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Computational Ecology and Software, 2015, 5(3): 222-238
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Article

Bifurcation and complex dynamics of a discrete-time predator-prey system involving group defense

S. M. Sohel Rana
Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh

Received 1 May 2015;Accepted 10 June 2015;Published online 1 September 2015
IAEES

Abstract
In this paper, we investigate the dynamics of a discrete-time predator-prey system involving group defense. The existence and local stability of positive fixed point of the discrete dynamical system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of R+2 by using bifurcation theory. Numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including phase portraits, period-7, 20-orbits, attracting invariant circle, cascade of period-doubling bifurcation from period-20 leading to chaos, quasi-periodic orbits, and sudden disappearance of the chaotic dynamics and attracting chaotic set. The Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors.

Keywords discrete-time predator-prey system;chaos;flip and Neimark-Sacker bifurcations;Lyapunov exponents.



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