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WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

In the previous post, we discussed the dramatic decline of the Atlantic cod (Gadus morhua) off Newfoundland over the past 60 years. I left us with the question of why, given the very limited catch sizes since the 1990’s, there was little evidence of population recovery (at least up until 2005). An Allee effect is a likely explanation for the failure of the population to recover during that extended period of reduced fishing pressure.

Beginning around 1994, the population may have become limited by an Allee phenomenon, or more appropriately mechanism, where a population’s size is limited far below the presumed carrying capacity, or observed maximum population size, because of reduced population size itself. Analogous to carrying capacity, where an upper limit is set on population growth rate by the effects of a relatively large population size, an Allee effect is an upper limit set by relatively small population size. Intuitive examples are easy to find, e.g. (1) species that require sufficient numbers for successful defense against predators will be increasingly limited by predation at low population size; (2) species for which habitat engineering by a sufficient number of individuals is necessary for offspring success; (3) species that depend on a minimum number of participants for the formation of successful mating assemblages. G. morhua, in which individual fecundity increases with age and body size (to a limit) (Fudge and Rose, 2008), is known to form, or have formed, large pelagic assemblages during spawning. Allee effects, therefore, describe situations where individual fitness depends on the presence of conspecifics, and is positively correlated with population size.

One vulnerability of populations subject to Allee effects is that small population size becomes an inescapable trap, with the likelihood of extinction increasing as population size declines. The reasons for this are twofold. First, if growth rates decline to zero or even become negative below an Allee threshold, then the state of zero population size becomes a stable state and extinction is assured. If you recall, our earlier models of population growth considered X= 0 (extinction) to be an unstable steady state; unstable because the addition of reproducing individuals to the population would result in divergence away from the zero state —population growth. Second, even if growth rate never becomes negative below the Allee threshold, a sufficiently large or sustained decline of population size increases the probability of extinction due to random events, a phenomenon termed stochastic extinction. Stochastic extinction, the probability of which could increase with deteriorating environmental conditions, is of interest to anyone studying extinction, including paleontologists, and will be discussed in a later section. Here, however, we will first explore several simple models of Allee effects.

Models of Allee effects

In the logistic model (Eq. 1 here), mortality rate increases as population size, X, approaches carrying capacity K, and population growth rate subsequently declines. The logistic model has two alternative steady states, X=K and X= 0, the latter of which is unstable as discussed above. The extinct state is a stable attractor, however, in the presence of an Allee effect. There are several simple models that demonstrate the effect, but to appreciate them, and the Allee effect itself, let us first examine the relationship between population size and growth rate under the logistic model. If we plot growth rate (dX/dt) against population size in the logistic model (Fig. 1), we see that the rate increases steadily at small population size, reaches a maximum when population size is half of the carrying capacity —X(t) =K/2— and declines steadily thereafter, reaching zero at carrying capacity. This value can be arrived at analytically because what we are visualizing is the rate of change of growth rate itself, technically the second derivative of the logistic growth equation. If we expand the logistic growth rate equation
\frac{dX}{dt} = rX\left ( 1-\frac{X}{K}\right )
\Rightarrow \frac{dX}{dt} = rX - \frac{rX^2}{K}
and take the derivative, we derive the acceleration (or deceleration) of the rate of change of population size as a function of population size itself.
\frac{d^2X}{dt^2} = r - \frac{2rX}{K}
Setting d2X/dt equal to zero —the point at which growth rate is neither accelerating nor decelerating— we get the maximum that is illustrated in Fig. 1.
\frac{d^2X}{dt^2} = r - \frac{2rX}{K} = 0
\Rightarrow X = \frac{K}{2}
The important thing to note here is that growth rate is always positive when 0<X(t)<K, that is, when population size lies between zero and the carrying capacity.

Fig. 1: The relationship between population growth rate and population size under a logistic model. In this example carrying capacity K=100.

There are several ways in which an Allee effect can be modelled in a logistically growing population. For example, if the Allee threshold is represented as a specific population size A, then the effect can be incorporated into the logistic formula as
\frac{dX}{dt} = rX\left( 1-\frac{X}{K}\right ) \left( \frac{X-A}{K}\right )
(Lewis and Kareiva, 1993; Boukal and Berec, 2002). The first term on the RHS of the equation is the logistic function, where growth declines to zero as X approaches K. The second term introduces the threshold, A, with growth rate declining if X < A, and increasing when X > A. Here, the effect is treated as the difference between population size and the threshold, taken as a fraction of carrying capacity, or maximum population size. Note that if A=0 —there is no Allee effect— the model reduces to the logistic growth model. A more nuanced model, where A must be greater than zero —an Allee effect always exists— treats the Allee threshold as equivalent yet opposite to K, representing a lower bound on growth rate (Courchamp et al., 1999).
\frac{dX}{dt} = rX\left( 1-\frac{X}{K}\right ) \left( \frac{X}{A}-1\right )
If A=1 —in which a population comprising a single individual is compromised under all circumstances— then the strength of the Allee effect depends on the size of the population. In both models, growth rate becomes negative below the threshold A, effectively dooming the population to extinction (Fig. 2). This condition is often termed a “strong” Allee effect.

Negative growth rates, a feature that is common to many models of the Allee effect, can be somewhat problematic from a conceptual viewpoint because of their determinism. We’ll pick this point up in the next post, and also discuss why paleontologists might care about both Allee effects, and model determinism.

Fig. 2: Two models of strong Allee effects illustrates as plots of population growth rate vs. population size. K=100. Red shows the first model where growth rate is relative to the Allee threshold A as a function of K. Blue shows the second model where growth rate is relative to the threshold A itself.

Vocabulary
Allee effect — A positive correlation between individual fitness, or population growth rate, and population size. This means that fitness and/or growth rates decrease with declining population size.
Second derivative — The derivative of a function’s derivative (the first derivative), thus the acceleration (deceleration) of a rate. E.g. the first derivative of a body in motion, described by position and time, is velocity or speed. The second derivative is acceleration, or the rate at which the speed is changing.
Stochastic extinction — A relationship between the probability of a population’s extinction, and population size and/or environmental variability. In general, the risk of extinction increases due to random fluctuations of either factor.
Strong Allee effect — Population growth rate becomes negative below some threshold of population size.

References
Boukal, D. S. and Berec, L. (2002). Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. Journal of Theoretical Biology, 218(3):375–394.
Courchamp, F., Clutton-Brock, T., and Grenfell, B. (1999). Inverse density dependence and the Allee effect. Trends in Ecology & Evolution, 14(10):405–410
Fudge, S. B. and Rose, G. A. (2008). Changes in fecundity in a stressed population: Northern cod (Gadus morhua) off Newfoundland. Resiliency of gadid stocks to fishing and climate change. Alaska Sea Grant, University of Alaska Fairbanks.
Lewis, M. and Kareiva, P. (1993). Allee dynamics and the spread of invading organisms.Theoretical Population Biology, 43(2):141–158