My research is about identifying features that allow species to persist in a variable environment. When the focus is on one species, the task is to determine which environmental features lead to extinction and which provide protection from extinction. When there are multiple species, the idea is the same; there are features of the environment that might promote extinction and features that allow species to avoid extinction. The trick is to recognize that when we consider multiple species, factors promoting or delaying extinction will depend on other species.
In any case, we need a measure that helps us understand the risk of species to going extinct. Here, theory is very helpful. Across a number of models, a reliable measure of persistence is what is known as the low density growth rate. One can measure the low density growth rate by perturbing a species to low density and measuring how fast it recovers. In models, we take the mathematical limit as population size goes to zero.
Logistic Model of Population Growth
The logistic model is the simplest model of population regulation. The population grows to carrying capacity, given by the letter K.
The low-density growth rate for this model is r. When r > 0, the population grows from low density. One can see this by dividing both sides by N and then setting N = 0 on the right-hand side.
The low density growth rate can be any number. When it is positive, it means that a population may increase in abundance when perturbed to low density. When it is negative, it means that we expect a population to go extinct.
When might one want to know what factors drive a species extinct? In population management, we generally want to avoid extinction and so a common technique is to use models to discern whether a particular human activity might cause the low density growth rate to go negative. If we can find as much, we have good reason to argue against that particular human activity.
In other instances, there are species that we might wish to drive extinct in particular contexts. Pests and invasive species are one such example. But there are many more. One potentially powerful application of these ideas is to models of human disease. For cancer therapies, models can help us understand how to drive cancer populations inside a human body to have negative low density growth rates. An analogous idea applies to infectious diseases. We used this idea and calculated the average low density growth rate in a model of infectious disease motivated by the COVID-19 pandemic.
Two species who are similar in all respects except their low density growth rates. The species with the large low density growth rate (black) recovers quickly when pushed to low density. The species with the small low density growth rate (gray) spends long time periods at low density. It is much more likely to go extinct.
The low density growth rate is also helpful for understanding interactions between species, especially competitors. When one species is at low density, its growth is determined solely by the environment and the effects of the other species. This is a type of manipulation that removes any effects of within-species density-dependence. Perturbing species to low density is common in mathematical studies of competition, but is also relevant in experimental studies, although it is rarely done at scale in nature.
The figure to the left shows two hypothetical species in the same environment. The species are identical except for their low density growth rate. The low density growth rate is a good predictor of extinction in this scenario because it predicts how much time the population spends with few individuals. This is precisely when extinction risk is high.