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Dynamics of a stochastic Lotka-Volterra model perturbed by white noise. (English) Zbl 1107.92038

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.

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.)
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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 · Zbl 0495.60005
[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) · Zbl 1048.92027
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