Local Dynamics of a Logistic Equation with Delay

S. V. Aleshin ; S. A. Kaschenko

Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 1, pp. 73-88.

Résumé

We considered a logistic equation with delay and studied its local dynamics. The critical cases have been found in the problem of the equilibrium state stability. We applied standard Andronov–Hopf biffurcation methods for delay differential equations and an asymptotic method, developed by one of the authors, based on the construction of special evolution equations that define the local dynamics equations with delay. It is shown that all solutions of the equation tend to an equilibrium state or result in a single stable cycle. The results of numerical modelling are presented in this paper. The study has proved that analytical and numerical modeling results have a good correlation.

Détail

Keywords: logistic equation, relaxation cycle, normal forn, equilibrium state.

Bibliographie

[1] S. Kakutani, L. Markus, “On the non-linear difference-differential equation $y'(t)=(a-by(t-\tau))y(t)$. Contributions to the theory of non-linear oscillations”, Ann. Math. Stud., IV, Princeton University Press, Princeton, 1958, 1–18 | MR

[2] G. S. Jones, “The existence of periodic solutions of $f'(x)=-\alpha f(x-1)[1+f(x)]$”, J. Math. Anal. and Appl., 5 (1962), 435–450 | DOI | MR | Zbl

[3] S. A. Kashchenko, “Asymptotics of the Solutions of the Generalized Hutchinson Equation”, Automatic Control and Computer Science, 47:7 (2013), 470–494 | DOI

[4] Kashchenko S. A., “O periodicheskikh resheniyakh uravneniya $x'(t)=-lx(t-1)[1+x(t)]$”, Issledovaniya po ustoychivosti i teorii kolebaniy, YarGU, Yaroslavl, 1978, 110–117 (in Russian)

[5] Kashchenko S. A., “Primenenie metoda normalizatsii k izucheniyu dinamiki differentsialno-raznostnykh uravneniy s malym mnozhitelem pri proizvodnoy”, Dif. uravneniya, 25:8 (1989), 1448–1451 (in Russian) | MR

[6] S. A. Kaschenko, “Normalisation in the systems with small diffusion”, Int. J. of Bifurcation and chaos, 6:7 (1996), 1093–1109 | DOI | MR | Zbl

[7] S. A. Kashchenko, “The Ginzburg–Landau equation as a normal form for a second-order difference-differential equation with a large delay”, Computational Mathematics and Mathematical Physics, 38:3 (1998), 443–451 | MR | Zbl

[8] I. S. Kashchenko, “Asymptotic analysis of the behavior of solutions to equations with large delay”, Doklady Mathematics, 78:1 (2008), 570–573 | DOI | MR | Zbl

[9] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Extremal dynamics of the generalized Hutchinson equation”, Computational Mathematics and Mathematical Physics, 49:1 (2009), 71–83 | DOI | MR | Zbl

[10] Glyzin S. D., “A registration of age groups for the Hutchinson's equation”, Modeling and Analysis of Information Systems, 14:3 (2007), 29–42 (in Russian) | MR

[11] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, 1977 | MR | Zbl

[12] Ph. Hartman, Ordinary differential equations, Wiley, New York, 1964 | MR | Zbl

[13] Bryuno A. D., Lokalnyy metod nelineynogo analiza differentsialnykh uravneniy, Nauka, M., 1979 (in Russian) | MR

[14] Glyzin S. D., Kolesov A. Yu., Lokalnye metody analiza dinamicheskikh sistem, uchebnoe posobie, YarGU, Yaroslavl, 2006, 92 pp. (in Russian)

[15] Bautin N. N., Leontovich E. A., Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti, Nauka, M., 1990 (in Russian) | MR | Zbl

[16] Glyzin S. D., “Difference approximations of “reaction-diffusion” equation on a segment”, Modeling and Analysis of Information Systems, 16:3 (2009), 96–116 (in Russian)

[17] E. Hairer, G. Wanner, S. P. Norsett, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics, 2ed., revised, Springer, 2008 | MR