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Volume 11,     Number 2,     Summer 2003

 

HOPF BIFURCATIONS IN DISCRETE MAY-LEONARD COMPETITION MODELS
LIH-ING W. ROEGER

Abstract. The discrete-time Hopf bifurcations at the interior equilibrium of four discrete-time May-Leonard (M-L) competition models are analyzed and compared with the continuous M-L model. Chi et al. [SIAM J. Appl. Math., 58:211-226, 1998] had shown that the Hopf bifurcation of the continuous M-L model is degenerate, neither supercritical nor subcritical. However, several discrete-time M-L models show the supercritical Hopf bifurcations. We used the method of center manifolds and normal forms to verify our results and showed that the discrete model derived using Kahan's method has vanishing first focal value, the same as the continuous M-L model. The model derived from annual plant competition possesses supercritical bifurcation. The M-L discrete model of the Ricker type and the model derived using Mickens' nonstandard method have subcritical or supercritical bifurcations depending on the parameters.

 

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