  Information theory provides a useful tool to understand the evolution of
complex nonlinear systems and their sustainability. In particular, Fisher
Information (FI) has been evoked as a useful measure of sustainability and the
variability of dynamical systems including self-organising systems. By
utilising FI, 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 feedbacks 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 FI 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 fluctuation in negative
feedback.
