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Unsustainable zip-bifurcation in a predator–prey model involving discrete delay

Published online by Cambridge University Press:  03 December 2013

Jocirei D. Ferreira  and
V. Sree Hari Rao
Jocirei D. Ferreira
Affiliation:
Federal University of Mato Grosso, Institute of Exact and Earth Science, Barra do Garcas, Mato Grosso, Brazil (jocirei@ufmt.br)
V. Sree Hari Rao
Affiliation:
Jawaharlal Nehru Technological University, Department of Mathematics, Hyderabad 500 085, India
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Abstract

A third-order system of ordinary differential equations, modelling two predators competing for a single prey species, is analysed in this paper. A delay term modelling the delayed logistic growth of the prey is included. Fixed points of the system are identified, and a linearized stability analysis is carried out. For some parameter regime, there exists a continuum of equilibria and these equilibria may undergo a zip bifurcation. The main results presented herein are that this zip bifurcation is ‘unsustainable’ for certain ranges of values of the time-delay parameter. Finally, spatial diffusion is incorporated in the delay differential equation model, and it is shown that the zip bifurcation remains unsustainable.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 143 , Issue 6 , December 2013 , pp. 1209 - 1236
Copyright
Copyright © Royal Society of Edinburgh 2013 

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