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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:
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