Noise suppresses or expresses exponential growth

In this paper we will show that noise can make a given system whose solutions grow exponentially become a new system whose solutions will grow at most polynomially. On the other hand, we will also show that noise can make a given system whose solutions are bounded become a new system whose solutions will grow exponentially. In other words, we reveal that the noise can suppress or expresses exponential growth

Asymptotic properties of stochastic population dynamics

In this paper we stochastically perturb the classical Lotka{Volterra model x_ (t) = diag(x1(t); ; xn(t))[b + Ax(t)] into the stochastic dierential equation dx(t) = diag(x1(t); ; xn(t))[(b + Ax(t))dt + dw(t)]: The main aim is to study the asymptotic properties of the solution. It is known (see e.g. [3, 20]) if the noise is too large then the population may become extinct with probability one. Our main aim here is to nd out what happens if the noise is relatively small. In this paper we will establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss the limit of the average in time of the sample paths