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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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