Skip to main content
Log in

Models for the effect of toxicant in single-species and predator-prey systems

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

Abstract

Models of single-species and predator-prey systems in a polluted closed environment are developed and partially analyzed. Three cases are considered: a single influx of toxicant, a constant influx of toxicant, and a periodic pollution of the environment. In the case of single-species growth we are able to determine some local and global dynamics. In the case of predator-prey systems, we investigate the existence of steady states for a small constant influx of toxicant.

On leave from Department of Mathematics, Indian Institute of Technology, Kanpur, India

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Google Scholar 

  • Luna, de J. T., Hallam, T. G.: Effects of toxicants on populations: a qualitative approach IV. Resource-consumer-toxicant models. Ecolog. Model. 35, 249–273 (1987)

    Google Scholar 

  • Freedman, H. I.: The implicit function theorem in the scalar case. Can. Math. Bull. 12, 721–732 (1969)

    Google Scholar 

  • Freedman, H. I.: Deterministic mathematical models in population ecology. Edmonton: HIFR Consulting Ltd. 1987

    Google Scholar 

  • Freedman, H. I., So, J. W.-H.: Global stability and persistence of simple food chains. Math. Biosci. 76, 69–86 (1985)

    Google Scholar 

  • Haas, C. N.: Application of predator-prey models to disinfection. J. Water Pollut. Control. Fed. 53, 378–386 (1981)

    Google Scholar 

  • Hallam, T. G., Clark, C. E.: Nonautonomous logistic equations as models of populations in a deteriorating environment. J. Theor. Biol. 93, 303–311 (1982)

    Google Scholar 

  • Hallam, T. G., Clark, C. E., Jordan, G. S.: Effects of toxicants on populations: a qualitative approach II. First order kinetics. J. Math. Biol. 18, 25–37 (1983)

    Google Scholar 

  • Hallam, T. G., Clark, C. E., Lassiter, R. R.: Effects of toxicants on populations: a qualitative approach I. Equilibrium environmental exposure. Ecolog. Model. 18, 291–304 (1983)

    Google Scholar 

  • Hallam, T. G., de Luna, J. T.: Effects of toxicants on populations: a qualitative approach III. Environmental and food chain pathways. J. Theor. Biol. 109, 411–429 (1984)

    Google Scholar 

  • Jensen, A. L., Marshall, J. S.: Application of a surplus production model to assess environmental impacts on exploited populations of Daphnia pluex in the laboratory. Environ. Pollut. (Ser. A) 28, 273–280 (1982)

    Google Scholar 

  • Lancaster, P. L., Tismenetsky, M.: The Theory of Matrices, Second Edition. New York: Academic Press 1985

    Google Scholar 

  • Nelson, S. A.: The problem of oil pollution of the sea. In: Advances in Marine Biology, pp. 215–306. London: Academic Press 1970

    Google Scholar 

  • Shukla, J. B., Freedman, H. I., Pal, V. N., Misra, O. P., Agarwal, M., Shukla, A.: Degradation and subsequent regeneration of a forestry resource: a mathematical model. Ecolog. Model. 44, 219–229 (1989)

    Google Scholar 

  • Woodman, J. N., Cowling, E. B.: Airborne chemicals and forest health. Environ. Sci. Technol. 21, 120–126 (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A4823

Rights and permissions

Reprints and permissions

About this article

Cite this article

Freedman, H.I., Shukla, J.B. Models for the effect of toxicant in single-species and predator-prey systems. J. Math. Biol. 30, 15–30 (1991). https://doi.org/10.1007/BF00168004

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00168004

Key words

Navigation