Coviability, Through the Lens of the Mathematical Theory of Viability



Viability and coviability are polysemous terms for which nobody can claim ownership. The (mathematical) co-evolution is defined here as “the joint evolution of a state and a given environment”. The first is described as a vector of a vector space, the second as a subset of this space, termed “environment”. Coviability means that whenever both state and environment evolve, the vector’s state always remains in the environment. The (mathematical) theory of viability studies both these evolutions on temporal windows, and proves whether or not evolutionary ‘engines’ provide coviable evolutions of both states and environments.

Mathematics is a logical process used to demonstrate that a set of hypotheses implies a set of conclusions. A theorem explains ‘how’ a conclusion answers the ‘why’ described by these hypotheses. At this stage, demonstrating a theorem is an intellectual activity and not a scientific one. It only becomes so when a mathematical metaphor of an assertion in a different field of knowledge is “validated”. This requires validation processes specific to these fields; physics requires experiments, other domains resort to historical validations or more laborious exercises of reflection.

This article describes concepts ‘motivated’ by different fields of life sciences and the ‘theorems’ that relate them. The article is concerned with “mathematical metaphors”, rather than their confirmation which is sometimes hard to justify. The mathematical results are mainly qualitative and different from those obtained with more usual tools motivated by inert matter’s sciences.

Since scientific concepts only make full sense within the confines of their origins, the history of this concept, motivated by environmental sciences since the 1970s, is broadly outlined,


Viability Coviability Environement Evolutionary systems Differential inclusion Mutational equations Tychastic uncertainty Contingent uncertainty Regulation maps Motivated mathematics 



The authors dedicate this work in memory of Jacques Weber (1946–2014), a faithful and enthusiastic participant of the seminar “viable development” and a witness of the historical development of coviability as a concept. He was a great source of inspiration and support.

This chapter was written during the course of the program ANR GAIA-TROP on the governance and viability of tropical agro-systems. Valérie Angeon, Samuel Bates, Anya Désilles, Jean-Louis Diman, Audrey Fanchone, Harry Ozier-Lafontaine, Sophie Martin and Patrick Saint-Pierre, are also among those involved in this reflexion on coviability.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.VIMADES, (Viabilité, Marchés, Automatique et Décision)ParisFrance
  2. 2.French Institute of Research for Sustainable Development (IRD), UMR GREDMontpellierFrance

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