Journal of Mathematical Biology

, Volume 14, Issue 2, pp 231–250 | Cite as

A predator prey model with age structure

  • J. M. Cushing
  • M. Saleem


A general predator-prey model is considered in which the predator population is assumed to have an age structure which significantly affects its fecundity. The model equations are derived from the general McKendrick equations for age structured populations. The existence, stability and destabilization of equilibria are studied as they depend on the prey's natural carrying capacity and the maturation periodm of the predator. The main result of the paper is that for a broad class of maturation functions positive equilibria are either unstable for smallm or are destabilized asm decreases to zero. This is in contrast to the usual rule of thumb that increasing (not decreasing) delays in growth rate responses cause instabilities.

Key words

Predator-prey Age structure Stability 


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  1. 1.
    Cushing, J. M.: An operator equation and bounded solutions of integrodifferential systems, SIAM J. Math. Anal.6, (No. 3) 433–445 (1975)CrossRefGoogle Scholar
  2. 2.
    Cushing, J. M.: Stability and instability in predator-prey models with growth rate response delays. Rocky Mountain J. Math.9, (No. 1) 43–50 (1979)Google Scholar
  3. 3.
    Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. In: Lecture notes in biomathematics, Vol. 20. Berlin: Springer 1977Google Scholar
  4. 4.
    Cushing, J. M.: Model stability and instability in age structured populations. J. Theoret. Biol.86, 709–730 (1980)CrossRefGoogle Scholar
  5. 5.
    Cushing, J. M.: Stability and maturation periods in age structured populations. In: Differential Equations and Applications in Ecology, Epidemics and Population Problems (Busenberg, S., and Cooke, K., ed.) New York: Academic Press 1981Google Scholar
  6. 6.
    Goursat, E.: A course in mathematical analysis, Vol. 1, pp. 45–46. New York: Dover 1959Google Scholar
  7. 7.
    Gurtin, M. E., MacCamy, R. C.: Non-linear age-dependent population dynamics. Arch. Rat. Mech. Anal.3, 281–300 (1974)Google Scholar
  8. 8.
    Gurtin, M. E., MacCamy, R. C.: Population dynamics with age-dependence. Research notes in mathematics,30, Vol. III, pp. 1–35. San Francisco: Pitman 1979Google Scholar
  9. 9.
    Hale, J. K.: Nonlinear oscillations in equations with delays. In: Nonlinear oscillations in biology, Lecture in applied mathematics, Vol. 17, Providence-Rhode Island: AMS 1979Google Scholar
  10. 10.
    Hoppensteadt, F.: Mathematical theories of populations: Demographics, genetics and epidemics. Reg. Conf. Series in Appl. Math., SIAM, Philadelphia, Pa., 1975Google Scholar
  11. 11.
    McKendrick, A. G., Pai, M. K.: The rate of multiplication of micro-organisms: A mathematical study. Proc. Roy. Soc. Edinburgh31, 649–655 (1910)Google Scholar
  12. 12.
    May, R. M.: Stability and complexity in model ecosystems, monographs. In: Population biology, Vol. 6 (Second edition). Princeton, New Jersey: Princeton University Press 1974Google Scholar
  13. 13.
    May, R. M., Conway, G. R., Hassell, M. P., Southwood, T. R. E.: Time delays, density-dependence, and single species oscillations. J. Anim. Ecol.43, (No. 3) 747–770 (1974)Google Scholar
  14. 14.
    Maynard Smith, J.: Models in ecology. Cambridge: Cambridge University Press 1974Google Scholar
  15. 15.
    Miller, R. K.: Asymptotic stability and perturbations for linear Volterra integrodifferential systems. In: Delay and functional differential equations and their applications, (Schmitt, K., ed.). New York: Academic Press 1972Google Scholar
  16. 16.
    Oster, G.: The dynamics of nonlinear models with age structure. Studies in mathematical biology, Part II: Populations and communities (Levin, S. A., ed.). MAA Studies in Mathematics16, 411–438 (1978)Google Scholar
  17. 17.
    Oster, G., Guckenheimer, J.: Bifurcation phenomena in population models. In: The Hopf bifurcation and its applications (Marsden, J. E., McCracken, M., eds.). Series in applied mathematical sciences, Vol. 19. New York: Springer 1976Google Scholar
  18. 18.
    Pianka, E. R.: Evolutionary ecology. New York: Harper and Row 1978Google Scholar
  19. 19.
    Pielou, E. C.: Population and community ecology: Principles and methods. New York: Gordon and Breach 1978Google Scholar
  20. 20.
    Ricklefs, R. E.: Ecology. Newton, MA: Chiron Press 1973Google Scholar
  21. 21.
    Rorres, C.: Stability of an age-specific population with density dependent fertility. Theoret. Population Biology10, 26–46 (1976)CrossRefGoogle Scholar
  22. 22.
    Rosenzweig, M. L.: Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time. Science171, 385–387 (1971)Google Scholar
  23. 23.
    Slobodkin, L. B.: Growth and regulation of animal populations. New York: Holt, Rinehart and Winston 1961Google Scholar
  24. 24.
    Volterra, V.: Comments on the note by Mr. Regnier and Miss Lambin. In: The golden age of theoretical ecology (Scudo, F. M., Ziegler, J. R., eds.). Lecture notes in biomathematics, Vol. 22, pp. 47–49. Berlin: Springer 1978Google Scholar
  25. 25.
    Foerster, H., von: The kinetics of cellular proliferation. (Stohlman, F., Jr., ed.) New York: Grune and Stratton, 382–407, 1959Google Scholar
  26. 26.
    Wang, F. J. S.: Stability of an age-dependent population. SIAM J. Math. Anal.11, (No. 4) 683–689 (1980)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • J. M. Cushing
    • 1
  • M. Saleem
    • 1
  1. 1.Department of Mathematics and Program in Applied MathematicsUniversity of ArizonaTucsonUSA

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