Bulletin of Mathematical Biology

, Volume 60, Issue 4, pp 703–719 | Cite as

Food chains in the chemostat: Relationships between mean yield and complex dynamics

  • Alessandra Gragnani
  • Oscar De Feo
  • Sergio Rinaldi


Atritrophic food-chain chemostat model composed of a prey with Monod-type nutrient uptake, a Holling Type II predator and a Holling Type II exploited superpredator is considered in this paper. The bifurcations of the model show that dynamic complexity first increases and then decreases with the nutrient supplied to the bottom of the food chain. Extensive simulations prove that the same holds for food yield, i.e., there exists an optimum nutrient supply which maximizes mean food yield. Finally, a comparative analysis of the results points out that the optimum nutrient supply practically coincides with the nutrient supply separating chaotic dynamics from high-frequency cyclic dynamics. This reinforces the idea, already known for simpler models, that food yield maximization requires that the system behaves on the edge of chaos.


Food Chain Bifurcation Diagram Strange Attractor Bifurcation Curve Homoclinic Bifurcation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Abrams, P. A. (1993). Effect of increased productivity on the abundances of trophic levels. Amer. Nat. 141, 351–371.CrossRefGoogle Scholar
  2. Abrams, P. A. and J. D. Roth (1994a). The effects of enrichment of three-species food chains with non-linear functional responses. Ecology 75, 1118–1130.CrossRefGoogle Scholar
  3. Abrams, P. A. and J. D. Roth (1994b). The responses of unstable food chains to enrichment. Evol. Ecol. 8, 150–171.CrossRefGoogle Scholar
  4. Allen, J. C., W. M. Schaffer and D. Rosko (1993). Chaos reduces species extinction by amplifying local population noise. Nature 364, 229–232.CrossRefGoogle Scholar
  5. Bader, F. G., H. M. Tsuchiya and A. G. Fredrickson (1976). Grazing of ciliates on bluegreen algae: effects of ciliate encystment and related phenomena. Biotechnol. Bioeng. 18, 311–332.CrossRefGoogle Scholar
  6. Boer, M. P., B. W. Kooi and S.A. Kooijman (1998). Food chain dynamics in the chemostat. Math. Biosci. in press.Google Scholar
  7. Butler, G. J., S. B. Hsu and P. Waltman (1983). Coexistence of competing predators in a chemostat. J. Math. Biol. 17, 133–151.MathSciNetCrossRefGoogle Scholar
  8. Canale, R. P. (1969). Predator-prey relationships in a model for the activated process. Biotechnol. Bioeng. 11, 887–907.CrossRefGoogle Scholar
  9. Canale, R. P. (1970). An analysis of models describing predator-prey interaction. Biotechnol. Bioeng. 12, 353–378.CrossRefGoogle Scholar
  10. Canale, R. P., T. D. Lustig, P. M. Kehrberger and J. E. Salo (1973). Experimental and mathematical modeling studies of protozoan predation on bacteria. Biotechnol. Bioeng. 15, 707–728.CrossRefGoogle Scholar
  11. Champneys, A. R. and Yu. A. Kuznetsov (1994). Numerical detection and continuation of codimension-two homoclinic bifurcations. Int. J. Bif. Chaos 4, 785–822.MathSciNetCrossRefGoogle Scholar
  12. Cunningham, A. and R. M. Nisbet (1983). Transients and oscillations in continuos culture, in Mathematics in Microbiology, M. Bazin (Ed.), London: Academic Press, pp. 77–103.Google Scholar
  13. De Feo, O. and S. Rinaldi (1997). Yield and dynamics of tritrophic food chains. Amer. Nat. 150, 328–345.CrossRefGoogle Scholar
  14. De Feo, O. and S. Rinaldi (1998). Singular homoclinic bifurcations in tri-trophic food chains. Math. Biosci. in press.Google Scholar
  15. Doedel, E. and J. Kernévez (1986). AUTO: software for continuation problems in ordinary differential equations with applications. California Institute of Technology Technical Report, Applied Mathematics, CALTEC, Pasadena.Google Scholar
  16. Drake, J. F. and H. M. Tsuchiya (1976). Predation on Escherichia coli by Colpoda stenii. Appl. Environ. Microbiol. 31, 870–874.Google Scholar
  17. Ferrière, R. and M. Gatto (1993). Chaotic population dynamics can result from natural selection. Proc. Roy. Soc. Lond. B251, 33–38.Google Scholar
  18. Gilpin, M. E. (1972). Enriched predator-prey systems: theoretical stability. Science 177, 902–904.Google Scholar
  19. Gragnani, A. and S. Rinaldi (1995). A universal bifurcation diagram for seasonally perturbed predator-prey models. Bull. Math. Biol. 57, 701–712.Google Scholar
  20. Guckenheimer, J. and P. Holmes (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, New York: Springer.Google Scholar
  21. Hastings, A. and T. Powell (1991). Chaos in a three-species food chain. Ecology 72, 896–903.CrossRefGoogle Scholar
  22. Hogeweg, P. and B. Hesper (1978). Interactive instruction on population interaction. Comp. Biol. and Med. 8, 319–327.CrossRefGoogle Scholar
  23. Jost, J. L., J. F. Drake, A. G. Fredrickson and H. M. Tsuchiya (1973). Interactions of Tetrahymena pyriformis, Escherichia coli, Azotobacter vinelandii, and glucose in a minimal medium. J. Bacteriol. 113, 834–840.Google Scholar
  24. Kauffman, S. A. (1993). Origin of Order: Self Organization and Selection in Evolution. Oxford: Oxford University Press.Google Scholar
  25. Klebanoff, A. and A. Hastings (1994). Chaos in three-species food chains. J. Math. Biol. 32, 427–451.MathSciNetCrossRefGoogle Scholar
  26. Kooi, B. W., M. P. Boer and S. A. Kooijman (1997). Complex dynamic behaviour of autonomous microbial food chains. J. Math. Biol. 36, 24–40.MathSciNetCrossRefGoogle Scholar
  27. Kot, M., G. S. Sayler and T. W. Schultz (1992). Complex dynamics in a model microbial system. Bull. Math. Biol. 54, 619–648.CrossRefGoogle Scholar
  28. Kuznetsov, Yu. A. (1995). Elements of Applied Bifurcation Theory, New York: Springer.Google Scholar
  29. Kuznetsov, Yu. A., S. Muratori and S. Rinaldi (1992). Bifurcations and chaos in a periodic predator-prey model. Int. J. Bif. Chaos 2, 117–128.MathSciNetCrossRefGoogle Scholar
  30. Kuznetsov, Yu. A. and S. Rinaldi (1996). Remarks on food chain dynamics. Math. Biosci. 134, 1–33.MathSciNetCrossRefGoogle Scholar
  31. May, R. M. (1972). Limit cycles in predator-prey communities. Science 177, 900–902.Google Scholar
  32. McCann, K. and P. Yodzis (1994). Biological conditions for chaos in a three-species food chain. Ecology 75, 561–564.CrossRefGoogle Scholar
  33. McCann, K. and P. Yodzis (1995). Bifurcation structure of three-species food chain model. Theoret. Pop. Biol. 48, 93–125.CrossRefGoogle Scholar
  34. Muratori, S. and S. Rinaldi (1992). Low-and high-frequency oscillations in threedimensional food chain systems, SIAM J. Appl. Math. 52, 1688–1706.MathSciNetCrossRefGoogle Scholar
  35. Oksanen, L., S. D. Fretwell, J. Arruda and P. Niemela (1981). Exploitation ecosystems in gradients of primary productivity. Amer. Nat. 118, 240–261.CrossRefGoogle Scholar
  36. Pavlou, S. and I. G. Kevrekidis (1992). Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally imposed frequencies. Math. Biosci. 108, 1–55.MathSciNetCrossRefGoogle Scholar
  37. Rapp, P. E., I. D. Zimmerman, A. M. Albano, G. C. de Guzman and N. N. Greenbaum (1985). Dynamics of spontaneous neural activity in the simian motor cortex: the dimension of chaotic neurons. Phys. Lett. 110A, 335–338.Google Scholar
  38. Rinaldi, S., S. Muratori and Yu. A. Kuznetsov (1993). Multiple attractors, catastrophes and chaos in periodically forced predator-prey communities. Bull. Math. Biol. 55, 15–35.CrossRefGoogle Scholar
  39. Rosenzweig, M. L. (1971). Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385–387.Google Scholar
  40. Sarkar, A. K., D. Mitra, S. Ray and A. B. Roy (1991). Permanence and oscillatory co-existence of a detritus-based prey-predator model. Ecol. Model. 53, 147–156.CrossRefGoogle Scholar
  41. Scheffer, M. (1991). Should we expect strange attractors behind plankton dynamics and if so, should we bother? J. Plank. Res. 13, 1291–1305.Google Scholar
  42. Smith, H. L. and P. Waltman (1995). The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge: Cambridge University Press.Google Scholar
  43. Waltman, P. (1983). Competition Models in Population Biology, Philadelphia, PA: Society for Industrial and Applied Mathematics.Google Scholar
  44. West, B. J. and A. L. Goldberger (1987). Physiology in fractal dimensions. Amer. Sci. 75, 354–365.Google Scholar

Copyright information

© Society for Mathematical Biology 1998

Authors and Affiliations

  • Alessandra Gragnani
    • 1
  • Oscar De Feo
    • 2
  • Sergio Rinaldi
    • 3
  1. 1.Dipartimento di Elettronica e InformazionePolitecnico di MilanoMilanoItaly
  2. 2.Swiss Federal Institute of TechnologyEPFL DE-CIRCLausanneSwitzerland
  3. 3.Dipartimento di Elettronica e InformazionePolitecnico di MilanoMilanoItaly

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