Mathematical Analysis of Some Resource-Prey-Predator Models: Application to a NPZ Microcosm Model

  • Thomas C. Gard
Part of the Lecture Notes in Biomathematics book series (LNBM, volume 52)


It has been proposed (Lassiter, 1978) that nutrient-phytoplankton-zooplankton (NPZ) submodels of ecosystem microcosm models exhibit the dynamics of a three-species food chain with Monod (Michaelis-Menton, Holling) functional response, except that the resource growth rate is a decreasing function of resource density. In this case, the basal prey species of the food chain is replaced by an abiotic resource which is supplied externally and is subject to density dependent dissipation as in a chemostat. Such a chemostat-chain, representing a sugar-bacteria-protozoa (SBP) system, has been studied qualitatively (local stability of equilibria) by Canale (1969,1970) and numerically by Jost et al. (1973).


Food Chain Positive Equilibrium Sustained Oscillation Abiotic Resource Resource Density 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andronov, A.A., E.A. Leontovich, I.I. Gordon, and A.G. Maier (1973): Qualitative Theory of Second Order Dynamical Systems, Wiley, New York.Google Scholar
  2. Bungay, H.R. III, and M.L. Bungay (1968): Microbial interactions in continuous cultures, Adv. AppZ. Microbiol. 10: 269–290.CrossRefGoogle Scholar
  3. Canale, R.P. (1969): Prey-predator relationships in a model for the activated process, Biotech. Bioengng. 11: 887–907.CrossRefGoogle Scholar
  4. Canale, R.P. (1970): An analysis of models describing predator-prey interaction, Biotech. Bioengng. 12: 353–378CrossRefGoogle Scholar
  5. Drake, J.F., J.L. Jost, A.G. Fredrickson, and H.M. Tsuchiya (1966): The food chain in bio-regenitive systems, NASA SP-165, Washington, D.C.Google Scholar
  6. Freedman, H.I. (1980): Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York.MATHGoogle Scholar
  7. Freedman, H.I., and P. Waltman (1977): Mathematical analysis of some three-species food-chain models, Math. Biosci. 33: 257–276.MathSciNetMATHCrossRefGoogle Scholar
  8. Gard, T.C. (1980): Persistence in food chains with general interactions, Math. Biosci. 51: 165–174.MathSciNetMATHCrossRefGoogle Scholar
  9. Gard, T.C., and T.G. Hallam (1979): Persistence in food webs: I. Lotka-Volterra food chains, Bull. Math. Biol. 41: 877–891.MathSciNetMATHGoogle Scholar
  10. Hale, J.K. (1969): Ordinary Differential Equations, Wiley-Interscience, New York.MATHGoogle Scholar
  11. Hallam, T.G. (1977): Controlled persistence in rudimentary plankton models, Proc. 1st. Int. Conf. Math. Modelling, St. Louis.Google Scholar
  12. Jost, J.L., J.F. Drake, H.M. Tsuchiya, and A.G. Fredrickson (1973): Microbial food chains and food webs, J. Theor. Biol. 41: 461–484.CrossRefGoogle Scholar
  13. Lassiter, R.R. (1978): Microcosms as ecosystems for testing ecological models, State-of-the-Art in Ecological Modelling, Vol. 7, Proc. 1st Int. Conf. Ecol. Modelling (ISEM) (S.E. Jorgensen, editor ), Copenhagen, pp. 127–161.Google Scholar
  14. Nemytskii, V.V. and V.V. Stepanov (1960): Qualitative Theory of Differential Equations, Princeton University Press, Princeton.MATHGoogle Scholar
  15. Saunders, P.T. and M.J. Bazin (1975): On the statbility of food chains, J. Theor. Biol. 52: 121–142.MathSciNetCrossRefGoogle Scholar
  16. Schoener, T.W. (1973): Population growth regulated by intraspecific competition for energy and time: Some simple representations, Theor. Pop. Biol. 4: 56–84.CrossRefGoogle Scholar
  17. Sell, G.R. (1977): What is a dynamical system? Studies in Ordinary Differential Equations (J. Hale, editor ), Math. Assoc. Amer.Google Scholar
  18. Tsuchiya, H.M., J.F. Drake, J.L. Jost, and A.G. Fredrickson (1972): Predator-prey interactions of dictyostelium-discoidium and escherichia-coli in continuous culture, J. Bact. 110: 1147–1153.Google Scholar
  19. Waltman, P.: Competition for a renewable resource, this Proceedings.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Thomas C. Gard

There are no affiliations available

Personalised recommendations