Population models with Allee effect: a new model

Abstract

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.

Global behavior of solutions of a periodically forced Sigmoid Beverton–Holt model

“…Lemma 9 Assume that {a n } is a positive periodic sequence with period p, δ ∈ (1, ∞), and let condition (14) hold. Assume that {ũ n } is a positive p-periodic solution of Equation (19) with (20).…”

“…In [32] several periodically forced discrete models exhibiting the Allee effect are studied, while a class of general unimodal maps with such properties has been investigated in [14]. The effects of harvesting in some periodically forced population models with the Allee effect are studied in [42].…”

“…Periodically forced population models exhibiting the Allee effect are relatively new in the literature [14,32,42]. In [32] several periodically forced discrete models exhibiting the Allee effect are studied, while a class of general unimodal maps with such properties has been investigated in [14].…”

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

“…In particular, our theoretical results are in agreement with the fact that the maximum tolerable depensation can vary with time in the study of fish populations and the observed Allee effect [5] in smooth cordgrass could be modelled with a periodic system such as the one studied here. 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].…”

A biographical sketch of Robert J. Sacker

“…The classical Lyapunov -Perron method offers a linkage between certain solutions of equation (14) and the existence of exponential dichotomies for equation (13). For example, if f ¼ f ðtÞ is in the space BðnÞ ¼ CðR; XÞ > L 1 ðR; XÞ of bounded continuous functions with the sup-norm, and equation (13) The use of several examples of admissible pairs plays a major role in this paper.…”