Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map

Abstract

A recurring problem in population biology – as well as other stochastic dynamical systems in biology, the physical and social sciences – is the distinction between the ‘true’ dynamics of a system and observational noise: i.e. can we from present data reliably infer e.g. biological mechanisms, or are signals swamped by noise. Here, we approach this problem using the canonical model for simple systems that exhibit complex behaviour, the logistic map. At each time-point noise is added, which allows us to study the long-term behaviour of a system which exhibits both non-linear dynamics and intrinsic noise. We show that the interplay between deterministic non-linear dynamics and simple Gaussian noise results in a perplexingly simple system when viewed statistically. In particular we show that for the case of Gaussian noise it is possible to derive at very reliable approximations for the time until the system has reached an absorbing state. This generic model allows us, for example, to study the life-time of molecular species involved in noisy feedback loops.