Identifying persistence in enzyme kinetics and growth dynamics using the a-m model

The $a-m$ model governed by the equation $ay^3+y=mx$ with $a>0$, traces S-curves on an $xy-$ plane. Their superposition has been used to model the protease kinetic activity on various substrates as well as growth of yeast cultures, bacterial colonies, plants, and animals. Enzyme kinetics and growth are dynamic processes that exhibit correlations across multiple temporal scales. However, concrete conclusions about the time-scale dependence using a generic model are absent in the literature. In this paper, we analyze the Fourier representation of the $a-m$ model superposition and observe the black noise spectrum, which shows extreme peaking at low frequencies. This reflects the observed self-organization nature of these dynamic processes with marked persistence. Based on the Fourier sum, we conclude that both dynamic processes are linear along with dominant slow-varying oscillations in time. In other words, both processes reach saturation at a constant rate with the added autocorrelated black noise.