Population dynamics with threshold effects give rise to a diverse family of Allee effects

Abstract

The Allee effect describes populations that deviate from logistic growth models and arises in applications including ecology and cell biology. A common justification for incorporating Allee effects into population models is that the population in question has altered growth mechanisms at some critical density, often referred to as a threshold effect. Despite the ubiquitous nature of threshold effects arising in various biological applications, the explicit link between local threshold effects and global Allee effects has not been considered. In this work, we examine a continuum population model that incorporates threshold effects in the local growth mechanisms. We show that this model gives rise to a diverse family of Allee effects and we provide a comprehensive analysis of which choices of local growth mechanisms give rise to specific Allee effects. Calibrating this model to a recent set of experimental data describing the growth of a population of cancer cells provides an interpretation of the threshold population density and growth mechanisms associated with the population.Here, C(t) ≥ 0 is the density at time t, dC(t)/dt is the growth rate, r > 0 is the low-density growth rate, C(0) is the initial density, and K > 0 is the carrying capacity density. Equation (1) has two equilibria: C * = 0 and C * = K, where an equilibrium is any value C * such that dC(t)/dt = 0 when C(t) ≡ C * . Since densities near C(t) ≡ K will approach K, while densities near C(t) ≡ 0 diverge away from zero, we say that C * = K is a stable equilibrium, while C * = 0 is an unstable equilibrium. This means that the logistic growth model implicitly assumes that all densities, no matter how small, eventually thrive, since lim t→∞ C(t) = K forMathematical models that include an Allee effect relax the assumption that all population densities will thrive and survive, which is inherent in (1) (Edelstein-Keshet, 2005;Murray, 2003;Stephens et al., 1999;Taylor and Hastings, 2005). Consequently, populations described using Allee effect models exhibit more complicated and nuanced dynamics, including reduced growth at low densities (Gerlee, 2013;Johnson et al., 2006;Neufeld et al., 2017) and extinction below a critical density threshold (Allee and Bowen, 1932;Courchamp et al., 1999;Taylor and Hastings, 2005). The phrase Allee effect can have many different interpretations in different parts of the literature. For instance, the Weak Allee effect is used to describe density growth rates that deviate from logistic growth, but do not include additional equilibria (Edelstein-Keshet, 2005;Murray, 2003;Stephens et al., 1999;Taylor and Hastings, 2005). A common mathematical description of the Weak Allee effect iswhere the factor 1 + C(t)/A represents the deviation from the classical logistic growth model. Despite the similarity between (1) and (2), it is not possible to write down an explicit solution for (2) in terms of C(t), like we can for (1). Despite this, we are still able to examine the equilibria of (2) to understand the salient features of the Weak Al...