arXiv subjects

Nancy Rodriguez

Publications and source records attributed to Nancy Rodriguez.

21 records · Page 2Linked to original sources

arXiv2015

Analysis of a model for the dynamics of riots

This paper is concerned with modeling the dynamics of social outbursts of activity, such as protests or rioting activity. In this sequel to our work in \cite{Berestycki2014}, written in collaboration with J-P. Nadal, we model the effect of restriction of information and explore the effects that it has on the existence of {\it upheaval waves}. The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. We prove the existence of global solutions to the Cauchy problem in $\mathbb{R}^d$ as well as the existence of {\it traveling wave solutions} under certain parameter regimes. We furthermore explore the effects of heterogeneities in the environment with the help of numerical simulations, which leads to pulsating waves in certain cases. We analyze the effects of periodic domains as well as the {\it barrier} problem with the help of numerical simulations and discuss many open problems.

math.AP↗

arXiv2013

Traveling Wave Solutions in a Reaction-Diffusion Model for Criminal Activity

We study a reaction-diffusion system of partial differential equations, which can be taken to be a basic model for criminal activity. We show that the assumption of a populations natural tendency towards crime significantly changes the long-time behavior of criminal activity patterns. Under the right assumptions on these natural tendencies we first show that there exists traveling wave solutions connecting zones with no criminal activity and zones with high criminal activity, known as hotspots. This corresponds to an invasion of criminal activity onto all space. Second, we study the problem of preventing such invasions by employing a finite number of resources that reduce the payoff committing a crime in a finite region. We make the concept of wave propagation mathematically rigorous in this situation by proving the existence of entire solutions that approach traveling waves as time approaches negative infinity. Furthermore, we characterize the minimum amount of resources necessary to prevent the invasion in the case when prevention is possible. Finally, we apply our theory to what is commonly known as the gap problem in the excitable media literature, proving existing conjectures in the literature.

math.AP↗

arXiv2011

Inhomogeneous Patlak-Keller-Segel models and Aggregation Equations with Nonlinear Diffusion in $\Real^d$

Aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with nonlinear diffusion are popular models for nonlocal aggregation phenomenon and are a source of a number of interesting mathematical problems in nonlinear PDE. The purpose of this work is twofold. First, we continue our previous work, which focused on nonlocal aggregation, modeled with a convolution. The goal was to unify the local and global theory of these convolution-type models, including the identification of a sharp critical mass; however, some cases involving unbounded domains were left open. In particular, the biologically relevant case $\Real^2$ was not treated. In this paper, we present an alternative proof of local existence, which now applies to $\Real^d$ for all $d \geq 2$ and give global results that were left open. The proof departs from previous work in that it uses a more direct and intuitive regularization that constructs approximate solutions on $\Real^d$ instead of on sequences of bounded domains. Second, this work develops the local, subcritical, and small data critical theory for a variety of Patlak-Keller-Segel models with spatially varying diffusion and decay rate of the chemo-attractant.

math.AP↗