Allee Effects, Invasion Pinning, and Species' Borders

Abstract

All species' ranges are the result of successful past invasions. Thus, models of species' invasions and their failure can provide insight into the formation of a species' geographic range. Here, we study the properties of invasion models when a species cannot persist below a critical population density known as an "Allee threshold." In both spatially continuous reaction-diffusion models and spatially discrete coupled ordinary-differential-equation models, the Allee effect can cause an invasion to fail. In patchy landscapes (with dynamics described by the spatially discrete model), range limits caused by propagation failure (pinning) are stable over a wide range of parameters, whereas, in an uninterrupted habitat (with dynamics described by a spatially continuous model), the zero velocity solution is structurally unstable and thus unlikely to persist in nature. We derive conditions under which invasion waves are pinned in the discrete space model and discuss their implications for spatially complex dynamics, including critical phenomena, in ecological landscapes. Our results suggest caution when interpreting abrupt range limits as stemming either from competition between species or a hard environmental limit that cannot be crossed: under a wide range of plausible ecological conditions, species' ranges may be limited by an Allee effect. Several example systems appear to fit our general model.

How Predation can Slow, Stop or Reverse a Prey Invasion

“…This bound is similar to, but tighter than, previously deduced bounds that ensure the prey-only system is stationary (Keitt et al, 2000). Figure 15 shows the above analytical bound, regions for invading and stationary solutions for the prey-only model (35) with v ≡ 0, and regions for invading, stationary and receding solutions for the full predator-prey system (35).…”

“…16 below). A similar phenomenon has been analysed by Keitt et al (2000) in an ecological context for the prey-only model. It also arises as so-called 'propagation failure' in models for excitable systems in physiology (Keener, 1987(Keener, , 1993.…”

Roles of parasites in animal invasions

“…because mating encounters are rare) [34]. Apparent competition: indirect interaction leading to negative effects of one species on another, mediated through a shared natural enemy, such as predators and parasites.…”

Allee Effect and Control of Lake System Invasion

“…We start analysis of the model (10) by observing that, provided (w in − w out ) is sufficiently small and the initial lake population is also small, then the invader population in the lake will remain small (see also [13]). Proposition 1.…”

“…The importance of Allee effect for spatio-temporal dynamics of biological invasions has been shown in [16,13,8]. It appears that in presence of the effect the invasion front can move slower, stop, or even reverse its direction.…”