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Codimension-two bifurcations of fixed points in a class of discrete prey-predator systems

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Abstract
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.
Keywords
MODEL, MAPS

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Please use this url to cite or link to this publication:

Chicago
Khoshsiar Ghaziani, Reza, Willy Govaerts, and Charlotte Sonck. 2011. “Codimension-two Bifurcations of Fixed Points in a Class of Discrete Prey-predator Systems.” Discrete Dynamics in Nature and Society.
APA
Khoshsiar Ghaziani, Reza, Govaerts, W., & Sonck, C. (2011). Codimension-two bifurcations of fixed points in a class of discrete prey-predator systems. DISCRETE DYNAMICS IN NATURE AND SOCIETY.
Vancouver
1.
Khoshsiar Ghaziani R, Govaerts W, Sonck C. Codimension-two bifurcations of fixed points in a class of discrete prey-predator systems. DISCRETE DYNAMICS IN NATURE AND SOCIETY. 2011;
MLA
Khoshsiar Ghaziani, Reza, Willy Govaerts, and Charlotte Sonck. “Codimension-two Bifurcations of Fixed Points in a Class of Discrete Prey-predator Systems.” DISCRETE DYNAMICS IN NATURE AND SOCIETY (2011): n. pag. Print.
@article{1894259,
  abstract     = {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.},
  articleno    = {862494},
  author       = {Khoshsiar Ghaziani, Reza and Govaerts, Willy and Sonck, Charlotte},
  issn         = {1026-0226},
  journal      = {DISCRETE DYNAMICS IN NATURE AND SOCIETY},
  keyword      = {MODEL,MAPS},
  language     = {eng},
  pages        = {24},
  title        = {Codimension-two bifurcations of fixed points in a class of discrete prey-predator systems},
  url          = {http://dx.doi.org/10.1155/2011/862494},
  year         = {2011},
}

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