UNSPECIFIED (1994) STABILITY OF THE UNFOLDING OF THE PREDATOR-PREY MODEL. DYNAMICS AND STABILITY OF SYSTEMS, 9 (3). pp. 179-195. ISSN 0268-1110

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Abstract

We prove a conjecture of Zeeman that any generic unfolding of the Volterra's original predator-prey model is stable. This well-known two-dimensional model has co-dimension one in the planar Lotka-Volterra system and all its orbits are closed in the region of physical interest. Any generic unfolding of the model locally induces a degenerate Hopf bifurcation, but the presence of a cycle of saddles makes the global stability analysis quite involved. We solve the problem by working in the equivalent replicator system. Our proof of stability uses a family of Lyapunov functions for the unfolding. There are two other co-dimension one bifurcations tit the planar replicator (equivalently Lotka-Volterra) system, which involve cycles of saddles and are therefore non-trivial. In one case we prove the stability of the bifurcation and in the other we determine a topologically versal unfolding of the co-dimension one flow. This then, together with previous work on the subject, completes the study of co-dimension one bifurcations of the system.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery
Journal or Publication Title:	DYNAMICS AND STABILITY OF SYSTEMS
Publisher:	CARFAX PUBL CO
ISSN:	0268-1110
Date:	1994
Volume:	9
Number:	3
Number of Pages:	17
Page Range:	pp. 179-195
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/20074

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