consider a five species example where all of the species are aligned on a circle, and each interacts only with the two neighbors on either side with strength and respectively. Thus, species 3 interacts only with species 2 and 4, species 1 interacts only with species 2 and 5, etc. The interaction matrix will now be If each species is identical in its interactions with neighboring species, then each row of the matrix is just a permutation of the first row. A simple, but non-realistic, example of this type of system has been characterized by Sprott et al. The coexisting equilibrium point for these systems has a very simple form given by the inverse of the sum of the row Lyapunov functions A Lyapunov function is a function of the system whose existence in a system demonstrates stability. It is often useful to imagine a Lyapunov function as the energy of the system. If the derivative of the function is equal to zero for some orbit not including the equilibrium point, then that orbit is a stable attractor, but it must be either a limit cycle or n-torus - but not a strange attractor (this is because the largest Lyapunov exponent of a limit cycle and n-torus are zero while that of a strange attractor is positive). If the derivative is less than zero everywhere except the equilibrium point, then the equilibrium point is a stable fixed point attractor. When searching a dynamical system for non-fixed point attractors, the existence of a Lyapunov function can help eliminate regions of parameter space where these dynamics are impossible. The spatial system introduced above has a Lyapunov function that has been explored by Wildenberg et al. If all species are identical in their spatial interactions, then the interaction matrix is circulant. The eigenvalues of a circulant matrix are given by for and where the Nth root of unity. Here is the jth value in the first row of the circulant matrix. The Lyapunov function exists if the real part of the eigenvalues are positive ( for ). Consider the system where , , , and . The Lyapunov function exists if for . Now, instead of having to integrate the system over thousands of time steps to see if any dynamics other than a fixed point attractor exist, one need only determine if the Lyapunov function exists (note: the absence of the Lyapunov