### Tuesday, June 28, 2016 (11:30 a.m. in Yost 306)

**Title: **Segregation Dynamics Motivated by Territorial Markings: The Transition from a Particle to a Continuum Model

**Student:** Abdulaziz Alsenafi

**Advisor:** Alethea Barbaro (Assistant Professor, Case Western Reserve University)

**Committee Members:** Erkki Somersalo, Wojbor Woyczynski, and Wanda Strychalski

**Abstract: **We present an agent-based model to simulate gang territorial development motivated by graffiti marking on a two-dimensional discrete lattice. Moreover, we study and analyse the dynamics and steady-state solutions of gang agents and their graffiti markings. For simplicity, we assume that there are only two rival gangs present, and they compete for territory. In this model, agents represent gang members who move according to a biased random walk. All agent interactions are indirect, with the interactions occurring through the graffiti field. Using numerical simulations, we show that gang segregation and territory formation may happen for different system parameters. We also show that a phase transition occurs between a well-mixed state and a segregated state. The numerical results show that the inverse temperature, decay rate and graffiti rate effect the phase transition. From the discrete model, we derive a continuum system for territorial development. Using the continuum equations, we perform a linear stability analysis to determine the stability of the equilibrium solutions. From the results of the linear stability analysis, we show that the critical values of both the discrete model and the continuum system have the same behavior.