Long memory and scaling for multiplicative stochastic processes with application to the study of population oscillations

Abstract

An analytically tractable multiplicative random process is introduced based on an analogy between the random phase modulation, wave propagation in a random medium and the population growth in a fluctuating environment. It is assumed that the process depends on a multiplicative random parameter which can be eliminated by introducing an intrinsic time scale; the relation between the intrinsic and the physical (watch) time scales is determined by the stochastic properties of the random parameter. The multitime joint probability densities of the state variables expressed in terms of the physical time can be computed in a closed form in terms of the corresponding joint probability densities expressed in the intrinsic time scale. The theory is applied to the study of age-dependent population oscillations in a random environment. In this case the intrinsic time scale is a biological time which is the same for any physical realization of the random environment. The random fluctuations of the environment lead to a decrease of the intrinsic rate of population growth and generate a temporal analogue of Anderson localization: due to fluctuations the population oscillations are damped. The asymptotic behavior of the process depends on the range of the memory 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; for 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 extinct.