Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cushing, J. M.: An operator equation and bounded solutions of integrodifferential systems, SIAM J. Math. Anal.6, (No. 3) 433–445 (1975)
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)
Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. In: Lecture notes in biomathematics, Vol. 20. Berlin: Springer 1977
Cushing, J. M.: Model stability and instability in age structured populations. J. Theoret. Biol.86, 709–730 (1980)
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 1981
Goursat, E.: A course in mathematical analysis, Vol. 1, pp. 45–46. New York: Dover 1959
Gurtin, M. E., MacCamy, R. C.: Non-linear age-dependent population dynamics. Arch. Rat. Mech. Anal.3, 281–300 (1974)
Gurtin, M. E., MacCamy, R. C.: Population dynamics with age-dependence. Research notes in mathematics,30, Vol. III, pp. 1–35. San Francisco: Pitman 1979
Hale, J. K.: Nonlinear oscillations in equations with delays. In: Nonlinear oscillations in biology, Lecture in applied mathematics, Vol. 17, Providence-Rhode Island: AMS 1979
Hoppensteadt, F.: Mathematical theories of populations: Demographics, genetics and epidemics. Reg. Conf. Series in Appl. Math., SIAM, Philadelphia, Pa., 1975
McKendrick, A. G., Pai, M. K.: The rate of multiplication of micro-organisms: A mathematical study. Proc. Roy. Soc. Edinburgh31, 649–655 (1910)
May, R. M.: Stability and complexity in model ecosystems, monographs. In: Population biology, Vol. 6 (Second edition). Princeton, New Jersey: Princeton University Press 1974
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)
Maynard Smith, J.: Models in ecology. Cambridge: Cambridge University Press 1974
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 1972
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)
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 1976
Pianka, E. R.: Evolutionary ecology. New York: Harper and Row 1978
Pielou, E. C.: Population and community ecology: Principles and methods. New York: Gordon and Breach 1978
Ricklefs, R. E.: Ecology. Newton, MA: Chiron Press 1973
Rorres, C.: Stability of an age-specific population with density dependent fertility. Theoret. Population Biology10, 26–46 (1976)
Rosenzweig, M. L.: Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time. Science171, 385–387 (1971)
Slobodkin, L. B.: Growth and regulation of animal populations. New York: Holt, Rinehart and Winston 1961
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 1978
Foerster, H., von: The kinetics of cellular proliferation. (Stohlman, F., Jr., ed.) New York: Grune and Stratton, 382–407, 1959
Wang, F. J. S.: Stability of an age-dependent population. SIAM J. Math. Anal.11, (No. 4) 683–689 (1980)
Author information
Authors and Affiliations
Additional information
Research supported by National Science Foundation Grant No. MCS-7901307-01
Research supported by National Scholarship for Study Abroad No. EDN/S-59/80 from the government of India
Rights and permissions
About this article
Cite this article
Cushing, J.M., Saleem, M. A predator prey model with age structure. J. Math. Biology 14, 231–250 (1982). https://doi.org/10.1007/BF01832847
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01832847