay we measured the highest rates of denitrification activity. This threshold was found whatever the site considered. Interestingly, at this European scale, silt & clay content was not considered in our model to be a significant variable to estimate denitrification in alluvial soils. Yet, the role of soil texture in controlling nitrogen cycling has been already documented in forests [START_REF] Pastor | Aboveground production and N and P cycling along a nitrogen mineralization gradient on Blackhawk Island[END_REF][START_REF] Mcclaugherty | Forest litter decompositionin relation to soil nitrogen dynamics and litter quality[END_REF], coastal forests [START_REF] Seely | Transformation and retention of nitrogen in a coastal forest ecosystem[END_REF], short-grass steppe [START_REF] Schimel | Biochemistry of C, N and P in a soil catena of the shortgrass steppe[END_REF], desert ecosystems [START_REF] Schlesinger | On the spatial pattern of soil nutrients in desert ecosystems[END_REF] and grassland [START_REF] Parton | Rates and pathways of nitrous oxide production in a shortgrass steppe[END_REF]. We know also that flooding indirectly affects nutrient cycling and especially denitrification in floodplain soils by influencing the soil structure and texture through sediment deposits [START_REF] Pinay | Nitrogen cycling in two riparian forest soils under different geomorphic conditions[END_REF][START_REF] Pinay | Geomorphic control of denitrification in large river floodplain soils[END_REF]. Hence, locally, floodplain and stream channel hydromorphological processes influence the sorting of flood sediment deposits on a grain size basis creating a mosaic of soils of different textures. However, sediment texture does not seem to be an important variable at the European scale.

Possible changes in denitrification activity patterns under climatic change scenario

Bacterial denitrification activity is a key process in the cycling of N in floodplain soils since it constitutes a major sink for nitrate, and as such is involved in the control of nitrogen fluxes along river systems. The denitrification rates measured in this pan-European study confirm also that this process is important in a wide range of climatic conditions. Yet, the construction of predictive models of N cycling in floodplain soils is extremely difficult because of the often episodic nature of flood and rainfall events. Moreover, the development of predictive models necessitates that a link be established between processes occurring at a microbial scale and landscape patterns visible at the floodplain scale. For instance, although the role of soil moisture, nitrate concentration and available carbon in regulating this microbial process is now well documented [START_REF] Rolston | Simulation of denitrification losses of nitrate fertilizer applied to uncropped, cropped and manure-amended field plots[END_REF][START_REF] Myrold | Establishment of denitrification capacity in soils: Effects of carbon nitrate and moisture[END_REF][START_REF] Davidson | Environmental parameters regulating gaseous nitrogen losses from two forested ecosystems via nitrification and denitrification[END_REF], these three factors do not explain fully the variation in the denitrification rates observed at the floodplain scale. Indeed, we found that the main driving factors of microbial denitrification in alluvial soils are soil moisture, temperature and to a lesser extend soil nitrate concentration. These three factors might well be affected by global change [START_REF] Shaver | Global warming and terrestrial ecosystems: a conceptual framework for analysis[END_REF][START_REF] Georgakakos | Soil moisture tendencies into the next century for the conterminous United States[END_REF]. Indeed geomorphological and hydrological processes can be altered both from the upslope by land use/land cover change and from the river discharge changes [START_REF] Nilsson | Alterations of riparian ecosystems caused by river regulation[END_REF][START_REF] Nijssen | Hydrologic sensitivity of global rivers to climate change[END_REF][START_REF] Burt | Water table fluctuations in the riparian zone: comparative results from a pan-European experiment[END_REF][START_REF] Pinay | Basic principles and ecological consequences of changing water regimes on nitrogen cycling in fluvial systems[END_REF] while temperature should rise under atmospheric carbon dioxide concentration increase (IPCC, 2001). Yet there remains a major difficulty under climate change scenarios to estimate the resulting hydrological changes since the modelling of their components, i.e. precipitation, evapotranspiration, soil moisture and runoff in considerably less reliable than modelling temperature and pressure for instance [START_REF] Jones | Modelling the impact of climate change on hydrological regimes[END_REF].

Moreover, the effect of a change in climate on the natural hydrological system is frequently non-linear due to the existence of critical thresholds [START_REF] Arnell | Thresholds and response to climate change forcing: the water sector[END_REF]. For instance, even as the Earth as a whole continues to warm gradually, large regions may experience a rapid shift into colder climate [START_REF] Peterson | INcreasing river discharge to the Arctic ocean[END_REF].

Despite these uncertainties, there is evidence from Quaternary records that impact of climate change on fluvial dynamics is more significant in the upper reaches of river networks [START_REF] Veldkamp | Registration of abrupt climate changes within fluvial systems: insights from numerical modelling experiments[END_REF]. Therefore, signs of hydrological changes due to climate change are expected to be less marked in the lower part of the Danube River than in the other study sites, for instance. The 4 scenarios of climatic change, i.e. UKHI [START_REF] Mitchell | Equilibrium climate change and its implication for the future[END_REF];

UKTR [START_REF] Murphy | Transient response of the Hadley Centre coupled ocean-atmosphere model to increasing carbon dioxide. part II: spatial and temporal structure of response[END_REF], CCC [START_REF] Boer | Green-house gas induced climate change simulated with the CCC second generation General Circulation Model[END_REF], GHGx, [START_REF] Johns | THe second Hadley Centre coupled ocean-atmosphere GCM: model description, spinup and validation[END_REF] used [START_REF] Arnell | The effect of climate change on hydrological regimes in Europe: a continental perspective[END_REF] to evaluate the hydrological change in Europe consistently assume an increase of precipitation and average annual runoff in northern Europe and a decrease of both precipitation and runoff in southern Europe. In cold regions, the timing of flow through the year should be unaffected but we can expect an increase in the spring streamflow peak in response to an increase winter precipitation stored as snow [START_REF] Nijssen | Hydrologic sensitivity of global rivers to climate change[END_REF]. This should potentially trigger both an increase of denitrification rates and duration in alluvial soils like in the Vindel River in Northern Sweden. A temperature rise will reduce the snow cover at the margin between maritime and continental regimes which will reduce spring flow and increase winter flows in these areas. This shift in the high-water period could reduce denitrification activity since waterlogging conditions would occur under colder temperature conditions in areas such as the Helge River region in the southern part of Sweden. In maritime regions such as in the Trent valley in the United Kingdom, an increase of both the frequency and magnitude of flooding events is expected [START_REF] Reynard | The flood characteristics of large U.K. rivers: potential effects of changing climate and land use[END_REF].

If there is still no firm evidence of change in flooding trends due to climate change, by contrast, a number of studies have suggest that rainfall becomes more variable and that rainfall intensity and frequency has increased [START_REF] Osborn | Evidence for trends in heavy rainfall events over the UK[END_REF][START_REF] Robson | Evidence for trends in UK flooding[END_REF][START_REF] Burt | The climate of Malham Tarn[END_REF]. This increase of rainfall variability should increase the frequency of aerobicanaerobic cycles in alluvial soils and therefore increase the denitrification activity. This high frequency of wet-dry cycles might increase the proportion of N 2 O emission by denitrification and contribute to the increase the greenhouse effect [START_REF] Blitcher-Mathiesen | Denitrification as a sink for dissolved nitrous oxide in freshwater riparian fen[END_REF][START_REF] Jacinthe | Bioremediation of nitrate-contaminated shallow soils and waters via water table management techniques: evolution and release of nitrous oxide[END_REF]. In warmer regions such as those represented by the Garonne River in South West France and the Po River in northern Italy, a decrease in annual rainfall is expected together with an increase of heavy summer rainfall events. Our results show that these alluvial soils, i.e. Garonne and Po River (Figs. 4 & 5) are highly reactive to such short-term rain events, which result in high rates of denitrification activity. One can expect that an increase of these summer rainfall events would increase the loss of nitrogen in alluvial soils during the vegetated period, might increase the competition for N and, in turn, alter the productivity of these riparian zones.

New challenge

It is now recognized that the importance of indirect effects of climate change such as hydrological changes on the structure and function of ecosystems may have more effects than the increase of temperature per se [START_REF] Vitousek | Beyond global warming: ecology and global change[END_REF][START_REF] Pace | Successes, limitations and frontiers in ecosystem science[END_REF]. In this study we showed that microbial denitrification, an important process of N cycling in alluvial soils, had a pattern of activity which varied widely with the regional climatic context. Even so, it was possible to identify the main control variables of denitrification at the European scale, i.e. soil moisture and temperature, and provide a model to forecast denitrification in alluvial soils which could be used to predict the response of alluvial soils to climatic and anthropogenic changes. For instance, two thirds of the freshwater flowing to the oceans is obstructed by large dams [START_REF] Nilsson | Alterations of riparian ecosystems caused by river regulation[END_REF]. This is especially true in the Northern Hemisphere and particularly in Europe and North America so that it is possible that hydraulic engineering has produced both local and global-scale impacts on the terrestrial water cycle [START_REF] Vörösmarty | Anthropogenic disturbance of the terrestrial water cycle[END_REF]. The new challenge today is to decouple the relative impact of direct human intervention from those effects due to climate change and determine the consequences of hydrological variations fostered by climate change on the resistance and resilience of biogeochemical cycles in river systems.

Figure captions Figure 1 :

 captions1 Figure captions

Figure 2 :

 2 Figure 2: Relationship between denitrification activity and alluvial soil variables. All the data from the different study sites were combined. A: Denitrification rates and soil moisture. B: Denitrification rates and soil texture expressed as percentage of silt plus clay. C: Denitrification rates and riparian soil nitrate content. D: Denitrification rates and herbaceous plant biomass in the riparian sites. E: Denitrification rates and soil temperature.

Figure 3 :

 3 Figure 3: Relationship between riparian soil temperature and denitrification activity in the different rivers under study.

Figure 4 :

 4 Figure 4: A: Monthly denitrification activity in the soils of the Garonne riparian zones -B: Monthly variations of the Garonne River discharge -C: Monthly variations of the riparian soil temperature -D: Monthly variations of rainfall in the study sites along the Garonne River. Box 1 underlines a period with moderate rainfall and high soil temperature. Box 2 underlines a high water period with low soil temperature.

Figure 5 :

 5 Figure 5: A: Monthly denitrification activity in the soils of the Po riparian zones -B: Monthly variations of the Po River discharge -C: Monthly variations of the riparian soil temperature -D: Monthly variations of rainfall in the study sites along the Po River. Box 1 underlines a period with rainfall and floods. Box 2 underlines summer rainfall event.

Table 1 :

 1 Main characteristics of the rivers under study

		_______________________________________________________
		Aar Danube	Garonne	Helge Po Trent Vindel
	_________________________________________________________________________
	Latitude N	47	42	44	56	44	52	64
	_________________________________________________________________________
	Mean annual T°C	8.1	10.6	13.1	6.7	14.5 10.0 1.3
	_________________________________________________________________________
	Mean January T°C 1.0	-1.3	5.5	-0.6 4	3.9	-12.8
	_________________________________________________________________________
	Mean July T°C	17.6 22.0	21.5	16.7 26	16.5 14.4
	_________________________________________________________________________
	Mean annual	1169 408	671	698 664 766 585
	Precipitation (mm)
	_________________________________________________________________________
	Month of max	Nov June	May	July Oct. Dec. Aug.
	Precipitation
	Mean (mm)	80	65	80	77	71	80	88
	_________________________________________________________________________
	Month of min	March Jan.	July	May Feb. July April
	Precipitation
	Mean (mm)	72	16	46	33	38	48	25
	_________________________________________________________________________
	Mean daily	247 6032	202	22	1500 81	180
	Discharge (m 3 s -1 )
	_________________________________________________________________________
	Month of max	June April	May	April May Feb. June
	Discharge
	Mean	686 8473	378	34	1960 153 562
	_________________________________________________________________________
	Month of min	Dec. Sept.	Sept.	Aug Aug Sept March
	Discharge
	Mean	62	4231	86	10	971 40	34
	_________________________________________________________________________

Table 2 :

 2 Ranking of predictors for denitrification by order of importance. The predictors are soil moisture (SM), temperature (TEMP), nitrate (N), herbaceous plant biomass (HPB), and soil texture (SSC). -GCV is the loss in Generalized Cross-Validation when all basis functions involving the corresponding predictor are removed from the final model. Importance (%) is the relative importance of predictors as compared to the best one (soil moisture here).

	__________________________________________________________________
	Predictor	Importance	-GCV
	__________________________________________________________________
	Soil Moisture (SM)	100	719.980
	Temperature (TEMP)	78.349	632.551
	Nitrate (N)	68.036	598.368
	Herbaceous Plant Biomass (HPB)	13.708	497.817
	Soil texture (SSC)	0	493.562
	__________________________________________________________________

Acknowledgements

These results arose from the European River Margins programme (ERMAS), which was supported by a European Grant from the Environment and Climate Program (ENV5V-CT92-0100). We would like to thank Dr. H. Barth, EU scientific advisor, for his support and our colleagues from the different teams in Europe which helped collecting and analysing the data. We also thank two anonymous reviewers who provided constructive comments on an earlier version of the manuscript.