A stochastic analogue of the Lotka –Volterra model for
predator–prey relationshipis obtained when the birth rate of the prey
and the death rate of the predator are perturbed by independent white noises
with intensities of order $\varepsilon^2$, where $\varepsilon>0$ is a small
parameter.The evolution of this system is studied on large time intervals of
$O(1/\varepsilon^2)$. It is shown that for small initial population sizes the
stochastic model is adequate, whereas for large initial population sizes it is
not as suitable, because it leads to ever-increasing fluctuations in population
sizes, although it still precludes extinction. New results for the classical
deterministic Lotka–Volterra model are obtained by a probabilistic
method; we show in particular 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.