Stability analysis of a time varying population model without migration

Abstract

In this paper, we analyse the population model (1) $\begin{array}{rl}{x}_{n+1}& =a{x}_n+b{y}_n{e}^{-y_{n-1}},\\ y_{n+1}& =c{y}_n+d{x}_n{e}^{-x_{n-1}},n=0,1,2,\dots .\end{array}$(1) without migration, where $a,b,c,d$ are positive real numbers and the initial conditions $x_{-1},x_0,y_{-1},y_0$ are arbitrary positive real numbers. Here $x_n$ and $y_n$ are considered as population densities of two species in a colony. The parameters a, c are considered as adult surviving rate and b, d are considered as the rate of increase in each species. Bifurcation diagrams are then presented to visualise the periodic and chaotic dynamics. Open problems are proposed to derive the conditions for the solution of the system to be 2,7,14-periodic and chaotic.

Keywords:

• Difference equation
• invariance
• basin of attraction
• global behaviour
• bifurcation

2020 Mathematics Subject Classification:

• 39A22

Acknowledgments

The authors are grateful to the reviewers for their comments and suggestions which help us to improve the presentation of this paper.

Disclosure statement

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