Approaching predator-prey Lotka-Volterra equations by simplicial linear differential equations

Abstract

Predator-prey Lotka-Volterra equations was one of the rst models reflecting interaction of different species and modeling evolution of respective populations. It considers a large population of hares (preys) which is depredated by an also large population of lynxes (predators). It proposes an increasing/decreasing law of the number of individuals in each population thus resulting in an apparently simple system of ordinary differential equations. However, the Lotka-Volterra equation, and most of its modi cations, is non-linear and its generalization to a larger number of species is not trivial. The present aim is to study approximations of the evolution of the proportion of species in the Lotka-Volterra equations using some simple model de ned in the simplex. Calculus in the simplex has been recently developed on the basis of the Aitchison geometry and the simplicial derivative. Evolution of proportions in time (or other parameters) can be represented as simplicial ordinary di erential equations from which the simpler models are the linear ones. Simplicial Linear Ordinary Di erential Equations are not able to model the evolution of the total mass of the population (total number of predators plus preys) but only the evolution of the proportions of the different species (ratio predators over preys). This way of analysis has been successful showing that the compositional growth of a population in the Malthusian exponential model and the Verhulst logistic model were exactly the same one: the rst order simplicial linear di erential equation with constant coeffcients whose solution is a compositional straight-line. This strategy of studying the total mass evolution and the compositional evolution separately is used to get a simplicial differential equation whose solutions approach suitably the compositional behavior of the Lotka-Volterra equations. This approach has additional virtues: it is linear and can be extended in an easy way to a number of species larger than two.