Allee effects, extinctions, and chaotic transients in simple population models

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible-infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.

Section: Introductionmentioning

confidence: 99%

Fatal disease and demographic Allee effect: population persistence and extinction

Friedman

Yakubu

2012

“…Surprisingly, the literature on mathematical modeling of the Allee effect is lagging behind. For the convenience of the reader we cite some of the recent mathematical papers on modeling the Allee effect: [5], [15], [18], [21], [24], [25].…”

Section: Introductionmentioning

confidence: 99%

Population models with Allee effect: a new model

Elaydi

Sacker

2009

In this paper we develop several mathematical models of Allee effects. We start by defining the Allee effect as a phenomenon in which individual fitness increases with increasing density. Based on this biological assumption, we develop several fitness functions that produce corresponding models with Allee effects. In particular, a rational fitness function yields a new mathematical model that is our focus of study. Then we study the dynamics of 2-periodic systems with Allee effects and show the existence of an asymptotically stable 2-periodic carrying capacity.

“…See [9] for a discussion of some new examples of models exhibiting the Allee effect and, similar to the Beverton-Holt model, having important biological quantities as parameters, for example, intrinsic growth rate, carrying capacity, Allee threshold, and a new parameter, the shock recovery parameter. Further references pertaining to the Allee effect can be found in [1,3,4,6,[10][11][12][16][17][18]23,26,31,32], and for references to the general theory of difference equations, see [7,20]. For a discussion on the use of the Sigmoid model, see [28, p. 82] In what follows, we show that under certain conditions on the coefficients, Equation (1) has an asymptotically stable p-periodic state and an unstable p-periodic Allee state.…”

Section: Introductionmentioning

confidence: 74%

Difference equations with the Allee effect and the periodic Sigmoid Beverton–Holt equation revisited

Gaut

Goldring

Grogan

et al. 2012

In this paper, we investigate the long-term behaviour of solutions of the periodic Sigmoid Beverton-Holt equationwhere the a n and δ n are p-periodic positive sequences. Under certain conditions, there are shown to exist an asymptotically stable p-periodic state and a p-periodic Allee state with the property that populations smaller than the Allee state are driven to extinction while populations greater than the Allee state approach the stable state, thus accounting for the long-term behaviour of all initial states. This appears to be the first study of the equation with variable δ. The results are discussed with possible interpretations in Population Dynamics with emphasis on fish populations and smooth cordgrass.