Abstract. 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.

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As a measure of sustainability, Fisher information is employed in the Gompertz growth model. The effect of different oscillatory modulations is examined on the system's evolution and Probability Density Function (PDF). For a sufficiently large frequency of periodic fluctuations occurring in both positive and negative feedbacks, the system maintains its initial conditions. A similar PDF is shown regardless of the initial values when there are periodic fluctuations in positive feedback. By periodic fluctuations in negative feedback, the Gompertz model can lose its self-organization. Finally, despite the fact that the Gompertz and logistic systems evolve differently over time, the results show that they are exceptionally similar in terms of information and sustainability.

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