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 three 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 non-autonomous 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.

Keywords:

• unimodal Allee maps
• threshold point
• carrying capacity
• composition map
• stability
• bifurcation

Notes

1. [email protected]

2. [email protected]

3. This work is part of the first author's PhD dissertation.

4. The command ‘resultant’ is a powerful tool that helps us in finding the implicit solutions for polynomial equations with low degree. We are not aware of similar techniques that work for non-polynomial equations such as the Ricker map .

Additional information

Notes on contributors

Saber Elaydi

1

Henrique Oliveira

2 3