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

Abstract

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.

Hierarchical competition models with the Allee effect II: the case of immigration

“…An example is an invasive species x that strongly competes with a native species y. The terms e −m 1 /(1+s 1 x) and e −m 2 /(1+s 2 y) represent the positive density dependence of x and y, respectively, which is the hallmark of the Allee effect caused by predator saturation [1,2,9,12,13,16,19,21,24,25,27]. The parameter m i represents the predation intensity, and s i is a parameter proportional to the 'handling time', that is, the duration of the capture and eating phase, i = 1, 2.…”

Catastrophes, connectivity and Allee effects in the design of marine reserve networks

“…1b). Alternative models of a Beverton–Holt model with Allee effects are also possible (Gaut et al 2012). The growth function for each patch is then:…”

“…1c). This results in a spatially‐implicit model as the spatial structure between the reserves is not considered (Perry and Enright 2012). We model dispersal success as an exponentially decaying function of distance, d , between reserves at a rate δ to yield the post‐dispersal population sizes: …”

Difference population equation with variable Allee effect and periodic carrying capacity

“…The new mathematical models were developed with the different types of the compensatory mechanism with respect to fluctuating carrying capacity and, therefore, with the globally or locally attractive periodic solutions which are attenuant, resonant or neither. To name just a few of the publications: [36], [35], [22], [23], [26], [1], [42], [27], [29], [9], [7], and [10].…”

“…Since with the defined set of parameters K, r, A, and C for the case of the weak Allee effect and Let us recall the criterion for the asymptotic stability of the equilibrium point: 27). Let x * be an equilibrium point of the difference equation…”