LONG MEMORY AND SCALING FOR MULTIPLICATIVE STOCHASTIC-PROCESSES WITH APPLICATION TO THE STUDY OF POPULATION OSCILLATIONS

An analytically tractable multiplicative random process is introduced based on an analogy between the random phase modulation, wave propagat ion in a random medium and the population growth in a fluctuating envi ronment. It is assumed that the process depends on a multiplicative ra ndom parameter which can be eliminated by introducing an intrinsic tim e scale; the relation between the intrinsic and the physical (watch) t ime scales is determined by the stochastic properties of the random pa rameter. The multitime joint probability densities of the state variab les expressed in terms of the physical time can be computed in a close d form in terms of the corresponding joint probability densities expre ssed in the intrinsic time scale. The theory is applied to the study o f age-dependent population oscillations in a random environment. In th is case the intrinsic time scale is a biological time which is the sam e for any physical realization of the random environment. The random f luctuations of the environment lead to a decrease of the intrinsic rat e of population growth and generate a temporal analogue of Anderson lo calization: due to fluctuations the population oscillations are damped . The asymptotic behavior of the process depends on the range of the m emory effects of the environmental fluctuations: for short memory the qualitative asymptotic behavior of the population size for large time is the same for a fluctuating as well as for a constant environment; f or slowly decaying correlations, however, the exponential increase of the population is outweighed by a compressed exponential decay due to environmental fluctuations and the population eventually becomes extin ct.