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Human societies are formed by different socio-economical classes which are characterized by their contribution to, and their share of, the common wealth available. Cheaters, defined as those individuals that do not contribute to the common wealth but benefit from it, have always existed, and are likely to be present in all societies in the foreseeable future. Their existence brings about serious problems since they act as sinks for the community wealth and deplete resources which are always limited and often scarce. To fight cheaters, a society can invest additional resources to pursue one or several aims. For instance, an improvement in social solidarity (e.g. by fostering education) may be sought. Alternatively, deterrence (e.g. by increasing police budget) may be enhanced. Then the following questions naturally arise: (i) how much to spend and (ii) how to allocate the expenditure between both strategies above. This paper addresses this general issue in a simplified setting, which however we believe of some interest. More precisely, we consider a society constituted by two productive classes and an unproductive one, the cheaters, and proposes a dynamical system that describes their evolution in time. We find it convenient to formulate our model as a three-dimensional ordinary differential equation (ODE) system whose variables are the cheater population, the total wealth and one of the productive social classes. The stationary values of the cheater population and the total wealth are studied in terms of the two parameters: phi (how much to invest) and s (how to distribute such expenditure). We show that it is not possible to simultaneously minimize the cheater population and maximize the total wealth with respect to phi and s. We then discuss the possibility of defining a compromise function to find suitable values of phi and s that optimize the response to cheating. In our opinion, this qualitative approach may be of some help to plan and implement social strategi, Italy-Spain Integrated Action, Ministerio de Ciencia e Innovación, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

Criminals are common to all societies. To fight against them the community takes different security measures as, for example, to bring about a police. Thus, crime causes a depletion of the common wealth not only by criminal acts but also because the cost of hiring a police force. In this paper, we present a mathematical model of a criminal-prone self-protected society that is divided into socio-economical classes. We study the effect of a non-null crime rate on a free-of-criminals society which is taken as a reference system. As a consequence, we define a criminal-prone society as one whose free-ofcriminals steady state is unstable under small perturbations of a certain socioeconomical context. Finally, we compare two alternative strategies to control crime: (i) enhancing police efficiency, either by enlarging its size or by updating its technology, against (ii) either reducing criminal appealing or promoting social classes at risk, Italy-Spain Integrated Action, Acción Especial UCM, MEC, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

Este Trabajo de Fin de Grado estudia un algoritmo basado en Physarum polycephalum (PPA). P.polycephalum es un organismo que ha sido descrito como un ejemplo de inteligencia celular. Los PPAs se han desarrollados en los últimos años debido a su capacidad para resolver problemas complejos, al igual que el organismo que los inspira. El objetivo de este trabajo es desarrollar un PPA y aplicarlo a la red de transporte público de Madrid (MPTN). Para ello se han utilizado mapas georreferenciados de la red de metro y cercanías de la ciudad. Se han usado diversos programas y lenguajes para el desarrollo del algoritmo, adaptación a las redes de transporte y análisis de los resultados. Las redes generadas por P.polycephalum se han comparado a las redes reales. Para ello, se usaron métricas de similaridad. Además, se ha estudiado el efecto de modificaciones en la red generada por P.polycephalum. La adaptación del organismo a la adición o eliminación de determinadas paradas de la red de transporte ha quedado plasmada en nuestro trabajo. Con todo ello, este trabajo muestra una aplicación de un algoritmo biológico en un campo que va más allá de los límites de la biología ABSTRACT This Bachelor End Project studies a Physarum polycephalum inspired Algorithm (PPA). P.polycephalum is an organism that has been described as an example of cellular intelligence. PPAs have been developed in recent years due to their ability to solve complex problems, just like the organism that inspires them. The aim of this work was to develop a PPA and apply it to Madrid Public Transport Network (MPTN). To this end, georeferenced maps of the metro and suburban train networks have been used. Different programs and languages have been applied for the development of the algorithm, adaptation to the transport networks and analysis of the results. The networks generated by P.polycephalum were compared to the real networks. For this purpose, similarity metrics were required. In addition, the effect of modifica

Network analysis is an essential tool for understanding complex systems through the relationships between their constituent elements. In this project, we map large tree population samples gathered from public data bases into either spatial networks (graphs) of individuals or networks (graphs) of species. In particular, spatial networks are obtained in terms of a distance threshold r: two trees are linked if they are apart less than r. The species network is built from aggregating individuals of the same species and linking two species when at least two or, in general Tp, individuals of each species are closer than r. The topological properties of these networks are studied as a function of r and Tp. Furthermore, this work presents a methodology that brings together various procedures: access to species distribution data, creation of efficient code to form the graphs and to study their properties, graphical or geographical representation of these, etc. which, as a whole, make it possible to carry out an exhaustive analysis of these types of relationships, providing information on both the structure of any tree population and the species coexistence An example is given by the application of this methodology to two Andalusian ecoregions: Andalusian Sierra Morena and Sierra Nevada; They are chosen due to the exceptional coverage of geolocation forest data in this autonomous community, thanks to the work of REDIAM. Despite being a sample of the real population, the high number of trees of the inventory makes the computation of the network properties in a personal computer a big challenge. This project shows that only the design of efficient strategies that combines various computational and mathematical tools can provide a solution.

We analyse the properties of real plant-based networks constructed from geospatial databases, in particular vineyards, olive trees, cork oak and prickly pear in the Iberian Pensinsula. Due to the large scale of such networks, we study the sampling dependence of the networks measures: giant component, clustering cofficient, modularity, diameter and degree distribution. To compare with, we review the sampling properties of classical synthetic networks, namely ramdom Erdos-Renyi networks and its spatial variant, the random geometric networks. Concerning sampling, the main results obtained were: 1) Clustering coefficient and Degree Distribution are independent of the sampling size, both on synthetic and real networks. 2) While modularity depends on the sampling size in synthetic networks, it behaves similarly for every size of the plant-based complex networks. Observed topological properties of real plant-based networks are framed and contextualized according to our results.

Los Toy models siempre han estado ligados a la ciencia, a través de los cuales se han podido hacer visibles ideas o conceptos difíciles de comprender. Los bloques de LEGO son uno de los juguetes de construcción más conocido a nivel mundial y ha sido utilizado numerosas veces como medio de construcción de estos Toy models. Este Trabajo de Fin de Grado (TFG) trata de trasladar a la arquitectura el concepto de sucesión matemática, generando geometrías y espacios nunca antes explorados. Se implementarán sucesiones matemáticas en forma de torres generadas a partir de módulos del tamaño de los bloques de LEGO. Estas torres serán analizadas geométrica y estructuralmente para asegurar su estabilidad. Más allá de la realización del TFG, se pretende construir torres reales de LEGO con patrones matemáticos y realizar una exposición con ellas. Para llevar a cabo el análisis de las torres se aplican programas informáticos del entorno de Rhinoceros 3D como Grasshopper y Karamba 3D, así como programación en Python con GhPython.

Mathematical modelling of bacterial communities has capital importance in ecology, as it provides of means to characterize such comunities, identify key species and establish in silico predictions. In reference to human microbiota, a proper model would allow, not only to determine certain diseases' e_ect on microbiota composition, but also to predict the impact of perturbations such as antibiotics or diet and reconduct an unhealthy microbiota to a healthy one. The aim of this work is to characterize the e_ect of the environment (host) on the viability of bacterial communities. Two types of models are being used by the scienti_c community to include the environment in the study of ecosystems: traditionally, the environment is assumed to be static, but lately the study of a changing environment is growing. In this work two models will be used: one based on Lotka-Volterra generalized model, and a second one based on evolutionary games theory. Communities of two and three species will be considered. This work also proposes the use of the term viability to refer to the equilibrium point of an ecosystem. This term includes both the considerations of feasibility and stability, given that, in order to have a biological meaning, an equilibrium point is required to meet both conditions. Following the Lotka-Volterra approach, random 2x2 and 3x3 matrices were modi_ed by the environment, having their viability studied before and after. We studied the parameter space of matrices that were made viable, as well as that of their corresponding environments. Our results show that an environment that promotes viability strengthens intraspeci_c competition, that is to say, reduces the species' growth rate, as well as reduces interspeci_c interactions, whether introducing spatial structure or functional redundancies. On the other hand, the evolutionary game model was based on that proposed by Tilman et al (2020), in which two payo_ matrices are de_ned for minimum and maximum values of a

Inspections are used to prevent tax evasion or any other unlawful behavior. ? The effect of inspections depends on the network topology and the contagion rule. ? The network is modeled as a Watts?Strogatz Small World that is tuned from regular to random. ? Two contagion rules are applied: continuous and discontinuous. ? The equilibrium populations of payers and evaders are obtained in terms of these system parameters.

We study the evolution of a finite size population formed by mutationally isolated lineages of error-prone replicators in a two-peak fitness landscape. Computer simulations are performed to gain a stochastic description of the system dynamics. More specifically, for different population sizes, we compute the probability of each lineage being selected in terms of their mutation rates and the amplification factors of the fittest phenotypes. We interpret the results as the compromise between the characteristic time a lineage takes to reach its fittest phenotype by crossing the neutral valley and the selective value of the sequences that form the lineages. A main conclusion is drawn: for finite population sizes, the survival probability of the lineage that arrives first to the fittest phenotype rises significantly

We investigate optimal strategies to defend valuable goods against the attacks of a thief. Given the value of the goods and the probability of success for the thief, we look for the strategy that assures the largest benefit to each player irrespective of the strategy of his opponent. Two complementary approaches are used: agent-based modeling and game theory. It is shown that the compromise between the value of the goods and the probability of success defines the mixed Nash equilibrium of the game, that is compared with the results of the agent-based simulations and discussed in terms of the system parameters.

Estudio del tiempo en que un fenotipo alcanza un estado estacionario

Criminals are common to all societies. To fight against them the community takes different security measures as, for example, to bring about a police. Thus, crime causes a depletion of the common wealth not only by criminal acts but also because the cost of hiring a police force. In this paper, we present a mathematical model of a criminal-prone self-protected society that is divided into socio-economical classes. We study the effect of a non-null crime rate on a free-of-criminals society which is taken as a reference system. As a consequence, we define a criminal-prone society as one whose free-of-criminals steady state is unstable under small perturbations of a certain socio-economical context. Finally, we compare two alternative strategies to control crime: (i) enhancing police efficiency, either by enlarging its size or by updating its technology, against (ii) either reducing criminal appealing or promoting social classes at risk

Human societies are formed by different socio-economical classes which are characterized by their contribution to, and their share of, the common wealth available. Cheaters, defined as those individuals that do not contribute to the common wealth but benefit from it, have always existed, and are likely to be present in all societies in the foreseeable future. Their existence brings about serious problems since they act as sinks for the community wealth and deplete resources which are always limited and often scarce. To fight cheaters, a society can invest additional resources to pursue one or several aims. For instance, an improvement in social solidarity (e.g. by fostering education) may be sought. Alternatively, deterrence (e.g. by increasing police budget) may be enhanced. Then the following questions naturally arise: (i) how much to spend and (ii) how to allocate the expenditure between both strategies above. This paper addresses this general issue in a simplified setting, which however we believe of some interest. More precisely, we consider a society constituted by two productive classes and an unproductive one, the cheaters, and proposes a dynamical system that describes their evolution in time. We find it convenient to formulate our model as a three-dimensional ordinary differential equation (ODE) system whose variables are the cheater population, the total wealth and one of the productive social classes. The stationary values of the cheater population and the total wealth are studied in terms of the two parameters: φ (how much to invest) and s (how to distribute such expenditure). We show that it is not possible to simultaneously minimize the cheater population and maximize the total wealth with respect to φ and s. We then discuss the possibility of defining a compromise function to find suitable values of φ and s that optimize the response to cheating. In our opinion, this qualitative approach may be of some help to plan and implement social strategies aga

This paper presents a model of a population of error-prone self-replicative species (replicators) that interact with its environment. The population evolves by natural selection in an environment whose change is caused by the evolutionary process itself. For simplicity, the environment is described by a single scalar factor, i.e. its temperature. The formal formulation of the model extends two basic models of Ecology and Evolutionary Biology, namely, Daisyworld and Quasispecies models. It is also assumed that the environment can also change due to external perturbations that are summed up as an external noise. Unlike previous models, the population size self-regulates, so no ad hoc population constraints are involved. When species replication is error-free, i.e. without mutation, the system dynamics can be described by an (n + 1)-dimensional system of differential equations, one for each of the species initially present in the system, and another for the evolution of the environment temperature. Analytical results can be obtained straightforwardly in low-dimensional cases. In these examples, we show the stabilizing effect of thermal white noise on the system behavior. The error-prone self-replication, i.e. with mutation, is studied computationally. We assume that species can mutate two independent parameters: its optimal growth temperature and its influence on the environment temperature. For different mutation rates the system exhibits a large variety of behaviors. In particular, we show that a quasispecies distribution with an internal sub-distribution appears, facilitating species adaptation to new environments. Finally, this ecologically inspired evolutionary model is applied to study the origin and evolution of public opinion.

This paper presents an extension of stoichiometric analysis in systems where the catalytic compounds (enzymes) are also intermediates of the metabolic network (dual property), so they are produced and degraded by the reaction network itself. To take this property into account, we introduce the definition of enzyme-maintaining mode, a set of reactions that produces its own catalyst and can operate at stationary state. Moreover, an enzyme-maintaining mode is defined as elementary with respect to a given reaction if the removal of any of the remaining reactions causes the cessation of any steady state flux through this reference reaction. These concepts are applied to determine the network structure of a simple self-maintaining system.

This paper is concerned with a quantitative model describing the interaction of three sociological species, termed as owners, criminals and security guards, and denoted by X, Y and Z respectively. In our model, Y is a predator of the species X, and so is Z with respect to Y . Moreover, Z can also be thought of as a predator of X, since this last population is required to bear the costs of maintaining Z. We propose a system of three ordinary differential equations to account for the time evolution of X(t), Y (t) and Z(t) according to our previous assumptions. Out of the various parameters that appear in that system, we select two of them, denoted by H, and h, which are related with the efficiency of the security forces as a control parameter in our discussion. To begin with, we consider the case of large and constant owners population, which allows us to reduce (3)–(5) to a bidimensional system for Y (t) and Z(t). As a preliminary step, this situation is first discussed under the additional assumption that Y (t) + Z(t) is constant. A bifurcation study is then performed in terms of H and h, which shows the key role played by the rate of casualties in Y and Z, that results particularly in a possible onset of bistability. When the previous restriction is dropped, we observe the appearance of oscillatory behaviours in the full two-dimensional system. We finally provide a exploratory study of the complete model (3)–(5), where a number of bifurcations appear as parameter H changes, and the corresponding solutions behaviours are described.