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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 \[ \dot{N}(t)=(r+\alpha\dot{B}(t))N(t)[1-(N(t)/K)], \] where \(N(0)\) is a random variable satisfying \(0<N(0)<K\) and \(B(t)\) 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.

34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34A55 Inverse problems involving ordinary differential equations
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[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 · Zbl 0752.34039
[3] Hale J K. Nonlinear oscillations in equations with delays, nonlinear oscillations in biology. Lect in Appl Math, 17: 157–185 (1979) · Zbl 0433.34052 · 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 · Zbl 0216.45001
[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 · Zbl 1014.62103
[9] Kendall M G. Advanced Theory of Statistics. Griffin: Charles, 1987 · Zbl 0621.62001
[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
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