Global dynamics, Neimark-Sacker bifurcation and hybrid control in a Leslie’s prey-predator model

Abstract

In the present study, we explore the topological classifications at fixed points, global dynamics, Neimark-Sacker bifurcation and hybrid control in the two-dimensional discrete-time Leslie’s prey-predator model. It is proved that for all involved parameters a,b,c,d,h and α, discrete-time Leslie’s prey-predator model has boundary and interior fixed points: Ex0(ab,0),Exy+aαcd+bα,adcd+bα respectively. Then by linear stability theory, local dynamics with different topological classifications are investigated at fixed points: Ex0(ab,0),Exy+aαcd+bα,adcd+bα. Further for the discrete-time Leslie’s prey-predator model, existence of periodic points are also investigated. By bifurcation theory, it is also proved that if (a,b,c,d,h,α)∈NSBExy+aαcd+bα,adcd+bα then at interior fixed point: Exy+aαcd+bα,adcd+bα, discrete Leslie’s prey-predator model undergo Neimark-Sacker bifurcation and no other bifurcation occurs at it. Moreover, hybrid control strategy is applied to control Neimark-Sacker bifurcation. Boundedness and global dynamics of the discrete-time Leslie’s prey-predator model are also investigated. Finally, obtained results are numerically verified.

Keywords

• 40A05
• 35B35
• 92D25
• 70K50
• 39A10