be bounded between 0 and 1 at all times (, for all ) as long as the populations started out positive. Smale showed that Lotka–Volterra systems that meet the above conditions and have five or more species (N ≥ 5) can exhibit any asymptotic behavior, including a fixed point, a limit cycle, an n-torus, or attractors. Hirsch proved that all of the dynamics of the attractor occur on a manifold of dimension N−1. This essentially says that the attractor cannot have dimension greater than N−1. This is important because a limit cycle cannot exist in fewer than two dimensions, an n-torus cannot exist in less than n dimensions, and chaos cannot occur in less than three dimensions. So, Hirsch proved that competitive Lotka–Volterra systems cannot exhibit a limit cycle for N < 3, or any torus or chaos for N < 4. This is still in agreement with Smale that any dynamics can occur for N ≥ 5. More specifically, Hirsch showed there is an invariant set C that is homeomorphic to the (N−1)-dimensional simplex and is a global attractor of every point excluding the origin. This carrying simplex contains all of the asymptotic dynamics of the system. To create a stable ecosystem the αij matrix must have all positive eigenvalues. For large-N systems Lotka–Volterra models are either unstable or have low connectivity. Kondoh and Ackland and Gallagher have independently shown that large, stable Lotka–Volterra systems arise if the elements of (i.e. the features of the species) can evolve in accordance with natural selection. 4-dimensional example A simple 4-dimensional example of a competitive Lotka–Volterra system has been characterized by Vano et al. Here the growth rates and interaction matrix have been set to with for all . This system is chaotic and has a largest Lyapunov exponent of 0.0203. From the theorems by Hirsch, it is one of the lowest-dimensional chaotic competitive Lotka–Volterra systems. The Kaplan–Yorke dimension, a measure of the dimensionality of the attractor, is 2.074. This value is not a whole number, indicative of the fractal structure inherent in a strange attractor. The coexisting equilibrium point, the point at which all derivatives are equal to zero but that is not the origin, can be found by inverting the interaction matrix and multiplying by the unit column vector, and is equal to Note that there are always equilibrium points, but all others have at least one species' population equal