Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model

Abstract

Abstract A discrete-time Holling-Tanner model with ratio-dependent functional response is examined. We show that the system experiences a flip bifurcation and Neimark-Sacker bifurcation or both together at positive fixed point in the interior of ℝ+2 $\mathbb {R}^{2}_{+}$ when one of the model parameter crosses its threshold value. We concentrate our attention to determine the existence conditions and direction of bifurcations via center manifold theory. To validate analytical results, numerical simulations are employed which include bifurcations, phase portraits, stable orbits, invariant closed circle, and attracting chaotic sets. In addition, the existence of chaos in the system is justified numerically by the sign of maximum Lyapunov exponents and fractal dimension. Finally, we control chaotic trajectories exists in the system by feedback control strategy.

Keywords

• Holling-Tanner model
• Ratio-dependent
• Bifurcations
• Chaos
• Feedback control