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Multiple bifurcation in a predator–prey system with nonmonotonic predator response

Published online by Cambridge University Press:  14 November 2011

Franz Rothe  and
Douglas S. Shafer
Franz Rothe
Affiliation:
Mathematics Department, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, U.S.A.
Douglas S. Shafer
Affiliation:
Mathematics Department, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, U.S.A.
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Synopsis

A model of a predator–prey system showing group defence on the part of the prey is formulated, and reduced to a three-parameter family of quartic polynomial systems of equations. Mathematically, this system contains the Volterra–Lotka system, and yields numerous kinds of bifurcation phenomena, including a codimension-two singularity of cusp type, in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation. Biologically, the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant, so that the system exhibits an enrichment paradox, and, for certain choices of parameters, coexistence of stable oscillation and a stable equilibrium.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 3-4 , 1992 , pp. 313 - 347
Copyright
Copyright © Royal Society of Edinburgh 1992

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