## Periodic Report Summary 1 - STREVCOMS (Structure and Evolution of Complex Systems with Applications in Physics and Life Sciences)

The aim of the project is to make a major step in developing novel effective tools for studying complex systems based on stochastic geometry and stochastic evolution methods, on appropriate methods of functional and complex analysis and combinatorics, as well as on numerical methods and computer simulations. The studied systems are supposed to consist of large (infinite) number of interacting entities and may evolve in continuous space and time. Both their structure and evolution are of interest. Such systems appear in broadly understood statistical physics, spatial ecology, evolutionary and population biology, epidemiology, etc. The planned research is divided into seven work packages (WPs).

WP1 is dedicated to the structure of systems related to physics, such as complex networks, polymers, irregular and random graphs, Gibbs random fields on graphs. In the period 1.01.2014–31.12.2015, we have performed the following work: (a) elaborating new techniques in modeling polymer networks; (b) introducing and studying new classes of random graphs, including those based on motifs and characterized by interactions between the edges and vertices, as well as some irregular graphs; (c) describing phase transitions in a number of models based on geometric random graphs, developing new techniques for studying Gibbs random fields on graphs. In particular, a cellular model of phase transitions in continuum systems has been introduced and investigated. The main achievement is making essential step towards elaborating a synthetic approach to solving these problems based on stochastic geometry and stochastic and combinatorial analysis. The final result – complete elaborating the mentioned approach – is expected to be of great use.

WP2 is dedicated to the microscopic modeling in the spatial and evolutionary biology. In the period 1.01.2014–31.12.2015, we have performed the following work. A general mathematical framework of modeling large populations of entities evolving in space and time has been elaborated. In this framework, a microscopic (individual-based ) model describing a non-immune disease spread has been introduced and studied by means of an algorithmic method based on the Monte Carlo simulations especially created for this purpose. A number of the results were obtained for the case of a quenched distribution of individuals and the one-parameter stochastic dynamics (synchronous SIS model with varying number of neighbors). A microscopic model of predation in marine ecology has also been elaborated and studied. In this model, animals of two species (predators and preys) appear as point particles that can jump and produce new particles. The main achievement is elaborating the general mathematical framework of modeling large populations of living entities evolving in space and time. It is planned to study a number of new microscopic models of animal movement, that can result in new level of understanding their behavior.

WP3 is dedicated to the development of a general microscopic dynamical theory of systems of interacting point particles distributed over continuum space and evolving in time. Such systems are studied both in physics and life sciences. In this theory, the individual motion of each particle is taken into account. In the period 1.01.2014–31.12.2015, we have preformed the following work. A general theory has been elaborated in which the system states are probability measures on the space of particle configurations the evolution of which is described as the evolution of the corresponding correlation functions. In this framework, the spatial logistic model, widely applied in e.g. plant ecology, has been described in which the particles give birth to new particles and die, also due to competition. Various types of the birth and competition kernels were considered. Similar models in which the particles perform jumps, coalescing jumps, fragmentation, possess additional characteristics such as age or life-time, have also been described. In studying holomorphic generating functionals, the common fixed point set of commuting holomorphic mappings in Cartesian products of Banach spaces were found and described. The main achievement is elaborating the general microscopic dynamical theory of infinite systems of interacting particles. It is expected that this theory will have a substantial impact on the theory of interacting particles and its applications.

WP4 is dedicated to the mesoscopic description of the dynamics of models studied in WP3. Here the particles are described not individually but in terms of their densities. The corresponding evolution equations are deduced from the microscopic theory by means of certain scaling procedures. One of the aims is then developing and studying such procedures. In the period 1.01.2014–31.12.2015 we have performed the following work: (a) studying various types of scaling techniques, including scaling of the death potential in the birth-and-death Markov generators; (b) elaborating the way of applying Ovcyannikov-type techniques to proving convergence of the rescaled correlation functions in the Vlasov scaling limit; (c) studying the solutions of the mesoscopic (kinetic) equations of the spatial logistic model, the Widom-Rowlinson jump model, coalescing jump models; (d) studying wave propagation in the spatial logistic model. The main achievement is elaborating the way of deriving mesoscopic theories from the corresponding microscopic ones. We expect that this will have a substantial impact on understanding the behavior of such systems.

WP5 is dedicated to the compartmental modeling systems of entities as in WP2, including non-Markovian evolutions. In the period 1.01.2014–31.12.2015, we have performed the following work: principles of the description of infinite systems of interacting entities undergoing non-Markovian evolution (e.g. with time delay), specific to each of the compartment, have been elaborated. A mechanistic stochastic Ricker model was studied analytically and numerically. It is one of the most important models used in this field which manifests a complex nonlinear behavior. Several analytical and precise numerical methods were employed to study the model, showing how each approach contributes to understanding its dynamics. Sustained oscillations in simple contagion models with relapse have also been found and described. In particular, it have been found that an intriguing phenomenon of some contagion models with re-infection. This is the possibility of endogenous periodic oscillations at the population level, i.e., the appearance of periodically recurring epidemics which do not decay to an equilibrium state. This phenomenon has not been noted before for this kind of models. The main significance of these results is that they uncover a simple mechanism for the generation of periodic epidemics, which could potentially be observed in the real world. We expect that this will have a substantial impact on the theory of such systems.

WP6 is dedicated to the evolution theory of complex shapes. In this direction during the period 1.01.2014–31.12.2015, we have preformed the following work: (a) a physical phenomenon of crumpled wires has been studied in its relation to the Liouville field theory; (b) versions of the stochastic Loewner evolution in multiply connected domains and their connections with Villat kernels were studied; (c) an explicit method for finding the extremal function and the 2-module of a foliated family of curves to general Euclidean spaces and polarizable groups making use of Fuglede's p-module of systems of measures. The extremal functions are identified and the p-module of systems of measures is computed in condensers of rather general type and in their images under homeomorphisms of certain regularity. The main achievement is elaborating a general approach in the theory of complex shapes. We expect that this will have a substantial impact on the theory of such systems.

WP7 is dedicated to the geometric theory of complex shapes and pattern recognition. In the period 1.01.2014–31.12.2015, we have performed the following work. There has been proposed an approach in which shapes are realized as conformal embeddings of canonical domains into the complex plane. Then a variational model for pattern recognition has been considered which leads to the Cauchy differential equation. Heisenberg type groups and algebras have been employed which admit lattices and rational structure, respectively. The main achievement is that our approach brought us to interesting applications in combinatorial designs and block space time coding in wireless communication systems. In particular, we applied our results to the square semiregular 1-factorization of a complete graph, and we fixed a Geramita-Seberry-Wallis problem on a maximal number of variables in an amicable orthogonal design, where orthogonality is understood with respect to a non-degenerate metric with the neutral signature. We expect that this will have a substantial impact on the theory of such systems.

WP1 is dedicated to the structure of systems related to physics, such as complex networks, polymers, irregular and random graphs, Gibbs random fields on graphs. In the period 1.01.2014–31.12.2015, we have performed the following work: (a) elaborating new techniques in modeling polymer networks; (b) introducing and studying new classes of random graphs, including those based on motifs and characterized by interactions between the edges and vertices, as well as some irregular graphs; (c) describing phase transitions in a number of models based on geometric random graphs, developing new techniques for studying Gibbs random fields on graphs. In particular, a cellular model of phase transitions in continuum systems has been introduced and investigated. The main achievement is making essential step towards elaborating a synthetic approach to solving these problems based on stochastic geometry and stochastic and combinatorial analysis. The final result – complete elaborating the mentioned approach – is expected to be of great use.

WP2 is dedicated to the microscopic modeling in the spatial and evolutionary biology. In the period 1.01.2014–31.12.2015, we have performed the following work. A general mathematical framework of modeling large populations of entities evolving in space and time has been elaborated. In this framework, a microscopic (individual-based ) model describing a non-immune disease spread has been introduced and studied by means of an algorithmic method based on the Monte Carlo simulations especially created for this purpose. A number of the results were obtained for the case of a quenched distribution of individuals and the one-parameter stochastic dynamics (synchronous SIS model with varying number of neighbors). A microscopic model of predation in marine ecology has also been elaborated and studied. In this model, animals of two species (predators and preys) appear as point particles that can jump and produce new particles. The main achievement is elaborating the general mathematical framework of modeling large populations of living entities evolving in space and time. It is planned to study a number of new microscopic models of animal movement, that can result in new level of understanding their behavior.

WP3 is dedicated to the development of a general microscopic dynamical theory of systems of interacting point particles distributed over continuum space and evolving in time. Such systems are studied both in physics and life sciences. In this theory, the individual motion of each particle is taken into account. In the period 1.01.2014–31.12.2015, we have preformed the following work. A general theory has been elaborated in which the system states are probability measures on the space of particle configurations the evolution of which is described as the evolution of the corresponding correlation functions. In this framework, the spatial logistic model, widely applied in e.g. plant ecology, has been described in which the particles give birth to new particles and die, also due to competition. Various types of the birth and competition kernels were considered. Similar models in which the particles perform jumps, coalescing jumps, fragmentation, possess additional characteristics such as age or life-time, have also been described. In studying holomorphic generating functionals, the common fixed point set of commuting holomorphic mappings in Cartesian products of Banach spaces were found and described. The main achievement is elaborating the general microscopic dynamical theory of infinite systems of interacting particles. It is expected that this theory will have a substantial impact on the theory of interacting particles and its applications.

WP4 is dedicated to the mesoscopic description of the dynamics of models studied in WP3. Here the particles are described not individually but in terms of their densities. The corresponding evolution equations are deduced from the microscopic theory by means of certain scaling procedures. One of the aims is then developing and studying such procedures. In the period 1.01.2014–31.12.2015 we have performed the following work: (a) studying various types of scaling techniques, including scaling of the death potential in the birth-and-death Markov generators; (b) elaborating the way of applying Ovcyannikov-type techniques to proving convergence of the rescaled correlation functions in the Vlasov scaling limit; (c) studying the solutions of the mesoscopic (kinetic) equations of the spatial logistic model, the Widom-Rowlinson jump model, coalescing jump models; (d) studying wave propagation in the spatial logistic model. The main achievement is elaborating the way of deriving mesoscopic theories from the corresponding microscopic ones. We expect that this will have a substantial impact on understanding the behavior of such systems.

WP5 is dedicated to the compartmental modeling systems of entities as in WP2, including non-Markovian evolutions. In the period 1.01.2014–31.12.2015, we have performed the following work: principles of the description of infinite systems of interacting entities undergoing non-Markovian evolution (e.g. with time delay), specific to each of the compartment, have been elaborated. A mechanistic stochastic Ricker model was studied analytically and numerically. It is one of the most important models used in this field which manifests a complex nonlinear behavior. Several analytical and precise numerical methods were employed to study the model, showing how each approach contributes to understanding its dynamics. Sustained oscillations in simple contagion models with relapse have also been found and described. In particular, it have been found that an intriguing phenomenon of some contagion models with re-infection. This is the possibility of endogenous periodic oscillations at the population level, i.e., the appearance of periodically recurring epidemics which do not decay to an equilibrium state. This phenomenon has not been noted before for this kind of models. The main significance of these results is that they uncover a simple mechanism for the generation of periodic epidemics, which could potentially be observed in the real world. We expect that this will have a substantial impact on the theory of such systems.

WP6 is dedicated to the evolution theory of complex shapes. In this direction during the period 1.01.2014–31.12.2015, we have preformed the following work: (a) a physical phenomenon of crumpled wires has been studied in its relation to the Liouville field theory; (b) versions of the stochastic Loewner evolution in multiply connected domains and their connections with Villat kernels were studied; (c) an explicit method for finding the extremal function and the 2-module of a foliated family of curves to general Euclidean spaces and polarizable groups making use of Fuglede's p-module of systems of measures. The extremal functions are identified and the p-module of systems of measures is computed in condensers of rather general type and in their images under homeomorphisms of certain regularity. The main achievement is elaborating a general approach in the theory of complex shapes. We expect that this will have a substantial impact on the theory of such systems.

WP7 is dedicated to the geometric theory of complex shapes and pattern recognition. In the period 1.01.2014–31.12.2015, we have performed the following work. There has been proposed an approach in which shapes are realized as conformal embeddings of canonical domains into the complex plane. Then a variational model for pattern recognition has been considered which leads to the Cauchy differential equation. Heisenberg type groups and algebras have been employed which admit lattices and rational structure, respectively. The main achievement is that our approach brought us to interesting applications in combinatorial designs and block space time coding in wireless communication systems. In particular, we applied our results to the square semiregular 1-factorization of a complete graph, and we fixed a Geramita-Seberry-Wallis problem on a maximal number of variables in an amicable orthogonal design, where orthogonality is understood with respect to a non-degenerate metric with the neutral signature. We expect that this will have a substantial impact on the theory of such systems.