Phase transitions and diversification in complex systems: The role of heterogeneites, adaptation a other essencial aspects of real systems

Abstract

Let us summarize the main issues treated in the different chapters of this thesis. In chapter 1 basic concepts needed in this thesis are summarized. Firstly, main features of continuous and discontinuous phase transitions are presented in order to a correct distinction between them (section 1.1). Then, prototypical models presenting these types of transition are described and analyzed in section 1.2. In chapter 2, we study the importance of demographic stochasticity and diffusion in a generic system subjected to a discontinuous transition in the mean-field approach. We investigate how the order of the transition would surprisingly depend on such mechanisms. Beside this, we also study the the unavoidable presence of spatial heterogeneity in real systems. In this case, a rounding phenomenon for low dimensional systems appears. The ideas presented here can help to further understand discontinuous transitions, and contribute to the discussion about the possibility of preventing these shifts in order to minimize their disruptive ecological, economic, and societal consequences. For a deeper understanding of some of the previous results, in chapter 3 we present a more technical and detailed study of the effect of spatial heterogeneity on a prototypical model exhibiting a discontinuous transition. Here we try to explain how, in analogy with what happens in problems of thermodynamic equilibrium, the existence of some form of spatial disorder implies that potentially discontinuous transitions are rounded-off, thus making the system critical (at low dimensions). In this context, in chapter 4 we wonder whether a structurally (and so spatially) disordered system would also present the same smoothing effect. An extensive analysis of all possible systems presenting this structural heterogeneity may constitute a thesis itself. As a consequence, we focus on the brain cortex, a system that is well described by models exhibiting discontinuous transitions at mean-field and which presents a complex and known network structure. Interestingly, criticality appears for small topological dimensions and so, a compatibility of integrative models of neural activity (exhibiting discontinuous transitions in mean-field), and the critical features experimentally measured in the cortex, is accomplished. The above chapters do not consider any type of mutation or variation of its individuals due to the fact that, in those cases, evolution usually takes place in longer times than the considered ones. However, apart from the previous inherent properties, adaptation is an essential feature of real systems. What would it happen if individuals rapidly evolve affecting community dynamics? In chapter 5 we propose a relatively simple computational eco-evolutionary model specifically devised to describe rapid phenotypic diversification in a particular experiment of species-rich communities [236]. Despite this, the model is easily generalizable to analyze different eco-evolutionary problems within a relatively simple and unified computational framework. We show that it captures the main phenomenology observed experimentally, and it also makes non-trivial predictions. Although, unlike it was awaited, no phase transition from poor to rich communities appear, in future we will investigate the needed mechanisms for which this phase transition occurs. Finally, thesis conclusions are presented in chapter chapter 6.