Allee effects in stochastic populations

Abstract

Dennis, B. 2002. Allee effects in stochastic populations. -Oikos 96: 389-401.The Allee effect, or inverse density dependence at low population sizes, could seriously impact preservation and management of biological populations. The mounting evidence for widespread Allee effects has lately inspired theoretical studies of how Allee effects alter population dynamics. However, the recent mathematical models of Allee effects have been missing another important force prevalent at low population sizes: stochasticity. In this paper, the combination of Allee effects and stochasticity is studied using diffusion processes, a type of general stochastic population model that accommodates both demographic and environmental stochastic fluctuations. Including an Allee effect in a conventional deterministic population model typically produces an unstable equilibrium at a low population size, a critical population level below which extinction is certain. In a stochastic version of such a model, the probability of reaching a lower size a before reaching an upper size b, when considered as a function of initial population size, has an inflection point at the underlying deterministic unstable equilibrium. The inflection point represents a threshold in the probabilistic prospects for the population and is independent of the type of stochastic fluctuations in the model. In particular, models containing demographic noise alone (absent Allee effects) do not display this threshold behavior, even though demographic noise is considered an "extinction vortex". The results in this paper provide a new understanding of the interplay of stochastic and deterministic forces in ecological populations.

The risk of establishment of aquatic invasive species: joining invasibility and propagule pressure

“…Following Leung et al (2004), we used a Weibull function to link propagule pressure to the probability of establishment for an invasible site (see also Dennis 2002),…”

Experimental demonstration of an Allee effect in microbial populations

“…A strong Allee effect occurs when the per capita growth rate is negative for some small population size, which gives rise to a critical density [5]. This critical density can be detected in the relationship between a population's probability of establishment and its initial size (see [8]). A strong Allee effect induces a sigmoidal relationship with an inflection point at the threshold [8].…”

“…This critical density can be detected in the relationship between a population's probability of establishment and its initial size (see [8]). A strong Allee effect induces a sigmoidal relationship with an inflection point at the threshold [8]. In contrast, for populations without a strong Allee effect, the probability of establishment will be a concave function of initial density owing to demographic stochasticity.…”

Allee effects and extinction in a lattice model

“…In real populations, the positive correlation between size and per capita growth rate of a population is known as the Allee effect (Allee, 1931), which has recently received much interest (e.g. Dennis, 2002;Hurford et al, 2006;Johnson et al, 2006). If the Allee effect is strong enough, the population size may even decrease for small population sizes as in our model.…”

“…Allen et al, 2005) or diffusion processes (e.g. Dennis, 2002), lattice based models have space, as well as time, as a variable and take into account individuals, rather than just the macroscopic view of the population. After introducing the model in the next section, we examine the Allee effects present in our model in Section 3, particularly with respect to a sudden decrease in population in Section 5.…”