Deterministic chaos versus stochastic oscillation in a prey-predator-top predator model

Mathematical Modelling and Analysis

Ranjit Kumar Upadhyay¹  Sharada Nandan Raw²  Rana Parshad³  Malay Banerjee⁴ 

[1] Department of Applied Mathematics, Indian School of Mines Dhanbad, Jharkhand-826004, India;Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand-826004, India;Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13676, USA;Department of Mathematics and Statistics, Indian Institute of Technology Kanpur-208016, India;

关键词: Deterministic chaos;    stochastic oscillation;    Hopf bifurcation;    Holling type IV functional response;   

DOI  :  10.3846/13926292.2011.601767

来源: DOAJ

【 摘 要 】

The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stability of first and second order moments. Exhaustive numerical simulations are carried out to validate the analytical findings.

【 授权许可】

Unknown