Bifurcation Analysis of a Discrete-Time Two-Species Model

• Published in: Discrete dynamics in nature and society Vol. 2020; no. 2020; pp. 1 - 12
• Main Author: Khan, A. Q.
• Format: Journal Article
• Language: English
• Published: Cairo, Egypt Hindawi Publishing Corporation 2020; Hindawi; John Wiley & Sons, Inc; Hindawi Limited
• Subjects: Bifurcation theory; Computer simulation; Liapunov exponents; Quasi-Periodic Oscillations
• ISSN: 1026-0226; 1607-887X
• DOI: 10.1155/2020/2954059

We study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant R + 2 . It is proved that model has two boundary equilibria: O 0,0 ,A ζ 1 − 1 / ζ 2 , 0 , and a unique positive equilibrium B r e r / e r − 1 , r under certain parametric conditions. We study the local dynamics along their topological types by imposing method of Linearization. It is proved that fold bifurcation occurs about the boundary equilibria: O 0,0 ,A ζ 1 − 1 / ζ 2 , 0 . It is also proved that model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium B r e r / e r − 1 , r and meanwhile stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasi-periodic oscillations between host and parasitoid populations. Some simulations are presented to verify theoretical results. Finally, bifurcation diagrams and corresponding maximum Lyapunov exponents are presented for the under consideration model.