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Making ecosystem models viable

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Abstract

Viability conditions permit to characterize all processes compatible with given constraints, notably of available food, so that there exists at least one possibility for the system to perpetuate itself forever. The concept of contingent cone to a set of constraints permits to identify two classes of corrections to apply to equations of natural growth. Usual basic models convey those corrections only in certain regions of the parameter space. A general model-building stemming from the constraints is presented. Experimental populations from historical case studies highlight the mathematical concept of viability corrections.

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References

  • Aubin, J.-P. (1991). Viability Theory, Boston: Birkhäuser.

    MATH  Google Scholar 

  • Aubin, J.-P. (1997). Dynamic Economic Theory: A Viability Approach, Boston, Basel: Springer, Birkhäuser.

    MATH  Google Scholar 

  • Aubin, J.-P. (2003). Regulation of the evolution of the architecture of a network by tensors operating on coalitions of actors. J. Evol. Econ. 13, 95–124.

    Article  Google Scholar 

  • Aubin, J.-P. and H. Frankowska (1990). Set-Valued Analysis, Boston, Basel: Birkhäuser.

    MATH  Google Scholar 

  • Auerbach, M. J. (1984). Stability, probability and the topology of food webs, in Ecological Communities, Conceptual Issues and the Evidence, D. R. Strong, Jr., D. Simberloff, L. G. Abele and A. B. Thistle (Eds), Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Boer, M. P., B. W. Kooi and S. A. L. M. Kooijman (2001). Multiple attractors and boundary crises in a tri-trophic food chain. Math. Biosci. 169, 109–128.

    Article  MathSciNet  MATH  Google Scholar 

  • Brown, J. S. and T. L. Vincent (1985). The evolutionary game. Theor. Popul. Biol. 31, 140–166.

    Article  MathSciNet  Google Scholar 

  • Brown, J. S. and T. L. Vincent (1987). Coevolution as an evolutionary game. Evolution 41, 66–79.

    Article  Google Scholar 

  • Caswell, H (1978). Predator-mediated co-existence: a nonequilibrium model. Am. Nat. 112, 127–154.

    Article  Google Scholar 

  • Drake, J. A. (1990). The mechanics of community assembly and succession. J. Theor. Biol. 147, 213–233.

    Google Scholar 

  • Gard, T. C. (1980). Persistence in food webs: Holling-type food chains. Math. Biosci. 49, 61–67.

    Article  MATH  MathSciNet  Google Scholar 

  • Gause, G. F. (1934). The Struggle for Existence, New York: Dover (reprinted 1971).

    Google Scholar 

  • Gause, G. F. (1935). Vérification expérimentale de la théorie mathématique de la lutte pour la vie. Actual. Sci. Ind. 277, Paris: Hermann éditeur, p. 62.

    Google Scholar 

  • Gomez, J. M. and R. Zamora (1994). Top-down effects in a tritrophic system: parasitoids enhance plant fitness. Ecology 75, 1023–1030.

    Article  Google Scholar 

  • Gutieriez, A. P., N. J. Mills and S. J. Schreiber Ellis (1994). A physiological based tritrophic perspective on bottom-up-top-down regulation of populations. Ecology 75, 2227–2242.

    Article  Google Scholar 

  • Hastings, A. (1988). Food web theory and stability. Ecology 69, 1665–1668.

    Article  Google Scholar 

  • Hawkins, B. A. (1992). Parasitoid-host food web and donor control. Oikos 65, 159–162.

    Google Scholar 

  • Hofbauer, J. and K. Sigmund (1988). The Theory of Evolution and Dynamical Systems, Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Holling, C. S. (1965). The functional response of predators to prey density and its role in mimicry and population regulation. Mam. Entomol. Soc. Can. 45, 5–60.

    Google Scholar 

  • Huffaker, C. B. (1958a). Experimental studies on predation: dispersion factors and predator-prey oscillations. Hilgardia 27, 343–383.

    Google Scholar 

  • Huffaker, C. B. (1958b). The concept of balance in nature. Proc. Xth Int. Congr. Ent. 2, 625–636.

    Google Scholar 

  • Krivan, V. (1991). Construction of population growth equations in the presence of viability constraints. J. Math. Biol. 29, 379–387.

    Article  MATH  MathSciNet  Google Scholar 

  • Krivan, V. (1995). Differential inclusions as a methodology tool in population biology, in Proceedings of the 1995 European Simulation Multiconference, M. Snorek, M. Sujansky and A. Verbraeck (Eds), San Diego: Computer Simulation International, pp. 544–547.

    Google Scholar 

  • Krivan, V. (1998). A Non-stochastic approach for modeling uncertainty in population dynamics. Bull. Math. Biol. 60, 721–751.

    Article  MATH  Google Scholar 

  • Lack, D. (1954). The Natural Regulation of Animal Numbers, Oxford: Clarendon Press.

    Google Scholar 

  • Law, R. and J. C. Blackford (1992). Self-assembling food webs: a global viewpoint of coexistence of species in Lotka-Volterra communities. Ecology 73, 567–578.

    Article  Google Scholar 

  • Law, R. and D. Morton (1996). Permanence and the assembly of ecological communities. Ecology 77, 762–775.

    Article  Google Scholar 

  • Lotka, A. J. (1934). Théorie analytique des associations biologiques, Actual. Sci. Ind. 187, Paris: Hermann.

    Google Scholar 

  • Luckinbill, L. S. (1973). Coexistence in laboratory populations of Paramecium aurelia and its predator Didinium nasutum. Ecology 54, 1320–1327.

    Article  Google Scholar 

  • Milne, A. (1957). The natural control of insect populations. Can. Ent. 89, 193–213.

    Article  Google Scholar 

  • Mitchell, P., W. Arthur and M. Farrow (1992). An investigation of population limitation using factorial experiments. J. Anim. Ecol. 61, 591–598.

    Google Scholar 

  • Morin, P. J. and S. P. Lawler (1995). Food web architecture and population dynamics: theory and empirical evidence. Ann. Rev. Ecol. Syst. 26, 505–529.

    Article  Google Scholar 

  • Muratori, S. and S. Rinaldi (1989). Remarks on competitive coexistence. SIAM J. Appl. Math. 49, 1462–1472.

    Article  MathSciNet  MATH  Google Scholar 

  • Nicholson, A. J. (1933). The balance of animal populations. J. Anim. Ecol. 2(Suppl.), 132–178.

    Google Scholar 

  • Pianka, E. R. (1978). Evolutionary Ecology, 2nd edn, New York: Harper and Row Publishers.

    Google Scholar 

  • Pimentel, D., W. P. Nagel and J. L. Madden (1963). Space-time structure of the environment and the survival of parasite-host systems. Am. Nat. 97, 141–167.

    Article  Google Scholar 

  • Pimentel, D. and F. A. Stone (1968). Evolution and population ecology of parasite-host system. Can. Ent. 100, 655–662.

    Article  Google Scholar 

  • Pimm, S. L. (1991). The balance of nature? Ecological Issues in the Conservation of Species and Communities, Chicago, IL: University of Chicago Press.

    Google Scholar 

  • Post, W. M. and S. L. Pimm (1983). Community assembly and food web stability. Math. Biosci. 64, 169–192.

    Article  MATH  Google Scholar 

  • Rockafellar, R. T. and R. Wets (1997). Variational Analysis, Berlin: Springer.

    Google Scholar 

  • Saint-Pierre, P. (1994). Approximation of the viability kernel. Appl. Math. Optim. 29, 187–209.

    Article  MATH  MathSciNet  Google Scholar 

  • Schoenly, K. and J. E. Cohen (1991). Temporal variation in food web structure: 16 empirical cases. Ecol. Monogr. 61, 267–298.

    Article  Google Scholar 

  • Schwerdtfeger, F. (1958). Is the density of animal populations regulated by mechanisms or by chance? Proc. 10th Int. Congr. Ent. 4, 115–122.

    Google Scholar 

  • Stiling, P. D. (1987). The frequency of density dependence in insect host-parasitoid systems. Ecology 68, 844–856.

    Article  Google Scholar 

  • Stiling, P. D. (1988). Density-dependent processes and key factors in insect populations. J. Anim. Ecol. 57, 581–593.

    Google Scholar 

  • Stiling, P. D., A. Throckmorton, J. Silvanima and D. R. Strong (1991). Does spatial scale affect the incidence of density dependence? A field test with insect parasitoids. Ecology 72, 2143–2154.

    Article  Google Scholar 

  • Thompson, W. R. (1956). The fundamental theory of natural and biological control. Ann. Rev. Ent. 1, 379–402.

    Article  Google Scholar 

  • Utida, S. (1957). Cyclic fluctuations of population density intrinsic to the host-parasite system. Ecology 38, 442–449.

    Article  Google Scholar 

  • Vincent, T. L. and J. S. Brown (1986). Evolution under nonequilibrium dynamics, Proceedings of the Fifth ICMM, In Math. Modelling in Science and Technology, pp. 29–31 and 766–771.

  • Vincent, T. L. and J. S. Brown (1989). The evolutionary response of systems to a changing environment. Appl. Math. Comput. 32, 185–206.

    Article  MathSciNet  MATH  Google Scholar 

  • Volterra, V. (1931). Leçons sur la théorie mathématique de la lutte pour la vie, M. Brélot (Ed.), Paris: Gauthiers-Villars (reprint Gabay, 1990).

    Google Scholar 

  • Williamson, M. (1987). Are communities ever stable? in Colonization, Succession and Stability, A. J. Gray, M. J. Crawley and P. J. Edwards (Eds), Oxford, England: Blackwell Scientific.

    Google Scholar 

  • Wilson, E. O. (1992). The Diversity of Life, Boston: Harvard University Press.

    Google Scholar 

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Authors and Affiliations

  1. Institut national des études démographiques, 133, Bld Davout, 75980, Paris Cedex 20, France

    Noël Bonneuil

  2. École des hautes études en sciences sociales, 54, Bld Raspail, 75006, Paris, France

    Noël Bonneuil

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  1. Noël Bonneuil
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Bonneuil, N. Making ecosystem models viable. Bull. Math. Biol. 65, 1081–1094 (2003). https://doi.org/10.1016/S0092-8240(03)00060-0

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  • DOI: https://doi.org/10.1016/S0092-8240(03)00060-0

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