Abstract
Logistic equations have numerous applications, especially in population dynamics. In Chap. 11, several modifications of delay logistic equations are considered, in particular, both additive and multiplicative generalizations.
In the study of the nonoscillation properties of nonlinear equations usually one of the following three methods is applied: first, a differential equation can be transformed into an operator equation with the following property: if the operator equation has a nonnegative solution, then the differential equation has a nonoscillatory solution, second, for the relevant operator equation either the Schauder Fixed-Point Theorem is applied, or convergence of monotone approximations to a solution is demonstrated, and, third, the connection of oscillation properties of nonlinear and linear delay differential equations can be employed. In this chapter, all the three methods are used. In the most difficult superlinear case the differential equation is transformed into an operator equation which incorporates both an increasing and a decreasing operators.
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References
Aiello, W.G.: The existence of nonoscillatory solutions to a generalized nonautonomous, delay logistic equation. J. Math. Anal. Appl. 149, 114–123 (1990)
Berezansky, L., Braverman, E.: On oscillation of a multiplicative delay logistic equation. Proceedings of the Third World Congress of Nonlinear Analysts, Part 2, Catania, 2000. Nonlinear Anal. 47, 1199–1209 (2001)
Berezansky, L., Braverman, E.: On oscillation of a generalized logistic equation with several delays. J. Math. Anal. Appl. 253, 389–405 (2001)
Chen, M.P., Lalli, B.S., Yu, J.S.: Oscillation and global attractivity in a multiplicative delay logistic equation. Differ. Equ. Dyn. Syst. 5, 75–83 (1997)
Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995)
Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht, Boston, London (1992)
Grace, S.R., Győri, I., Lalli, B.S.: Necessary and sufficient conditions for the oscillations of a multiplicative delay logistic equations. Q. Appl. Math. 53, 69–79 (1995)
Grove, E.A., Ladas, G., Qian, C.: Global attractivity in a “food-limited” population model. Dyn. Syst. Appl. 2, 243–249 (1993)
Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991)
Győri, I., Pituk, M.: Comparison theorems and asymptotic equilibrium for delay differential and difference equations. Dyn. Syst. Appl. 5, 277–303 (1996)
Jones, J.S.: On the nonlinear differential difference equation \(\dot{f}(x)=f(x-1)[1+f(x)]\). J. Math. Anal. Appl. 4, 440–469 (1962)
Kakutani, S., Markus, L.: On the nonlinear difference differential equation \(\dot{y}(t)=[A-By(t-\tau)]y(t)\). Contrib. Theory Nonlinear Oscil. 4, 1–18 (1958)
Kuang, Y., Zhang, B.G., Zhao, T.: Qualitative analysis of nonautonomous nonlinear delay differential equation. Tohoku Math. J. 43, 509–528 (1991)
Ladas, G., Qian, C.: Oscillation and global stability in a delay logistic equation. Dyn. Stab. Syst. 9, 153–162 (1994)
Lenhart, S.M., Travis, C.C.: Global stability of a biological model with time delay. Proc. Am. Math. Soc. 96, 75–78 (1986)
Lin, L., Wang, G.: On oscillation of first order nonlinear neutral equations. J. Math. Anal. Appl. 186, 605–618 (1994)
May, R.M.: Time delay versus stability in population models with two or three trophic levels. Ecology 54, 315–325 (1973)
Olach, R.: Oscillation and nonoscillation of first order nonlinear delay differential equations. Acta Math. Univ. Ostrav. 12, 41–47 (2004)
Palaniswami, S.C., Ramasami, E.K.: Nonoscillation of generalized nonautonomous logistic equation with multiple delays. Differ. Equ. Dyn. Syst. 4, 379–385 (1996)
Wang, Z., Yu, J.S., Huang, L.H.: Nonoscillatory solutions of generalized delay logistic equations. Chin. J. Math. 21, 81–90 (1993)
Wright, E.M.: A nonlinear difference-differential equation. J. Reine Angew. Math. 194, 66–87 (1955)
Yan, J.R.: Oscillation of solution of first order delay differential equations. Nonlinear Anal. 11, 1279–1287 (1987)
Yang, Z.Q.: Necessary and sufficient conditions for oscillation of delay-logistic equations. J. Biomath. 7, 99–109 (1992)
Zhang, B.G., Gopalsamy, K.: Oscillation and nonoscillation in a nonautonomous delay-logistic equation. Q. Appl. Math. 46, 267–273 (1988)
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Agarwal, R.P., Berezansky, L., Braverman, E., Domoshnitsky, A. (2012). Nonlinear Models—Modifications of Delay Logistic Equations. In: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3455-9_11
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