## Allee Effects at the Population Level

Single Populations

Allee argued that benefits of sociality would put small populations at risk because they would receive fewer benefits of interactions compared to larger populations. This idea has prompted the concern among population biologists that Allee effects induce thresholds below which a population will crash, a phenomenon sometimes called critical depensation. To see why depensation occurs, imagine two populations: one experiences survival according to the unimodal relationship in Figure 2, while a second experiences only competitive interactions (dashed line in Figure 2). Surviving individuals in each population produce five offspring and then die. We can calculate the change in population size as A = Nt + 1/N„ where Nt and Nt +1 are the populations sizes in the tth and following generation, respectively. A population declines if A < 1, grows if A > 1, and replaces itself when A = 1. The population experiencing only competition starts with a very high growth rate that declines as population size increases, crossing the line of replacement at a high population size. At larger population sizes the population decreases. This point (A in Figure 3) often referred to as the carrying capacity, is therefore a stable equilibrium.

Now consider a population with Allee effects also included. At high population sizes, this population has the same pattern as a population with only competitive

Figure 3 Change in population size (a) as a function of initial population size (Nt, log-transformed to emphasize small population sizes) resulting from a unimodal fitness relationship (solid line) and a purely competitive relationship (dashed line). The dotted line represents population replacement (a = 1). Arrows represent population trajectories. White dots are stable equilibria, while black dots are unstable equilibria. See text for other details.

Figure 3 Change in population size (a) as a function of initial population size (Nt, log-transformed to emphasize small population sizes) resulting from a unimodal fitness relationship (solid line) and a purely competitive relationship (dashed line). The dotted line represents population replacement (a = 1). Arrows represent population trajectories. White dots are stable equilibria, while black dots are unstable equilibria. See text for other details.

Figure 4 Unimodal fitness functions in relation to the line of population replacement. Populations characterized by fitness functions crossing the line of replacement have unstable thresholds (black dots) at small population sizes, while those characterized by relationships above the line of replacement do not (unstable threshold occurs only in an extinct population).

interactions. However, the population growth line also crosses the line of replacement at low population sizes (point B). The population grows when Nt is greater than at point B, but below this point the population declines. Point B is therefore an unstable equilibrium, and represents a threshold below which the population is at high risk of extinction. As the fitness curve sinks further and further below the line of replacement, Allee effects create extinction thresholds at larger population sizes (Figure 4). Note, however, that the fitness curve could in theory lie entirely above the line of replacement. In such cases, Allee effects exist but the theoretical unstable equilibrium occurs at Nt = 0 (total extinction).

Researchers have in fact found extinction thresholds in a number of species. For example, in several flowering plant species pollinated by animals, small isolated populations have low fertilization success, set fewer seeds, and are at risk of extinction, whereas larger populations have

higher pollination rates, seed set, and tend to expand. The theoretical comparisons in Figure 4 indicate that some species might be particularly vulnerable to population crashes resulting from Allee effects, while others might not. Sensitive species might include schooling fish, which require large numbers of conspecifics to form a functional school. Insensitive species would presumably have high fecundity or other traits enabling them to exponentially increase at low population sizes. Strictly speaking, such populations would also have to be parthenogenetic or clonal in order to escape mate-finding limitations at very low population densities.

### Spatial Ecological Aspects

The theoretical examples used thus far assume that the population is isolated and limited by a carrying capacity, as might happen if it occurred on an island. In reality, populations often spread after surpassing carrying capacity, as individuals disperse from their natal population. In such circumstances, the population is characterized by a region of high density (the population center) and a region of low density (the population front). Theoretical studies show that Allee effects can delay rates of spread because the same threshold at low population sizes limits spatial spread at the front. Whereas traditional diffusion models often ignore the initial delays in population spread and sometimes have relegated them to an 'establishment phase', inclusion of Allee effects link establishment directly with population spread, resulting in a better ability to predict the spatial dynamics of invading species.

By reducing rates of spread and by intensifying con-specific attraction, Allee effects further consolidate individuals and thereby tend to increase the clumping and subdivision of a population. Theoretical models incorporating subdivision of populations connected by random dispersal indicate that the existence of multiple subpopulations can buffer declining populations because large subpopulations above the extinction threshold can rescue smaller ones. Whereas single-population models incorporating an Allee effect result in two stable equilibria (population at carrying capacity or extinction), the existence of a second population can result in two additional stable equilibria in which one population (a source) is near carrying capacity while the other population (a sink) is below the extinction threshold but persists due to dispersal from the first population (Figure 5a).

In addition, models that incorporate conspecific attraction can improve persistence even more. If individuals assess local population density during settlement, they should prefer populations at low-to-medium densities over high-density populations because fitness at low

(a) Random settlement

Habitat selection

(a) Random settlement

Habitat selection

Figure 5 Trajectories of the size of two populations (N1 and N2, represented as proportions of carrying capacity) with an extinction threshold at either 40% (a, b) or 60% (c, d) of carrying capacity, under either random settlement (a, c) or density-dependent habitat selection (b, d). Arrows represent population trajectories, while lines represent boundaries of population dynamics. White dots are stable equilibria, black dots are unstable equilibria, and the hatched areas represent combinations of N1 and N2 resulting in extinction of the entire population. Density-dependent habitat selection reduces the number of possible combinations resulting in total extinction compared to random settlement, while at the same time reducing the number of extant populations.

Figure 5 Trajectories of the size of two populations (N1 and N2, represented as proportions of carrying capacity) with an extinction threshold at either 40% (a, b) or 60% (c, d) of carrying capacity, under either random settlement (a, c) or density-dependent habitat selection (b, d). Arrows represent population trajectories, while lines represent boundaries of population dynamics. White dots are stable equilibria, black dots are unstable equilibria, and the hatched areas represent combinations of N1 and N2 resulting in extinction of the entire population. Density-dependent habitat selection reduces the number of possible combinations resulting in total extinction compared to random settlement, while at the same time reducing the number of extant populations.

densities increases as a function of density. It is at low-to-medium densities at which populations may be subject to threshold population declines. Therefore, as long as a pool of settlers exists, a population near the extinction threshold can be buffered by density-dependent habitat selection (Figure 5b). These effects can be particularly strong when more than two local populations exist and Allee effects are large relative to carrying capacity (compare Figures 5b and 5d).

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