# Analyzing the factors driving the adaptability and robustness of mixed ruminant herds in grassland systems

Thomas Puech, Fabien Stark, Rodolphe Sabatier

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Thomas Puech, Fabien Stark, Rodolphe Sabatier. Analyzing the factors driving the adaptability and robustness of mixed ruminant herds in grassland systems. Agricultural Systems, 2025, 228, pp.104371. 10.1016/j.agsy.2025.104371. hal-05092891

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Research Paper

# Analyzing the factors driving the adaptability and robustness of mixed ruminant herds in grassland systems

T. Puech<sup>a, *</sup>, F. Stark<sup>b</sup>, R. Sabatier<sup>c</sup>

## HIGHLIGHTS

* Global change raises questions about the ability of grazing livestock systems to cope with uncertainties.

* Using theory viability, we have modelled the robustness and adaptability of mixed dairy-cattle and suckler-sheep herds.

* The economic constraint is the main factor driving the ability of mixed systems to cope with hazards.

* The size and composition of herds may have synergistic or antagonistic effects depending on the economic constraints.

## GRAPHICAL ABSTRACT

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

Editor: Paul Crosson

**Keywords:**

Viability theory

Mixed ruminants herds

Environmental hazard

Agroecology

France

## ABSTRACT

**CONTEXT:** Diversified systems that work to agroecology principles are a pathway worth exploring. However, the complexity of these systems and their high dependency on environmental conditions creates issues around the methods needed to assess them and the proxies needed to design them.

**OBJECTIVE:** The aim of this article is to characterize the robustness and adaptability of mixed herds in the face of environmental hazards. It also aims to identify the structural drivers (herd size and composition) of these two properties.

**METHODS:** Here we used the viability theory modelling approach calibrated on data from a long-term experiment to investigate the adaptability and robustness of mixed ruminant herds to meteorological and economic hazards, and their structural drivers. We applied our model to grass-based dairy-cattle and suckler-sheep herds.

**RESULTS AND CONCLUSIONS:** Results show that expected economic constraint is a determinant factor in the shape and composition of viable herds. Herd size and proportion of adult cattle in the herd are drivers of robustness in situations of uncertainty. The results also show that mixed herds are particularly valuable in situations with low economic requirements, especially in terms of herd adaptability to environmental hazards.

**SIGNIFICANCE:** Our results are consistent with existing mixed systems in western Europe but call for a change in the scale of analysis to include farm-level dynamics, associated management practices (land-use trade-offs,

<footnote><sup>a</sup> INRAE, UR ASTER, 88500 Mirecourt, France</footnote>

<footnote><sup>b</sup> INRAE, Institut Agro Montpellier, University Montpellier, CIRAD, UMR SELMET, 34060 Montpellier, France</footnote>

<footnote><sup>c</sup> INRAE, UR Ecodéveloppement, 84000 Avignon, France</footnote>

<footnote>* Corresponding author.
E-mail address: thomas.puech@inrae.fr (T. Puech).</footnote>
forage management, etc.) and uncertainties. This work questions the specialization of livestock farms and public policies to support agroecological transition and emergency aid for farmers.

# 1. Introduction

Farms and rural areas in Western Europe have become increasingly specialized over the second half of the 20th century. This specialization process has been based on transposing the industrial model to agriculture in order to increase productivity across the agricultural sector (Jepsen et al., 2015), via (i) strong public policy intervention and (ii) artificialization of the environment through increased use of inputs facilitated by low energy prices (and by-products such as nitrogen fertilizers). Changes in European public policies in the 1990s (end of market intervention and then milk quotas) followed more recently by geopolitical and economic crises have led to price volatility in agricultural products. Specialized systems are particularly sensitive to these fluctuations (Chatellier, 2011; De Roest et al., 2018). In addition, the increasing scarcity of fossil fuels (Campbell et al., 2017; Pinsard, 2022), ongoing climate change (Lee et al., 2023), and changing societal expectations around agriculture have converged to challenge the ability of agricultural systems to cope and adapt to these new production contexts.

Furthermore, given the negative externalities of agriculture (water quality, erosion of biodiversity, etc.), a number of institutions have called for Europe's agricultural systems to make an agroecological transition (Altieri et al., 2012; Gliessman, 2004; Tomich et al., 2011; Wezel et al., 2009). Diversification and the search for complementarities between crops and livestock are among the avenues to be explored. Integrating animals, especially ruminants, into farming systems can work toward these goals if the livestock can make the most of grassland or rangeland resources that are less competitive with human food (Barbieri et al., 2022; Benoit and Mottet, 2023; Mottet et al., 2017; Van Zanten et al., 2019). Combining different animal species on the same farm is one way to improve farm sustainability (Martin et al., 2020). For example, some authors have shown that combining mixed-species herds with grazing is a way to reduce health risks (Joly et al., 2023) and increase productivity per unit area (Martin et al., 2020) while also limiting economic risks by reducing variability in the performance of the farming system (Bowman and Zilberman, 2013). This interest is particularly evident in agroecological systems, as they are strongly based on ecological processes and are thus dependent on or even more vulnerable to climatic hazards. For example, Perrin et al. (2020) showed that climatic and economic hazards are the main factors that disrupt organic dairy farming systems.

The complexity associated with the diversity of system components and their integration means that agricultural systems take on emergent properties (Buchmann, 2001). Assessing these emergent properties is a complex task, because (i) they result from indirect processes that are difficult to observe in the field, and (ii) these systems are based on ecological processes which make them fraught with uncertainty given the diversity of hazards associated with the environments in which they evolve (Prost et al., 2023). The methods and tools used to evaluate diversified farming systems now need to be revisited and redesigned (Magne et al., 2024). Emerging properties relating to productive capacity (productivity or efficiency - Bonaudo et al., 2014) are commonly used in agronomy to assess farming systems and have been the subject of numerous methodological developments (Battheu-Noirfalise et al., 2023; Godinot et al., 2015), but little efforts were made on assessing the capacity of farming systems to cope with uncertainty while these properties are becoming at least as important as assessing their productive performances (Prost et al., 2023; Sabatier and Mouysset, 2018; Urruty et al., 2016).

The aim of this paper is to investigate the emerging properties that shape the ability of systems to cope with uncertainty, and their determinants of these properties in the context of mixed ruminant herds.

We address the ability of mixed herds to cope with various hazards by assessing their robustness and adaptability with a modelling approach based on the Viability Theory (Aubin, 1991). The 'robustness' of a system is a measure of its ability to maintain performances when exposed to perturbations (Accatino et al., 2014; Urruty et al., 2016), while the 'adaptability' of a system is a measure of the set of possible management decisions needed to maintain a system within acceptable conditions against an uncertain environment (Accatino et al., 2014; Gallopín 2006). These two dynamic properties contribute to the resilience of mixed herds (Holling, 1973). Analysis of these emergent properties requires us to explore all possible hazards and adaptation options. Viability theory is a mathematical framework adapted to the study of dynamic systems. This 'state-control' approach have been used to study sustainability issues at the interface between economy and environment (Krawczyk and Pharo, 2013; Oubraham and Zaccour, 2018). The interest of these methods lies in the fact that they serve to identify not just optimal situations (Pareto fronts) but all states and decision sequences that respect a set of constraints in the long term, even in situations of uncertainty in decision-making. However, these methods are costly in terms of computational time because they require exploring a large number of modalities (curse of dimensionality; Bellman, 1961) and have therefore been little used to study diversified farming systems.

We applied this research question to mixed herds of dairy cattle and suckler sheep. The association between these two species is of interest because of their different biological cycles (short cycle for sheep, long cycle for cattle) and the nature of their production (mainly milk for cattle, exclusively meat for sheep). This type of cattle-sheep association is common in certain regions of the world, especially in extrem climatic contexts (Manoli et al., 2014; Sabatier et al., 2017), but remains marginal in temperate regions. Furthermore, here we set out to characterize the structural determinants of adaptability and robustness, both in terms of the size and composition of mixed herds, with the aim of generating useful knowledge for designing and supporting sustainable livestock systems.

# 2. Materials and methods

## 2.1. Case study

This work is based on a diversified farming system developed at the INRAE ASTER experimental station (PAPILLE project – "Self-sufficient crop-livestock farming system, designed step-by-step for human food, with local resources"; Mirecourt, France). This farming system is certified to organic farming specifications. It started in 2016 and aims to use the land as directly as possible for human consumption (Barbieri et al., 2022; Mottet et al., 2017; Van Zanten et al., 2019). To achieve this goal, the system is managed with a view to self-sufficiency and diversification, using the principles of agroecology (Altieri et al., 2012; Dumont et al., 2014; Gliessman, 2004). The system consists of annual crops grown exclusively for human consumption (about 70 ha), grassland (35 ha of temporary grassland in rotation, and 135 ha of permanent grassland), ruminants (dairy cattle and suckler sheep) and pigs (Puech and Stark, 2023). The ruminants are reared on a strict herbivorous diet to limit competition with human food. Given the objectives of this article, here we focus on studying the two ruminant herds.

The dairy herd consists of dairy cows and two cohorts of heifers: heifers aged 0–1 years reared on nurse cows (Spengler Neff et al., 2023) with the aim of calving at 24 months, and heifers aged 1–2 years. Calving is grouped at the end of winter to synchronize the peak of milk production with the high availability of grass in spring. Dairy cows are milked once a day throughout the year. The cattle graze for most of the
year when the clayey soil conditions allow. Sheep are reared to complement the dairy cows in terms of available forage resources and outdoor grazing. Ewe lambs birth at 12 months (cross-bred lambs are not kept for replacement). Lamb births are grouped in April to make the most of summer grazing. The lambs are fattened on grass only.

Datas collected on the experimental farm are structured into broad categories: technical performances (animal growth, milk production), economic (selling prices, operating costs) and meteorological datas. These datas are structured in dedicated information systems (Trommenschlager et al., 2010), enabling their long-term evolution to be assessed. These datas were used to calibrate the model. Interviews with experimenters were conducted to estimate (i) the working time required for breeding activities and (ii) the causal relationships between environmental hazards (meteorological, economic) and performances (technical and economic).

## 2.2. Model

The model is of 'state-control' type. It requires the identification of descriptors of the system ('states') and possible actions for the manager of the system ('controls'), which guides the dynamics of the system. This system is subject to environmental hazards (meteorological, economic): the model is stochastic in the sense that it incorporates environmental hazards. The viability algorithm aims to identify the viability kernel, defined as the set of states-controls that respect short and long-term constraints.

The model was developed in discrete time, with an annual time-step based on a cropping season time horizon. In this paper we assume that the cropping season starts in spring (when cattle are turned out to grass) and ends at the end of winter. Livestock numbers are those considered at the start of the cropping season (March 15th).

### 2.2.1. States - controls

In this work, we consider a state for the main physiological stages of the animals in the herd (3 for cattle, 2 for sheep). States are shown in blue color in Fig. 1:

- $x_1$: Dairy cows. Dairy cows give birth in late winter and produce milk for all or part of the crop year. Male calves are systematically sold off.

- $x_2$: Heifers aged between 0 and 1 year may be kept for replacement. These animals are reared throughout the grazing season by nurse cows (the milk produced by these cows is not sold). The heifers are weaned at 9 months when they enter the barn.

- $x_3$: Heifers aged 1–2 years old, kept for renewal. The heifers are inseminated during the year. Animals that are not pregnant at the end of the campaign are considered in the same category the following year.

- $x_4$: Ewes. The ewes give birth to lambs (group lambing in April). Females can be kept for replacement purposes, and the males are systematically sold off.

- $x_5$: Replacement ewe lambs. Lambs are born in spring, but all lambs born to a ewe lamb (male and female) are sold for slaughter (terminal cross). Only female lambs born to a ewe may be kept for replacement. Ewe lambs that are not pregnant at the end of the season are culled.

Each of these states is expressed in Livestock Units (LU), and a co-efficient specific to each physiological type is used to correlate with the number of animals (see Table 2 in Appendix 1).

For the purposes of this study, we have selected 4 controls that correspond to the main decisions taken by the farmer to manage the demography of the herd between a state at time *t* and a state at time *t* + 1. Controls are shown in orange color in Fig. 1:

<figure><img src="image_3.png" /><figcaption>Fig. 1. Conceptual model and the livestock dynamics modelled.</figcaption></figure>
- $u_1(t)$ and $u_3(t)$ represent renewal rates for cattle and sheep, respectively, which corresponds to the proportion of females (0–1-year-old heifers and renewal ewe lambs) bred for renewal in the herd. It varies between 0 (no females kept for renewal) and 1 (all females bred)

- $u_2(t)$ and $u_4(t)$ represent culling rates for cattle and sheep, respectively. Culling rate is bounded by 0.1 and 1 (sale of the whole herd). We assume that at least 10% of the adult animals are sold each year, i.e. animals that are not adapted to the system (non-pregnant females, health problems, lameness, etc.). These culling rates do not apply to heifers and ewe lambs unless all adults are sold.

Controls are limited to the ‘natural’ demographic management of the herd: we do not consider controls linked to purchases of animals (e.g. pregnant heifers).

### 2.2.2. Hazards

Several authors single out climatic and economic hazards as the two main hazards faced by organic dairy systems (McNamara and Weiss, 2005; Perrin et al., 2020). We propose to distinguish between these two categories of hazards (green color in Fig. 1), which we will describe below in the form of a short qualitative narrative. The quantitative dimension, and in particular the impacts of these hazards on animal performances, is based on observations at the experimental farm between 2016 and 2023 and is detailed in Appendix 1. Hazards are explored by the viability algorithm, with no assumption about their frequency.

#### 2.2.2.1. Meteorological hazards.
The nature of the production and management decisions (see 2.1) means that production is highly seasonal and that technical performances (health, reproduction, growth, milk production) are dependent on environmental conditions. We have chosen to consider 3 different seasons:

- Spring, which covers the period from mid-March to the end of June. This is the period of maximum grass growth, and the quality and availability of grazed fodder is relatively stable from one year to the next. Spring marks the lambing season, peak lactation for cows, and the breeding season for cattle.

- Summer covers the period from July to September. Summer is a critical period for lamb growth and milk production. The quality and availability of forage varies from year to year depending on soil moisture conditions.

- Autumn-winter covers the period from October to mid-March. In autumn (October–November), lambs for slaughter are finished and sold, ewes and ewe lambs are bred with rams, heifers are weaned, and all the cattle are moved into barns. The mid-winter period (mid-December to mid-March) is characterized by highly regular inter-annual conditions (cattle indoors on winter rations, sheep outdoors on extensive grazing), but weather conditions in the first part of the period are decisive both for finishing lambs, mating ewes, and the length of time the cattle can graze (limited by the carrying capacity of the soils).

We consider two different conditions for each season. This binary characterization of the seasons is restrictive given the diversity of situations observed, but it allows an approximation of the variability of environmental conditions to accommodate the methodological challenge of limiting the dimensionality of the viability problem. These conditions are expressed as follows:

- Dry conditions: These conditions are relatively unfavorable to grass growth, especially in summer and autumn. This reduction in grass availability creates a need to provide hay during grazing period and leads to a relative reduction in milk production. Parasite pressure is relatively low, especially for sheep (cattle are less susceptible). Soil

conditions are good in early spring and late autumn, allowing an extension of the grazing period for cattle. Possible heatwaves in late spring lead to a decrease in reproductive performance (Wolfenson and Roth, 2019), particularly due to a decrease in the expression of estrus in cows (reproduction only by artificial insemination).

- Wet conditions: These conditions are favorable to grass growth, whatever the season. The hay requirements during grazing are lower (summer, autumn), but the grazing period for the cattle is shorter (early spring or late autumn due to the low carrying capacity of soils). In general, the relatively good quality of grazed fodder limits the drop in milk production in the second half of the cow's lactation (low feeding). Parasite pressure (intestinal strongyles, tapeworms, myiasis) is higher than in dry conditions (especially in summer). This leads to an increase in the frequency of care interventions and mortality (especially in sheep) and a reduction in the growth of lambs and heifers (which affects their reproductive performance). Autumn rains, on the other hand, are favorable to better-quality fodder (regrowth of meadows, plant cover), which is beneficial for sheep reproduction (flushing).

Using long-term data collected on the farm, we estimate the range of variation in performances in the face of meteorological hazards. From a numerical point of view, we propose to use an index of technical performance variability linked to meteorological hazards ($\omega^W(t)$). This index is used to weight the ‘standard’ value of each technical performances, i.e. milk production, growth, pregnancy rate, fertility, health aspects, and mortality (Tables 1 & 2 in Appendix 1). We assume that weather conditions are season-independent. However, there may be a memory effect of environmental conditions on animal performances from one season to the next. We therefore propose a composite indicator resulting from the interaction between the seasons, formalized by multiplying each seasonal coefficient together. A value of 1 means that the season has no effect on performance (for example, we consider cattle mortality to be independent of environmental conditions). A value less than 1 means the seasonal factor reduces performance (for example, the pregnancy rate of ewe lambs is penalized by a wet summer, with the underlying mechanism being a reduction in growth combined with an increase in parasite pressure). A value greater than 1 indicates favorable seasonal conditions (e.g. ewes fertility is increased by wet autumn conditions, which are favorable to the growth of grazed plant cover and consequently have a flushing effect on reproduction). This technique enables compensation/reinforcement mechanisms between seasons (see Table 3 in Appendix 1).

#### 2.2.2.2. Economic hazards.
Management of the system, based on maximizing grazing, requires little purchase of inputs specific to livestock production (no feed purchases). The main operating costs of the herds are related to reproduction and veterinary interventions. Consequently, this study does not consider hazards related to input prices, and so we only propose hazards on the sale prices of each animal product (milk, beef, and sheep meat). Based on prices charged by the organic operators the experimental farm works with (who are also major operators in the French organic sectors concerned), we consider 3 levels of uncertainty for each of these products (low, medium, and high selling prices). The low/high values are calculated as averages of the 20 % lowest/highest prices observed over the 2021–2023 period. The numerical values observed are rounded in order to symmetrize hazards (see Table 4 in Appendix 1).

### 2.2.3. Dynamics

In general, a dynamical system is characterized by a set of dynamics $f$ reflecting the time-course evolution of the states of the system ($X \in \mathcal{X}$) under the effect of controls ($U \in \mathcal{U}$) and uncertainties ($\omega \in \Omega$). Eq. 1 defines these dynamics in the general case:
Equation 1: General dynamics.

$$X(t+1) = f(X(t), U(t), \omega(t))$$

Eq. 2 presents the system of equations for herd demography (see P1, S1... in Fig. 1). More specifically, we consider that when culling rate is equal to 1 ($u_2(t) = 1$ for cattle, $u_4(t) = 1$ for sheep), the whole herd is culled, including replacement lambs or heifers. The dynamics of the adult animals (dairy cows and ewes, respectively) are governed by a survival function ($S_1$ and $S_4$, respectively; Appendix 1.b.), an ‘enter production’ function for young animals ($P_1$ and $P_4$, respectively; Appendix 1.c.) and a culling function ($R_1$ and $R_4$, respectively; Appendix 1. d.). It is assumed that non-pregnant adults are culled by priority. However, non-pregnant cows can be kept in extended lactation (Sehested et al., 2016) for a second crop year without a decrease in annual production level (the decrease in average production level is compensated by a longer lactation period). The dynamics of the first renewal generation (0-1-year-old heifers and renewal ewe lambs) is determined by the dynamics of the adults and renewal decisions ($P_2$ and $P_5$, respectively; Appendix 1.c.). Finally, the dynamics of heifers aged 1 to 2 years is defined by the dynamics of heifers of the previous age class ($P_3$) and by a simplification of reproductive failures ($G_3$; Appendix 1.e.). Survival functions are only explained for adult animals but are included in the production functions for young animals. Other than the culling functions, all other functions depend partly on climatic hazards.

Equation 2: Herds dynamics.

$$ \left\{ \begin{array}{l} \text{Dairyherd} \left\{ \begin{array}{l} x_1(t+1) = x_2(t+1) = x_3(t+1) = 0 \\ x_1(t+1) = S_1(x_1, \omega, t) - R_1(x_1, u_2, t) + P_1(x_3, \omega, t) \\ x_2(t+1) = P_2(x_1, x_3, u_1, \omega, t) \\ x_3(t+1) = P_3(x_2, \omega, t) + G_3(x_3, \omega, t) \end{array} \right\} \text{if } u_2(t) < 1 \\ \\ \text{Sheepherd} \left\{ \begin{array}{l} x_4(t+1) = S_4(x_4, \omega, t) - R_4(x_4, u_4, t) + P_4(x_5, \omega, t) \\ x_5(t+1) = P_5(x_4, x_5, u_3, \omega, t) \end{array} \right\} \text{if } u_4(t) < 1 \end{array} \right. $$

## 2.3. Viability constraints and metrics of robustness and adaptability

### 2.3.1. Viability constraints

We define two constraints (Eq. 3) that correspond to the maximum herd size (C1) and an economic constraint (C(t)) on the difference between income and breeding costs (C2).

Equation 3: Set of constraints.

$$ K := \{(x_1, x_2, x_3, x_4, x_5)(t)\} \quad \left[ \begin{array}{l} C1: x_{min}^* \le x_1(t) + x_2(t) + x_3(t) + x_4(t) + x_5(t) \le x_{max}^*, \forall t \in (0 : T) \\ C2: C(t) \ge C^*, \forall t \in (0 : T), C^* \in [15, 30, 50, 70, 90, 110, 130, 150, 175, 200] \end{array} \right. $$

Equation 4: Definition of Income and Breeding costs.

$$C(t) = \text{Income}(t) - \text{Breeding costs}(t)$$

$$ \left\{ \begin{array}{l} \text{Income}(t) = \text{Income}_{\text{milk}}(t) + \text{Income}_{\text{meat-beef}}(t) + \text{Income}_{\text{meat-sheep}}(t) \\ \text{Breeding costs}(t) = (\text{TC}_{\text{dairy cattle}}(t) + \text{TC}_{\text{sheep herd}}(t)) + (\text{LC}_{\text{dairy cattle}}(t) + \text{LC}_{\text{sheep herd}}(t)) \end{array} \right. $$

We consider a first constraint related to the maximum stocking rate, which is the threshold beyond which problems posed by the physical capacity of the livestock buildings would be difficult to overcome, particularly in periods of poor soil-bearing capacity. As this work focuses on the herd, we do not consider constraints related to the relationship between forage resources and herd size requirements, which includes components linked to levels of organization other than that of the herd (grassland areas, forage stocks, etc.). Consequently, this work is carried out under the assumption of non-limiting forage resources (see the discussion section).

The second constraint (subsequently referred to as an “economic constraint”) concerns the capacity of the herd to generate a minimum amount of money (C*), which is calculated as total sales of animal products minus operating and labor costs. We consider that C* can be used at farm level in addition to other income (sale of crop products, public subsidies, etc.) to cover structural costs (mechanization, energy, land, etc.) or annual payments. We studied different economic constraints values in order to explore a variety of situations (Eq. 4) covering the diversity found in the French Livestock Institute case studies for grassland-based dairy cattle and suckler sheep systems (Seegers and Sarzeaud, 2023).

### 2.3.1.1. Products sold.
Products sold are defined by Eq. 4 (see Appendix 1.f.i.j. for further details). Milk sold is equal to milk produced minus milk consumed by calves. The sale of cattle corresponds to the sale of cull cows plus all calves sold and all females not selected for replacement. Similarly, the sale of sheep is made up of the sale of cull ewes and lambs (except replacement lambs ewes).

### 2.3.1.2. Breeding costs.
We consider two types of costs for breeding activities: technical costs (TC) and labor costs (LC) (see Eq. 4). This is a simplified approach compared to the approaches classically employed in livestock economics (Mosnier et al., 2009; Veysset et al., 2015) (see Appendix 1.k.l for more details).

Technical costs are considered proportional to the number of animals. They correspond to veterinary costs, reproduction costs (insemination doses, purchase of breeding stock, pregnancy diagnostics), feed minerals, carcass disposal costs, and all costs related to milking (disinfectants, maintenance of milking equipment, etc.). Some costs per LU may vary according to weather conditions, particularly in relation to health issues. Fixed costs (e.g. annual amortization of investments) are not included at this level.

Labor costs correspond to the cost of the work involved in rearing the two species. The amount of work is quantified using the Work Assessment Method (Cournut et al., 2018). For sheep, we use the French
Livestock Institute references (Béguin et al., 2010) because they describe farming practices very similar to those studied here. Farm work consists of seasonal work (moving animals, shearing, weaning) and on-call work (feeding, lambing, pasture management, etc.). On-call work in sheep farming offers economies of scale. We performed a logarithmic regression on on-call work for the references used in our case study, which transforms the data from Béguin et al. (2010) into continuous variable and thus avoids threshold effects, which would make little sense in this work (see details in Appendix 1.m.).

On the other hand, the particularities of cattle management (heifer calves reared under nurses, once-a-day milking) make it inappropriate to use national references as we had done for sheep. We have therefore subdivided working time into different activities (milking, reproduction, winter management of livestock buildings, grazing, animal monitoring, distribution of feed, etc.). Some activities are proportional to number of animals in the herd (e.g. reproduction, cleaning buildings, milking) whereas others are proportional to a number of batches of animals (e.g. monitoring grazing) and others are independent of the number of animals (e.g. cleaning and maintenance of the milking system). Details are given in Appendix 1.n.

By integrating labor costs into the cost of rearing livestock, we consider that the remuneration of rearing labor cannot be an adjustment variable for other expenses.

### 2.3.2. Viability kernel

The assessment of the robustness and adaptability of mixed herds is calculated using the conceptual framework of viability theory (Aubin, 1991). Once the states, controls and dynamics have been defined, we define a set K of constraints that corresponds to a set of limits on different dimensions that we want the system to respect through time, at least for one hazard. Once this set K is defined, viability algorithms make it possible to calculate the ‘viability kernel’ (Viab(t)). The viability kernel is defined as the set of states from which starts at least one viable trajectory (Eq. 5 and Fig. 2). The robustness and adaptability metrics are calculated based on the viability kernel.

<caption>Equation 5: Definition of the viability kernel.</caption>

$$ \left\{ \begin{array}{l} \text{Viab}(t) = \left\{ X(t); \exists(U, \omega)(t) \mid \begin{cases} (X, U, \omega)(t) \in K \\ X(t+1) \in \text{Viab}(t+1) \end{cases} \right\}; \forall t \in [0; T] \\ \text{Viab}(T) = \{X(T); \exists(U, \omega)(T) \mid \{(X, U, \omega)(T) \in K\} \} \end{array} \right. $$

### 2.3.3. Adaptability and robustness of mixed herds: Metrics and drivers

Adaptability and Robustness metrics are calculated from the stable viability kernels, i.e. defined by $t = T$ such that $Viab(T+1) = Viab(T)$ (Eq. 5). Robustness reflects the ability of a mixed herd to maintain its characteristics in response to an uncertain situation, while adaptability reflects the ability of a system to adapt to a hazard (Accatino et al., 2014). We propose to measure the robustness (Eq. 6) of a state as the proportion of hazards for which there is at least one viable control, where maintenance of properties is understood in terms of compliance with viability constraints. We define adaptability as the proportion of controls for which there is at least one viable hazard (Eq. 7). We propose a hybrid metric, termed ‘robust adaptability’, that reflects the ability of a mixed herd to maintain its properties regardless of the hazard and accounts for the proportion of controls that remain viable whatever the hazard (Eq. 8). The set of robust controls is therefore a subset of viable controls. A simplified example is shown in Fig. 2 (see figure legend).

<caption>Equation 6: Definition of Robustness<fnref n="1" /></caption>

$$ \text{Rob}_{(X,C')}(T) = \frac{\text{Card} \left( \omega(T) \mid \exists U(T) \left\{ \begin{array}{l} (X, U, \omega)(T) \in K \\ X(T+1) \in \text{Viab}(T) \end{array} \right\} \right)}{\text{Card}(\omega(T))} $$

<caption>Equation 7: Definition of Adaptability<fnref n="1" /></caption>

$$ \text{Adapt}_{(X,C')}(T) = \frac{\text{Card} \left( U(T) \mid \exists \omega(T) \left\{ \begin{array}{l} (X, U, \omega)(T) \in K \\ X(T+1) \in \text{Viab}(T) \end{array} \right\} \right)}{\text{Card}(U(T))} $$

<caption>Equation 8: Definition of Robust adaptability<fnref n="1" />.</caption>

$$ \text{Adapt}^{\text{Rob}}_{(X,C')}(T) = \frac{\text{Card} \left( (U)(T) \mid \forall \omega(T) \left\{ \begin{array}{l} (X, U, \omega)(T) \in K \\ X(T+1) \in \text{Viab}(T) \end{array} \right\} \right)}{\text{Card}(U(T))} $$

Once these metrics are computed, drivers of emergent properties are evaluated using structural indicators for mixed herds. To do this, we compare the structure of the 5 % best performing herd configurations for each of the 3 metrics. These drivers are calculated from the size (total number of animals) and composition of the herds via (i) the proportion of each species in the herd and (ii) the proportion of adults within intraspecific herds.

## 2.4. Resolution, statistical processing, and formatting

The viability kernels were computed in C. The discretization step was set to 10 LU for the states space and 10 % for the controls space. These values are the result of a trade-off between (i) the granularity of the discretization grid, (ii) the number of economic constraints to explore, and (iii) the time required to compute the 10 viability kernels (one kernel per value of economic constraint). Statistical analyses of the drivers of adaptability and robustness were performed using permutation tests in R (version 4.3.1, PermTest function in the pgirmess library; Giraudoux, 2005). Graphical representations were produced using Python (Matplotlib library; Hunter, 2007) to simplify visualization per species (cattle and sheep).

# 3. Results

## 3.1. Economic constraint: The main constraint for mixed herds

Simulations show that kernels are non-empty. This means that there is at least one herd configuration and one management option that meet the constraints, at least for one hazard. Furthermore, the volume of the viability kernel decreases as the economic constraint increases (Fig. 3): increasing the minimum economic constraint decreases the range of viable herd combinations. The higher the constraint, the higher the number of animals required to ensure viability. The effect of this constraint is asymmetrical: only herds with a high proportion of cattle are viable in a situation of high economic constraint requirements. Conversely, a smaller herd size or higher proportion of sheep makes these herds less viable in a situation of high economic constraint.

Failure to meet the economic constraint is the main cause of non-viability for mixed herds, whatever its value. This cause represents 63 % of the causes of non-viability for the lowest level of economic constraint modelled (€15 k) and 95 % at the highest level of C* (€200 k). Beyond an economic constraint of €50 k, the constraint on maximum

<footnote n="1">¹ Card means cardinality, i.e. the number of hazards (Eq.6) or controls (Eq.7 and Eq.8)</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Viability theory: concepts and examples of dynamics and emergent properties. The set of constraints defines the constraint domain. The viability kernel corresponds to the set of initial states for which there is at least one control and one hazard that allows the final state to satisfy the constraints (while B respects the constraints, it is not in the viability kernel). The robustness of A corresponds to the number of hazards for which there is at least one viable control: $ω_1$ and $ω_2$ are viable therefore $Rob_{(A)} = 2/2 = 1$. The adaptability of A corresponds to the number of controls for which there is at least one viable hazard: U and $U''$ are viable ($U'$ is not viable whatever $ω_1$ or $ω_2$) therefore $Adapt_{(A)} = 2/3$. Finally, robust adaptability of A corresponds to the number of viable controls regardless of the hazard: only U is viable whatever $ω_1$ or $ω_2$ therefore $Adapt_{(A)}^{Rob} = 1/3$.</figcaption></figure>

Viability kernel (maximum value of economic constraint)

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Evolution of the volume of the viability kernel with economic constraint.</figcaption></figure>

number of animals is rarely a limiting factor (< 10 % of situations). Non-reproducibility of the system ($X(t + 1) \notin Viab(t + 1)$) represents 6 %–15 % of reasons for non-viability, depending on the economic constraint.

In the following sections, we move on to study the emergent properties of mixed herds and their structural drivers, based on 3 contrasting situations presented below. We therefore focus the remainder of the results on the properties of herds and the drivers of these properties that have to respond to a low (€30 k), medium (€90 k) and high (€150 k) economic constraint. Details are presented in Appendix 2.

## 3.2. Drivers of robustness

The robustness (proportion of hazards for which there is at least one viable control) of mixed herds is promoted by a large herd size (p-value

<10–3, symbolized by ‘***’ below). The lower the economic constraint, the wider the range of viable herd compositions (Fig. 4). Note too that the economic constraint is expressed asymmetrically between the two species. Below €70 k, mixing does not play a significant role in herd robustness (Fig. 5). However, above €90 k, herd mixing is a negative factor for robustness: robustness is conferred by a predominance of cattle in the herd (***). Small herds or herds dominated by sheep become less robust to hazards at higher economic constraint.

Composition of the different physiological types in each herd plays an important role: the presence of adults in the cattle herd (i.e. dairy cows) is a factor of robustness regardless of the economic constraint (***). Similarly, for sheep, the most robust flock configurations have a higher proportion of adults (i.e. ewes) up to constraint values of €130 k (beyond which proportion of adults is not a significant driver of robustness).

## 3.3. Drivers of adaptability

The effect of herd size on the adaptability property (proportion of controls that are viable for at least one hazard) depends on the expected level of economic constraint (Fig. 6a). Herd size is a factor of adaptability in situations where the economic constraint is greater than €90 k: the most adaptable herds have a larger number of animals than all viable herds (***; Fig. 7a). Conversely, in situations of low economic requirements (C* ≤ €70 k), the most adaptive herds are smaller than viable herds (***).

Mixed breeding is a strong factor for adaptability in situations with low economic constraint (Fig. 6a.α; Fig. 7a). In fact, the highest adaptive value for monospecific herds is 90 % (given the 10 % discretization step, the only non-viable control is the one that leads to culling all animals). Monospecific (or quasi-monospecific) herds are thus limited by the adaptability of the dominant species. On the other hand, when the two species are co-dominant, each of the two species is able to respect the minimum economic constraint (Fig. 6a.α). This means that almost 100 % of the controls are viable for at least 1 hazard (the only non-viable control is the one that leads to culling all animals from each of the two species). The value of mixed herds (synergy between the two
<figure><img src="image_6.png" /><figcaption>Fig. 4. Robustness at the 3 levels of economic constraints.</figcaption></figure>

<table><caption>Fig. 5. Drivers of robustness according to economic constraints. “+” (resp. “-”) means that the herd structure indicator is a driver (respectively a barrier) of robustness. “NS” indicates no statistically significant impact on robustness.</caption><thead><tr><th>Herd structure indicator</th><th>Impact on robustness according to economic constraint<br>C* ε [15, 30, 50, 70, 90, 110, 130, 150, 175, 200]</th></tr></thead><tbody><tr><td>Herd size</td><td><div style="width: 100px; height: 10px; background-color: #90ee90; margin: 0 auto;"></div></td></tr><tr><td>Herd mix (majority species = max ⅜ of the herd)</td><td><div style="width: 100px; height: 10px; background-color: #ffa500; margin: 0 auto;"></div>NS</td></tr><tr><td>Proportion of adults in the cattle herd</td><td><div style="width: 100px; height: 10px; background-color: #90ee90; margin: 0 auto;"></div></td></tr><tr><td>Proportion of adults in the sheep herd</td><td><div style="width: 100px; height: 10px; background-color: #90ee90; margin: 0 auto;"></div>NS</td></tr></tbody></table>

species) is maintained up to an economic constraint of €90 k. Beyond this level, the most adaptable combinations are found in two situations. The first situation concerns herds of more than 150 LU specializing in sheep for meat (with a high proportion of ewe lambs). The advantage of these herds disappears above an economic constraint of €130 k (Fig. 6a.β). The second combination corresponds to a more heterogeneous herd composition, consisting of a herd dominated by cattle (more than 70 %, ****) and with a minimum size that increases with higher economic constraint (120 LU for C* = 90 k - Fig. 6a.β, 150 LU for C* = 150 k; Fig. 6a.γ).

Analysis of the composition of the most adaptable mixed herds shows that the presence of adult cattle in the cattle herd is a factor of adaptability, whatever the value of the economic constraint (***). Conversely, for the sheep herd, it is the presence of ewe lambs that stands out as a factor of adaptability (whatever the level of economic constraint, **), probably due to the methodological choices made for management of the sheep herd (all terminal-crossed lambs born to ewe lambs are sold, with no renewal).

Robust controls (viable whatever the hazards, and whose robust adaptability measures frequency; Fig. 6b-7b) is a subset of viable controls. Herd size and the proportion of dairy cows in the dairy herd are drivers of robust adaptability whatever the economic constraint (***). Mixed herds (where the dominant herd represents at most 2/3 of the herd) are a barrier to robust adaptability, but there is a synergy between the two herds in low-economic-constraint situations (C* < €50 k) when average proportion of cattle is 70%-80% (Fig. 6b.α). Above this value, the best-performing configurations are composed almost exclusively of cattle (> 85% - Fig. 6b.β,γ). As a result, the proportion of adult sheep in

the flock is no longer a determining factor. For the highest value of the economic constraint (C* = €200 k), no control is robust. A corollary of these two results is that (i) sheep-dominated herds are highly adaptive but are also less robust to hazards than cattle herds (robustness and robust adaptability are low), and (ii) robust adaptability is mainly provided by the dairy cow herd.

# 4. Discussion

## 4.1. Results validity

The approach developed in this paper is of interest for assessing the behavior of mixed-species livestock farming systems subject to the uncertainties of the environment in which they evolve (Magne et al., 2024). The 3 metrics presented here can be used to assess crucial properties that are difficult to measure ‘in vivo’. Characterizing the drivers of these properties using routine field variables (herd size, mix, renewal rate approximated by proportion of adults) can be used to design agroecological farming systems (Meynard et al., 2023). The use of biotechnical data from long term experiments to calibrate the model (in particular the variability of technical performances according to environmental conditions) may give credibility to the adoption of modelling among livestock farmers and stakeholders in agricultural development.

In parallel to this work, we studied the structure of French farms with mixed herds (i.e. with at least 1 dairy cow and 1 suckler ewe) and dominated by grassland (i.e. with at least 85 % grass in the utilized agricultural area). A summary of this work, carried out on more than 1500 farms through the 2020 French agricultural census, is presented in Appendix 3. This work shows that (i) ¾ of these farms have less than 110 LU, (ii) these herds are relatively specialized in the sense that the dairy herd represents more than 2/3 of the livestock in more than 3 out of 4 farms. Although little information is available on farming practices, particularly for cattle (once-a-day milking, rearing method for heifers), these observations are consistent with our results: the dairy herd is an important factor of stability in mixed ruminant farming systems. In the type of system modelled (grassland based), the nature of cattle production (mainly milk) generates more capital than sheep production for an equivalent number of animals. The breakdown of costs is also different: the nature of the technical management of sheep (full open-air) requires high labor costs but low structural costs. Conversely, for an equivalent number of animals, beef herds require more structural costs (livestock and milking buildings) and less labor than sheep herds. Working assumptions about herd costs, which include labor costs but not structural costs (see 2.3.1), may explain our results on viability of herds mainly composed of cattle. More broadly, results are in line with observations from regions exposed to severe environmental constraints (Manoli et al., 2014; Sabatier et al., 2017). The longer cycle of dairy
a. Adaptability

<figure><img src="image_7.png" /></figure>

b. Robust adaptability

<figure><img src="image_8.png" /><figcaption>Fig. 6. (a) Adaptability and (b) robust adaptability at the 3 levels of economic constraints.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Drivers of (a) adaptability and (b) robust adaptability according to economic constraint. “+” (resp. “-”) means that the herd structure indicator is a driver (respectively a barrier). “NS” indicates no statistically significant impact on robustness. There is no robust control for C* = €200 k, and so no statistics can be calculated.</figcaption></figure>

cows and the nature of production (milk and meat) may be better able to buffer environmental fluctuations, although the inertia of production (first calving at 2 years or more) makes it more difficult to adapt to market and weather variations in the short term.

Furthermore, although the proportion of adult animals in the cattle herd varies on these farms, more than 75 % of the sheep herds are made up of ewes. The omnipresence of ewes observed in mixed herds nuances

some of our results on the robustness and adaptability of sheep herds composed mainly of ewe lambs. In fact, the change in status of ewe lambs after lambing makes the long-term management of herds composed mainly of ewe lambs of little practical relevance, particularly in the modelled system here where all lambs born to ewe lambs are sold (male and female).
## 4.2. A model to be completed in terms of animal physiology

Agriculture, and particularly forms of agriculture based on agroecological principles, relies on biological processes that are highly sensitive to biotic and abiotic perturbations associated with the environment in which they evolve. Accounting for these is becoming essential to assess farming systems in the face of future changes in the agricultural environment (Lee et al., 2023). Viability theory and related tools (Sabatier et al., 2017) are of particular interest in this perspective, but they require simplification to make them processable or even operational. The biological processes and mechanisms at work in agroecological systems are often complex and poorly documented (or at levels of organization that are difficult to reconcile with approaches linked to the practices and management methods used). As a result, modelling them requires compromises and simplifications in the description of how they work. For example, the model does not take into account fine processes involved in feeding animals and their relationship with farming practices (co-grazing) and technical performances (milk production, growth, health). In fact, co-grazing by cattle and sheep is of particular interest as there is little overlap between cattle and sheep in terms of feeding, provided there is sufficient forage available (Walker, 1994). Nozieres et al. (2011) showed that multispecies herd can use resources more efficiently than a monospecific herd. Likewise, we did not model the finer processes of parasite dilution during co-grazing (Joly et al., 2023; Martin et al., 2020), which are closely linked to grazing practices. However, as these practices (and their diversity) are not the focus of this work, we only considered proxies based on actual observations of the experimental farm's operations, which we used to calibrate this model. The benefits of mixed herds therefore go beyond the results obtained with this model: a more detailed consideration of the advantages and disadvantages of mixed livestock farming and the associated grazing practices would provide a deeper understanding of the benefits of species mixing (Marley et al., 2006).

Moreover, in our model, we propose possible cumulative effects of seasonal impacts on animal performances. Our work incorporates effects of interaction between contrasting seasons within the same farming cycle (although a binary approach of dry/wet seasons). However, multi-year mechanisms that could affect biotechnical performance were not integrated into the model. These mechanisms particularly affect long-cycle species modelled in this article (cattle) and may result from farming-system choices (such as genetic selection) or environmental conditions (drought, high temperatures; Lee et al., 2023). Indeed, we assumed that the lactation of non-pregnant dairy cows at the end of the breeding season was extended to allow them to be bred the following year. Prolonged lactations can impact both the amount of milk produced per day of lactation and the reproductive performance of dairy cows (Sehested et al., 2019; Van Loon et al., 2023). Similarly, health issues or nutritional deficiencies (and associated consequences particularly on reproduction) can affect an animal's entire career (Wathes et al., 2014). Reproduction is a key factor in these self-sufficient systems, where environmental conditions (feed quality, temperature) are crucial to success. Extending lactations or catching up on reproduction are technical levers that we were unable to model due to computational time constraints for numerical resolution.

## 4.3. The farm as a space for trade-offs in resource use

We chose to focus on the demographic dynamics of mixed herds, on the assumption that forage was not a limiting resource (that was not a constraint on herd size). However, forages are one of the main limiting resources in herbivorous ruminants systems, particularly when these systems are pasture-based and aim for self-sufficiency. Given the forage area of the farm in this study, the stocking rate could reach up to 1.4 LU.ha⁻¹ for a herd of 230 LU. While this stocking rate is not impossible (10 % of the mixed-livestock farms presented in Appendix 3 have a stocking rate greater than 1.45 LU.ha⁻¹), it must be interpreted in relation to the

environment, and thus question trade-offs between herd size, production output, and forage self-sufficiency. Note that the median stocking rate for pasture-based mixed-farming systems in France is 0.75 LU.ha⁻¹.

Analyzing trade-offs between herd size and forage production requires efforts to explore other productive dimensions of the farms (forges, annual crops, and so on), as well as regulating mechanisms in response to environmental uncertainties (cash flow, forage stocks, etc.) and the associated agricultural practices (typically trade-offs in resource allocation). Viability theory provides a pointful framework for studying the dynamic properties of systems. However, technical difficulties in implementing the variability theory framework for diversified systems (exponential increase in calculation time; Bellman, 1961) require specific methodological developments. One way of solving this problem is to explore the modularity of diversified farming systems. De Lapparent et al. (2024) have developed an approach combining communicating modules at the same level of organization. In this article, we explore a first step in a scale nesting approach between the “herd” and “farm” levels of organization. We plan to explore a multi-scale approach (Sabatier et al., 2015) where the mixed herd studied in this paper represents one of many other components (grasslands, forage stocks...) of diversified systems. The future directions of this work aim to extend the emerging properties of mixed herds (robustness and adaptability) to other properties of interest for agricultural systems (productivity, efficiency, resilience) at the farm level. This effort will also help identify the regulatory levers between livestock and forage systems (Martin and Magne, 2015; Nozieres et al., 2011) as well as relationships or trade-offs between emerging properties.

## 4.4. Mixity: An interesting model for small farms?

Western European agricultural systems face a number of challenges, ranging from price volatility in agricultural products to high levels of debt and low levels of attractiveness. These difficulties are particularly evident in livestock systems: in France, there has been 10 % to 20 % decline in the number of farms with livestock (depending on the type of species raised; AGRESTE, 2022). In this article, we chose to explore different values for the economic constraint. This constraint represents a proxy for the level of investment in farms: the money generated by the herd explored in this work can be used alongside other farm-level resources (annual crops, secondary activities, public subsidies) to meet farms' needs (equipment, buildings, land, energy, and so on). By including the cost of labor in livestock production costs, we implicitly assume that the remuneration of livestock production labor cannot be considered as a compromise with structural costs when studying the viability of livestock production systems. A corollary of our results is that breeding systems with high structural loads have few alternative options to specialization. Structural costs and especially levels of investment reflect the farms' development and underlying production strategies that De Roest et al. (2018) simplify into two archetypes. The first strategy focuses on the intensification and expansion of livestock systems in a globalized economy. It relies on economies of scale, high levels of investment, and typically production specialization, particularly in livestock. The results of our work show that in situations of high investment, strategies involving large herds and low species diversity are the most robust and adaptable to meteorological and economic hazards. To some extent, our results confirm (i) that mixed breeding is poorly suited to this type of strategy (for which the forage system is generally less pasture-based than the one modelled) and (ii) that these systems engage in the ‘get big or get out’ vicious cycle of expansion and specialization described by Bouttes et al. (2019), reinforced by decades of public policies in Europe (Gorton et al., 2009). A second strategy relies on economies of scope achieved through production diversification, especially in livestock. The results presented in this paper highlight the specific benefits of mixed herds when economic constraints (and, by extension, investment levels) do not exceed €70 k–90 k. Our findings point to the existence of a virtuous cycle (in terms of adaptability and
robustness metrics) for mixed herds conditioned by low economic con-
straints (regardless of labor remuneration, which is integrated into our
model). Such situations might be found in modest-sized farms such as
those described in Appendix 3. As a result, this work can contribute to
research and policy support for agriculture. While some ‘crisis’ agri-
cultural policies (and associated subsidies) provide temporary and
necessary support to farms (cash-flow support, Emergency fund to
support organic farms in difficulty announced by the French government
for 2024), the anticipated global changes over the coming decades raise
questions about the long-term relevance of this type of ‘insurance’ policy
(Dumont et al., 2022). This work thus refocuses debate around the na-
ture of public policies that could guide farms toward mixed models,
which are structurally more resilient to climatic and economic shocks
and also more environment-friendly (biodiversity, landscapes, water
quality, etc.) due to the pasture-based systems they rely on.

5. Conclusion

In this article, we analyzed two emerging properties, adaptability and robustness, that are tightly related to the capacity of agricultural systems to cope with uncertainty, and the structural drivers of these properties. We applied this work to mixed ruminant herds in pasture-based systems. We showed that expected economic performances are a key factor in shaping viable herd configurations, and that the economic constraint is expressed asymmetrically between suckler sheep and dairy cattle herds. The drivers of adaptability and robustness play contrasting roles depending on the properties observed and the expected economic performances. Herd size is a driver of robustness whatever the economic constraints and a driver of adaptability only in situations with high economic demands (but a barrier to adaptability in situations of lower economic requirements). The mixed-livestock herd is only advantageous for herd adaptability with low economic requirements. A high proportion of adult cows in the cattle herd benefits all properties, whereas the proportion of ewes in sheep herd plays a role that diverges between robustness and adaptability (mainly due to management choices). Viability theory is a valuable tool for studying the dynamics of agricultural systems managed in uncertain environments. This work opens up new perspectives for resituating the farm level as a space where properties of interest for agricultural systems (productivity, efficiency, resilience) can emerge, and a space for trade-offs in resource allocation (food crops, fodder, herd), and even trade-offs between these properties.

CRediT authorship contribution statement

T. Puech: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Project administration, Methodology, Funding acquisition, Formal analysis, Data curation, Conceptualization. F. Stark: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Methodology, Funding acquisition, Formal analysis, Conceptualization. R. Sabatier: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Funding acquisition, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

Acknowledgments

The authors thank the INRAE's ACT department for funding this
research project. We also thank the ASTER technical team for their help
and support with data acquisition and calibrating the technical part of
the mixed farming model (S. Ditsch, R. Lavé, F. Laurent, E. Fontana, T.

Hance, A. Durpoix, and L. Brunet). Authors also thanks Metaform Lan-
gues to English translation.

**Appendix A. Supplementary data**

Supplementary data to this article can be found online at https://doi.
org/10.1016/j.agsy.2025.104371.

Data availability

The source code is available in the INRAE institutional archives
(Thomas Puech. Viability algorithm to assess the robustness and
adaptability of mixed herds. 2024. https://hal.inrae.fr/hal-04731209).

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