Population collapse to extinction: the catastrophic combination of parasitism and Allee effect

Abstract

Infectious diseases are responsible for the extinction of a number of species. In conventional epidemic models, the transition from endemic population persistence to extirpation takes place gradually. However, if host demographics exhibits a strong Allee effect (AE) (population decline at low densities), extinction can occur abruptly in a catastrophic population crash. This might explain why species suddenly disappear even when they used to persist at high endemic population levels. Mathematically, the tipping point towards population collapse is associated with a saddle-node bifurcation. The underlying mechanism is the simultaneous population size depression and the increase of the extinction threshold due to parasite pathogenicity and Allee effect. Since highly pathogenic parasites cause their own extinction but not that of their host, there can be another saddle-node bifurcation with the re-emergence of two endemic equilibria. The implications for control interventions are discussed, suggesting that effective management may be possible for R 0 1.

Fatal disease and demographic Allee effect: population persistence and extinction

“…Using a similar SI model with the strong Allee effect, Thieme et al [16] established by mathematical theorems that the transition from population decline to population collapse is mediated by a Hopf bifurcation and a heteroclinic orbit. The SI models of Hilker et al and Thieme et al are structurally similar to predator-prey models that have an Allee effect in the prey population and a linear functional response (prey eaten per predator per unit time) [2,3,6,10,[12][13][14]17,18]. However, most of these studies on the interplay ofAllee effects and infectious diseases, especially those in the conservation biology literature, seem to be largely concerned with the role of the Allee effect at small population densities [9].…”

“…In these models, when the strong Allee effect is missing in the host demographics, the transition from host persistence to host extinction occurs 'gradually' or 'slowly' while the model parameters vary. However, in the presence of a strong Allee effect, the transition from host persistence to extinction is much more abrupt [12].…”

Allee effects in a predator--prey system with a saturated recovery function and harvesting

“…It has been shown in a large number of theoretical works that replacing the logistic growth function with an Allee-type function having a threshold (the strong Allee effect) could dramatically change the patterns of dynamics ( [19,20,21,22,23,24]. Interestingly, even in the case where the Allee effect is weak, the model predictions on ecosystem behaviour can be rather different compared to the same systems with logistic growth [25,26,27].…”

Dynamics of SI epidemic with a demographic Allee effect

“…This threshold quantity R(p) is defined to be the average number of secondary infections produced by an infective individual during the entire infectious period Hethcote (2000). It should be noted that some authors use the term reproduction number/ratio instead of replacement number as in (Hilker et al, 2009;Hilker, 2010). Since in the absence of the disease the population being above the Allee threshold will settle at its carrying capacity then setting p = 1 in Eq.…”