# American Institute of Mathematical Sciences

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

 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.
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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