Mathematical Models and Methods in Applied Sciences

We extend an agent-based model of crime-pattern formation initiated in Short et al. by incorporating the effects of law enforcement agents. We investigate the effect that these agents have on the spatial distribution and overall level of criminal activity in a simulated urban setting. Our focus is on a two-dimensional lattice model of residential burglaries, where each site (target) is characterized by a dynamic attractiveness to burglary and where criminal and law enforcement agents are represented by random walkers. The dynamics of the criminal agents and the target-attractiveness field are, with certain modifications, as described in Short et al. Here the dynamics of enforcement agents are affected by the attractiveness field via a biasing of the walk, the detailed rules of which define a deployment strategy. We observe that law enforcement agents, if properly deployed, will in fact reduce the total amount of crime, but their relative effectiveness depends on the number of agents deployed, the deployment strategy used, and spatial distribution of criminal activity.

For certain policing strategies, continuum PDE models can be derived from the discrete systems. The continuum models are qualitatively similar to the discrete systems at large system sizes.

The analysis of criminal behavior with mathematical tools is a fairly new idea, but one which can be used to obtain insight on the dynamics of crime. In a recent work,³⁴ Short et al. developed an agent-based stochastic model for the dynamics of residential burglaries. This model produces the right qualitative behavior, that is, the existence of spatio-temporal collections of criminal activities or "hotspots", which have been observed in residential burglary data. In this paper, we prove local existence and uniqueness of solutions to the continuum version of this model, a coupled system of partial differential equations, as well as a continuation argument. Furthermore, we compare this PDE model with a generalized version of the Keller–Segel model for chemotaxis as a first step to understanding possible conditions for global existence versus blow-up of the solutions in finite time.

In this paper, we provide the O(ε) corrections to the hydrodynamic model derived by Degond and Motsch from a kinetic version of the model by Vicsek and co-authors describing flocking biological agents. The parameter ε stands for the ratio of the microscopic to the macroscopic scales. The O(ε) corrected model involves diffusion terms in both the mass and velocity equations as well as terms which are quadratic functions of the first-order derivatives of the density and velocity. The derivation method is based on the standard Chapman–Enskog theory, but is significantly more complex than usual due to both the non-isotropy of the fluid and the lack of momentum conservation.

Animal groups represent magnificent archetypes of self-organized collective behavior. As such, they have attracted enormous interdisciplinary interest in the last years. From a mechanistic point of view, animal aggregations remind physical systems of particles or spins, where the individual constituents interact locally, giving rise to ordering at the global scale. This analogy has fostered important research, where numerical and theoretical approaches from physics have been applied to models of self-organized motion. In this paper, we discuss how the physics methodology may provide precious conceptual and technical instruments in empirical studies of collective animal behavior. We focus on three-dimensional groups, for which empirical data have been extremely scarce until recently, and describe novel experimental protocols that allow reconstructing aggregations of thousands of individuals. We show how an appropriate statistical analysis of these large-scale data allows inferring important information on the interactions between individuals in a group, a key issue in behavioral studies and a basic ingredient of theoretical models. To this aim, we revisit the approach we recently used on starling flocks, and apply it to a much larger data set, never analyzed before. The results confirm our previous findings and indicate that interactions between birds have a topological rather than metric nature, each individual interacting with a fixed number of neighbors irrespective of their distances.

Phylogenetic methods have recently been rediscovered in several interesting areas among which immunodynamics, epidemiology and many branches of evolutionary dynamics. In many interesting cases the reconstruction of a correct phylogeny is blurred by high mutation rates and/or horizontal transfer events. As a consequence, a divergence arises between the true evolutionary distances and the distances between pairs of taxa as inferred from the available data, making the phylogenetic reconstruction a challenging problem. Mathematically this divergence translates in the non-additivity of the actual distances between taxa and the quest for new algorithms able to efficiently cope with these effects is wide open. In distance-based reconstruction methods, two properties of additive distances were extensively exploited as antagonist criteria to drive phylogeny reconstruction: on the one hand a local property of quartets, i.e. sets of four taxa in a tree, the four-point condition; on the other hand, a recently proposed formula that allows to write the tree length as a function of the distances between taxa, the Pauplin's formula. A deeper comprehension of the effects of the non-additivity on the inspiring principles of the existing reconstruction algorithms is thus of paramount importance. In this paper we present a comparative analysis of the performances of the most important distance-based phylogenetic algorithms. We focus in particular on the dependence of their performances on two main sources of non-additivity: back-mutation processes and horizontal transfer processes. The comparison is carried out in the framework of a set of generative algorithms for phylogenies that incorporate non-additivity in a tunable way.

We consider a self-propelled interacting particle system for the collective behavior of swarms of animals, and extend it with an attraction term called roosting force, as it has been suggested in Ref. 30. This new force models the tendency of birds to overfly a fixed preferred location, e.g. a nest or a food source. We include roosting to the existing individual-based model and consider the associated mean-field and hydrodynamic equations. The resulting equations are investigated analytically looking at different asymptotic limits of the corresponding stochastic model. In addition to existing patterns like single mills, the inclusion of roosting yields new scenarios of collective behavior, which we study numerically on the microscopic as well as on the hydrodynamic level.