Well, this applies generally to all these models—why should it look like an exponential or a power law at all to begin with? These are simplifications that are born out of the fact that we can write out these very simple ODE models that reasonably approximate the dynamics and produce meaningful trajectories.
However I think “sigmoid” is definitely the most likely pattern, if we broaden that term to mean not strictly just the logistic function (which is the solution of y′=y(1−y) ) but also any other kind of similar function that has an S shape, possibly not even symmetric. “Running into a wall” is much more unphysical—it implies a discontinuity in the derivative that real processes never exhibit.
Also you could see it as this: all these are special cases of a more general y′=P(y), where P is any polynomial. And that means virtually any analytical function, since those can be Taylor-expanded into polynomials reaching arbitrary accuracy in the neighbourhood of a specific point. So really the only assumptions baked in there are:
the rate of growth is analytical (no weird discontinuities or jumps; reasonable)
the rate of growth does not feature an explicit time dependence (also sensible, as these phenomena should happen equally regardless of which year they were kickstarted in)
Within this framework, the exponential growth is the result of a first order expansion, and the logistic is a second order expansion (under certain conditions for the coefficients). Higher orders, if present, could give rise to more complex models, but generally speaking as far as I can tell they’ll all tend to either converge to a given value (a root of the polynomial) or diverge to infinity. It would be interesting to consider the conditions under which convergence occurs I guess; it should depend on the spectrum of the polynomial but it might have a more physical interpretation.
Well, this applies generally to all these models—why should it look like an exponential or a power law at all to begin with? These are simplifications that are born out of the fact that we can write out these very simple ODE models that reasonably approximate the dynamics and produce meaningful trajectories.
However I think “sigmoid” is definitely the most likely pattern, if we broaden that term to mean not strictly just the logistic function (which is the solution of y′=y(1−y) ) but also any other kind of similar function that has an S shape, possibly not even symmetric. “Running into a wall” is much more unphysical—it implies a discontinuity in the derivative that real processes never exhibit.
Also you could see it as this: all these are special cases of a more general y′=P(y), where P is any polynomial. And that means virtually any analytical function, since those can be Taylor-expanded into polynomials reaching arbitrary accuracy in the neighbourhood of a specific point. So really the only assumptions baked in there are:
the rate of growth is analytical (no weird discontinuities or jumps; reasonable)
the rate of growth does not feature an explicit time dependence (also sensible, as these phenomena should happen equally regardless of which year they were kickstarted in)
Within this framework, the exponential growth is the result of a first order expansion, and the logistic is a second order expansion (under certain conditions for the coefficients). Higher orders, if present, could give rise to more complex models, but generally speaking as far as I can tell they’ll all tend to either converge to a given value (a root of the polynomial) or diverge to infinity. It would be interesting to consider the conditions under which convergence occurs I guess; it should depend on the spectrum of the polynomial but it might have a more physical interpretation.