Invariant Manifolds for Competitive Systems in the Plane

Let $T$ be a competitive map on a rectangular region $\mathcal{R}\subset \mathbb{R}^2$, and assume $T$ is $C^1$ in a neighborhood of a fixed point $\bar{\rm x}\in \mathcal{R}$. The main results of this paper give conditions on $T$ that guarantee the existence of an invariant curve emanating from $\bar{\rm x}$ when both eigenvalues of the Jacobian of $T$ at $\bar{\rm x}$ are nonzero and at least one of them has absolute value less than one, and establish that $\mathcal{C}$ is an increasing curve that separates $\mathcal{R}$ into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. Several applications to planar systems of difference equations with non-hyperbolic equilibria are given.Comment: 20 pages, 2 figure

Global dynamics of a rational system of difference equations in the plane

We investigate the global stability properties and asymptotic behavior of solutions of the system of difference equations Xn+1 = Xn/a+Yn2, Yn+1 = Yn/b+Xn2, n = 0, 1,... where the parameters a and b are positive numbers, and the initial conditions x0 and y0 are arbitrary nonnegative numbers