Abstract
The existence of a stable positive equilibrium state for the density ρ of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ρ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.
Similar content being viewed by others
References
Clark, C. W.: Mathematical Bioeconomics. New York, Wiley 1976
Coleman, B. D.: On growth of populations with narrow spread in reproductive age, J. Math. Biol. 6, 1–19 (1978)
Cushing, J. M.: Model stability and maturation periods in age structured populations, J. Theo. Biol. 86, 709–730 (1980)
Cushing, J. M.: Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics, Comput. Math. Appl. 9, 459–478 (1983)
Cushing, J. M.: Existence and stability of equilibria in age-structured population dynamics. J. Math. Biol. 20, 259–276 (1984)
Cushing, J. M.: Global branches of equilibrium solutions of the McKendrick equations of age-structured population growth. To appear in Comput. Math. Appl. (1985)
Cushing, J. M.: Periodic McKendrick equations for age-structured population growth. To appear in Comput. Math. Appl. (1986)
Cushing, J. M.: Volterra Integrodifferential Equations in Population Dynamics. In: (Iannelli, M. ed.) Mathematics in Biology. Editore, Naples: Liguori 1981
DiBlasio, G.: Nonlinear age-dependent population growth with history dependent birth rate. Math. Biosci. 46 279–291 (1979)
DiBlasio, G.: Asymptotic behavior of an age-structured fish population. Comput. Math. Appl. 9, 377–381 (1983)
DiBlasio, G, Iannelli, M., Sinestrari, E.: Approach to equilibrium in age structured populations with an increasing recruitment process. J. Math. Biol. 13, 371–382 (1982)
Diekmann, O., Nisbet, R. M., Gurney, W. S. C., van den Bosch, F.: Simple mathematical models for cannibalism: a critique and a new approach. Centre for Math. Comput. Sci., Report no. AM-R8505. Amsterdam 1985
Dietz, K.: Transmission and control of arbovirus diseases Proc. SIMS Conf. on Epidemics. Alta Utah 1974
Frauenthal, J. C., Swick, K. E.: Stability of biochemical reaction tanks. Comput. Math. Appl., 9, 499–506 (1983)
Getz, W.: The ultimate sustainable yield problem in nonlinear age-structured populations. Math. Biosci. 48, 279–292 (1980)
Gurtin, M, MacCamy, R. C.: Nonlinear age dependent population dynamics. Arch. Rat. Mech. Anal. 54, 281–300 (1974).
Gurtin, M., MacCamy, R. C.: Some simple models for nonlinear age-dependent population dynamics. Math. Biosci. 43, 199–211 (1979)
Gurtin, M., Levine, D. S.: On populations which cannibalize their young. SIAM J. Appl. Math. 42, 94–108 (1982)
Gurtin, M., Levine, D. S.: On predator-prey interactions with predation dependent on age of prey. Math. Biosci. 47, 207–219 (1979).
Gurtin, M. E.: The Mathematical Theory of Age-Structured Populations. Preprint
Hoppenstaedt, F.: Mathematical Theories of Populations: Demographics, Genetics and Epidemics. SIAM Conf. Series Appl. Appl. Math., Philadelphia 1975
Levine, D. S.: Models of age-dependent predation and cannibalism via the McKendrick equation. Comput. Math. Appl. 9, 403–414 (1983)
Levine, D. S.: On the stability of a predator-prey system with egg-eating predators. Math. Biosci. 56, 27–46 (1981)
Marcati, P.: Asymptotic behavior in age dependent population dynamics with hereditary renewal law. SIAM J. Math. Anal. 12, 904–916 (1981)
Marcati, P.: On the global stability of the logistic age dependent population equation, J. Math. Biol. 15, 215–226 (1982)
Marcati, P.: Some considerations on the mathematical approach to nonlinear age dependent population dynamics. Comput. Math. Appl. 9, 361–370 (1983)
Murphy, L.: Density dependent cellular growth in an age structured colony. Comput. Math. Appl. 9, 383–392 (1983)
Nisbet, R. M., Gurney, W. S. C.: Modelling Fluctuating Populations. New York, Wiley 1982
Oster, G.: Lectures in Population Dynamics. Appearing in Lec. Appl. Math. 16, AMS 149–170 (1977)
Prüss, J.: Equilibrium solutions of age-specific population dynamics of several species. J. Math. Biol. 11, 65–84 (1981)
Prüss, J.: On the qualitative behaviour of populations with age-specific interactions. Comput. Math. Appl. 9, 327–339 (1983)
Prüss, J.: Stability analysis for equilibria in age-specific population dynamics. Nonl. Anal. Th. Math. Appl. 7, 1291–1313 (1983)
Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. Func. Anal. 1, 487–513 (1971)
Rorres, C.: Stability of an age specific population with density dependent fertility. Theor. Pop. Biol. 10, 24–46 (1976)
Rorres, C.: Local stability of a population with density dependent fertility. Theor. Pop. Biol. 16, 283–300 (1979)
Rubinow, S. I.: A maturity-time representation for cell population. Biophys. J. 8, 1055–1073 (1968)
Rubinow, S. I.: Age-structured equations in the theory of cell populations. Studies in Math. Biol. II, MAA, Washington, 1978
Saleem, M.: Predator-prey relationships: egg-eating predators. J. Math. Biol. 65, 187–197 (1983)
Sinestrari, E.: Non-linear age-dependent population growth. J. Math. Biol. 9, 331–345 (1980)
Sinestrari, E.: Local and global stability for the solutions of a nonlinear renewal equation. Comput. Math. Appl. 9, 353–360 (1983)
Sinko, J. W., Streifer, W.: A new model for the age-size structure of a population. Ecol. 48, 910–918 (1967)
Streifer, W.: Realistic models in population ecology. Appearing in: Macfayden, A. (ed.) Advances in Ecology Research, 8. New York: Academic Press 1974
Swick, K. E.: A nonlinear age-dependent model of single species population dynamics. SIAM J. Appl. Math. 22, 488–498 (1977)
von Foerster, H.: Some remarks on changing populations. Appearing in: Strohlman, Jr., F. (ed.) The Kinetics of Cellular Proliferation, 382–407, New York: Gruen and Stratton 1959
Wang, F. J. S.: Stability of an age-dependent population. SIAM J. Math. Anal. 11, 683–689 (1980)
Webb, G. F.: Theory of Nonlinear Age-Dependent Population Dynamics. Monographs in Pure and Applied Mathematics Series, 89 New York: Marcel Dekker 1985
Webb, G. F.: Logistic models of structured population growth. Preprint
Author information
Authors and Affiliations
Additional information
This research was done while the author was on leave at the Lehrstuhl für Biomathematik, Universität Tübingen, Auf der Morgenstelle 10, 7400 Tübingen 1, Federal Republic of Germany
Rights and permissions
About this article
Cite this article
Cushing, J.M. Equilibria in structured populations. J. Math. Biology 23, 15–39 (1985). https://doi.org/10.1007/BF00276556
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00276556