Global bifurcation for discrete competitive systems in the plane 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) Abstract: A global bifurcation result is obtained for families of competitive systems of difference equations
$x_{n+1} = f_\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 nondecreasing in $x$ and nonincreasing in $y$, and $g_\alpha(x, y)$ is nonincreasing in $x$ and nondecreasing in $y$.
A unique interior fixed point is assumed
for all values of the parameter $\alpha$. $x_{n+1} = F_\alpha(x_n, x_{n1}), \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, periodtwo solution.
Received: September 2008; Revised: February 2009; Published: May 2009. 
2012 Impact Factor.88
