Global higher bifurcations in coupled systems of nonlinear eigenvalue problems

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Abstract

Coexistent steady-state solutions to a Lotka-Volterra model for two freely-dispersing competing species have been shown by several authors to arise as global secondary bifurcation phenomena. In this paper we establish conditions for the existence of global higher dimensional n-ary bifurcation in general systems of multiparameter nonlinear eigenvalue problems which preserve the coupling structure of diffusive steady-state Lotka-Volterra models. In establishing our result, we mainly employ the recently-developed multidimensional global multiparameter theory of Alexander-Antman. Conditions for ternary steady-state bifurcation in the three species diffusive competition model are given as an application of the result.

Original languageEnglish (US)
Pages (from-to)113-120
Number of pages8
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume106
Issue number1-2
DOIs
StatePublished - 1987

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Nonlinear Eigenvalue Problem
Coupled System
Lotka-Volterra Model
Bifurcation
Competing Species
Competition Model
Steady-state Solution
Ternary
High-dimensional

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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abstract = "Coexistent steady-state solutions to a Lotka-Volterra model for two freely-dispersing competing species have been shown by several authors to arise as global secondary bifurcation phenomena. In this paper we establish conditions for the existence of global higher dimensional n-ary bifurcation in general systems of multiparameter nonlinear eigenvalue problems which preserve the coupling structure of diffusive steady-state Lotka-Volterra models. In establishing our result, we mainly employ the recently-developed multidimensional global multiparameter theory of Alexander-Antman. Conditions for ternary steady-state bifurcation in the three species diffusive competition model are given as an application of the result.",
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AB - Coexistent steady-state solutions to a Lotka-Volterra model for two freely-dispersing competing species have been shown by several authors to arise as global secondary bifurcation phenomena. In this paper we establish conditions for the existence of global higher dimensional n-ary bifurcation in general systems of multiparameter nonlinear eigenvalue problems which preserve the coupling structure of diffusive steady-state Lotka-Volterra models. In establishing our result, we mainly employ the recently-developed multidimensional global multiparameter theory of Alexander-Antman. Conditions for ternary steady-state bifurcation in the three species diffusive competition model are given as an application of the result.

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