Local stability and Chaotic Dynamics of Discrete Time Population Model with and Without Allee Effect

Abstract

Background: In addressing the shortcomings identified in the work of Gumus, this paper focuses on rectifying the mathematical concepts employed in This paper. Furthermore, an extension of the paper's core idea is undertaken for the nth generation of discrete time population models, incorporating considerations with and without Allee effect.

Methods: The correction of mathematical concepts is executed with precision, and the extension to nth generation models is rigorously explored. Analytical techniques are deployed to assess the local stability around fixed points in each generation, with a specific emphasis on the impact of Allee effect on stability regions. Findings: In the 2nd generation, it is empirically observed that the presence of Allee effect significantly diminishes the region of local stability surrounding the equilibrium point. Notably, this reduction is more pronounced compared to the generation without Allee effect. For the 3rd generation, a parallel trend is identified, revealing a shrinking stability region and the subsequent emergence of chaos beyond this region. Results for the nth generation echo those of the 2nd generation, both in terms of stability and instability regions. Novelty and applications: The identified trends in stability and chaos provide valuable information for policymakers and researchers involved in population modelling, aiding in the development of more accurate and predictive models for ecological and demographic studies. The analytical results are further validated through numerical simulations, enhancing the credibility of the findings and facilitating a comprehensive comparison between the 1st and 2nd generations.