Structured Equations for Complex Living Systems - Modeling, Asymptotics and Numerics

Complex living systems differ from those systems whose evolution is well described by the laws of Classical Physics. In fact, they are endowed with self-organizing abilities that result from the interactions among their constituent individuals, which behave according to specific functions, strategies or traits. These functions/strategies/traits can evolve over time, as a result of adaptation to the surrounding environment, and are usually heterogeneously distributed over the individuals, so that the global features expressed by the system as a whole cannot be reduced to the superposition of the single functions/strategies/traits. Quoting Aristotle, we can say that, within these systems, “the whole is more than the sum of its parts”. As a result, when we study the dynamics of complex living systems, there are new concepts that come into play, such as adaptation, herding and learning, which do not belong to the traditional vocabulary of physical sciences and make the dynamics of these systems hardly to be forecast. Moving from the above considerations, the subject of my PhD was the development and the study of structured equations for population dynamics (partial differential equations and integro-differential equations) applied to modelling the evolution of complex living systems. In particular, I designed models for multicellular systems, living species and socio-economic systems with the aim of inspecting mechanisms underlying the emergence of collective behaviors and self-organization. In the framework of structured equations, individuals belonging to a given system are divided into different populations and heterogeneously distributed characteristics are modelled by suitable independent variables, the so-called structuring variables. For each population, a function describing the distribution of the individuals over the structuring variables is introduced, which evolves through a partial differential equation, or an integro-differential equation, whose parameter functions are defined according to the phenomena under study. I decided to use such mathematical framework since it makes possible to effectively model the afore mentioned complexity aspects of living systems and provides an efficient way to reduce complexity in view of the mathematical formalization. With particular reference to multicellular systems, I focused on the design and the study of mathematical models describing the evolutionary dynamics of cancer cell populations under the selective pressures exerted by therapeutic agents and the immune system. Proliferation, mutation and competition phenomena are included in these models, which rely on the idea that the process leading to the emergence of resistance to anti-cancer therapies and immune action can be considered, at least in principles, as a Darwinian micro-evolution. It is worth noting that most of these models stem from direct collaborations with biologists and clinicians. Besides local and global existence results for the mathematical problems linked to the models, my PhD thesis presents results related to concentration phenomena arising in phenotype-structured equations and opinion-structured equations (i.e., the weak convergence of the solutions to sums of Dirac masses), and with the derivation of macroscopic models from space-velocity structured equations. From the applicative standpoint such concentration phenomena provide a possible mathematical formalization of the selection principle in evolutionary biology and the emergence of opinions; macroscopic models, instead, offer an overall view of the systems at hand. Numerical simulations are performed with the aim of illustrating, and extending, analytical results and verifying the consistency of the model with empirical data.