STABILITY ANALYSIS AND FLIP BIFURCATION OF A DISCRETE-TIME PREY-PREDATOR MODEL WITH PREDATOR IMMIGRATION
F˙IGEN KANGALG˙IL *
Dokuz Eyl¨ul University, Bergama Vocational School, ˙Izmir, Turkey.
N˙IL ¨UFER TOPSAKAL
Department of Mathematics, Sivas Cumhuriyet University, Sivas, Turkey.
*Author to whom correspondence should be addressed.
Abstract
Aims: In this paper, a discrete-time prey-predator model with predator immigration has been considered.
Methodology: The model was mathemati-cally analysed and simulated in Mapple.
Results: The complex dynamical behavior of the presented model has been analyzed. Stability and existence of coexistence positive fixed point have been investigated. Moreover, using bifurcation theory, it has been shown that the model undergoes Flip bifurcation. Also, direction of Flip bifurcation has been given. Some numerical simulations including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the model have been given to support of the
analytical finding. The computation of the maximum Lyapunov exponents has confirmed the presence of chaotic behavior in the system.
Conclusion: The model was successfully built, analysed and simulated showing results that matched the theory.
Keywords: Predator-Prey model, fixed point, stability, immigration flip bifurcation.