Advertisement

Bulletin of Mathematical Biology

, Volume 45, Issue 6, pp 991–1004 | Cite as

The trade-off between mutual interference and time lags in predator-prey systems

  • H. L. Freedman
  • V. Sree Hari Rao
Article

Abstract

We present a Gause predator-prey model incorporating mutual interference among predators, a density-dependent predator death rate and a time lag due to gestation. It is well known that mutual interference is stabilizing, whereas time delays are destabilizing. We show that in combining the two, a long time-lag usually, but not always, destabilizes the system. We also show that increasing delays can cause a bifurcation into periodic solutions.

Keywords

Periodic Solution Hopf Bifurcation Functional Differential Equation Negative Real Part Positive Real Part 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, V. D., D. L. De Angelis and R. A. Goldstein. 1980. “Stability Analysis of the Time Delay in a Host-Parasitoid Model.”J. theor. Biol. 83, 43–62.CrossRefGoogle Scholar
  2. Arditi, R., J.-M. Abillon and J. Vieira da Silva. 1977. “The Effect of a Time-delay in a Predator-Prey Model.”Math. Biosci. 33, 107–120.zbMATHCrossRefGoogle Scholar
  3. Bellman, R. and K. L. Cooke. 1963.Differential-difference Equations. New York: Academic Press.Google Scholar
  4. Bounds, J. M. and J. M. Cushing. 1975. “On the Behavior of Solutions of Predator-Prey Equations with Hereditary Terms.”Math. Biosci. 26, 41–54.MathSciNetCrossRefGoogle Scholar
  5. Brauer, F. 1977. “Periodic Solutions of Some Ecological Models.”J. theor. Biol. 69, 143–152.MathSciNetCrossRefGoogle Scholar
  6. —, 1979. “Characteristic Return Times for Harvested Population Models with Time Lag.”Math. Biosci. 45, 295–311.zbMATHMathSciNetCrossRefGoogle Scholar
  7. Cooke, K. L. and J. A. Yorke, 1973. “Some Equations Modelling Growth Processes and Gonorrhea Epidemics.”Math. Biosci. 16, 75–101.zbMATHMathSciNetCrossRefGoogle Scholar
  8. Cushing, J. M. 1976a. “Forced Asymptotically Periodic Solutions of Predator-Prey Systems with or without Hereditary Effects.”SIAM J. appl. Math.,30, 665–674.zbMATHMathSciNetCrossRefGoogle Scholar
  9. —. 1976b. “Periodic Solutions of Two Species Interaction Models with Lags.”Math. Biosci. 31, 143–156.zbMATHMathSciNetCrossRefGoogle Scholar
  10. —. 1976c. “Predator-Prey Interaction with Time Delays.”J. math. Biol. 3, 369–380.zbMATHMathSciNetGoogle Scholar
  11. Freedman, H. I. 1969. “The Implicit Function Theorem in the Scalar Case.”Can. math. Bull. 12, 721–732.zbMATHGoogle Scholar
  12. —. 1979. “Stability Analysis of a Predator-Prey System with Mutual Interference and Density-dependent Death Rates.”Bull. math. Biol. 41, 67–78.zbMATHMathSciNetCrossRefGoogle Scholar
  13. —. 1980.Deterministic Mathematical Models in Population Ecology. New York: Marcel Dekker.Google Scholar
  14. Hale, J. 1977.Theory of Functional Differential Equations. New York: Springer Verlag.Google Scholar
  15. Hassell, M. P. 1971. “Mutual Interference between Searching Insect Parasites.”J. Anim. Ecol. 40, 473–486.CrossRefGoogle Scholar
  16. Levin, S. A. 1977. “A More Functional Response to Predator-Prey Stability.”Am. Nat. 111, 381–383.CrossRefGoogle Scholar
  17. — and R. M. May. 1976. “A Note on Difference-delay Equations.”Theor. pop. Biol. 9, 178–187.zbMATHMathSciNetCrossRefGoogle Scholar
  18. May, R. M. 1973. “Time-delay versus Stability in Population Models with Two and Three Trophic Levels.”Ecology 54, 315–325.CrossRefGoogle Scholar
  19. Reddingius, J. 1963. “A Mathematical Note on a Model of a Consumer-Food Relation in which the Food is Continually Replaced.”Acta biotheor. 16, 183–198.CrossRefGoogle Scholar
  20. Rogers, D. J. and M. P. Hassell. 1974. “General Models for Insect Parasite and Predator Searching Behaviour: Interference.”J. Anim. Ecol. 43, 239–253.CrossRefGoogle Scholar
  21. Rosenzweig, M. L. and R. H. MacArthur. 1963. “Graphical Representation and Stability Conditions of Predator-Prey Interactions.”Am. Nat. 47, 209–223.CrossRefGoogle Scholar
  22. Smith, R. H. and R. Mead. 1974. “Age Structure and Stability in Models of Prey-Predator Systems.”Theor. pop. Biol. 6, 308–322.CrossRefGoogle Scholar
  23. Taylor, C. E. and R. R. Sokal. 1976. “Oscillations in Housefly Population Sizes due to Time Lags.”Ecology 57, 1060–1067.CrossRefGoogle Scholar
  24. Thingstad, T. F. and T. I. Langeland. 1974. “Dynamics of Chemostat Culture: the Effect of a Delay in Cell Response.”J. theor. Biol. 48, 149–159.CrossRefGoogle Scholar
  25. Veilleux, B. G. 1979. “An Analysis of the Predatory Interaction betweenParamecium andDidinium.”J. Anim. Ecol. 48, 787–803.CrossRefGoogle Scholar
  26. Wangersky, P. J. and W. J. Cunningham. 1957. “Time Lag in Population Models.”Cold Spring Harb. Symp. qual. Biol. 22, 329–338.Google Scholar

Copyright information

© Society for Mathematical Biology 1983

Authors and Affiliations

  • H. L. Freedman
    • 1
  • V. Sree Hari Rao
    • 1
  1. 1.Department of MathematicsUniversity of AlbertaEdmontonCanada

Personalised recommendations