THE DECOUPLING & SOLUTION OF LOGISTIC & CLASSICAL TWO-SPECIES LOTKA-VOLTERRA DYNAMICS WITH VARIABLE PRODUCTION RATES

Central to the dynamics of population biology are various versions of the Lotka-Volterra equations. Particular cases may be used to model competitive, commensal, predatory and other behaviour. Similar equations describe macro-economic interactions, epidemics and other processes of mass action. Refinements of many versions of these equations have been exhibited in the BIOMAT meetings to describe new biological features. The solutions to such equations may display a variety of forms. Broadly speaking, quite a lot of qualitative information may often be obtained about the solutions. In many cases it is convenient to eliminate time from the equations, so obtaining equations for the joint values of population sizes. By contrast, determining explicit expressions for the evolution with time of the component population sizes frequently appears to be infeasible, although numerical procedures may be available. This is less than fully satisfactory, as numerical work with many joint choices of the driving parameters, involving a number of dimensions, may be needed to obtain any sort of collective overview of the population dynamics. Perhaps surprisingly, in a number of general situations it is in fact possible to derive explicit solutions for the population sizes as functions of time. We may indeed decouple the basic equations and perform explicit quadratures. Several techniques are available for our purposes, such as Euler substitution and separation and the methods of Gambier and Painlevé. The working is sometimes heavy but often manageable. We illustrate these techniques by considering a number of well-known classical biological models from the literature.