Nicola Bellomo

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics

This paper proposes a systems approach to social sciences based on mathematical framework derived from a generalization of the mathematical kinetic theory and on theoretical tools of game theory. Social systems are modeled as a living evolutionary ensemble composed by many individuals, who express specific strategies, cooperate, compete and might aggregate into groups which pursue a common interest. A critical analysis on the complexity features of social system is developed and a differential structure is derived to provide a general framework toward modeling.

On the interplay between mathematics and biology: hallmarks toward a new systems biology.

This paper proposes a critical analysis of the existing literature on mathematical tools developed toward systems biology approaches and, out of this overview, develops a new approach whose main features can be briefly summarized as follows: derivation of mathematical structures suitable to capture the complexity of biological, hence living, systems, modeling, by appropriate mathematical tools, Darwinian type dynamics, namely mutations followed by selection and evolution. Moreover, multiscale methods to move from genes to cells, and from cells to tissue are analyzed in view of a new systems biology approach.

Human behaviours in evacuation crowd dynamics: From modelling to “big data” toward crisis management

This paper proposes an essay concerning the understanding of human behaviours and crisis management of crowds in extreme situations, such as evacuation through complex venues. The first part focuses on the understanding of the main features of the crowd viewed as a living, hence complex system. The main concepts are subsequently addressed, in the second part, to a critical analysis of mathematical models suitable to capture them, as far as it is possible. Then, the third part focuses on the use, toward safety problems, of a model derived by the methods of the mathematical kinetic theory and theoretical tools of evolutionary game theory. It is shown how this model can depict critical situations and how these can be managed with the aim of minimizing the risk of catastrophic events.

A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up

ABSTRACT This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated Keller–Segel system, with the main novelty being that diffusion is nonlinear with flux delimiter features. More precisely, as a prototypical representative of this class we study radially symmetric solutions of the parabolic–elliptic system under the initial condition and no-flux boundary conditions in balls Ω⊂ℝn, where χ>0 and . The main results assert the existence of a unique classical solution, extensible in time up to a maximal Tmax∈(0,∞] which has the property that The proof of this is mainly based on comparison methods, which first relate pointwise lower and upper bounds for the spatial gradient ur to L∞ bounds for u and to upper bounds for ; second, another comparison argument involving nonlocal nonlinearities provides an appropriate control of z+ in terms of bounds for u and |ur|, with suitably mild dependence on the latter. As a consequence of (⋆), by means of suitable a priori estimates, it is moreover shown that the above solutions are global and bounded when either with if χ>1 and mc: = ∞ if χ≤1. That these conditions are essentially optimal will be shown in a forthcoming paper in which (⋆) will be used to derive complementary results on the occurrence of solutions blowing up in finite time with respect to the norm of u in L∞(Ω).

A Quest Towards a Mathematical Theory of Living Systems

y The special volume offers a global guide to new concepts and approaches concerning the following topics: reduced basis methods, proper orthogonal decomposition, proper generalized decomposition, approximation theory related to model reduction, learning theory and compressed sensing, stochastic and high-dimensional problems, system-theoretic methods, nonlinear model reduction, reduction of coupled [...]

Crowd dynamics and safety: Reply to comments on “Human behaviours in evacuation crowd dynamics: From modelling to “big data” toward crisis management”

The survey [13] presents an overview and critical analysis of the existing literature on the modeling of crowd dynamics related to crisis management toward the search of safety conditions. Out of this general review some rationale on research perspectives have been brought to the attention of the reader. The content of this paper is also related to the authors’ knowledge acquired in the EU project [50] which focuses on security problems during evacuation from complex venues. Crisis management should assure that the evacuation process occurs in a reasonably short time and that the local density of the people involved in the evacuation remains below a safety threshold.

Macroscopic First Order Models of Multicomponent Human Crowds with Behavioral Dynamics

This paper presents a new approach to the behavioral dynamics of human crowds. Macroscopic first order models are derived based on mass conservation at the macroscopic scale, while methods of the kinetic theory are used to model the decisional process by which walkers select their velocity direction. The present approach is applied to describe the dynamics of a homogeneous crowd in venues with complex geometries. Numerical results are obtained using a finite volume method on unstructured grids. Our results visualize the predictive ability of the model. Solutions for heterogeneous crowd can be obtained by the same technique where crowd heterogeneity is modeled by dividing the whole system into subsystems identified by different features.

On the derivation of angiogenesis tissue models: From the micro-scale to the macro-scale

This paper deals with the derivation of mathematical models at the macroscale of biological tissues corresponding to angiogenesis phenomena. The derivation is obtained by mathematical description delivered at the microscale of cells using a kinetic theory approach. A classical Chapman–Enskog expansion properly truncated is used to obtain the desired result. It is shown that the approach is general enough to describe a broad variety of different angiogenesis models corresponding to well-defined assumptions on interactions at the cellular scale.

Methods and tools of mathematical kinetic theory towards modelling complex biological systems

Methods of mathematical kinetic theory have been recently developed to describe the collective behavior of large populations of interacting individuals such that their microscopic state is identified not only by a mechanical variable (typically position and velocity), but also by a biological state (or sociobiological state) related to their organized, somehow intelligent, behavior. The interest in this type of mathematical approach is documented in the collection of surveys edited in [1], in the review papers [2], [3], and in the book [4].