Predator-Prey Dynamics for Rabbits, Trees, and Romance

The Lotka-Volterra equations represent a simple nonlinear model for the dynamic interaction between two biological species in which one species (the predator) benefits at the expense of the other (the prey). With a change in signs, the same model can apply to two species that compete for resources or that symbiotically interact. However, the model is not structurally stable, since persistent time-dependent (oscillatory) solutions occur for only a single value of the parameters. This paper considers structurally stable variants of the Lotka-Volterra equations with arbitrarily many species solved on a homogeneous two-dimensional grid with coupling between neighboring cells. Interesting, biologically-realistic, spatio-temporal patterns are produced. These patterns emerge from random initial conditions and thus exhibit self-organization. The extent to which the patterns are self-organized critical (spatially and temporally scale-invariant) and chaotic (positive Lyapunov exponent) will be examined. The same equations, without the spatial interactions, can be used to model romantic relationships between individuals. Different romantic styles lead to different dynamics and ultimate fates. Love affairs involving more than two individuals can lead to chaos. The likely fate of couples with different romantic styles will be described.