Lotka-Volterra predator-prey models analytic and numerical methods.

Abstract

The Lotka-Volterra equations are a classical model of the populations of interacting species. In the case of two interacting species, we present a closed parametric solution to a particular case of the Lotka-Volterra model. We also determine closed expressions for the branch points, bounds on the parameter, amplitude of the oscillation of the prey and predator populations, and period of this model in terms of the Lambert W function. In the case of three interacting species, under certain conditions solutions are again periodic. However, standard numerical methods often fail to preserve this periodicity, as well as other important properties of the model. The underlying geometry of the three-species predator-prey model is developed through the framework of Poisson dynamics. It is shown that the system is bi-Poisson and possesses two independent first integrals. Numerical methods for approximating solutions to the model are constructed which incorporate the underlying Poisson geometry of the continuous system. These methods preserve the periodicity of solutions, and the error in the first integrals remains bounded. Simulations are used to show that these methods produce more accurate results than standard numerical methods which do not consider the Poisson structure of the equations.