DUGi

Approaching predator-prey Lotka-Volterra Equations by Simplicial Linear Differential Equations

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Predator-prey Lotka-Volterra equations was one of the first models reflecting interaction of different species andmodeling evolution of respective populations. It considers a large population of hares (preys) which is depredatedby an also large population of lynxes (predators). It proposes an increasing/decreasing law of the number ofindividuals in each population thus resulting in an apparently simple system of ordinary differential equations.However, the Lotka-Volterra equation, and most of its modifications, is non-linear and its generalization to alarger number of species is not trivial. The present aim is to study approximations of the evolution of theproportion of species in the Lotka-Volterra equations using some simple model defined in the simplex.Calculus in the simplex has been recently developed on the basis of the Aitchison geometry and the simplicialderivative. Evolution of proportions in time (or other parameters) can be represented as simplicial ordinarydifferential equations from which the simpler models are the linear ones. Simplicial Linear Ordinary DifferentialEquations are not able to model the evolution of the total mass of the population (total number of predators pluspreys) but only the evolution of the proportions of the different species (ratio predators over preys). This way ofanalysis has been successful showing that the compositional growth of a population in the Malthusian exponentialmodel and the Verhulst logistic model were exactly the same one: the first order simplicial linear differentialequation with constant coefficients whose solution is a compositional straight-line. This strategy of studying thetotal mass evolution and the compositional evolution separately is used to get a simplicial differential equationwhose solutions approach suitably the compositional behavior of the Lotka-Volterra equations. This approachhas additional virtues: it is linear and can be extended in an easy way to a number of species larger than two

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada