A Mathematical Approach to Agroecosystem Coviability

Studying the coviability of farming systems implicates three main challenges: (i) ensuring a minimum level of performance on the multiple functions of the system, (ii) ensuring the reproducibility of the system in time and (iii) ensuring adaptability to an uncertain context. The objective of this chapter is to show how the mathematical framework of viability theory is a relevant tool for handling these three issues so as to assess the coviability of farming systems. After briefly presenting the main concepts associated to viability theory, we illustrate (based on three examples), how this framework can be used to consider the three issues mentioned. A first example based on a bio-economic model developed at the national level (France) illustrates how viability theory makes it possible to consider, simultaneously, multiple dimensions of performance (in this case biodiversity and farmers’ income) without having to weight or rank these dimensions. A second example, based on a model linking bird dynamics to grassland management by grazers in a marshland area (Marais Poitevin, France), illustrates how the algorithms associated to viability theory make it possible to find the fine adjustment of grazing sequences, ensuring suitable habitat characteristics for the birds both in the short term and the long term. The last example, based on a model of multi-species livestock farming system (Bolivia), illustrates how management strategies robust to environmental uncertainty and that buffer environmental uncertainty can be found. We conclude on the advantages of viability theory in the search for new solutions in comparison with more classical simulation approaches based on a limited number of scenarios.