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July  2010, 14(1): 233-250. doi: 10.3934/dcdsb.2010.14.233

The saddle-node-transcritical bifurcation in a population model with constant rate harvesting

Kie Van Ivanky Saputra 1, , Lennaert van Veen 2, and  Gilles Reinout Willem Quispel 3,

1. 

Faculty of Sciences and Mathematics, Universitas Pelita Harapan, Jl. M.H. Thamrin Boulevard, Tangerang, 15811, Indonesia
2. 

Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe St. N., L1H 7K4 Oshawa, Ontario, Canada
3. 

Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

Received  April 2009 Revised  March 2010 Published  April 2010

We study the interaction of saddle-node and transcritical bifurcations in a Lotka-Volterra model with a constant term representing harvesting or migration. Because some of the equilibria of the model lie on an invariant coordinate axis, both the saddle-node and the transcritical bifurcations are of codimension one. Their interaction can be associated with either a single or a double zero eigenvalue. We show that in the former case, the local bifurcation diagram is given by a nonversal unfolding of the cusp bifurcation whereas in the latter case it is a nonversal unfolding of a degenerate Bogdanov-Takens bifurcation. We present a simple model for each of the two cases to illustrate the possible unfoldings. We analyse the consequences of the generic phase portraits for the Lotka-Volterra system.
Keywords: degenerate Bogdanov-Takens bifurcation, Transcritical bifurcation, nonversal unfolding, Lotka-Volterra model..
Mathematics Subject Classification: Primary: 37H20, 37L10, 37N2.
Citation: Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
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