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Coviability, Through the Lens of the Mathematical Theory of Viability

  • Jean-Pierre Aubin
  • Marie-Hélène Durand
Chapter

Abstract

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,

Keywords

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

Notes

Acknowledgements

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.

References

  1. Alvarez I, Martin S, Dordan O, Litrico X, Saint-Pierre P (2013) Indicateurs de sécurité et de restauration dynamiques. In: Ancey V, Avelange I, Dedieu B (eds) Agir en situation d’incertitude en agriculture, regards pluridisciplinaires au Nord et au Sud. Peter Lang, Bruxelles, pp 309–326Google Scholar
  2. Aubin J-P (1985) Motivated mathematics. SIAM News 18:1, 2 & 3Google Scholar
  3. Aubin J-P (1991) Viability theory. Birkhäuser, Boston/BerlinGoogle Scholar
  4. Aubin J-P (2000) Mutational and morphological analysis: tools for shape regulation and morphogenesis. Birkhäuser, BostonGoogle Scholar
  5. Aubin J-P (2010) La mort du devin, l’émergence du démiurge. Essai sur la contingence, la viabilité et l’inertie des systèmes. Éditions Beauchesne, ParisGoogle Scholar
  6. Aubin J-P, Bayen A, Saint-Pierre P (2011) Viability theory: new directions. Springer, Heidelberg/New YorkCrossRefGoogle Scholar
  7. Aubin J-P (2013a) Conjurer l’angoisse d’un futur inconnu. In: Bouamrane M, Antona M, Barbault R, Cormier M-C (eds) Rendre possible, Jacques Weber, itinéraire d’un économiste passe-frontières. Éditions Quae, Versailles, pp 157–166Google Scholar
  8. Aubin J-P (2013b) Time and money, how long and how much money is needed to regulate a viable economy. Lecture Notes in Economics and Mathematical Systems, 670, SpringerGoogle Scholar
  9. Aubin J-P, Catté F (2002) Bilateral fixed-point and algebraic properties of viability kernels and capture basins of sets. Set Valued Anal 10:379–416CrossRefGoogle Scholar
  10. Aubin J-P, Dordan O (2016) A survey on Galois stratifications and measures of viability risk. J Convex Anal 23:181–225Google Scholar
  11. Aubin J-P, Frankowska H (1990) Set-valued analysis. Birkhäuser, BostonGoogle Scholar
  12. Aubin J-P (to appear) La valeur n’existe pas. À moins que [...]. Essai sur le temps, l’argent et le hasardGoogle Scholar
  13. Aubin J-P, Lesne A (2006) Analyse morphologique et mutationnelle: des outils pour la morphogenèse. In Bourgine P, Lesne A (eds) Morphogenèse. Belin, pp 162–177Google Scholar
  14. Aubin J-P, Bayen A, Saint-Pierre P (2011) Viability theory, new directions. Springer, Heidelberg and New York, Springer-VerlagCrossRefGoogle Scholar
  15. Barbault R, Weber J (2010) La Vie, quelle entreprise! Pour une révolution écologique de l’économie. Edition Le Seuil, ParisGoogle Scholar
  16. Béné C, Doyen L, Gabay D (2001) A viability analysis for a bio-economic model. Ecol Econ 36:385–396CrossRefGoogle Scholar
  17. Bernard C, Martin S (2013) Comparing the sustainability of different action policy possibilities: application to the issue of both household survival and forest preservation in the corridor of Fianarantsoa. Math Biosci 245(2):322–330CrossRefGoogle Scholar
  18. Brahic E, Terreaux J-P (2009) Évaluation économique de la biodiversité. Edition Quae, ParisCrossRefGoogle Scholar
  19. Cury P, Mullon C, Garcia S, Shannon L (2005) Viability theory for an ecosystem approach to fisheries. ICES J Mar Sci 62(3):577–584CrossRefGoogle Scholar
  20. d’Holbach (1770) Système de la nature ou des lois du monde physique et du monde moral, Corpus des œuvres de philosophie. Fayard (réédition 1990), ParisGoogle Scholar
  21. De Lara M, Doyen L (2008) Sustainable management of natural resources, mathematical models and methods, Environmental science and engineering. Springer, Berlin/HeidelbergCrossRefGoogle Scholar
  22. De Lara M, Martinet V (2009) Multi-criteria dynamic decision under uncertainty: a stochastic viability analysis and an application to sustainable fishery management. Math Biosci 218(2):118–124CrossRefGoogle Scholar
  23. Domenech PA, Saint-Pierre P, Zaccour Z (2011) Forest conservation and CO2 emissions: A viable approach. Environ Model Assess 16:519–539CrossRefGoogle Scholar
  24. Dordan O (1995) Analyse qualitative. Masson, MilanGoogle Scholar
  25. Doyen L, De Lara M, Ferraris J, Pelletier D (2007) Sustainability of exploited marine ecosystems through protected areas: a viability model and a coral reef case study. Ecol Model 208:353–366CrossRefGoogle Scholar
  26. Doyen L, Cissé A, Gourguet S, Mouysset L, Hardy P-Y, Béné C, Blanchard F, Jiguet F, Pereau J-C, Thébaud O (2013) Ecological-economic modelling for the sustainable management of biodiversité. Comput Manag Sci 10:353–364CrossRefGoogle Scholar
  27. Durand M-H, Desilles A, Fronville A (2013) Incertitude contingente, adversité tychastique. In: Ancey V, Avelange I, Dedieu B (eds) Agir en situation d’incertitude en agriculture, regards pluridisciplinaires au Nord et au Sud. Peter Lang, Bruxelles, pp 297–308Google Scholar
  28. Durand M-H, Desilles A, Saint-Pierre P, Angeon V, Ozier-Lafontaine H (2017) A viability approach for agro-ecological transition, the example of soil preservation in French West Indies. Nat Resour Model 30(3):e12134CrossRefGoogle Scholar
  29. Gourguet S, Thébaud O, Jennings S, Little LR, Dichmont CM, Pascoe S, Deng RA, Doyen L (2015) The cost of co-viability in the Australian Northern Prawn Fishery. Environ Model Assess 21:1–19Google Scholar
  30. Griffon M, Griffon L (2010) L’homme viable. Edition Odile Jacob, ParisGoogle Scholar
  31. Griffon M, Griffon L (2011) Pour un monde viable: Changement global et viabilité planétaire. Edition Odile Jacob, ParisGoogle Scholar
  32. Hardy P-Y, Béné C, Doyen L, Schwarz A-M (2013) Food security versus environment conservation: a case study of Solomon Islands’ small-scale fisheries. Environ Develop 8:38–56CrossRefGoogle Scholar
  33. Hayek F (1978) Coping with ignorance, the Ludwig von Mises memorial lecture, Imprimis 7. Hillsdale College, HillsdaleGoogle Scholar
  34. Hayek FA (1993) La présomption fatale. Presses Universitaires de FranceGoogle Scholar
  35. Le Moigne J-L (1983) La Théorie du Système Général, Théorie de la Modélisation. PUF, ParisGoogle Scholar
  36. Longo G, Montévil M (2013) Biological time, symmetries and singularities. Springer, BerlinGoogle Scholar
  37. Lorenz T (2010) Mutational analysis, a joint framework for Cauchy problems in and beyond vector spaces, Lecture notes in mathematics, vol 1996. Springer, BerlinGoogle Scholar
  38. Martinet V, Thebaud O, Doyen L (2007) Defining viable recovery paths toward sustainable fisheries. Ecol Econ 64:411–422CrossRefGoogle Scholar
  39. Monod J (1971) Le hasard et la nécessité. Édition du Seuil, ParisGoogle Scholar
  40. Mouysset L, Doyen L, Jiguet F (2014) From population viability analysis to co-viability of farmland biodiversity and agriculture. Conserv Biol 28:187–201CrossRefGoogle Scholar
  41. Mullon C (2013) Network economics of marine ecosystems and their exploitation. CRC Press, Boca RatonCrossRefGoogle Scholar
  42. Mullon C, Cury P, Shannon L (2004) A viability model of trophic interactions in marine ecosystems. Nat Resour Model 17(1):71–102CrossRefGoogle Scholar
  43. Pavé A (2007) La nécessité du hasard: Vers une théorie synthétique de la biodiversité. EDP Sciences, Les UlisGoogle Scholar
  44. Peirce C-S (1893) Evolutionary love, The Monist, 3, 176–200. Repris dans Hartshorne and Weiss (eds) (1958) Collected papers of Charles Sanders Peirce, 6. Harvard University Press, Cambridge, MAGoogle Scholar
  45. Pereau JC, Doyen L, Little R, Thebaud O (2012) The triple bottom line: meeting ecological, economic and social goals with individual transferable quotas. J Environ Econ Manag 63(3):419–434CrossRefGoogle Scholar
  46. Rapaport A, Terreaux J-P, Doyen L (2006) Sustainable management of renewable resource: a viability approach. Math Comput Model 43:466–483CrossRefGoogle Scholar
  47. Regnier E, De Lara M (2015) Robust viable analysis of a harvested ecosystem model. Environ Model Assess 20(6):687–698CrossRefGoogle Scholar
  48. Rojey P (2013) Entre utopie et principe de réalité. L’Harmattan, ParisGoogle Scholar
  49. Sabatier R, Doyen L, Tichit M (2010) Modelling trade-offs between livestock grazing and water conservation in a grassland agroecosystem. Ecol Model 221:1292–1300CrossRefGoogle Scholar
  50. Sabatier R, Oates LG, Jackson RD (2015) Management flexibility of a grassland agroecosystem: a modeling approach based on viability theory. Agric Syst 139:76–81CrossRefGoogle Scholar
  51. Sueur C (2012) Viability of decision-making systems in human and animal groups. J Theor Biol 306:93–103CrossRefGoogle Scholar
  52. Taleb N (2008) Le Cygne noir: la puissance de l’imprévisible. Belles Lettres, ParisGoogle Scholar
  53. Terreaux J-P (2018) N′oublions pas le futur. Valeurs, justice et taux d′actualisation. Ethics and Econ 15(1):66–80Google Scholar
  54. Tichit M, Doyen L, Lemel JY, Renault O (2007) A co-viability model of grazing and bird community management in farmland. Ecol Model 206(3–4):277–293CrossRefGoogle Scholar
  55. Von Bertalanffy L (1968) Théorie générale des systèmes. Dunod, Paris, 2012Google Scholar
  56. Wei W, Alvarez I, Martin S (2013) Sustainability analysis: viability concepts to consider transient and asymptotical dynamics. Ecol Model 251:103–113CrossRefGoogle Scholar
  57. Weigel E (Erhardi VVeigelii) (1669) Idea matheseos universæ cum speciminibus inventionum mathematicarumGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Jean-Pierre Aubin
    • 1
  • Marie-Hélène Durand
    • 2
  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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