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Asymptotic properties of stochastic population dynamics. (English) Zbl 1171.34038

The authors study asymptotic properties of a solution of the stochastically perturbed classical Lotka-Volterra model

$dx\left(t\right)=\text{diag}\left({x}_{1}\left(t\right),...,{x}_{n}\left(t\right)\right)\left[\left(b+Ax\left(t\right)\right)dt+\beta dw\left(t\right)\right]$

which describes the interactions between $n$ species ($\beta$ is called the intensity of noise).

In the first part, there are presented two important results in this area, given by A. Bahar and X. Mao [J. Math. Anal. Appl. 292, No. 2, 364–380 (2004; Zbl 1043.92034)] which explain the important fact that the environmental noise can make the population extinct.

The next section is concerned with asymptotic properties. The main results are given in Theorem 3.1 and Theorem 3.2. The first theorem says that with probability one the solution will not grow faster that ${t}^{1+\epsilon }$. In the second theorem is proved (using some additional conditions) that the solution will not decay faster than ${t}^{-\left(\theta +\epsilon \right)}$ for a specified $\theta$. Using these two results the authors prove in Theorem 3.3 that the average in time of the norm of the solution is bounded with probability one.

In the last section two examples are given: the one-dimensional case and the multi-dimensional system of facultative mutualism (the coefficients satisfy the conditions: ${a}_{ii}<0$ and ${a}_{ij}\ge 0$, $1\le i,j\le n$, $i\ne j$).

##### MSC:
 34F05 ODE with randomness 60H10 Stochastic ordinary differential equations 34C60 Qualitative investigation and simulation of models (ODE) 92D25 Population dynamics (general) 34D05 Asymptotic stability of ODE