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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.

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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