Abstract
In food chain models the lowest trophic level is often assumed to grow logistically. Anomalous behaviour of the solution of the logistic equation and problems with the introduction of mortality have recently been reported. As predation on the lowest trophic level is a kind of mortality, one expects problems with these food chain models. In this paper we compare two formulations for the lowest trophic level: the logistic growth formulation and the mass balance formulation with resources modelled explicitly. We examine the effects of both models on the dynamic behaviour of a tri-trophic microbial food chain in a chemostat. For this purpose bifurcation diagrams, which give the existence and stability of the equilibria of the nonlinear dynamic system, are used. It turns out that the dynamic behaviours differ in a rather large region of the control parameter space spanned by the dilution rate and the concentration of the resources in the reservoir. We urge that mass balance equations should be used in modelling food chains in chemostats as well as in ecosystems.
Similar content being viewed by others
References
Blanco, J. M. (1993). Relationship between the logistic equation and the Lotka-Volterra models. Ecological Modelling. 66, 301–303.
Cunningham, A. and R. M. Nisbet (1983). Transients and oscillations in continuous culture, in Mathematical Methods in Microbiology, M. J. Bazin (Ed.), London: Academic Press, pp. 77–103.
DeAngelis, D. L. (1992). Dynamics of Nutrient Cycling and Food Webs, London: Chapman and Hall.
Edelstein-Keshet, L. (1988). Mathematical Models in Biology, New York: Random House.
Ginzburg, L. R. (1992). Evolutionary consequences of basic growth equations. Trends in Ecology and Evolution. 7, 133.
Hallam, T. G. and C. E. Clark (1981). Non-autonomous logistic equations as models of population in a deteriorating environment. J. theor. Biol. 93, 303–311.
Hastings, A. and T. Powell (1991). Chaos in a three-species food chain. Ecology. 72, 896–903.
Hutchinson, G. E. (1978). An Introduction to Population Ecology, New Haven: Yale University Press.
Klebanoff, A. and A. Hastings (1994a). Chaos in one-predator, two-prey models: General results from bifurcation theory. Mathematical Biosciences. 122, 221–233.
Klebanoff, A. and A. Hastings (1994b). Chaos in three-species food chain. J. Math. Biol. 32, 427–451.
Kooi, B. W., M. P. Boer and S. A. L. M. Kooijman (1997). Mass balance equation versus logistic equation in food chains. J. Biol. Systems. 5, 77–85.
Kooi, B. W. and S. A. L. M. Kooijman (1994). Existence and stability of microbial prey-predator systems. J. theor. Biol. 170, 75–85.
Kooi, B. W. and S. A. L. M. Kooijman (1995). Many limiting behaviour in microbial food chains, in Mathematical Population Dynamics: Analysis of Heterogeneity, Arino O., Axelrod D. and Kimmel M. (Eds), Winnipeg, Canada: Wuerz, vol. 2, pp. 131–148.
Kuno, E. (1991). Some strange properties of the logistic equations defined with r and K: Inherent defects or artifacts? Researches on Population Ecology. 33, 33–39.
Kuznetsov, Y. A. and S. Rinaldi (1996). Remarks on food chain dynamics. Mathematical Biosciences. 124, 1–33.
May, R. M. (1976). Theoretical Ecology, Principles and Applications, Philadelphia: Saunders.
McCann, K. and P. Yodzis (1994). Biological conditions for chaos in a tree-species food chain. Ecology. 75, 561–564.
McCann, K. and P. Yodzis (1995). Bifurcation structure of a tree-species food chain model. Theor. Pop. Biol. 48, 93–125.
Nisbet, R. M., E. McCauley, A. M. de Roos, W. W. Murdoch and W. S. C. Gurney (1991). Population dynamics and element recycling in an aquatic plant-herbivore system. Theoretical Population Biology. 40, 125–147.
Pielou, E. C. (1977). Mathematical Ecology, New York: John Wiley.
Schoener, T. W. (1973). Population growth regulated by intraspecific competition for energy of time: Some simple representations. Theo. Pop. Biol. 4, 56–84.
Smith, H. L. and P. Waltman (1994). The Theory of the Chemostat. Cambridge: Cambridge University Press.
Waltman, P. (1983). Competition Models in Population Biology vol. 45, Philadelphia: Society for Industrial and Applied Mathematics.
Williams, F. M. (1967). A model of cell growth dynamics. J. theor. Biol. 15, 190–207.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kooi, B.W., Boer, M.P. & Kooijman, S.A.L.M. On the use of the logistic equation in models of food chains. Bull. Math. Biol. 60, 231–246 (1998). https://doi.org/10.1006/bulm.1997.0016
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1006/bulm.1997.0016