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Long term behavior of solutions of the Lotka-Volterra system under small random perturbations. (English) Zbl 1061.34513

This paper deals with a stochastic analogue of the Lotka-Volterra model for predator-prey systems where the birth rate of the prey and the death rate of the predator are perturbed by independent white noises with intensities of order ${\epsilon }^{2}$ ($\epsilon$ is a small parameter). Such a model can be described by the stochastic differential equations

$d{X}_{t}={X}_{t}\left(a-b{Y}_{t}\right)dt+\epsilon {\sigma }_{1}{X}_{t}d{W}_{t}^{1},\phantom{\rule{1.em}{0ex}}d{Y}_{t}={Y}_{t}\left(-c+d{X}_{t}\right)dt+\epsilon {\sigma }_{2}{Y}_{t}d{W}_{t}^{2},\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $X$ and $Y$ are the population sizes of prey and pradator, respectively, $a,\phantom{\rule{0.166667em}{0ex}}b,\phantom{\rule{0.166667em}{0ex}}c,\phantom{\rule{0.166667em}{0ex}}d,\phantom{\rule{0.166667em}{0ex}}{\sigma }_{1},\phantom{\rule{0.166667em}{0ex}}{\sigma }_{2}$ are positive constants, and the stochastic perturbations ${W}_{t}^{1}$, ${W}_{t}^{2}$ may be either of Itô or Stratonovich type.

The authors show that for small initial population sizes such stochastic model is adequate, whereas for large initial population sizes it seems to be not so suitable, because it leads to ever increasing fluctuations in population sizes, although it still precludes extinction. In particular, the authors establish that large population sizes of predator and prey coexist only for a very short time, and most of the time, one of the populations is exponentially small.

This paper is a continuation of previous investigations of R. Z. Hasminskij [Kybernetica, Praha 4, 260–279 (1968; Zbl 0231.60045)] and extends also results by L. Arnold, W. Horsthemke and J. W. Stuckl [Biom. J. 21, 451–471 (1979; Zbl 0433.92019)].

MSC:
 34F05 ODE with randomness 34C60 Qualitative investigation and simulation of models (ODE) 34E10 Perturbations, asymptotics (ODE) 60H10 Stochastic ordinary differential equations 92D25 Population dynamics (general)