Reaction-diffusion models on a network: stochastic and deterministic pattern formation.

This thesis deals with the study of pattern formation on complex networks, a topic of paramount importance in different fields of broad applied and fundamental interest. Starting from a prototypical reaction-diffusion model, two main directions of investigation have been explored: on the one side, we have examined the system in its deterministic limit. Partial differential equations hence govern the evolution of the concentrations of the interacting species. In the second part of the thesis, we have conversely adopted a stochastic viewpoint to the scrutinized problem. Both in the deterministic and in the stochastic settings, the species are assumed to populate a complex graph, which provide the spatial backbone to the inspected model. Diffusion is allowed between neighbouring nodes, as designated by the associated adjacency matrix. According to the deterministic formulation, a small perturbation of a homogeneous fixed point can spontaneously amplify in a reaction-diffusion system, as follow a symmetry breaking instability and eventually yield asymptotically stable non homogeneous patterns. These are the Turing patterns. Travelling waves can also develop as follows an analogous dynamical instability. In this Thesis we have considered the peculiar setting where the spatial support is a directed network. Due to the structure of the network Laplacian, the dispersion relation has both real and imaginary parts, at variance with the conventional case for a symmetric network. The homogeneous fixed point of the system can turn unstable because of the topology of the hosting network. This observation motivates the introduction of a new class of instabilities, termed topology driven, which cannot be induced on undirected graphs. A linear stability calculation enables one to analytically trace the boundaries of the instabilities in the relevant parameters plane. Numerical simulations show that the instability can lead to travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph. Another scenario where topology matters is that of multi-layered networks, also known as multiplex networks. We have shown in this Thesis that the emergence of self-organized patterns on a multiplex can be instigated by a constructive interference between layers. It can be in fact proven that patterns can emerge for a reaction-diffusion system defined on a multiplex, also when the Turing-like instability is prevented to occur on each single layer taken separately. In other cases inter-layer diffusion can have a destructive influence on the process of pattern formation, as we will discuss in details in this Thesis work. Beyond the deterministic scenario, single individual effects can also impact the process of pattern formation. Stochastic fluctuations, originating from finite size populations, can in fact significantly modify the mean-field predictions and drive the emergence of regular macroscopic patterns, in time and space, outside the region of deterministic instability. In the second part of the Thesis we have studied the dynamics of stochastic reaction-diffusion models defined on a network. A formal approach to the problem has been developed which makes use of the Linear Noise Approximation (LNA) scheme. Simulations based on the Gillespie algorithm were performed to test the analytical results and analyzed via a generalized Fourier transform which is defined using the eigenvectors of the discrete graph Laplacian. Travelling waves as well as stationary patterns reminiscent of the Turing instability are shown to develop as mediated by the discreteness of the stochastic medium. As a final point we considered the case of a general stochastic reaction-diffusion system, where the activator is solely allowed to diffuse. Working under the LNA, we proved that stochastic Turing like pattern can develop, an observation which marks a striking difference with the conventional, customarily adopted, deterministic scenario.