Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 862494, 27 pages
Research Article

Codimension-Two Bifurcations of Fixed Points in a Class of Discrete Prey-Predator Systems

1Department of Applied Mathematics and Computer Science, Shahrekord University, P.O. Box 115, Saman Road, Shahrekord 88186-34141, Iran
2Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, 9000 Gent, Belgium

Received 9 December 2010; Revised 7 March 2011; Accepted 20 March 2011

Academic Editor: Juan J. Nieto

Copyright © 2011 R. Khoshsiar Ghaziani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The dynamic behaviour of a Lotka-Volterra system, described by a planar map, is analytically and numerically investigated. We derive analytical conditions for stability and bifurcation of the fixed points of the system and compute analytically the normal form coefficients for the codimension 1 bifurcation points (flip and Neimark-Sacker), and so establish sub- or supercriticality of these bifurcation points. Furthermore, by using numerical continuation methods, we compute bifurcation curves of fixed points and cycles with periods up to 16 under variation of one and two parameters, and compute all codimension 1 and codimension 2 bifurcations on the corresponding curves. For the bifurcation points, we compute the corresponding normal form coefficients. These quantities enable us to compute curves of codimension 1 bifurcations that branch off from the detected codimension 2 bifurcation points. These curves form stability boundaries of various types of cycles which emerge around codimension 1 and 2 bifurcation points. Numerical simulations confirm our results and reveal further complex dynamical behaviours.