`a`
Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Global bifurcation for discrete competitive systems in the plane

Pages: 133 - 149, Volume 12, Issue 1, July 2009

doi:10.3934/dcdsb.2009.12.133       Abstract        Full Text (799.7K)       Related Articles

M. R. S. Kulenović - University of Rhode Island, Kingston, RI 02881, United States (email)
Orlando Merino - University of Rhode Island, Kingston, RI 02881, United States (email)

Abstract: 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.

Keywords:  bifurcation, competitive, map, global stable manifold, monotonicity, period-two solution.
Mathematics Subject Classification:  Primary: 37G35 Secondary: 39A10, 39A11.

Received: September 2008;      Revised: February 2009;      Published: May 2009.