Positive feedback with resource constraint (logistic dynamics)

Ball describes autocatalysis as a positive feedback process in which the more is made, the faster it appears. He says this lets small random fluctuations grow into large non-uniformities, the patches that Turing's process can generate.³ He also states the limit. Unchecked, the feedback runs away until the ingredients that feed it are exhausted.³ For chemical reactions, he says the outcome depends on a balance between reaction rates and diffusion rates, because fresh reagents arrive by diffusion.³ The constraint is therefore part of the account, not an extra.

Sorek, Balaban and Loewenstein (2013) argue that memory needs a long time-scale, which multistable dynamics provide. They state that bistability emerges when positive feedback and saturation are both present.¹ Their gene model has a protein that binds cooperatively to its own promoter and so drives its own synthesis.² The authors note that many gene regulatory networks meet both conditions, and they cite the lactose promoter in E. coli as a bistable example.¹

Pietak and Levin (2016) conclude that positive feedback can generate strong membrane voltage gradients across a cell collective. They suggest this may be one way bioelectric pre-patterns are generated and manipulated.⁴ Watson and colleagues (2016) describe a different loop. Correlated behaviours strengthen network interactions, and stronger interactions produce more correlated behaviours.⁵ Reid (2023) reviews earlier studies finding that ants relying more on positive feedback converge quickly on an optimal maze solution. The cost is lower adaptability to change than in Physarum, which likely relies on it less.⁸

Zakirov and colleagues (2021) use the logistic shape as a clue in angiogenesis. They say logistic-like shapes commonly come from positive feedback under a global resource constraint, here the limited actin that filopodia use to extend.⁶ Their simulations predict hysteresis and bistability in endothelial cell states. They take the logistic-like intensity curve as evidence of this feedback. They write that the shape may indicate positive feedback, and they report finding evidence of it.⁶⁷