Bifurcation Analysis of a Discrete-Time Two-Species Model

Abstract EN

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.