Skip to main content
SpringerLink
Log in

Predator-mediated competition in the chemostat

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Aris, R., Humphrey, A. E.: Dynamics of a chemostat in which two organisms compete for a common substrate. Biotechnol. Bioeng. 19, 1375–1386 (1977)

    Google Scholar 

  2. Armstrong, R. A., McGehee, R.: Competitive exclusion. Am. Nat. 115, 151–170 (1980)

    Google Scholar 

  3. Bungay, H. R., Bungay, M. L.: Microbial interactions in continuous culture. Adv. Appl. Microbiol. 10, 269–312 (1968)

    Google Scholar 

  4. Butler, G. J., Hsu, S. B., Waltman, P.: Coexistence of competing predators in a chemostat. J. Math. Biology 17, 133–151 (1983)

    Google Scholar 

  5. Butler, G. J., Wolkowicz, G. S. K.: A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM J. Appl. Math. 45, 138–151 (1985)

    Google Scholar 

  6. Butler, G. J., Freedman, H. I., Waltman, P.: Uniformly persistent systems. Proc. Amer. Math. Soc. 96, 425–430 (1986)

    Google Scholar 

  7. Canale, R. P.: An analysis of models describing predator-prey interaction. Biotech. Bioeng. 17, 353–378 (1970)

    Google Scholar 

  8. Coppel, W. A.: Stability and asymptotic behaviour of differential equations. D. C. Heath, New York 1965

    Google Scholar 

  9. Cramer, N. F., May, R. M.: Interspecific competition, predation, and species diversity: a comment J. Theoret. Biol. 34, 289–293 (1972)

    Google Scholar 

  10. Dayton, P. K.: Competition, disturbance, and community organization: the provision and subsequent utilization of space in a rocky intertidal community. Ecol. Mongr. 41, 351–389 (1971)

    Google Scholar 

  11. Dayton, P. K., Robilliard, G. A., Paine, R. T., Dayton, L. B.: Biological accommodation in benthic community at McMurdo Sound, Antarctica. Ecol. Monogr. 44, 105–128 (1974)

    Google Scholar 

  12. Drake, J. F., Tsuchiya, H. M.: Predation of Escherichia coli by Colpoda stenil. Appl. Envir. Microbiol. 31, 870–874 (1976)

    Google Scholar 

  13. Freedman, H. I.: Deterministic mathematical models in population ecology. Marcel Dekker, Inc., New York 1980

    Google Scholar 

  14. Freedman, H. I., Waltman, P.: Mathematical analysis of some three-species food-chain models. Math. Biosci. 33, 257–276 (1977)

    Google Scholar 

  15. Freedman, H. I., Waltman, P.: Persistence in models of three interacting predator-prey populations. Math. Biosci. 68, 213–231 (1984)

    Google Scholar 

  16. Gard, T. C.: Persistence in food chains with general interactions. Math. Biosci. 51, 165–174 (1980)

    Google Scholar 

  17. Gard, T. C.: Mathematical analysis of some resource-prey-predator models: application to a NPZ microcosm model. In: Freedman, H. I., Strobeck, C. (eds.) Population biology proceedings. Edmonton 1982, Lecture Notes Biomath. 52, Springer, Berlin, Heidelberg, New York 1982

  18. Gilpin, M. E.: Spiral chaos in a predator-prey model. Am. Nat. 113, 306–308 (1978)

    Google Scholar 

  19. Hallam, T. G., Svoboda, L. J., Gard, T. C.: Persistence and extinction in three species Lotka-Volterra competitive systems. Math. Biosci. 46, 117–124 (1979)

    Google Scholar 

  20. Hansen, S. R., Hubbell, S. P.: Single-nutrient microbial competition: qualitative agreement between experimental and theoretical forecast outcomes. Science 213, 972–979 (1981)

    Google Scholar 

  21. Hsu, S. B., Hubbell, S., Waltman, P.: A mathematical theory of single-nutrient competition in continuous cultures for microorganisms. SIAM J. Appl. Math. 32, 366–383 (1977)

    Google Scholar 

  22. Hutson, V., Vickers, G. T.: A criterion for permanent coexistence of species, with an application to a two-prey one predator system. Math. Biosci. 63, 253–269 (1983)

    Google Scholar 

  23. Jost, J. L., Drake, J. F., Tsuchiya, H. M., Fredrickson, A. G.: Microbial food chains and food webs. J. Theor. Biol. 41, 461–484 (1973)

    Google Scholar 

  24. Jost, J. L., Drake, J. F., Fredrickson, A. G., Tsuchiya, H. M.: Interactions of Tetrahymena pyriformis, Escherichia coli, Azotobacter vinelandii, and glucose in a minimal medium. J. Bacteriol. 113, 834–840 (1973)

    Google Scholar 

  25. Kubitschek, H. E.: Introduction to research with continuous cultures. Prentice-Hall, New York 1970

    Google Scholar 

  26. Levin, S. A.: Community equilibria and stability, and an extension of the competitive exclusion principle. Am. Nat. 104, 413–423 (1970)

    Google Scholar 

  27. May, R. M.: Stability in multispecies community models. Math. Biosci. 12, 59–79 (1971)

    Google Scholar 

  28. M'Kendrick, A. G., Pai, M. K.: The rate of multiplication of microorganisms: a mathematical study. Proc. R. Soc. Edinb. 31, 649–655 (1910)

    Google Scholar 

  29. Monod, J.: Recherches sur la croissance des cultures bacteriennes. Herman et Cie., Paris 1942

    Google Scholar 

  30. Neill, W. E.: Experimental studies of microcrustacean competition, community composition and efficiency of resource utilization. Ecology 56, 809–826 (1975)

    Google Scholar 

  31. Novick, A., Szilard, L.: Description of the chemostat. Science 112, 715–716 (1950)

    Google Scholar 

  32. Paine, R. T.: Food web complexity and species diversity. Am. Nat. 100, 65–75 (1966)

    Google Scholar 

  33. Paine, R. T.: Intertidal community structure. Experimental studies on the relationship between a dominant competitor and its principal predator. Oecologia 15, 93–120 (1974)

    Google Scholar 

  34. Porter, J. W.: Predation by acenthaster and its effect on coral species diversity. Am. Nat. 106, 487–492 (1972)

    Google Scholar 

  35. Rosenzweig, M. L.: Paradox of enrichment. Destabilization of exploitation ecosystems in ecological time. Science 171 385–387 (1971)

    Google Scholar 

  36. Saunders, P. T., Bazin, M. J.: On the stability of food chains. J. Theor. Biol. 52, 121–142 (1975)

    Google Scholar 

  37. Sell, G.: What is a dynamical system? In: Hale, J. K (ed.) Studies in ordinary differential equations, MAA Studies in Mathematics 14, (1977)

  38. Slobodkin, L. B.: Ecological populations of Hydrida. J. Anim. Ecol. 33 (Suppl.) 131–148 (1964)

    Google Scholar 

  39. Smale, S.: On the differential equations of species competition. J. Math. Biol. 3, 5–7 (1976)

    Google Scholar 

  40. So, J. W. H.: A note on the global stability and bifurcation phenomenon of a Lotka-Volterra food chain. J. Theor. Biol. 80, 185–187 (1979)

    Google Scholar 

  41. Taylor, P. A., Williams, P. J. LeB.: Theoretical studies on the coexistence of competing species under continuous-flow conditions. Can. J. Microbiol. 21, 90–98 (1975)

    Google Scholar 

  42. Tsuchiya, H. M., Drake, S. F., Jost, J. L., Fredrickson, A. G.: Predator-prey interaction of Dictoyostelium discordeum and Escherichia coli in continuous culture. J. Bacteriol. 110, 1147–1153 (1972)

    Google Scholar 

  43. Vance, R. R.: Predation and resource partitioning in a one predator-two prey model community. Am. Nat. 112, 797–813 (1978)

    Google Scholar 

  44. Veldkamp, H.: Ecological studies with the chemostat. Adv. Microbiol. Ecol. 1, 59–95 (1977)

    Google Scholar 

  45. Waltman, P., Hubbell, S. P., Hsu, S. B.: Theoretical and experimental investigation of microbial competition in continuous culture. In: Burton, T. (ed.) Modelling and differential equations in biology. Marcel Dekker, New York 1980

    Google Scholar 

  46. Wolkowicz, G. S. K.: An analysis of mathematical models related to the chemostat. Ph.D. thesis, University of Alberta 1984

  47. Yang, R. D., Humphrey, A. E.: Dynamics and steady state studies of phenol biodegradation in pure and mixed cultures. Biotechnol. Bioeng. 17, 1211–1235 (1975)

    Google Scholar 

  48. Zaret, T. M.: Predator-prey interaction in a tropical lacustrine ecosystem. Ecology 53, 248–257 (1972)

    Google Scholar 

Download references

Author information

Author notes

  1. G. S. K. Wolkowicz

    Present address: Division of Applied Mathematics, Brown University, 02912, Providence, R.I., USA

Authors and Affiliations

  1. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Canada

    G. J. Butler

  2. Department of Mathematics, Emory University, 30322, Atlanta, GA, USA

    G. S. K. Wolkowicz

Authors
  1. G. J. Butler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. S. K. Wolkowicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research partially supported by NSERC of Canada

Research partially supported by an NSERC of Canada Postdoctoral Fellowship. This represents part of this author's Ph.D. thesis submitted to the University of Alberta.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Butler, G.J., Wolkowicz, G.S.K. Predator-mediated competition in the chemostat. J. Math. Biology 24, 167–191 (1986). https://doi.org/10.1007/BF00275997

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00275997

Keywords

Search

Navigation