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Existence, uniqueness, and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation. (English) Zbl 1136.34324

Noting that population systems are often subject to environmental noise, the authors consider the random logistic equation

$\stackrel{˙}{N}\left(t\right)=\left(r+\alpha \stackrel{˙}{B}\left(t\right)\right)N\left(t\right)\left[1-\left(N\left(t\right)/K\right)\right],$

where $N\left(0\right)$ is a random variable satisfying $0 and $B\left(t\right)$ is a 1-dimensional standard Brownian motion. The existence, uniqueness and global attractivity of positive solutions are investigated, and maximum likelihood estimators of the parameters are found.

##### MSC:
 34F05 ODE with randomness 60H10 Stochastic ordinary differential equations 34A55 Inverse problems of ODE
##### References:
 [1] May R M. Stability and Complexity in Model Ecosystems. Princeton: Princeton University Press, 1973 [2] Gopalsamy K. Stability and Oscillations in Delay Differential Equations of Population Dynamics. London: Kluwer Academic Publishers, 1992 [3] Hale J K. Nonlinear oscillations in equations with delays, nonlinear oscillations in biology. Lect in Appl Math, 17: 157–185 (1979) · doi:10.1007/BFb0064317 [4] Mao X, Marion G, Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics. Stoc Proc and Appl, 97: 95–110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0 [5] Arnold L. Stochastic Differential Equations: Theory and Applications. New York: Wiley, 1972 [6] Fredman A. Stochastic Differential Equations and Their Applications. San Diego: Academic Press, 1976 [7] Fan J, Zhang C. A reexamination of diffusion estimators with applications to financial model validation. J Am Stat Assoc, 98: 118–134 (2003) · Zbl 1073.62571 · doi:10.1198/016214503388619157 [8] Fan J, Yao Q. Nonlinear Time Series, Nonparametric and Parametric Methods. New York: Springer, 2003 [9] Kendall M G. Advanced Theory of Statistics. Griffin: Charles, 1987 [10] Gilpin M E, Ayala F G. Global models of growth and competition, Proc Nat Acad Scis, 70: 3590–3593 (1973) · Zbl 0272.92016 · doi:10.1073/pnas.70.12.3590 [11] Gilpin M E, Ayala F G. Schoenner’s model and drosophila competition, Theor Popul Biol, 9: 12–14 (1976) · doi:10.1016/0040-5809(76)90031-9