Difference population equation with variable Allee effect and periodic carrying capacity

Abstract

In this paper, we develop a population equation based on the Ricker model with periodic carrying capacity and embedded mechanisms of both weak and strong types of Allee effect: $x(n+1)=x\left(n\right)\exp \left(r(1-\frac{x\left(n\right)}{K\left(n\right)})(1-\frac{A+C}{x\left(n\right)+C})\right),n\in {\mathbb{N}}_0.$ We prove a persistence property and existence of periodic solutions in each case of Allee effect. We find sufficient conditions for the periodic solution to be globally asymptotically stable and sufficient conditions for this solution to become unstable. We show that the mechanism of weak Allee effect increases the range of stability for the periodic solution.

Keywords:

• Difference equation
• Allee effect
• periodic carrying capacity
• asymptotic stability
• periodic solutions

2010 Mathematics Subject Classifications:

• 39A23
• 93D20
• 92D25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Government of Alberta, Canada, under Queen Elizabeth II Graduate Scholarship.