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July  2008, 10(1): 109-128. doi: 10.3934/dcdsb.2008.10.109

An investigation of the global properties of a two-dimensional competing species model

Patrick J. Johnson 1, and  Mark E. Burke 1,

1. 

Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland, Ireland

Received  July 2007 Revised  December 2007 Published  April 2008

In this paper we demonstrate how the global dynamics of a biological model can be analysed. In particular, as an example, we consider a competing species population model based on the discretisation of the original Lotka-Volterra equations. We analyse the local and global dynamic properties of the resulting two-dimensional noninvertible dynamical system in the cases when the interspecific competition is considered to be “weak”, “strong” and “mixed”. The main results of this paper are derived from the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated by the use of critical curves, a powerful tool for the analysis of the global properties of noninvertible two-dimensional maps.
Keywords: Global bifurcations; Noninvertible maps; Critical curves; Competing species..
Mathematics Subject Classification: 37N25, 92B0.
Citation: Patrick J. Johnson, Mark E. Burke. An investigation of the global properties of a two-dimensional competing species model. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 109-128. doi: 10.3934/dcdsb.2008.10.109
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