Premium
Supercritical Neimark‐Sacker bifurcation of a discrete‐time Nicholson‐Bailey model
Author(s) -
Khan A. Q.
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4934
Subject(s) - mathematics , quasiperiodic function , bifurcation , supercritical fluid , equilibrium point , discrete time and continuous time , parametric statistics , mathematical analysis , focus (optics) , statistical physics , nonlinear system , differential equation , physics , statistics , quantum mechanics , optics , thermodynamics
We study the local dynamics and supercritical Neimark‐Sacker bifurcation of a discrete‐time Nicholson‐Bailey host‐parasitoid model in the interior ofR + 2 . It is proved that if α >1, then the model has a unique positive equilibrium point Pα α − 1ln ( α )2 ,ln ( α )2, which is locally asymptotically focus, unstable focus and nonhyperbolic under certain parametric condition. Furthermore, it is proved that the model undergoes a supercritical Neimark‐Sacker bifurcation in a small neighborhood of the unique positive equilibrium point Pα α − 1ln ( α )2 ,ln ( α )2, and meanwhile, the stable closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasiperiodic oscillations between host and parasitoid populations. Some numerical simulations are presented to verify theoretical results.