Nguyen Huu Du; Vu Hai Sam Dynamics of a stochastic Lotka-Volterra model perturbed by white noise. (English) Zbl 1107.92038 J. Math. Anal. Appl. 324, No. 1, 82-97 (2006). It is shown that less restrictive hypotheses can be used in the derivation of certain well-known estimates of the upper growth rates of the solutions of the stochastic Lotka-Volterra differential equation \[ dx (t)=\text{diag}\bigl(x_1(t), x_2(t),\dots,x_n(t)\bigr)\bigl[b+Ax(t)+ \sigma x(t)dW(t)\bigr],\;t\geq 0, \] with \(x(0)=x_0\in\mathbb R^n_+\). Then lower growth rates are addressed by showing that solutions vanish at a rate greater than \(1/t^{1+\varepsilon}\) but smaller than \(1/\sqrt{\ln t}\), where \(\varepsilon\) is an arbitrary positive number. Reviewer: Melvin D. Lax (Long Beach) Cited in 69 Documents MSC: 92D25 Population dynamics (general) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) Keywords:Lotka-Volterra model; Brownian motion; stochastic differential equation; asymptotic behavior PDFBibTeX XMLCite \textit{Nguyen Huu Du} and \textit{Vu Hai Sam}, J. Math. Anal. Appl. 324, No. 1, 82--97 (2006; Zbl 1107.92038) Full Text: DOI References: [1] Gihman, I. I.; Skorohod, A. V., The Theory of Stochastic Processes (1979), Springer-Verlag: Springer-Verlag Berlin · Zbl 0404.60061 [2] Gopalsamy, K., Global asymptotic stability in a periodic Lotka-Volterra system, J. Aust. Math. Soc. Ser. B, 27, 66-72 (1988) · Zbl 0588.92019 [3] Ikeda, N.; Wantanabe, S., Stochastic Differential Equations and Diffusion Processes (1981), North-Holland: North-Holland Amsterdam [4] Khas’minskii, R. Z., Stochastic Stability of Differential Equations (1981), Sijthoff & Noordhoff: Sijthoff & Noordhoff Rockville, MD · Zbl 0441.60060 [5] Lipshter, R. S.; Shyriaev, A. S., Statistics of Stochastic Processes (1974), Nauka: Nauka Moscow [6] Mao, X., Stochastic Differential Equations and Applications (1997), Ellis Horwood: Ellis Horwood Chichester · Zbl 0874.60050 [7] Mao, X.; Sabais, S.; Renshaw, E., Asymptotic behavior of stochastic Lotka-Volterra model, J. Math. Anal., 287, 141-156 (2003) 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.