July  2009, 12(1): 133-149. doi: 10.3934/dcdsb.2009.12.133

Global bifurcation for discrete competitive systems in the plane

1. 

University of Rhode Island, Kingston, RI 02881, United States, United States

Received  September 2008 Revised  February 2009 Published  May 2009

A global bifurcation result is obtained for families of competitive systems of difference equations

$x_{n+1} = f_\alpha(x_n,y_n) $
$y_{n+1} = g_\alpha(x_n,y_n)$

where $\alpha$ is a parameter, $f_\alpha$ and $g_\alpha$ are continuous real valued functions on a rectangular domain $\mathcal{R}_\alpha \subset \mathbb{R}^2$ such that $f_\alpha(x,y)$ is non-decreasing in $x$ and non-increasing in $y$, and $g_\alpha(x, y)$ is non-increasing in $x$ and non-decreasing in $y$. A unique interior fixed point is assumed for all values of the parameter $\alpha$.
    As an application of the main result for competitive systems a global period-doubling bifurcation result is obtained for families of second order difference equations of the type

$x_{n+1} = F_\alpha(x_n, x_{n-1}), \quad n=0,1, \ldots $

where $\alpha$ is a parameter, $F_\alpha:\mathcal{I_\alpha}\times \mathcal{I_\alpha} \rightarrow \mathcal{I_\alpha}$ is a decreasing function in the first variable and increasing in the second variable, and $\mathcal{I_\alpha}$ is a interval in $\mathbb{R}$, and there is a unique interior equilibrium point. Examples of application of the main results are also given.

Citation: M. R. S. Kulenović, Orlando Merino. Global bifurcation for discrete competitive systems in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 133-149. doi: 10.3934/dcdsb.2009.12.133
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