Abstract
A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.
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References
Allee, W. C.: The social life of animals. New York: W. W. Norton and Co. 1983
Bellman, R. Cooke, K.: Differential difference equations. New York: Academic Press 1963
Busenberg, S., Cooke, K.: Models of vertically transmitted diseases with sequential-continuous dynamics. In: Nonlinear phenomena in mathematical sciences, V. Lakshmikantham (eds) 179–187, New York: Academic Press 1983
Busenberg, S., Iannelli, M.: “A class of nonlinear diffusion problems in age-dependent population dynamics” J. Nonlinear Anal. T.M.A. 7, 501–529 (1983)
Busenberg, S, Iannelli, M.: “A degenerate nonlinear diffusion problem in age-structured population dynamics”, J. Nonlinear Anal. T.M.A. 7 1411–1429 (1983)
Busenberg, S., Iannelli, M.: Nonlinear diffusion problems in age-structured population dynamics. Preprint
Coleman, C. S., Frauenthal, J. C.: Satiable egg eating predators, Math. Biosci. 63, 99–119 (1983)
Cushing, J., Saleem, M., A predator prey model with age structure. J. Math. Biol. 14, 231–250 (1982)
Cushing, J., Saleem, M.: A competition model with age structure. Preprint
Feller, W.: On the integral equation of renewal theory. Ann. Math. Stat. 12, 243–267 (1941)
Gurtin, M.: The mathematical theory of age-structured populations, manuscript, Department of Mathematics, Pittsburgh: Carnegie-Mellon University, Second draft (1982)
Gurtin, M. E., MacCamy, R. C.: Some simple models for non-linear age-dependent population dynamics. Math. Biosci. 43, 199–211 (1979)
Gurtin, M. E., MacCamy, R. C.: “Population dynamics with age dependence”, Nonlinear analysis and mechanics: heriot-watt symposium III. Boston: Pitman 1979
Gyllenberg, M.: Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures Math. Biosci. 62, 45–74 (1982)
Gyllenberg, M.: Stability of a nonlinear age-dependent population model containing a control variable. Preprint
Hadeler, K. P., Dietz, K.: Population dynamics of killing parasites which reproduce in the host. Preprint
Hastings, A.: Age dependent predation is not a simple process. Preprint
Holling, C. S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Canada 45, 5–60 (1965)
Hoppensteadt, F.: Mathematical theories of populations: demographics, genetics, and epidemics SIAM Reg. Conference in Applied Math. Philadelphia (1975)
Hsu, S. B., Hubbell, S. P., Waltman, P.: Competing predators, SIAM J. Appl. Math. 35, 617–625 (1978)
Hsu, S. N., Hubell, S. P., Waltman, P.: A mathematical theory for single-nutrient competition in continuous culture micro-organisms SIAM J. Appl. Math 32, 366–383 (1977)
La Salle, J. P. The stability of dynamical systems SIAM Regional Conference in Applied Math. Philadelphia (1976)
Lotka, A. J. The stability of the normal age distribution. Proc. Nat. Acad. Science 8, 339–345 (1922)
Levine, D. S. Bifurcation of periodic solutions for a class of age-structured predator-prey systems. Bull. Math. Biosc. 45, 901–915 (1983)
Markus, L.: Asymptotically autonomous differential systems. Contributions to Nonlinear Oscillations, vol. III, S. Lefschetz (eds) Princeton: Princeton U. Press 1–29 (1956)
Prüss, J.: On the qualitative behavior of populations with age-specific interactions. Comput. Math. Appl. 9, 327–339 (1983)
Smith, H. L.: The interaction of steady state on Hopf bifurcations in a two-predator-one-prey competition model. SIAM J. Appl. Math. 42, 27–43 (1982)
Webb, G. F.: Theory of Nonlinear Age-Dependent Population Dynamic manustript, Department of Mathematics, Memphis: Vanderbilt University, (1983)
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Busenberg, S., Iannelli, M. Separable models in age-dependent population dynamics. J. Math. Biology 22, 145–173 (1985). https://doi.org/10.1007/BF00275713
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DOI: https://doi.org/10.1007/BF00275713