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STABILITY OF THE UNFOLDING OF THE PREDATOR-PREY MODEL
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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 | ||||
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Subjects: | Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery |
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Journal or Publication Title: | DYNAMICS AND STABILITY OF SYSTEMS | ||||
Publisher: | CARFAX PUBL CO | ||||
ISSN: | 0268-1110 | ||||
Official Date: | 1994 | ||||
Dates: |
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Volume: | 9 | ||||
Number: | 3 | ||||
Number of Pages: | 17 | ||||
Page Range: | pp. 179-195 | ||||
Publication Status: | Published |
Data sourced from Thomson Reuters' Web of Knowledge
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