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Doctoral Thesis
DOI
https://doi.org/10.11606/T.76.2007.tde-18122007-135412
Document
Author
Full name
Fabio Campos Poderoso
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2007
Supervisor
Committee
Fontanari, José Fernando (President)
Moussa, Miled Hassan Youssef
Prado, Carmen Pimentel Cintra do
Theumann, Walter Karl
Yokoi, Carlos Seihiti Orii
Title in Portuguese
Ecossistemas de replicadores: uma abordagem via mecânica estatística de sistemas desordenados
Keywords in Portuguese
Método de réplicas
Temperado
Vidro de spin
Abstract in Portuguese
Nesta tese utilizamos o modelo do replicador aleatório, proposto por Diederich e Opper, para analisar as propriedades de equilíbrio de ecossistemas complexos (formados por um grande número de espécies) em três situações distintas. Na primeira parte desta tese, investigamos os efeitos de interações variáveis sobre a estrutura do ecossistema, utilizando o método de réplicas generalizado, introduzido por Penney et al. Este formalismo propõe uma nova interpretação para o índice de réplicas n, a saber, como sendo a razão entre duas temperaturas características: a temperatura relacionada aos acoplamento e a temperatura associada às variáveis de spin. Empregando t\'ecnicas de campo m\'edio de Mec\^anica Estat\'{\i}stica e tamb\'em simula\c\~oes num\'ericas tratamos, em particular, do estado fundamental ($\beta ightarrow + \infty$). Encontramos dois regimes distintos, um onde prevalece a coopera\c\~ao entre as diferentes esp\'ecies ($\beta^{\prime} > 0$) e outro no qual a competi\c\~ao \'e predominante ($\beta^{\prime} < 0$). No primeiro caso temos uma transi\c\~ao descont\'{\i}nua para um regime de diversidade nula e no outro temos uma m\'axima diversidade das esp\'ecies. Na segunda parte desta tese \cite, estudamos as implica\c\~oes de uma temperatura finita sobre a estrutura do ecossistema. Utilizamos a regra de Hebb para descrever as intera\c\~oes entre as diferentes esp\'ecies. A temperatura surge, no modelo, atrav\'es de um ru\'{\i}do gaussiano introduzido na equa\c\~ao estoc\'astica que rege a din\^amica do processo. Tratamos analiticamente o caso recozido ({\em annealed}), no qual as caracter\'{\i}sticas das esp\'ecies evoluem t\~ao rapidamente quanto suas concentra\c\~oes, e o caso temperado ({\em quenched}), onde tais caracter\'{\i}sticas est\~ao fixas. Conclu\'{\i}mos que h\'a uma transi\c\~ao de fase descont\'{\i}nua entre um estado onde a competi\c\~ao prevalece, determinando baixa diversidade, para outro estado onde predomina a coopera\c\~ao. Por fim \cite, analisamos as poss\'{\i}veis consequ\^encias de uma interven\c\~ao humana sobre as propriedades de equil\'{\i}brio do ecossistema. Admitimos o princ\'{\i}pio da exclus\~ao competitiva para modelar os acoplamentos entre as diferentes esp\'ecies, a regra de Hebb. Interferimos na comunidade impondo que um conjunto de caracteres, selecionados previamente, esteja presente em uma fra\c\~ao bem definida dos seus membros. O principal resultado deste estudo revela, desde que o par\^ametro de competi\c\~ao entre indiv\'{\i}duos semelhantes n\~ao seja muito pequeno, que o efeito de uma tal manipula\c\~ao conduz a perda da diversidade e, portanto, ao empobrecimento do ecossistema.
Title in English
Replicators ecosystems: a statistical mechanics of disordered systems approach
Keywords in English
Quenched
Replica method
Spin glass
Abstract in English
In this thesis we use the random replicator model, proposed by Diederich and Opper \cite, to analyse the equilibrium properties of complex ecosystems (formed by a large number of species) in three distinct situations. In the first part of this thesis \cite, we investigate the effects of variable interactions upon ecosystem structure, using the generalized replica method, introduced by Penney et al \cite{penn93,cool93}. In this formalism we find a new interpretation for the replica number $n$ as the ratio between two characteristic temperatures: the temperature related to the couplings ($\beta^{\prime}$) and the temperature associated to the spin variables ($\beta$). We approach the problem using mean field methods of statistical mechanics and intensive numerical simulations; in particular we are concerned with the ground state ($\beta ightarrow + \infty$). We find two distinct regimes, one where cooperation between different species prevails ($\beta^{\prime} > 0$) and the other in which competition is predominant ($\beta^{\prime} < 0$). In the first case we have a discontinuous transition to the zero diversity regime and in the other we have the maximum species diversity. In the second part of this thesis \cite, we examine the finite temperature implications upon ecosystem structure. Through the Hebb rule we can describe the interactions between different species. With the aid of a Gaussian noise in the stochastic equation, that governs the temporal evolution, we have a way to introduce the finite temperature in the model. We treat analytically the annealed case, in which the species characteristics evolve so fast as its concentrations, as well as the quenched case, in which such characteristics are fixed. We conclude that there is a discontinuous phase transition between a state where competition prevails, implying low diversity, to another state in which cooperation is stronger. At last \cite, we analyse the possible consequences of human intervention upon the equilibrium properties of the ecosystem. We assume the competitive exclusion principle to model the couplings between different species, the Hebb rule. We interfere in the community by imposing that a set of characters, previously selected, be present in a well defined fraction of its members. The main result of this study reveals, provided that the intraspecies competition parameter is not too weak, that the effect of such a manipulation leads to the impoverishment of the ecosystem.
 
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Publishing Date
2008-01-15
 
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