to zero. The eigenvalues of the system at this point are 0.0414±0.1903i, −0.3342, and −1.0319. This point is unstable due to the positive value of the real part of the complex eigenvalue pair. If the real part were negative, this point would be stable and the orbit would attract asymptotically. The transition between these two states, where the real part of the complex eigenvalue pair is equal to zero, is called a Hopf bifurcation. A detailed study of the parameter dependence of the dynamics was performed by Roques and Chekroun in. The authors observed that interaction and growth parameters leading respectively to extinction of three species, or coexistence of two, three or four species, are for the most part arranged in large regions with clear boundaries. As predicted by the theory, chaos was also found; taking place however over much smaller islands of the parameter space which causes difficulties in the identification of their location by a random search algorithm. These regions where chaos occurs are, in the three cases analyzed in, situated at the interface between a non-chaotic four species region and a region where extinction occurs. This implies a high sensitivity of biodiversity with respect to parameter variations in the chaotic regions. Additionally, in regions where extinction occurs which are adjacent to chaotic regions, the computation of local Lyapunov exponents revealed that a possible cause of extinction is the overly strong fluctuations in species abundances induced by local chaos. Spatial arrangements Background There are many situations where the strength of species' interactions depends on the physical distance of separation. Imagine bee colonies in a field. They will compete for food strongly with the colonies located near to them, weakly with further colonies, and not at all with colonies that are far away. This doesn't mean, however, that those far colonies can be ignored. There is a transitive effect that permeates through the system. If colony A interacts with colony B, and B with C, then C affects A through B. Therefore, if the competitive Lotka–Volterra equations are to be used for modeling such a system, they must incorporate this spatial structure. Matrix organization One possible way to incorporate this spatial structure is to modify the nature of the Lotka–Volterra equations to something like a reaction–diffusion system. It is much easier, however, to keep the format of the equations the same and instead modify the interaction matrix. For simplicity,