Date Published: March 23, 2017
Publisher: Public Library of Science
Author(s): Lijuan Qin, Feng Zhang, Wanxiong Wang, Weixin Song, Gui-Quan Sun.
Understanding Allee effect has crucial importance for ecological conservation and management because it is strongly related to population extinction. Due to various ecological mechanisms accounting for Allee effect, it is necessary to study the influence of multiple Allee effects on the dynamics and persistence of population. We here focus on organism-environment feedback which can incur strong, weak, and fatal Allee effect (AE-by-OEF), and further examine their interaction with the Allee effects caused by other ecological mechanisms (AE-by-OM). The results show that multiple Allee effects largely increase the extinction risk of population either due to the enlargement of Allee threshold or the change of inherent characteristic of Allee effect, and such an increase will be enhanced dramatically with increasing the strength of individual Allee effects. Our simulations explicitly considering spatial structure also demonstrate that local interaction among habitat patches can greatly mitigate such superimposed Allee effects as well as individual Allee effect. This implies that spatially structurized habitat could play an important role in ecological conservation and management.
Allee effect, the positive relationship between per capita growth rate (individual’s fitness) and population density at low population density , has received considerable attention in ecology and conservation because it is directly related to population extinction [2–4] and can incur complicated spatial pattern [5, 6]. Many mechanisms may account for Allee effect such as organism-environment feedback [7, 8], mate-finding limitations [9–11], cooperative defense [12, 13], social dysfunction [1, 14, 15], inbreeding depression [16, 17], and predator avoidance or defense [1, 18, 19]. Allee effect also occurs at a metapopualtion level similar to local populations, which may be because of dispersal cost or colonization difficulty [15, 20–23]. Due to various ecological mechanisms leading to Allee effect, it is necessary to study the influence of multiple Allee effects on population dynamics and persistence. Berec et al.  has suggested that two or more Allee effects could affect the dynamics of a population, and more empirical evidence for multiple Allee effects have been provided . We here studied organism-environment positive feedback when Allee effect caused by other ecological mechanism is involved in also.
Organism-environment positive feedback will individually incur Allee effect (i.e. AE-by-OEF) when λ > (μ + d)2 / d, of which inherent characteristic depends on model parameters (Eqs 1 or 2 with a = 0). For example, the population will be subjected to weak, strong, and fatal Allee effect in turn with increasing habitat destruction rate d (see Table 2, Figs 1 and 2). Individual AE-by-OM in our model is strong when a < (δ–β/(β + 1))2/(4δ) and δ < β/(β + 1) (where δ = (e + d)/c, α = λ/d, β = μ/d), and fatal otherwise (Fig 1). This system becomes more complicated when the two types of Allee effects are superimposed and interacted. Through analyzing per capita growth rate when habitat dynamics is at equilibrium (i.e. dh/dt = 0), we found that the combinations of strong AE-by-OM and either strong or weak AE-by-OEF, or even organism-environment feedback without Allee effect (λ > (μ + d)2/d), are all able to lead to strong Allee effect with higher Allee threshold (Fig 3A, 3C and 3E). These combinations can also incur fatal Allee effect which means population extinction in any case (Fig 3B, 3D and 3F). Phase plane analysis also confirmed these results (Fig 4).
Organism-environment positive feedback is ubiquitous in real world, especially in severe environment such as dry land [42–44] and intertidal zone . Allee effect is a distinct feature caused by the positive feedback. Due to diverse mechanism leading to Allee effect, it is necessary to estimate the effect of Allee effect caused by the organism-environment feedback on population dynamics and persistence when another mechanism generating Allee effect also act on the population [10, 24]. We here study theoretically the interaction of the two type of Allee effect to reveal their effect on population dynamics and persistence. The combination of multiple Allee effects could largely increase the extinction risk of population either due to the enlargement of Allee threshold or the change of inherent characteristic of Allee effect (Fig 3). We further found that the spatial range of the organism-environment feedback also profoundly affect the appearance of Allee effect. Both combined and individual Allee effect greatly mitigates when interaction is limited locally (Fig 5). This implies spatial scale could play an important role in ecological conservation [12, 15, 39, 45].
When a = 0, the Eq 2 returns to Eq 1. When habitat dynamics is at equilibrium (i.e. dh/dt = 0), the per capita growth rate of population (i.e.1pdpdt) is
From the condition of Fig 2, we deduce that organism-environment positive feedback will individually incur Allee effect (AE-by-OEF) when λ > (μ + d)2/d. We summarize parameter conditions of various Allee effects (Details are shown in Table 2). When λ = 0, the AE-by-OEF disappears and the population is only subjected to AE-by-OM in Eq 2. Through simple calculation, the interior equilibrium must be the solution of following equation
Thus, When λ = 0, Eq 2 has at most two interior equilibriums, and necessary conditions for the existence of two interior equilibriums are a < (δ–β/(β + 1))2/(4δ) and δ < β/(β + 1). When δ > β/(β + 1), there is no interior equilibrium.