Jiang, Daqing; Shi, Ningzhong A note on nonautonomous logistic equation with random perturbation. (English) Zbl 1076.34062 J. Math. Anal. Appl. 303, No. 1, 164-172 (2005). Consider the nonautonomous extension of randomized logistic equation \[ dN(t) = N(t) [(a(t)-b(t) N(t)) dt + \alpha (t) dB(t)], \quad N(0)=N_0 > 0,\;t \geq 0, \] driven by a \(1\)-dimensional Brownian motion \(B\) on a probability space \((\Omega,{\mathcal F},({\mathcal F}_t)_{t \geq 0}, P)\). Suppose that the coefficients \(a, b, \alpha\) are continuous, \(T\)-periodic functions satisfying \[ a(t) > 0, \quad b(t) > 0, \quad \int^T_0 [a(s)-\alpha^2(s)] \,ds > 0 . \] This note shows that \(E [1/N(t)]\) has a unique positive \(T\)-periodic solution under these conditions as a natural extension of a well-known property of the underlying deterministic model. Reviewer: Henri Schurz (Carbondale) Cited in 115 Documents MSC: 34F05 Ordinary differential equations and systems with randomness 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 37H10 Generation, random and stochastic difference and differential equations 60H30 Applications of stochastic analysis (to PDEs, etc.) 92D25 Population dynamics (general) Keywords:stochastic differential equations; nonautonomous logistic equation; existence and uniqueness; periodic solutions PDFBibTeX XMLCite \textit{D. Jiang} and \textit{N. Shi}, J. Math. Anal. Appl. 303, No. 1, 164--172 (2005; Zbl 1076.34062) Full Text: DOI References: [1] May, R. M., Stability and Complexity in Model Ecosystems (1973), Princeton Univ. Press [2] Globalism, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic: Kluwer Academic London [3] Burton, T. A., Volterra Integral and Differential equations (1983), Academic Press: Academic Press New York · Zbl 0515.45001 [4] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97, 95-110 (2002) · Zbl 1058.60046 [5] Arnold, L., Stochastic Differential Equations: Theory and Applications (1972), Wiley: Wiley New York [6] Freedman, A., Stochastic Differential Equations and Their Applications, vol. 2 (1976), Academic Press: Academic Press San Diego [7] Gilpin, M. E.; Ayala, F. G., Global models of growth and competition, Proc. Nat. Acad. Sci. USA, 70, 3590-3593 (1973) · Zbl 0272.92016 [8] Gilpin, M. E.; Ayala, F. G., Schoenner’s model and Drosophila competition, Theor. Popul. Biol., 9, 12-14 (1976) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.