BIFURCATION AND CHAOS IN A DISCRETE TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE WITH GENERALIZED HOLLING TYPE Ⅲ FUNCTIONAL RESPONSE | Zendy

Ali Atabaigi | Zendy

Open Access

BIFURCATION AND CHAOS IN A DISCRETE TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE WITH GENERALIZED HOLLING TYPE Ⅲ FUNCTIONAL RESPONSE

Author(s) -

Ali Atabaigi

Publication year - 2017

Publication title -

journal of applied analysis and computation

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.55

H-Index - 21

eISSN - 2158-5644

pISSN - 2156-907X

DOI - 10.11948/2017026

Subject(s) - mathematics , bifurcation , center manifold , lyapunov exponent , saddle node bifurcation , period doubling bifurcation , bifurcation theory , transcritical bifurcation , discrete time and continuous time , functional response , bifurcation diagram , control theory (sociology) , chaotic , fixed point , mathematical analysis , hopf bifurcation , nonlinear system , physics , predation , computer science , predator , paleontology , statistics , control (management) , quantum mechanics , artificial intelligence , biology

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 R+. 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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