×

Global stability results for a generalized Lotka-Volterra system with distributed delays. Applications to predator-prey and epidemic systems. (English) Zbl 0716.92020

Summary: The paper contains an extension of the general ODE system proposed in previous papers by the same authors [see e.g. the first two authors’ paper, Comput. Math. Appl., Part A 12, 677-694 (1986; Zbl 0622.92016)], to include distributed time delays in the interaction terms. The new system describes a large class of Lotka-Volterra like population models and epidemic models with continuous time delays. Sufficient conditions for the boundedness of solutions and for the global asymptotic stability of nontrivial equilibrium solutions are given. A detailed analysis of the epidemic system is given with respect to the conditions for global stability. For a relevant subclass of these systems an existence criterion for steady states is also given.

MSC:

92D30 Epidemiology
92D25 Population dynamics (general)
34C11 Growth and boundedness of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations

Citations:

Zbl 0622.92016
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Anderson, J. M.: The eutrophication of lakes. In: Meadows, D. L. and Meadows D. H. (eds.) Toward global equilibrium. Collected papers. Bristol Boston: Wright-Allen 1973
[2] Caperon, J.: Time lag in population growth response of Isochrysis galbana to a variable nitrate environment. Ecology 50, 188-192 (1969)
[3] Caswell, H.: A simulation study of a time lag population model. J. Theor. Biol. 34, 419-439 (1972)
[4] Cunningham, A.; Nisbet, R. M.: Time lag and co-operativity in the transient growth dynamics of microalgae. J. Theor. Biol. 84, 189-203 (1980)
[5] Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. Lect. Notes Biomath., vol. 20. Berlin Heidelberg New York: Springer 1977 · Zbl 0363.92014
[6] D’Ancona, v.: The struggle for existence. Leiden: E. J. Brill 1954
[7] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press 1981 · Zbl 0474.34002
[8] Hastings, A.: Multiple limit cycles in predator-prey models. J. Math. Biol. 11, 51-63 (1981) · Zbl 0471.92022
[9] Hsu, S. B., Hubbell, S.; Waltman, P.: A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms. SIAM J. Appl. Math. 32, 366-383 (1977) · Zbl 0354.92033
[10] MacDonald, N.: Time lags in biological models. Lect Notes Biomath., vol 27. Berlin Heidelberg New York: Springer 1978 · Zbl 0403.92020
[11] Nisbet, R. M., Gurney, W. S. C.: Model of material cycling in a closed ecosystem. Nature 264, 633-635 (1976)
[12] Roy, A. B.; Solimano, F.: Global stability of partially closed food-chains with resources. Bull. Math. Biol. 48, 455-468 (1986) · Zbl 0613.92026
[13] Stépán, G.: Great delay in a predator-prey model, Nonlinear Anal. 10, 913-929 (1986) · Zbl 0612.92016
[14] Waltman, P.; Hubbell, S. P.; Hsu, S. B.: Theoretical and experimental investigations of microbial competition in continuous culture. In: Burton, T. (ed.) Modeling and differential equations in biology. New York: Dekker 1980 · Zbl 0442.92015
[15] Whittaker, R. H.: Communities and ecosystems. New York: Macmillan 1975
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.