Stochastic Approach to Population Dynamics

We analyze a stochastic approach to population dynamics in which the number of individuals in each class is treated as a stochastic variable. The description of the dynamics is based on the Fokker-Planck equation from which we show that the time evolution of the average number of individuals are identified as the differential equations of the deterministic approach. We also show by calculating the time correlation function that the stochastic fluctuations as well as the stochastic oscillations in the number of individuals are proportional to the square root of the whole size of the population. We have applied the present approach to predator-prey or food chain models with three and four biological species and shown that both models display stochastic oscillations depending on the constant rates. The stochastic approach predicts extinction of species by the stochastic fluctuations in populations with small number of individuals.