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 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.
Highlights
Mathematical modeling is a promising approach to understand and analyze the dynamics of ecological systems
The Leslie type predator-prey model has received more interest to investigate the dynamical behaviors between the species
A number of famous ecologist and mathematician have been given attention and investigated extensively Holling-Tanner models [1,2,3]. Their empirical works found complex dynamical behaviors including stable or unstable limit cycle, stability states around positive equilibrium. They showed that the asymptotic stability of the positive equilibrium does not imply the global stability
Summary
Mathematical modeling is a promising approach to understand and analyze the dynamics of ecological systems. A number of famous ecologist and mathematician have been given attention and investigated extensively Holling-Tanner models [1,2,3] Their empirical works found complex dynamical behaviors including stable or unstable limit cycle, stability states around positive equilibrium. A discrete-time Holling and Leslie type predator-prey system with constant-yield prey harvesting analyzed in [16], in [17] the authors investigated a discrete Holling-Tanner model and a discrete predator-prey model with modified Holling-Tanner functional response discussed in [18] These studies paid their attention to determine the stability and directions of flip and Neimark-Sacker bifurcations via use of center manifold theory.
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