Growth & Saturation

  Logistic growth is the most natural model for population dynamics

 Many variants of Generalized Logistic Growth are widely used (e.g. during Covid-19 pandemic)

 What can justify a non-analytic saturation law, as opposed to higher order analytic terms?

The measured quantity is typically the sum (average) of local populations distributed in space:   

 Allowing for migration/dispersal/diffusion, as well as spatio-temporal (seascape) noise, leads to the set of equations:

+

        †

 † (Ito interpretation of multiplicative noise)

 Note that seascape noise (spatio-temporal growth fluctuations) is distinct from reproductive stochasticity (demographic noise): 

Next (Mean-field) Outline Growth & saturation Mean field analysis Extinction on a seascape Growth on a seascape Diversity vs Stability Coexistence on a seascape Phase behavior Summary

 Demographic noise leads to extinction via the Directed Percolation universality class  [Janssen+Tauber (2005), ...]

Including both forms of stochasticity in real space with diffusion leads to:

  †

The linear version (a=0) describes the evolving weight of directed polymers in random media (in the KPZ universality class).

[Tu+Grinstein+Munoz (1997); Munoz+Hwa (1998); ...]