  We consider the problem of finding optimal strategies that maximize the
average growth-rate of multiplicative stochastic processes. For a geometric
Brownian motion the problem is solved through the so-called Kelly criterion,
according to which the optimal growth rate is achieved by investing a constant
given fraction of resources at any step of the dynamics. We generalize these
finding to the case of dynamical equations with finite carrying capacity, which
can find applications in biology, mathematical ecology, and finance. We
formulate the problem in terms of a stochastic process with multiplicative
noise and a non-linear drift term that is determined by the specific functional
form of carrying capacity. We solve the stochastic equation for two classes of
carrying capacity functions (power laws and logarithmic), and in both cases
compute optimal trajectories of the control parameter. We further test the
validity of our analytical results using numerical simulations.
