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Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation. (English) Zbl 1140.60032
This paper deals with stochastic differential equation $dN(t)=N(t)[(a(t)-b(t)N(t))dt+\alpha(t)dB(t)]$, where $B(t)$ is the one-dimensional standard Brownian motion, $N(0)=N\sb{0}>0$. It is assumed that $a(t),b(t),\alpha(t)$ are continuous $T$-periodic functions, $a(t)>0$, $b(t)>0$ and $\min\sb{t\in [0,T]}a(t)>\max\sb{t\in[0,T]}\alpha\sp2(t)$. The authors show that considered equation is stochastically permanent and the positive solution $N\sb{p}(t)$ is globally attractive. The similar results for a generalized non-autonomous logistic equation $dN(t)=N(t)[(a(t)-b(t)N\sp{\theta}(t))dt+\alpha(t)dB(t)]$, where $\theta>0$ is an odd number, are presented.

##### MSC:
 60H10 Stochastic ordinary differential equations
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##### References:
 [1] May, R. M.: Stability and complexity in model ecosystems, (1973) [2] Globalism, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039 [3] Fan, M.; Wang, K.: Optimal harvesting policy for single population with periodic coefficients, Math. biosci. 152, 165-177 (1998) · Zbl 0940.92030 · doi:10.1016/S0025-5564(98)10024-X [4] Burton, T. A.: Volterra integral and differential equations, (1983) · Zbl 0515.45001 [5] Eisen, M.: Mathematical models in cell biology and cancer chemotherapy, Lecture notes in biomathematics 30 (1979) · Zbl 0414.92005 [6] Swan, G. W.: Optimization of human cancer radiotherapy, Lecture notes in biomathematics 42 (1981) · Zbl 0464.92002 [7] Michelson, S.; Miller, B. E.; Glicksman, A. S.; Leith, J. T.: Tumor micro-ecology and competitive interactions, J. theoret. Biol. 128, No. 2, 233-246 (1987) [8] Michelson, S.; Leith, J. T.: Dormancy, regression and recurrence: towards a unifying theory of tumor growth control, J. theoret. Biol. 169, No. 4, 327-338 (1994) [9] Krebs, C. J.: Ecology: the experimental analysis of distribution and abundance, (2001) [10] Mao, X.; Marion, G.; Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics, Stochastic process. Appl. 97, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0 [11] Arnold, L.: Stochastic differential equations: theory and applications, (1972) · Zbl 0216.45001 [12] Freedman, A.: Stochastic differential equations and their applications, vol. 2, (1976) · Zbl 0346.15003 [13] Jiang, D. Q.; Shi, N. Z.: A note on non-autonomous logistic equation with random perturbation, J. math. Anal. appl. 303, 164-172 (2005) · Zbl 1076.34062 · doi:10.1016/j.jmaa.2004.08.027 [14] Mao, X.: Stochastic versions of the lassalle theorem, J. differential equations 153, 175-195 (1999) · Zbl 0921.34057 [15] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus, (1991) · Zbl 0734.60060 [16] Friedman, A.: Stochastic differential equations and their applications, (1976) · Zbl 0323.60057 [17] Mao, X.: Stochastic differential equations and applications, (1997) · Zbl 0892.60057 [18] Barbǎlat, I.: Systems d’equations differential d’oscillations nonlineairies, Rev. roumaine math. Pures appl. 4, 267-270 (1959) [19] Gilpin, M. E.; Ayala, F. G.: Global models of growth and competition, Proc. natl. Acad. sci. USA 70, 3590-3593 (1973) · Zbl 0272.92016 · doi:10.1073/pnas.70.12.3590 [20] Gilpin, M. E.; Ayala, F. G.: Schooner’s model and drosophila competition, Theoret. population biol. 9, 12-14 (1976)