DYNAMICS OF RANDOM-DISTURBED VERHULST EQUATION AND THE METHOD OF MAXIMUM ENTROPY 

Zhuravlev Viktor Mikhaylovich, Doctor of physical and mathematical sciences, sub-department of theoretical physics, Ulyanovsk State University (Ulyanovsk, 42 L. Tolstogo str.), zhvictorm@gmail.com
Mironov Pavel Pavlovich, Postgraduate student, Research Technological Institute, Ulyanovsk State University (Ulyanovsk, 42 L. Tolstogo str.), museum86@mail.ru 

534.04: 536.12: 51-7 

The article analyzes the behavior of one-dimensional random-disturbed systems the dynamics of which is described by the Verhulst equation. The analysis is carried out on the basis of the Reynolds method and the maximum entropy principle. The authors consider interpretation from the point of view of models of the disturbed oscillator with attenuation and kinetic model of population. The Reynolds method is applied to Verhulst equation. The received average equations are isolated with the help of the maximum entropy method. The researchers establish a conservation law of specific entropy. The stability of station point of an average model is
analysed. The analytical solution of average Verhulst model is obtained. The authors reveal general features of dynamics on the basis of analytical solution of an average system of equations. It is obtained that the dynamics of Verhulst equation essentially depends on the value of noise dispersion. For small dispersions the model on the average evolves near the value that satisfies the undisturbed Verhulst equation. It is shown that all conditions with nonzero dispersions are unstable generally already in the first order of the theory of perturbations, which means that they rapidly turn to the initial undisturbed condition. It is also shown that the analytical solution is stable according to Lyapunov. 

random-disturbed Verhulst equation, Reynolds method, maximum entropy method. 

1. Riznichenko G.Yu. Lektsii po matematicheskim modelyam v biologii [Lectures on mathematical problems in biology]. Moscow;Izhevsk:Nauchno-izdatel'skiy tsentr «Regulyarnaya i khaoticheskaya dinamika»,2002,pt.1,231p.
2. Arnol'd V. I. «Zhestkie» i «myagkie» matematicheskie modeli [Rigid and soft mathematical models]. Moscow: MTsNMO, 2000, 32 p.
3. Monin A. S., Yaglom A. M. Statisticheskaya gidromekhanika [Statistical hydromechanics]. Moscow: Nauka, 1967, pt. 1, 639 p. ; 1969, pt. 2, 720 p.
4. Zhuravlev V. M., Shlyapin V. A. Nelineynyy mir [Nonlinear world]. 2008, vol. 6, no. 7, pp. 352–363.
5. Zhuravlev V. M. Zhurnal teoreticheskoy fiziki [Theoretical physics journal]. 2009, vol. 79, no. 1, pp. 16–27.
6. Zhuravlev V. M., Shlyapin V. A. Prikladnaya matematika i mekhanika [Applied mathematics and mechanics]. Ulyanovsk: UlGTU 2009, pp. 72–88.
7. Zhuravlev V. M., Mironov P. P. Nelineynyy mir [Nonlinear world]. 2011, vol. 9, no. 4, pp. 201–212.
8. Klimontovich Yu. L. Vvedenie v fiziku otkrytykh sistem [Introduction into physics of open systems]. Moscow: Yanus-K, 2002, 284 p.
9. Friden B. R. Trudy instituta inzhenerov po elektrotekhnike i radioelektronike [Proceedings of the institute of engineers in electrical engineerings and radio electronics]. 1985, vol. 73, no. 12, p. 78.
10. Stratanovich R. L. Teoriya informatsii [Theory of information]. Moscow: Sov. radio, 1975, 424 p.
11. Bazykin A. D. Nelineynaya dinamika vzaimodeystvuyushchikh populyatsiy [Nonlinear dynamics of interacting populations]. Moscow ; Izhevsk: IKI, 2003, 368 p.