Analysis of Oscillatory Patterns of a Discrete-Time Rosenzweig-MacArthur Model

作者：Wu Xiaoqin P; Wang Liancheng^*

来源：International Journal of Bifurcation and Chaos, 2018, 28(6): 1850075.

DOI：10.1142/S021812741850075X

• Rosenzweig-MacArthur model
• flip bifurcation
• Neimark-Sacker bifurcation
• stability
• 1:2
• 1:3
• 1:4 strong resonances

This article investigates the oscillatory patterns of the following discrete-time Rosenzweig-MacArthur model
{x(1)[n + 1] = (1+lambda)x(1)[n] - ax(1)(2) [n] - bx(1)[n]x(2)[n]/1 + mx(1)[n],
x(2)[n + 1] = (1+ mu)x(2)[n] + cx(1)[n]x(2)[n]/1 + mx(1)[n].
The system describes the evolution and interaction of the populations of two associated species (prey and predator) from generation to generation. We show that this system can exhibit co-dimension-1 bifurcations (flip and Neimark-Sacker bifurcations) as. crosses some critical values and codimension-2 bifurcations (1: 2, 1: 3, and 1: 4 resonances) for certain critical values of (lambda, mu) at the positive equilibrium point. The normal form theory and the center manifold theorem are used to obtain the normal forms. For codimension-2 bifurcations, the bifurcation diagrams are established by using these normal forms along the orbits of differential equations. Numerical simulations are presented to confirm the theoretical results.

• 出版日期2018-6-15