Simple dynamic models of the Allee effect

It is interesting to explore the consequences of depensation on population dynamics. First, consider a continuous model of density dependence

$$\dot N = R(N)N.$$

The presence of an Allee effect implies that the per capita growth rate $R(N)$ is not steadily decreasing with $N$, rather, for low $N$ values, it is an increasing function of density. For high values ​​of $N$, it is logical to assume that the phenomenon of intraspecific competition is anyway operating so that $R(N)$is decreasing with density (see Fig. 1 which is a revised version of the original Allee figure). As a result, the per capita rate of demographic increase is a unimodal function of density, as shown in Fig. 6(A). Sometimes $R(N)$ might even be negative for small $N$: in this case depensation is termed critical depensation (see again Fig. 6(A)), because at low density the mortality rate is larger than the birth rate.

In the case of non-critical depensation there exists only one non vanishing equilibrium population indicated with the label $K$ in the figure. It is easy to understand that this equilibrium population is also stable ($dN/dt> 0$ if $N<K$, while $dN/dt<0$ if $N>K$) and coincides with the carrying capacity. Therefore the effect of depensation operates only during the transient phase of population growth: if the population were reduced to small numbers, the demographic recovery would be very slow at the beginning and therefore this will increase the probability that the population is drawn into an extinction vortex due to the phenomena we are going to study later (demographic, environmental, genetic stochasticity, etc.), however, if these phenomena did not occur, the population could slowly recover and reach the carrying capacity.

Figure 6: Behaviour of the per capita growth rate ($R(N)$, panel A) and the rate of population growth ($N R(N)$, panel B) as a function of density $N$ in the case of non-critical depensation (dashed curve) and critical depensation (solid curve).

Figure 7: Demographic growth rates of a population in which the functional response of generalist predators can cause depensation. (A) For low predator density, depensation is not critical; (B) for high predator density, depensation is critical. The dashed curve is the population growth rate without predation, the orange curve is the predation rate.

If depensation is critical there exist two non-null equilibrium values, indicated in the figure with the labels $K$ and $K'$. Because in this case $R(N)N$ has the shape shown as a solid curve in Fig. 6(B), it turns out that:

$$\begin{array}{rcl} \dfrac{dN}{dt} < 0 & \text{if} & N < K' \text... ...} N > K\\ &&\\ \dfrac{dN}{dt} > 0 & \text{if} & K'< N < K .\\ \end{array}$$

Therefore $K'$ is an unstable equilibrium, while $K$ and the null population (extinction) are stable. The important result is thus that the Allee effect can produce a population threshold $K'$ below which extinction is certain.

We already mentioned that one of the causes of the effect can be the increased risk of predation to which small populations are exposed (like in the common guillemot example previously mentioned ). It is easy to use the theory of the predator's functional response (Holling, 1966) to obtain this intuitive result. Suppose that the population under study does not display inverse density dependence when predators are absent. We can assume for example that the population has logistic demography. Consider then the case in which predators are present and assume that these predators have a considerable variety of available prey, so their density (which we denote by $Y$) does not depend on the density of the organisms we are considering. If predators are generalists, such as seagulls that prey on guillemots, we can assume that $Y$, at least in the area we are studying, is practically constant. Indicate the predator's functional response by $p(N)$ and the per capita growth rate with no predators by $R^*(N)$; then the dynamics of $N$ is given by the equation

$$\dot{N} = NR^*(N) - Yp(N).$$

If the demography is logistic and the functional response is of the second type, the equation becomes

$$\dot{N} = rN\left( {1 - \frac{N}{K}} \right) - Y\dfrac{{\alpha N}}{{N + \beta }}$$

where $r$ is the intrinsic rate of demographic increase , $\beta$ the semi-saturation constant and $\alpha$ the maximum rate of consumption of each predator. Therefore the per capita growth rate $R(N)$ that actually governs the population in the area where $Y$ predators are present is given by

$$R(N) = r\left( {1 - \frac{N}{K}} \right) - Y\dfrac{{\alpha }}{{N + \beta }.}$$

The possible shapes of the population growth rates $N R(N)$ are reported in Fig. 7. The shape of $R(N)$ is actually unimodal and therefore gives rise to depensation. It is easy to understand that the growth rate is lower for small populations because the mortality rate due to predation is given by

$$\dfrac{Y\alpha}{{N + \beta }}$$

and thus the risk of death decreases with $N$. This depensation is critical if $Y$ exceeds a certain threshold (see panel B in Fig. 7). It is easy to compute the threshold by requiring that $R(0)$ be negative, namely that

$$r - Y\dfrac{{\alpha }}{\beta } < 0 .$$

We obtain the result that depensation is critical for $Y > \beta r/\alpha$, namely when the number of predators is high enough.