More and more I’m thinking that our continual attempt to try to ignore time (that things/events are born, live and die and no objects are eternal) is detrimental and that it’s better to describe all things as transient, even if it means modifying some of our mathematical descriptions of things.
Non-autonomy: explicit time dependence
Another important aspect of biological regulatory processes, which receives surprisingly little attention, is the explicit time dependence of these systems. As soon as we consider cellular dynamics, development, or evolution over a large-enough time span, the organisation of the underlying regulatory system starts to change. This affects the parameters—not just the state variables—of the system. Such explicitly time-dependent dynamical systems are called non-autonomous [32,81,82]. Time-dependent signalling cues and environmental conditions have long been known to shape many processes in the fields of evolutionary and developmental biology. Obvious examples of such phenomena are inductive processes or external (e. g. seasonal) cues that are essential to trigger many developmental pathways (as described in standard textbooks such as [83,84]), or evolutionary dynamics driven by changing environmental conditions (examples, based on the simulation of gene regulatory network models, can be found in [85-88]).
Still, it is rare to find studies based on explicitly non-autonomous models in the literature, and most authors avoid the challenge of dealing with dynamical systems where the parameters representing external cues are time-dependent. This is the case in the study of gap domain shifts by Manu and colleagues [35,89], where maternal morphogen gradients providing regulatory input to the system were assumed to reach steady state before gap gene boundary positioning was analysed. Such simplifications can be risky, especially when describing biological phenomena where the time scales of change in parameters and state variables are of similar order. In such cases, time scales should not be separated, nor quasi-steady states considered since it is easy for dynamical properties and behaviours of the system to be missed or misinterpreted under these conditions.
Time Dependent Saddle Node Bifurcation: Breaking Time and the Point of No Return in a Non-Autonomous Model of Critical Transitions
“For a system with a small λV, there exists a significant window of opportunity (τ^,τ∗) during which rapid reversal of the environment can save the system from catastrophe”