| Volume 7, Number 2, 2017, Pages 411-426 DOI：10.11948/2017026 |
| Bifurcation and chaos in a discrete time predator-prey system of Leslie type with generalized Holling type III functional response |
| Ali Atabaigi |
| Keywords:Predator-prey system, discrete time dynamical system, period doubling bifurcation, Neimark-Sacker bifurcation, chaotic dynamic, chaos Control. |
| Abstract: |
| This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type III functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of $\mathbb{R}_+^2$. Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-$2$, $4$, $8$, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method. |
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