Non-autonomous periodic systems with Allee effects

Abstract

A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with tree fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper the properties and stability of the three fixed points are studied in the setting of nonautonomous periodic dynamical systems or difference equations. Finally we investigate the bifurcation of periodic systems/difference equations when the system consists of two unimodal Allee maps.

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

“…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

“…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.…”

On Functions Attracting Positive Entropy

“…The concept of a stable point of a function or dynamical system is considered in [15,16]. Obviously if x 0 is a stable point of f then x 0 is a fixed point of f (the set of all fixed points of f will be denoted by Fix( f )).…”