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## Methods Model overview

The model relies on a state-control approach that represents a grassland agroecosystem which is the breeding habitat of a bird species, the lapwing, and the feeding resource for domestic cattle. It is an adaptation of the model of Sabatier, Doyen & Tichit (2010). It focuses on a single permanent grassland patch. This permanent grassland patch is one of the feeding resources available for cattle feeding. We thus assumed that when cattle do not graze the grassland patch, they are fed elsewhere with other ressources (either on temporary grasslands or indoor). The model is a discrete time model linking grass dynamics to bird population dynamics (Figure 8). Time step is defined on a monthly basis as most farmers implement middle term grazing sequences (three weeks to several months). In the grazed grass sub-model, biomass is harvested through grazing. The bird sub-model simulates population changes over time in response to the direct and indirect effects of grazing on bird life traits. Grazing intensity has a direct effect on clutch size through nest trampling by cattle. Grass height (i.e. habitat quality) is generated by grazing and impacts juvenile survival. The model computes two indicators summarizing the ecological and productive performance of each grazing strategy.

We studied the co-viability of the grassland agroecosystem in three scenarios (action-oriented, habitat-oriented and result-oriented scenario) by looking for viable management strategies that satisfy both ecological and productive constraints. Productive constraints are defined similarly in the three scenarios. Ecological constraints differ in each scenario. In the action-oriented scenario, constraints correspond to thresholds on maximal stocking density during the nesting period. In habitat-oriented scenario, they combine thresholds on maximal stocking density during nesting and on grass heights during the chick rearing period. In the result-oriented scenario, constraints correspond to a threshold on minimal bird population size throughout time. For each scenario, the model computed viable grazing strategies meeting the constraints.

**Grazed grass state and dynamics**

The first state of the system represents a grass biomass vector B(t) considered monthly and partitioned into live and standing dead grass (BL(t) , BD(t)) both expressed in organic matter (g OM ha-1). Grass dynamics is controlled by the timing and intensity of grazing u(t), expressed in livestock unit per ha (LU ha-1). The dynamics of the grass biomass B(t) controlled by grazing intensity u(t) is summarized as follows: B(t+1)=A(t,B(t))B(t)-G(u(t),B(t)) for t = 0,1,…,T (eqn 1) where matrix A is a time dependent matrix that encompasses the transition rates from t to t+1. It includes grass growth, senescence and decay rates that are time dependent on a monthly basis. G is a vector representing biomass harvest through grazing. The state of the biomass is linked to grass height through a linear function h(B). Databases from the Ouest-du-Lay marsh were used to parameterize the grazed grass dynamics (Sabatier, Doyen & Tichit 2010; appendix S3). For further details on biomass dynamics and parameter values, see Appendix S1 and Table S1 in Supporting Information.

### Bird state and dynamics

The second state of the system describes the lapwing life cycle. By contrast with Sabatier, Doyen & Tichit (2010), the bird model is deterministic and represents the female portion of a single class population. During the nesting period, cattle trampling impacts clutch size and during the chick rearing period grass height is a variation factor of juvenile survival. Assuming a pre-breeding census, the monthly dynamics of birds N(t) from t to t+1 reads as follows: N(t+1) = M(t,u(t),B(t),N(t)) N(t) (eqn 2) where t* is the nesting month, s2 adult survival, α the proportion of breeding females, f (u) the clutch size depending on cattle density u(t), σ the primary sex ratio and s1 (h(F(B,u))) the chick survival that depends on grass height h(B) at time t*+1. Grass height depends on grass biomass B(t*+1) and therefore on F(B(t*),u(t*)). We consider that breeding success is affected by an intra-specific competition. We use a Beverton-Holt-like density dependence function to model this competition in which c measures the strength of competition. A full description of the bird model along with parameter values are given in Appendix S1 and Table S2.

**Viability constraints**

Three types of viability constraints formalize the multiple roles played by the grazed grassland. Constraints applied to the three scenarios are listed in Table 5.

Tableau 5 Constraint sets taken into account in the three scenarios. Productive performance constraint imposes that productive performance P(u,T) stays over a minimal productive performance Pmin is the minimal annual number of grazing days per hectare associated with a grazing strategy u. Cattle feeding requirement constraint imposes that cattle demand q.u(t) is always lower than the available biomass B*(t). Cattle density constraint is an upper threshold u* on cattle density u(t) during the nesting month. A habitat quality constraint imposes that grass height to stay within a minimal hmin and maximal hmax grass heights during chick rearing. Population size constraint imposes that populations size N(t) stays over a minimum population size Nmin throughout time.

### Cattle feeding requirement constraint

Given a monthly biomass demand per livestock unit q, the feeding requirement constraint is defined as follows: (a) q u(t) < B*(t) for t=0,1,..,T. (eqn 6)

This feeding requirement constraint limits stocking density which cannot exceed the available biomass B*(t). It assumes that cattle cannot graze below a minimal biomass threshold and situations where insufficient grass availability could lead to a poorer body condition of livestock are not considered.

**Productive constraint**

A second constraint defines a minimal level of productive performance necessary for the farmer. Productive performance P(u,T) corresponds to the number of grazing days (simplifying to 30 days per month) associated with a grazing strategy u=[u(0),…,u(T)]. The model does not incorporate any spatial scale but the quantification of the productive performance is given for one hectare. The productive constraint reads as follows: (b) P(u,T ) =∑30 × ut ³ Pmin (eqn 7) where Pmin is the threshold of minimal productive performance. Its value was defined by the 10% lower quantile of a dataset of 344 real grazing strategies recorded on our study site (Durant et al., 2008a).

**Ecological constraints**

Ecological constraints are defined in three different ways so as to capture the three kinds of scenarios.

**Action-oriented scenario (AO)**

The model includes a single ecological constraint. It is related to clutch size in relation with trampling mechanisms. An upper threshold u* is imposed on cattle density during the nesting month t*: (c) u(t) ≤ u* for t= t* (eqn 8)

**Habitat-oriented scenario (HO)**

In addition to the previous constraint (eqn 8), the model also includes a constraint on habitat quality. It is imposed on grass height during the first month following chick birth (t*+1) in order to ensure suitable habitat for chicks. It is bounded by minimal and maximal grass heights as follows: (d) h min ≤ h (t) ≤ h max for t=t*+1 (eqn 9)

#### Result-oriented scenario (RO)

The model involves a single ecological constraint that imposes a minimum population size Nmin throughout time: (e) N(t) ≥ Nmin (eqn 10)

In the action-oriented scenario, the constraint on control variable reflects a strictly action oriented management scenario. In habitat-oriented scenario, ecological constraints combine both control and state constraints. It still limits grazing intensity to ensure a good clutch size and also focuses on an intermediate management objective linked with grass height to achieve a good juvenile survival. In the result-oriented scenario, no constraint is set either on grazing intensity or grass height and the only ecological constraint correspond to a state constraint on the final management goal which is the bird population size in the long term. Using such a state constraint relaxes all management restriction for the farmer.

**Co-viability analysis**

The viability framework is used to identify combinations of biomass B(.), population size N(.) and cattle density u(.) that satisfy viability constraints throughout time. It relies on the computation of the so called viability kernel (Aubin 1991). In the present case, this viability kernel depends on time and we prefer to speak of a viability corridor Viab(t). In this section we will refer to three concepts: the viability corridor, the viable grazing strategy and the viability tube.

**Viability corridor**

The viability corridor Viab (t) is the set of grass biomass conditions and bird population sizes (states, BL(t), BD(t) and N(t)) from which at least one grazing strategy is viable. The Viability corridor at t = t0 (Viab(t0)) is thus defined differently in each scenario:

**Action-oriented scenario:**

ViabAO (t0) = { (B(t0), N(t0)) | there exists grazing u(t) and a sequence of states (B(t),N(t)) starting from (B(t0), N(t0)) satisfying constraint (a) for any time t = t0,…,T, satisfying constraints (c) for any time t>12 and satisfying constraints (b) at time T } (eqn 11)

**Habitat-oriented scenario:**

ViabHO (t0) = { (B(t0), N(t0)) | there exists grazing u(t) and a sequence of states (B(t),N(t)) starting from (B(t0), N(t0)) satisfying constraint (a) for any time t = t0,…,T, satisfying constraints (c) and (d) for any time t>12 and satisfying constraints (b) at time T } (eqn 12)

**Result-oriented scenario:**

ViabRO (t0) = { (B(t0), N(t0)) | there exists grazing u(t) and a sequence of states (B(t),N(t)) starting from (B(t0), N(t0)) satisfying constraints (a) for any time t = t0,…,T, satisfying constraint (e) for any time t>12 and satisfying constraints (b) at time T} (eqn 13)

For the three scenarios, constraints (c) to (e) were not taken into account the first year (t<12) so as to enable a transition of the grazed system toward AES. This choice reflects a conventionally driven system in which AES would be introduced at the end of the first year.

#### Viable grazing strategies

Once the viability corridor has been found, we compute the viable grazing strategies that make it possible to verify the different constraints over the period of time involved. Such U exist as long as the state (B(t),N(t)) lies within the viability corridor Viab(t). We thus consider the set of the viable grazing strategy at time t for a given viable state (B(t), N(t)). From the management point of view, a viable grazing strategy is a temporal sequence of grazing intensities that keeps the whole system within the constraint set. To each viable grazing strategy corresponds a viable state trajectory defined in terms of grass biomass and population size.

**Table of contents :**

INTRODUCTION

1. EMERGENCE DE LA MULTIFONCTIONNALITE EN AGRICULTURE

2. ENJEUX SCIENTIFIQUES ET PROBLEMATIQUE : APPREHENDER LES DIMENSIONS ECOLOGIQUE ET PRODUCTIVE DE LA MULTIFONCTIONNALITE

a. Diversifier les modes de gestion pour répondre à des objectifs multiples

b. L’approche de viabilité pour appréhender la durabilité

c. La modélisation pour lier les dimensions productive et écologique des agroécosystèmes

d. Domaine d’application

3. QUESTIONS DE RECHERCHE

4. DEMARCHE

a. Cas d’étude

b. Stratégie de recherche

c. Cadre mathématique de la théorie de la viabilité

5. ORGANISATION DU MANUSCRIT

**CHAPITRE 1** RELATION ENTRE PRODUCTION AGRICOLE ET CONSERVATION DE LA BIODIVERSITE, ROLE DES SEQUENCES DE PATURAGE

**CHAPITRE 2 ** MESURES AGROENVIRONNEMENTALES A OBLIGATION DE RESULTATS POUR AUGMENTER LA FLEXIBILITE DES MODES DE GESTION

**CHAPITRE 3** LA PROPORTION DES USAGES AGRICOLES, FACTEUR DETERMINANT DE LA RELATION ENTRE PRODUCTION ET CONSERVATION

**CHAPITRE 4** COMPLEXIFIER LA STRUCTURE DES PAYSAGES POUR FAVORISER LES MECANISMES DE COMPLEMENTARITE ENTRE USAGES AGRICOLES

DISCUSSION GENERALE

**1. UNE DOUBLE FONCTION DE PRODUCTION ET DE CONSERVATION POUR LES PAYSAGES AGRICOLES **

**2. A QUELLES ECHELLES CONCILIER PRODUCTION ET CONSERVATION ? **

**3. QUELLE PORTEE POUR LES RESULTATS? **

a. Limites

b. Validation

c. Généricité

d. Modèle de recherche ou modèle d’aide à la décision ?

**4. LA DIVERSITE, MECANISME CENTRAL DE LA CONCILIATION ENTRE PRODUCTION ET CONSERVATION**

a. L’arbitrage entre production et conservation résulte d’un choix de mode d’exploitation

b. Variations interannuelles des modes d’exploitation pour s’adapter au climat

c. Interaction entre modes d’exploitation au sein d’une exploitation agricole

d. Structure spatiale des paysages pour favoriser les mécanismes de complémentarité

e. De la compréhension des mécanismes d’interaction à l’identification des leviers d’action pour la gestion

**5. PERSPECTIVES : COMPLEXIFIER LA COMPOSANTE DECISIONNELLE**

a. Contraintes spatiales limitant l’ensemble des paysages réalisables

b. Stratégies de gestion sub-optimale et flexibilité du système

**CONCLUSION **

**BIBLIOGRAPHIE**