The Complexity–Stability Debate, Chemical Organization Theory, and the Identification of Non-classical Structures in Ecology

  • Tomas VelozEmail author


We present a novel approach to represent ecological systems using reaction networks, and show how a particular framework called chemical organization theory (COT) sheds new light on the longstanding complexity–stability debate. Namely, COT provides a novel conceptual landscape plenty of analytic tools to explore the interplay between structure and stability of ecological systems. Given a large set of species and their interactions, COT identifies, in a computationally feasible way, each and every sub-collection of species that is closed and self-maintaining. These sub-collections, called organizations, correspond to the groups of species that can survive together (co-exist) in the long-term. Thus, the set of organizations contains all the stable regimes that can possibly happen in the dynamics of the ecological system. From here, we propose to conceive the notion of stability from the properties of the organizations, and thus apply the vast knowledge on the stability of reaction networks to the complexity–stability debate. As an example of the potential of COT to introduce new mathematical tools, we show that the set of organizations can be equipped with suitable joint and meet operators, and that for certain ecological systems the organizational structure is a non-boolean lattice, providing in this way an unexpected connection between logico-algebraic structures, popular in the foundations of quantum theory, and ecology.


Ecological modeling Complexity stability debate Reaction networks Chemical organization theory Non-boolean lattice 



This work was supported by the postdoctoral Project Fondecyt 3170122.


  1. Aerts, D. (2002). Being and change: Foundations of a realistic operational formalism. In Probing the structure of quantum mechanics (pp. 71–110). World Scientific.Google Scholar
  2. Aerts, D., Broekaert, J., Czachor, M., Kuna, M., Sinervo, B., & Sozzo, S. (2014). Quantum structure in competing lizard communities. Ecological Modelling, 281, 38–51.CrossRefGoogle Scholar
  3. Aerts, D., Czachor, M., & Sozzo, S. (2010). A contextual quantum-based formalism for population dynamics. In 2010 AAAI fall symposium series.Google Scholar
  4. Beltrametti, E. G., & Maczyński, M. J. (1995). On the range of non-classical probability. Reports on Mathematical Physics, 36(2–3), 195–213.CrossRefGoogle Scholar
  5. Birkhoff, G., & Von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37(4), 823–843. Scholar
  6. Centler, F., & Dittrich, P. (2007). Chemical organizations in atmospheric photochemistries—A new method to analyze chemical reaction networks. Planetary and Space Science, 55(4), 413–428.CrossRefGoogle Scholar
  7. Centler, F., Speroni di Fenizio, P., Matsumaru, N., & Dittrich, P. (2007) Chemical organizations in the central sugar metabolism of Escherichia coli. In Mathematical modeling of biological systems (Vol. I, pp. 105–119). Boston: BirkhäuserGoogle Scholar
  8. Chen, Y., & Zhou, Z. (2003). Stable periodic solution of a discrete periodic LotkaVolterra competition system. Journal of Mathematical Analysis and Applications, 277(1), 358–366.CrossRefGoogle Scholar
  9. Contreras, D., Pereira, U., Hernández, V., Reynaert, B., & Letelier, J. C. (2011). A loop conjecture for metabolic closure. Advances in artificial life, ECAL 2011: Proceedings of the 11th European conference on the synthesis and simulation of living systems. MIT PressGoogle Scholar
  10. Dittrich, P., & Speroni Di Fenizio, P. (2007). Chemical organisation theory. Bulletin of Mathematical Biology, 69(4), 1199–1231.CrossRefGoogle Scholar
  11. Dittrich, P., & Winter, L. (2005). Reaction networks as a formal mechanism to explain social phenomena. In Proceedings of 4th international workshop on agent-based approaches in economics and social complex systems (AESCS 2005) (pp. 9–13).Google Scholar
  12. Dittrich, P., & Winter, L. (2008). Chemical organizations in a toy model of the political system. Advances in Complex Systems, 11(04), 609–627.CrossRefGoogle Scholar
  13. Donohue, I., Hillebrand, H., Montoya, J. M., Petchey, O. L., Pimm, S. L., Fowler, M. S., et al. (2016). Navigating the complexity of ecological stability. Ecology Letters, 19(9), 1172–1185.CrossRefGoogle Scholar
  14. Dunne, J., Williams, R., & Martinez, N. (2002). Network structure and biodiversity loss in food webs: Robustness increases with connectance. Ecology Letters, 5(4), 558–567.CrossRefGoogle Scholar
  15. Fellermann, H., & Cardelli, L. (2014). Programming chemistry in DNA-addressable bioreactors. Journal of the Royal Society Interface, 11(99), 20130987.CrossRefGoogle Scholar
  16. Finke, D. L., & Denno, R. F. (2004). Predator diversity dampens trophic cascades. Nature, 429(6990), 407–410.CrossRefGoogle Scholar
  17. Florian, C., Kaleta, C., Speroni di Fenizio, P., & Dittrich, P. (2008). Computing chemical organizations in biological networks. Bioinformatics, 24(14), 1611–1618.CrossRefGoogle Scholar
  18. Fontaine, C., Guimarães, P. R., Kéfi, S., Loeuille, N., Memmott, J., Van Der Putten, W. H., et al. (2011). The ecological and evolutionary implications of merging different types of networks. Ecology Letters, 14(11), 1170–1181.CrossRefGoogle Scholar
  19. GarcíaCallejas, D., MolownyHoras, R., & Araújo, M. B. (2018). Multiple interactions networks: Towards more realistic descriptions of the web of life. Oikos, 127(1), 5–22.CrossRefGoogle Scholar
  20. Garg, V. K. (2015). Introduction to lattice theory with computer science applications. Hoboken: Wiley.CrossRefGoogle Scholar
  21. Gil, B., Centler, F., Dittrich, P., Flamm, C., Stadler, B., & Stadler, P. (2009). A topological approach to chemical organizations. Artificial Life, 15(1), 71–88. ((2010): 853-856).CrossRefGoogle Scholar
  22. Grimm, V., Revilla, E., Berger, U., Jeltsch, F., Mooij, W. M., Railsback, S. F., et al. (2005). Pattern-oriented modeling of agent-based complex systems: Lessons from ecology. Science, 310(5750), 987–991.CrossRefGoogle Scholar
  23. Harley, J. (1959). The biology of mycorrhiza. London: Leonard Hill.Google Scholar
  24. Heiner, M., Gilbert, D., & Donaldson, R. (2008). Petri nets for systems and synthetic biology. In International school on formal methods for the design of computer, communication and software systems (pp. 215–264). Berlin: Springer.Google Scholar
  25. Heylighen, F., Beigi, S., & Veloz, T. (2015). Chemical organization theory as a modeling framework for self-organization, autopoiesis and resilience. International Journal Of General Systems (submitted).Google Scholar
  26. Hordijk, W., Hein, J., & Steel, M. (2010). Autocatalytic sets and the origin of life. Entropy, 12(7), 1733–1742.CrossRefGoogle Scholar
  27. Hordijk, W., Steel, M., & Dittrich, P. (2018). Autocatalytic sets and chemical organizations: Modeling self-sustaining reaction networks at the origin of life. New Journal of Physics, 20(1), 015011.CrossRefGoogle Scholar
  28. Hucka, M., Finney, A., Sauro, H. M., Bolouri, H., Doyle, J. C., Kitano, H., et al. (2003). The systems biology markup language (SBML): A medium for representation and exchange of biochemical network models. Bioinformatics, 19(4), 524–531.CrossRefGoogle Scholar
  29. Janssen, M., & Ostrom, E. (2006). Empirically based, agent-based models. Ecology and Society, 11(2), 1–13.Google Scholar
  30. Jopp, F., Breckling, B., & Reuter, H. (Eds.). (2010). Modelling complex ecological dynamics. Berlin: Springer.Google Scholar
  31. Kaleta, C., Centler, F., & Dittrich, P. (2006). Analyzing molecular reaction networks. Molecular Biotechnology, 34(2), 117–123.CrossRefGoogle Scholar
  32. Kaleta, C., Centler, F., Speroni di Fenizio, P., & Dittrich, P. (2008). Phenotype prediction in regulated metabolic networks. BMC Systems Biology, 2(1), 37.CrossRefGoogle Scholar
  33. Kaleta, C., Richter, S., & Dittrich, P. (2009). Using chemical organization theory for model checking. Bioinformatics, 25(15), 1915–1922.CrossRefGoogle Scholar
  34. Kondoh, M. (2003). Foraging adaptation and the relationship between food-web complexity and stability. Science, 299(5611), 1388–1391.CrossRefGoogle Scholar
  35. Kreyssig, P., Escuela, G., Reynaert, B., Veloz, T., Ibrahim, B., & Dittrich, P. (2012). Cycles and the qualitative evolution of chemical systems. PloS ONE, 7(10), e45772.CrossRefGoogle Scholar
  36. Kreyssig, P., Wozar, C., Peter, S., Veloz, T., Ibrahim, B., & Dittrich, P. (2014). Effects of small particle numbers on long-term behaviour in discrete biochemical systems. Bioinformatics, 30(17), i475–i481.CrossRefGoogle Scholar
  37. Lacroix, V., Cottret, L., Thébault, P., & Sagot, M. F. (2008). An introduction to metabolic networks and their structural analysis. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), 5(4), 594–617.CrossRefGoogle Scholar
  38. Landi, P., Minoarivelo, H. O., Brännström, Å., Hui, C., & Dieckmann, U. (2018). Complexity and stability of ecological networks: A review of the theory. Population Ecology, 60(4), 319–345.CrossRefGoogle Scholar
  39. Lurgi, M., Montoya, D., & Montoya, J. M. (2016). The effects of space and diversity of interaction types on the stability of complex ecological networks. Theoretical Ecology, 9(1), 3–13.CrossRefGoogle Scholar
  40. Mackey, G. W. (2013). Mathematical foundations of quantum mechanics. North Chelmsford: Courier Corporation.Google Scholar
  41. Matsumaru, N., Hinze, T., & Dittrich, P. (2011). Organization-oriented chemical programming of distributed artifacts. In Theoretical and technological advancements in nanotechnology and molecular computation: Interdisciplinary gains (pp. 240–258). IGI Global.Google Scholar
  42. Matsumaru, N., Speroni di Fenizio, P., Centler, F., & Dittrich, P. (2006) On the evolution of chemical organizations. In Proceedings of the 7th German workshop of artificial life (pp. 135–146).Google Scholar
  43. May, R. (1972). Will a large complex system be stable? Nature, 238, 413–414.CrossRefGoogle Scholar
  44. May, R. (1973). Stability and complexity in model ecosystems (Vol. 6). Princeton: Princeton University Press.Google Scholar
  45. McCann, K. (2000). The diversitystability debate. Nature, 405(6783), 228–233.CrossRefGoogle Scholar
  46. Melián, C. J., Bascompte, J., Jordano, P., & Krivan, V. (2009). Diversity in a complex ecological network with two interaction types. Oikos, 118(1), 122–130.CrossRefGoogle Scholar
  47. Melkikh, A. V., & Khrennikov, A. (2015). Nontrivial quantum and quantum-like effects in biosystems: Unsolved questions and paradoxes. Progress in Biophysics and Molecular Biology, 119(2), 137–161.CrossRefGoogle Scholar
  48. Montoya, J., Pimm, S., & Solé, R. (2006). Ecological networks and their fragility. Nature, 442(7100), 259–264.CrossRefGoogle Scholar
  49. Olff, H., Alonso, D., Matty, P., Berg, B., Eriksson, K., Loreau, M., et al. (2009). Parallel ecological networks in ecosystems. Philosophical Transactions of the Royal Society B: Biological Sciences, 364(1524), 1755–1779.CrossRefGoogle Scholar
  50. Orth, J. D., Thiele, I., & Palsson, B. Ø. (2010). What is flux balance analysis? Nature Biotechnology, 28(3), 245.CrossRefGoogle Scholar
  51. Peter, S., & Dittrich, P. (2011). On the relation between organizations and limit sets in chemical reaction systems. Advances in Complex Systems, 14(01), 77–96.CrossRefGoogle Scholar
  52. Peter, S., Veloz, T., & Dittrich, P. (2011). Feasibility of organizations—A refinement of chemical organization theory with application to p systems. In Membrane computing (pp. 325–337). Berlin: SpringerGoogle Scholar
  53. Pilosof, S., Porter, M. A., Pascual, M., & Kéfi, S. (2017). The multilayer nature of ecological networks. Nature Ecology & Evolution, 1(4), 0101.CrossRefGoogle Scholar
  54. Pimm, S. (1984). The complexity and stability of ecosystems. Nature, 307(5949), 321–326.CrossRefGoogle Scholar
  55. Pimm, S. L. (1982). Food webs. In Food webs (pp. 1–11). Dordrecht: Springer.Google Scholar
  56. Piron, C. (1976). On the foundations of quantum physics. In Quantum mechanics, determinism, causality, and particles (pp. 105–116). Dordrecht: Springer.Google Scholar
  57. Razeto-Barry, P. (2012). Autopoiesis 40 years later. A review and a reformulation. Origins of Life and Evolution of Biospheres, 42(6), 543–567.CrossRefGoogle Scholar
  58. Real, R., Barbosa, A. M., & Bull, J. W. (2016). Species distributions, quantum theory, and the enhancement of biodiversity measures. Systematic Biology, 66(3), 453–462.Google Scholar
  59. Schuster, S., Dandekar, T., & Fell, D. A. (1999). Detection of elementary flux modes in biochemical networks: A promising tool for pathway analysis and metabolic engineering. Trends in Biotechnology, 17(2), 53–60.CrossRefGoogle Scholar
  60. Speroni di Fenizio, P. (2015). The lattice of chemical organisations. In Artificial Life (pp. 242–248).Google Scholar
  61. Strogatz, S. (2014). Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. Boulder: Westview Press.Google Scholar
  62. Svozil, K. (2009). Quantum scholasticism: On quantum contexts, counterfactuals, and the absurdities of quantum omniscience. Information Sciences, 179(5), 535–541.CrossRefGoogle Scholar
  63. Thébault, E., & Fontaine, C. (2010). Stability of ecological communities and the architecture of mutualistic and trophic networks. Science, 329(5993), 853–856.CrossRefGoogle Scholar
  64. Varela, F., Maturana, H., & Uribe, R. (1974). Autopoiesis: The organization of living systems, its characterization and a model. Biosystems, 5(4), 187–196.CrossRefGoogle Scholar
  65. Velegol, D., Suhey, P., Connolly, J., Morrissey, N., & Cook, L. (2018). Chemical game theory. Industrial & Engineering Chemistry Research, 57(41), 13593–13607.CrossRefGoogle Scholar
  66. Veloz, T. (2010). A computational study of algebraic chemistry. M.Sc. thesis. University of ChileGoogle Scholar
  67. Veloz, T. (2013). Teoría de organizaciones qumicas: Un lenguaje formal para la autopoiesis y el medio ambiente. In P. Razeto-Barry & R. Ramos-Jiliberto (Eds.), Autopoiesis. Un concepto vivo. Editorial Nueva Civilizacin (pp. 229–245). Chile: Santiago.Google Scholar
  68. Veloz, T., Bassi, A., & Maldonado, P. (2019). A novel and efficient approach to compute closure in reaction networks. Soft Computing (submitted).Google Scholar
  69. Veloz, T., & Flores, D. (2019). A reaction network model of endosymbiotic interactions. Soft-Computing (submitted).Google Scholar
  70. Veloz, T., & Razeto-Barry, P. (2017a). Reaction networks as a language for systemic modeling: Fundamentals and examples. Systems, 5(1), 11.CrossRefGoogle Scholar
  71. Veloz, T., & Razeto-Barry, P. (2017b). Reaction networks as a language for systemic modeling: On the study of structural changes. Systems, 5(2), 30.CrossRefGoogle Scholar
  72. Veloz, T., Razeto-Barry, P., Dittrich, P., & Fajardo, A. (2014). Reaction networks and evolutionary game theory. Journal of Mathematical Biology, 68(1–2), 181–206.CrossRefGoogle Scholar
  73. Veloz, T., Reynaert, B., Rojas, D., & Dittrich, P. (2011) A decomposition theorem in chemical organizations. In Proceedings of European conference in artificial life. LNCS SpringerGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center Leo Apostel for Interdisciplinary StudiesVrije Universiteit BrusselBrusselsBelgium
  2. 2.Instituto de Filosofía y Ciencias de la Complejidad and Fundación DICTASantiagoChile
  3. 3.Departamento Ciencias Biológicas Facultad Ciencias de la VidaUniversidad Andres BelloSantiagoChile

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