Sustainable theory of a logistic model - Fisher information approach.

Information theory provides a useful tool to understand the evolution of complex nonlinear systems and their sustainability. In particular, Fisher information has been evoked as a useful measure of sustainability and the variability of dynamical systems including self-organising systems. By utilising Fisher information, we investigate the sustainability of the logistic model for different perturbations in the positive and/or negative feedback. Specifically, we consider different oscillatory modulations in the parameters for positive and negative feedback and investigate their effect on the evolution of the system and Probability Density Functions (PDFs). Depending on the relative time scale of the perturbation to the response time of the system (the linear growth rate), we demonstrate the maintenance of the initial condition for a long time, manifested by a broad bimodal PDF. We present the analysis of Fisher information in different cases and elucidate its implications for the sustainability of population dynamics. We also show that a purely oscillatory growth rate can lead to a finite amplitude solution while self-organisation of these systems can break down with an exponentially growing solution due to the periodic fluctuations in negative feedback.