How to integrate integrable three-species Lotka-Volterra systems
Rob Maier
Department of Mathematics, University of Arizona
Date and time:
Thursday, March 14, 2013 - 3:15pm
Abstract:
Lotka-Volterra (LV) systems are quadratic dynamical systems of a special type. They arise in population biology, in chemical kinetics, and in multimode coupling. Well-known cases include ABC and May-Leonard systems. LV systems are typically not integrable, and even if conserved quantities exist, it is difficult to construct useful closed-form solutions. But integrable LV systems are prime examples of integrable non-Hamiltonian systems; thus the singularity structure of solutions in the complex time domain is interesting from the point of view of Painleve analysis. In this talk we explain how the general solutions of many integrable two and three-species LV systems can be constructed with the aid of special functions. Often the solutions are parametric, in that the time variable and the system state are expressed in terms of a new independent variable (a `new time’). As a function of the new time, the old time satisfies a nonlinear third-order ODE. We exploit Carton-LeBrun’s classification of many such ODE’s with the Painleve property, in order to find examples of three-species LV systems that have the property. Several of our explicit solutions involve Painleve transcendents.