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

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

2010-02-22

arxiv.org/abs/1002.4150v1

Mathematics

Dynamical Systems

Scientific paper

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.

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