Abstract
Applying an analytical method and several limiting equations arguments, some sufficient conditions are provided for the existence of a unique positive equilibriumK for the delay differential equationx=−γx+D(x t ), which is the general form of many population models. The results are concerned with the global attractivity, uniform stability, and uniform asymptotic stability ofK. Application of the results to some known population models, which shows the effectiveness of the methods applied here, is also presented.
Similar content being viewed by others
References
Arino, O., and Kimmel, M. (1986). Stability analysis of models of cell production systems.Math. Model. 7, 1269–1300.
Glass, L., Beuter, A., and Larocque, D. (1988). Time delays, oscillations, and chaos in physiological control systems.Math. Biosci. 90, 111–125.
Gopalsamy, K., Kulenović, M. R. S., and Ladas, G. (1990). Oscillations and global attractivity in models of haematopoiesis.J. Dynam. Diff. Eq. 2, 117–132.
Gurney, W. S. C., Blythe, S. P., and Nisbet, R. M. (1980). Nicholson's blowflies revisited.Nature 287, 17–21.
Karakostas, G. (1982). Causal operators and topological dynamics.Ann. Mat. Pura Appl. 131, 1–27.
Kulenović, M. R. S., and Ladas, G. (1987). Linearized oscillations in the population dynamics.Bull. Math. Biol. 49, 615–627.
Kulenović, M. R. S., Ladas, G., and Sficas, Y. G. (1989). Global attractivity in population dynamics.Comput. Math. Appl. 18, 825–925.
Kulenović, M. R. S., Ladas, G., and Sficas, Y. G. (1991). Global attractivity in Nicholson's blowflies.Applicable Anal. (in press).
Mackey, M. C., and Glass, L. (1977). Oscillation and chaos in physiological control systems.Science 197, 287–289.
Mallet-Paret, J., and Nussbaum, R. D. (1986). Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation.Ann. Mat. Pura Appl. 145, 33–128.
Sell, G. R. (1971).Topological Dynamics and Ordinary Differential Equations, Van Nostrand-Reinhold, London.
Sibirsky, K. R. (1975).Introduction to Topological Dynamics, Noordhoff International, Leyden.
Wazewska-Czyzewska, M., and Lasota, A. (1976). Mathematical problems of the red-blood cell system (in Polish),Ann. Polish Math. Soc. Ser. III Appl. Math. 6, 23–40.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Karakostas, G., Philos, C.G. & Sficas, Y.G. Stable steady state of some population models. J Dyn Diff Equat 4, 161–190 (1992). https://doi.org/10.1007/BF01048159
Issue Date:
DOI: https://doi.org/10.1007/BF01048159