Growth/decay on a seascape

 Variations at each node are governed by time-scales from growth, migration, and fluctuation

 However, the variations of the mean population     are not directly sensitive to migration and stochasticity

Assuming a separation of time scale in which evolution of      is slow enough for    to reach a quasi-state state, such that

 

There is a mechanism to generate the empirical Richard's growth equation for sufficiently strong seascape stochasticity

Note that the above conclusion holds if the node-level logistic growth equations are replaced with any analytic function of

 It is likely that some of these conclusions apply beyond the specific mean-field model, as:

 Fitness stochasticity (seascape) leads to very large local fluctuations,

Migration between sites somewhat smoothens the distribution, and

if the overall population changes slowly, can lead to broad (power-law) quasi-steady states of the form

 Averaging any analytical local growth rule then leads to the Richard's equation for the variations of the mean population

A potentially testable link between the local population distribution, and the global growth exponent.