All other things being equal, populations cannot grow indefinitely. Eventually, they will hit up against their enviornment’s capacity constraints.
The customary way of modeling capacity-constrained population growth is by way of the logistic function. The logistic function is a growth rate that allows a population level to grow exponentially when this level is far from the environmental carrying capacity. Growth slows as the population level approaches capacity and eventually levels off, thereby preventing overpopulation. Growth, so modeled, presents the appearance of an “S” curve.
The literature on the logistic function is interesting. Some if it has the intent of demonstrating tempered, S curve growth. Others, such as Marcus du Sautoy and his The Great Unknown: Seven Journeys to the Frontiers of Science, focus on the idea that logistic population growth exhibits chaotic behavior, as in chaos theory. Specifically, du Sautoy and others argue that population levels vary chaotically after a short period of time when the net reproduction rate (the difference between the natural birth and death rates) is sufficiently high. On this view, deterministic population growth models cannot produce reliable predictions of future population levels after just a short period of time.
Clearly, these two approaches are not compatible. And the latter flies in the face of many efforts to predict population levels.
In Logistic Growth and its accompanying appendix, I explore this issue. My findings can be summarized as follows:
- Claims about logistic population growth depend on how time is treated in the formal model.
- When time is treated in continuous terms, the resulting model, the logistic equation, produces non-chaotic S curve growth.
- When time is treated discretely, the resulting model, the logistic map, produces chaotic change after just a few time periods once the net reproduction rate becomes sufficiently large.
- The literature on the logistic function evinces considerable indifference with respect to the dependence of its conclusions on how time is modeled. Some of it seems unaware of the dependence. Others are aware, but do not appear to see the need to justify one choice over the other. Neither approach is tenable. Given the difference in results, and the gravity of this difference, a defensible justification for the approach taken to modeling time is not optional.
- I argue that time is best modeled in continuous terms in formal models of population growth. The chaotic implications of discrete logistic growth models are interesting, counterintuitive, and disturbing. But they are not meaningful when considered as empirical predictions.
- More importantly, the mechanism at the heart of the issue, the logistic function, is a poor model of capacity-constrained population growth. I say this because it implies that a given species has some sort of built-in self-correcting adjustment mechanism with respect to its procreative behavior. This is not a plausible claim because it lacks an empirical referent.
- I present an alternative model of capacity-constrained population growth that is not reliant on the dubious logistic function. Instead, it uses an empricially plausible brute force starvation mechanim to keep population levels in check. This alternative produces S curve growth. And crucially, it does so when time is modeled in both continuous and discrete terms. No chaos is evident.
In short, much ado about nothing.
I conclude with a brief discussion of the implications of my findings for realistic population growth models and for the methodology of dynamic process models as such.
Once again, the details of my argument can be found in the following notes:
