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

Predator-prey Lotka-Volterra equations was one of the first 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 modifications, 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 defined 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 differential equations from which the simpler models are the linear ones. Simplicial Linear Ordinary Differential 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 first order simplicial linear differential equation with constant coefficients 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

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada

Other contributions: Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada
Author: Jarauta Bragulat, Eusebio
Egozcue, Juan José
Date: 2011 May 11
Abstract: Predator-prey Lotka-Volterra equations was one of the first 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 modifications, 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 defined 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 differential equations from which the simpler models are the linear ones. Simplicial Linear Ordinary Differential 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 first order simplicial linear differential equation with constant coefficients 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
Format: application/pdf
Document access: http://hdl.handle.net/10256/13617
Language: eng
Publisher: Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada
Collection: CoDaWork 2011. The 4th International Workshop on Compositional Data Analysis
Rights: Tots els drets reservats
Subject: Anàlisi multivariable -- Congressos
Multivariate analysis -- Congresses
Equacions diferencials -- Congressos
Differential equations -- Congresses
Depredació (Biologia) -- Models matemàtics -- Congressos
Predation (Biology) -- Mathematical models -- Congresses
Title: Approaching predator-prey Lotka-Volterra Equations by Simplicial Linear Differential Equations
Type: info:eu-repo/semantics/conferenceObject
Repository: DUGiDocs

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