A Convection-Diffusion Model for Gang Territoriality
AA Convection-Diffusion Model for Gang Territoriality
Abdulaziz Alsenafi ∗ and Alethea B. T. Barbaro † Kuwait University, Faculty of Science,Department of Mathematics, P.O. Box 5969, Safat -13060, Kuwait Department of Mathematics, Case Western Reserve University,10900 Euclid Ave, Yost Hall, Cleveland, OH 44106, USA (Dated: February 15, 2018)
Abstract
We present an agent-based model to simulate gang territorial development motivated by graffitimarking on a two-dimensional discrete lattice. For simplicity, we assume that there are two rivalgangs present, and they compete for territory. In this model, agents represent gang membersand move according to a biased random walk, adding graffiti with some probability as they moveand preferentially avoiding the other gang’s graffiti. All agent interactions are indirect, with theinteractions occurring through the graffiti field. We show numerically that as parameters vary, aphase transition occurs between a well-mixed state and a well-segregated state. The numericalresults show that system mass, decay rate and graffiti rate influence the critical parameter. Fromthe discrete model, we derive a continuum system of convection-diffusion equations for territorialdevelopment. Using the continuum equations, we perform a linear stability analysis to determinethe stability of the equilibrium solutions and we find that we can determine the precise location ofthe phase transition in parameter space as a function of the system mass and the graffiti creationand decay rates.
Keywords: agent-based model, phase transition, segregation model, crime modeling ∗ Alsenafi@sci.kuniv.edu.kw † [email protected] (corresponding author) a r X i v : . [ n li n . C G ] F e b . INTRODUCTION Many of the world’s countries face violence related to gangs. In the United States ofAmerica alone, it is estimated that there are 1.4 million gang members, and gangs areresponsible for 48% of violence in most jurisdictions and up to 90% in some jurisdictions[12]. Gangs identify themselves by distinctive graffiti, clothing and handshakes [6, 13]. Inmany regions such as Los Angeles, California and Phoenix, Arizona, it has been found thatgangs claim territory through graffiti markings [1, 20]. Because of the widespread natureand societal impact, it is an important sociological question to understand how gangs formand operate.Recently, the physics and mathematics community has taken an interest in crime mod-eling. Much of this research was influenced by Schelling’s seminal work on segregationdynamics [33–35]; there is widespread interest in modeling social segregation, especiallysince many of these models can be viewed from a physical or mathematical perspective[3, 32, 40, 46]. Game theory has been widely used to study population-level effects ofcriminality [37, 41, 42], sometimes including a spatio-temporal or agent-based frameworkinto the game [7, 30]. Predator-prey-type dynamical systems have also been used in thesepopulation-level models [29].Other methodologies in crime modeling connect the models to data, such as clusteringmethods to deduce community affiliations among gang members in Los Angeles [43], self-exciting point processes to study the temporal patterns of residential burglary in Los Angeles[25], scaling laws for homicide in Brazilian cities [2], and an epidemiological model for the2005 rioting in France [4]. Several agent-based models for burglary and gang dynamicsare discussed in further detail below [16, 17, 36, 38, 39]. In fact, crime modeling has beenso effective that crime modeling has intersected with a new area, predictive policing, andmuch of the literature is now intertwined. Many of these models have proven effective atpredicting violence and at geographically profiling offenders [23, 24, 26] and there are manyapplications of crime modeling to assisting with police districting [9, 10, 21]. For a morethorough overview of the literature, the interested reader is directed to [14] and [11].In a seminal paper on crime modeling [38], Short et al. created a lattice model for bur-glary based on the assumption that burglars often return to homes that have previouslybeen successfully burglarized or ones that are close to it, a well-known phenomenon of-2en referred as the ‘broken windows’ effect [45]. They derived a continuum system fromthe discrete model, consisting of two coupled reaction-diffusion equations that describe thespatio-temporal evolution of agents and attractiveness densities, and identified under whichconditions crime hot spots occur. Several agent-based models have been proposed to ana-lyze these crime hot spots [16, 36]. Related models have been developed for policing suchhotspots [17], and for the mathematical analysis of the continuum model [31].Our work considers the formation of gang territories due to graffiti markings, hearkeningback to the mathematical ecology literature, where researchers first discovered the role ofscent marking in territorial development for wolf packs [22]. Researchers then created modelsfor the territorial behavior of coyotes and wolves using scent marking [19, 44]. Many of thesemodels include outside information such as a home den or information about the terrain[27, 28]. In some of these models, the authors give both discrete and continuum versionsof their model [5]. While similar in spirit to the model developed and analyzed here, themain focus of these papers is on simulating the real-world territorial dynamics of the coyotesand wolves. Therefore, the assumptions made in these models and the continuum limits aresignificantly different from those presented here.Researchers in crime modeling have considered graffiti markings instead of scent marking,and have applied these same ideas to gang dynamics. In [39], Smith et al. develop a modelto describe the equilibrium densities of gangs and their graffiti in the policing division ofHollenbeck by combining the wolf and coyote models [19, 27, 28, 44] with the biased L´evywalk with networks model by Hegemann et al. [15]. Taking a different approach, in [3],Barbaro et al. use a two-dimensional spin model to examine the problem of the developmentand formation of gang territory based on graffiti, proving that the system undergoes a phasetransition. Like [3], our model is premised on the fact that gangs avoid rival gangs’ graffitiand put down graffiti of their own as they move. However, unlike any previous agent-basedmodels known to the authors, in this paper we consider the temporal evolution of bothagent and graffiti densities where agent dynamics are designed to follow observed behaviorof gang members, allowing the results to be much more understandable and applicable in acriminological setting.Our paper is organized as follows: in Sec. II A, we present the agent-based model thatprovides a basis for the rest of the paper. In Sec. II B, we examine the different phasesthat this discrete model exhibits and offer an order parameter to aid in the analysis of the3hase transition. In Sec. III A, we present simulations of the discrete model and numericallyexplore the phase transition. In Sec. III B, we formally derive a set of four continuumequations from the discrete model. In Sec. III C, we find a steady-state for the continuummodel and perform a stability analysis to find where the well-mixed solution loses stability;we then compare this with the numerically computed critical parameter values from Sec.III A. In Sec. IV, we conclude with a discussion of the main results of this work.
II. THEORY AND CALCULATIONSA. Discrete Model
We begin with a two-dimensional L × L square lattice denoted by S . We assume thatthere are two gangs, red and blue, denoted by A and B ; we further assume that the numberof agents belonging to the red and blue gangs are equal and are denoted by N A and N B ,respectively. The total number of agents in the system henceforth will be denoted by N , sothat N = N A + N B . Initially, the agent locations are randomly distributed on S using themultivariate uniform distribution.In this model, multiple agents of any color can occupy the same location. We denotethe number of red and blue agents at a site ( x, y ) at time t by n A ( x, y, t ) and n B ( x, y, t ),respectively. Agents add graffiti markings of their own color with some probability, andthey move to avoid graffiti of the opposing color. The amount of red graffiti is denoted g A ( x, y, t ) and the blue graffiti by g B ( x, y, t ). Initially, we assume that the lattice is devoidof all graffiti. We assume periodic boundary conditions throughout. The precise descriptionof the model is given below.Gang members prefer to be in territory occupied by their own gang, and avoid rivals’territories except under exceptional circumstances [20]. Therefore, in our model, agentspreferentially avoid the opposing color graffiti, performing a biased random walk once thereis graffiti on the lattice. We assume that all agents move at each time step to one of theirfour neighboring lattice sites, so that an agent at site ( x, y ) must move to one of the foursites { ( x + l, y ) , ( x − l, y ) , ( x, y + l ) , ( x, y − l ) } , where l is the lattice spacing. We assignthe probability of a red agent to move from site s = ( x , y ) ∈ S to a neighboring site4 = ( x , y ) ∈ S to be M A ( x → x , y → y , t ) : = e − βl g B ( x ,y ,t ) (cid:80) (˜ x, ˜ y ) ∼ ( x ,y ) e − βl g B (˜ x, ˜ y,t ) , (1)where β is parameter that controls the strength of avoidance of blue graffiti and (˜ x, ˜ y ) ∼ ( x , y ) denotes the four neighbors of site ( x , y ); M B is defined similarly, with g B changedfor g A . Considering the amount of graffiti belonging to the red and blue gangs as densities ξ A = g A l and ξ B = g B l allows us to reformulate (1) as follows: M A ( x → x , y → y , t ) = e − βξ B ( x ,y ,t ) (cid:80) (˜ x, ˜ y ) ∼ ( x ,y ) e − βξ B (˜ x, ˜ y,t ) . (2)Since our model assumes that all agents must move at every time step, it follows triviallythat (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) M A ( x → ˜ x, y → ˜ y, t ) = 1 . After the agents have moved, the expected number of agents at site ( x, y ) ∈ S is n A ( x, y, t + δt ) = n A ( x, y, t ) + (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) n A (˜ x, ˜ y, t ) M A (˜ x → x, ˜ y → y, t ) − n A ( x, y, t ) (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) M A ( x → ˜ x, y → ˜ y, t ) , where the first sum represents the agents arriving at site ( x, y ) and the second sum representsthe agents leaving site ( x, y ). Similarly to the graffiti, we convert the number of red agents n A into a density ρ A = n A /l , which brings us to our discrete update rule for the agentdensity: ρ A ( x, y, t + δt ) = ρ A ( x, y, t ) + (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) ρ A (˜ x, ˜ y, t ) M A (˜ x → x, ˜ y → y, t ) − ρ A ( x, y, t ) (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) M A ( x → ˜ x, y → ˜ y, t ) . (3) The update rule for the density of the blue agents is defined analogously.As the agents move, they add graffiti to the lattice. At each time step, each agent has aprobability of γ , scaled by time step δt , of adding its own color graffiti at its current location.At each time step, the graffiti also decays at a rate λ >
0, similarly scaled by δt . Therefore,the amount of red graffiti at site ( x, y ) ∈ S at time t + δt is: g A ( x, y, t + δt ) = g A ( x, y, t ) − ( λ · δt ) g A ( x, y, t ) + ( γ · δt ) n A ( x, y, t ) . l to convert g i ( x, y, t ) into density ξ i ( x, y, t ) for i ∈ { A, B } , we arrive atthe graffiti density update rules: ξ A ( x, y, t + δt ) = ξ A ( x, y, t ) − ( λ · δt ) ξ A ( x, y, t ) + ( γ · δt ) ρ A ( x, y, t ) (4) ξ B ( x, y, t + δt ) = ξ B ( x, y, t ) − ( λ · δt ) ξ B ( x, y, t ) + ( γ · δt ) ρ B ( x, y, t ) . (5) B. Phases and an Order Parameter
When our model is simulated, we observe two possible states: a well-mixed state whereagents and graffiti of both colors are uniformly distributed throughout the lattice, anda segregated state where red agents and red graffiti separate from blue agents and bluegraffiti. In this section, we explore these two possible phases for our model, defining anorder parameter to quantify the distinction between the two states.
1. Expected Agent Density
In the well-mixed state , the agents and graffiti are roughly uniformly distributed on thelattice S , and each location ( x, y ) is likely to have both red and blue agents and graffitipresent. The movement of agents in a well-mixed state resembles an unbiased random walk.In the segregated state , each gang clusters together to form territories, and the movement ofagents becomes a biased random walk. In the segregated state, at each site ( x, y ), there areusually agents of only one color present.In a well-mixed state with N i agents from gang i ∈ { A, B } , the agents are uniformlydistributed over the L × L lattice, and we can compute the expected agent density for gang i ∈ { A, B } at any point ( x, y ) on the lattice: E ( ρ i ( x, y )) = (cid:88) ( x,y ) ∈ S ρ i ( x, y ) × L = (cid:88) ( x,y ) ∈ S n i ( x, y ) l × l = (cid:88) ( x,y ) ∈ S n i ( x, y )= N i . (6)
6n contrast, in the segregated state, the agents are separated into territories by color. Wemake the assumption, numerically borne out in Sec. III A, that the agents are uniformlydistributed within those territories. We consider all of the red territory as a sublatticedenoted by S A ⊂ S ; the area of the sublattice S A is denoted by R A . We note that thesublattice S A may not be connected. Similarly, we assume that blue territory considered alltogether is sublattice S B with area R B . We further assume that in a well-segregated statethere are no empty sites, so that S = S A ∪ S B . These assumptions are reasonable from ournumerical simulations; see Sec. III A for details.Under these assumptions, in the segregated state, the expected density of of red agentsis: E ( ρ A ( x, y )) = (cid:88) ( x,y ) ∈ S ρ A ( x, y ) × L . Splitting the lattice into sublattices S A and S B gives: E ( ρ A ( x, y )) = (cid:88) ( x,y ) ∈ S A ρ A ( x, y ) R A + (cid:88) ( x,y ) ∈ S B ρ A ( x, y ) R B = (cid:88) ( x,y ) ∈ S A ρ A ( x, y ) R A = (cid:88) ( x,y ) ∈ S A n A ( x, y ) l R A = N A l R A , where ( x, y ) ∈ S A . The same argument holds for blue agents. Hence, for i ∈ { A, B } : E ( ρ i ( x, y )) = N i l R i , ( x, y ) ∈ S i , ( x, y ) / ∈ S i . (7)Finally, we note that in our model, the areas dominated by gangs A and B are nearly equalif we begin with N A = N B . Thus, the assumption that R i = L is reasonable, giving: E ( ρ i ( x, y )) = N i , ( x, y ) ∈ S i , ( x, y ) / ∈ S i . (8) . An Order Parameter To examine the phase transition, we define an order parameter at time t : E ( t ) = (cid:18) LN (cid:19) (cid:88) ( x,y ) ∈ S (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) ( ρ A ( x, y, t ) − ρ B ( x, y, t )) ( ρ A (˜ x, ˜ y, t ) − ρ B (˜ x, ˜ y, t )) . (9) This order parameter is akin to a magnetization for the system, and is similar in formto the Hamiltonian for the Ising Model. The summand is positive if neighboring sites aredominated by the same color, and negative if they are dominated by the opposite colors.The coefficient at the front normalizes the sum, so that the maximum value is 1 independentof the lattice size and number of agents. In the segregated state, agents from each colorcluster together to form territories, and at each site ( x, y ) there is only one color present.This forces the term inside each of the sets of parentheses in equation (9) to be large inmagnitude, with the sign in both cases very likely to be identical; once multiplied together,the result would be large and positive. However, in the well-mixed state, the agents of bothgangs are uniformly distributed over all sites. This means that the terms inside the firstand second brackets of equation (9) both tend to be very small, and the signs are unlikelyto agree. Hence, once multiplied together, the result is very small in magnitude and, aftersummation, the order parameter is very close to zero.We now calculate an approximation of the order parameter for the well-mixed and seg-regated states. Similar to the expected value approximation and for the same reasons, wedrop the time t from the notation. Starting with the well-mixed case, the approximatedorder parameter is: E = (cid:18) LN (cid:19) (cid:88) ( x,y ) ∈ S (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) ( ρ A ( x, y ) − ρ B ( x, y )) ( ρ A (˜ x, ˜ y ) − ρ B (˜ x, ˜ y )) . Using equation (6), the assumption that in a well-mixed state the distribution of agents isequal for all sites, and that each site has four neighbors, E = (cid:18) LN (cid:19) (cid:88) ( x,y ) ∈ S ( N A − N B ) (4 N A − N B )= (cid:18) LN (cid:19) (cid:88) ( x,y ) ∈ S ( N A − N B ) = (cid:18) LN (cid:19) L ( N A − N B ) = 1 N ( N A − N B ) = 0 , (10) N A = N B . Therefore, the order parameterin a well-mixed state is approximately zero.The order parameter of the system in a completely segregated state can be derived simi-larly. Splitting the sum (9) over the regions S A and S B , we see that: E = (cid:18) LN (cid:19) (cid:34) (cid:88) ( x,y ) ∈ S A (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) ( ρ A ( x, y ) − ρ B ( x, y )) ∗ ( ρ A (˜ x, ˜ y ) − ρ B (˜ x, ˜ y ))+ (cid:88) ( x,y ) ∈ S B (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) ( ρ A ( x, y ) − ρ B ( x, y )) ∗ ( ρ A (˜ x, ˜ y ) − ρ B (˜ x, ˜ y )) (cid:35) . We substitute in the expectation of ρ A and ρ B for each sublattice; letting R i = | S i | andignoring the boundaries of the regions, we find: E ≈ (cid:18) LN (cid:19) (cid:34) (cid:88) ( x,y ) ∈ S A (cid:18) N A l R A (cid:19) (cid:18) N A l R A (cid:19) + (cid:88) ( x,y ) ∈ S B (cid:18) N B l R B (cid:19) (cid:18) N B l R B (cid:19) (cid:35) = 14 (cid:18) lN (cid:19) l (cid:34) (cid:88) ( x,y ) ∈ S A (cid:18) N A R A (cid:19) + (cid:88) ( x,y ) ∈ S B (cid:18) N B R B (cid:19) (cid:35) = (cid:18) LN (cid:19) (cid:34) R A (cid:18) N A R A (cid:19) + R B (cid:18) N B R B (cid:19) (cid:35) = (cid:18) LN (cid:19) (cid:18) N A R A + N B R B (cid:19) . By assumption, we have perfect segregation with N A = N B , thus E ≈ (cid:18) LN A N (cid:19) (cid:18) R A + 1 R B (cid:19) = (cid:18) L (cid:19) (cid:18) R A + 1 R B (cid:19) . We substitute R A = R B = L to find: E ≈ . (11) Therefore, the order parameter parameter ranges from close to zero in a well-mixed stateto close to one in a completely segregated state.9
II. RESULTS AND DISCUSSIONA. Simulations of the Discrete Model
1. Well-Mixed State
We first consider the system in a well-mixed phase, with a small β value. We let β =1 × − , and evolve the system according to the discrete model; the resulting lattices areshown in Fig. 1. The lattices in top row of Fig. 1 represent the agent density over time,while the bottom lattices represent the temporal evolution of the graffiti territory. The agentplots show how many agents of each gang are on the site; the higher the ratio of gang A togang B, the more red the site appears, and the higher the ratio of gang B to gang A, themore blue it appears. When the ratio is close to one, the site appears green. The graffititerritory plots show which gang has more graffiti on a particular site; if there is more graffitifrom gang A than gang B on a site, then the site will be marked by the color red, and inthe opposite situation, the site is marked by the color blue. We also assign the color greenif there is exactly the same amount of graffiti present from both gangs at a site.It is evident from Fig. 1 that we do not have segregation for β = 1 × − , as we neithersee red nor blue patches developing. In fact, the movement of each agent in this simulationresembles an unbiased random walk on a two-dimensional lattice. The agents’ random walksare a direct result of the definition of the gang movement (2). If β is sufficiently small, thenthe probability of an agent moving from ( x , y ) to ( x , y ) approaches , and thereforethe movement approximates an unbiased random walk. In effect, the parameter β dampensagents’ response to variations in the graffiti density ξ . Larger β values amplify the variationsof the graffiti density, leading to a phase transition. This phase transitions is studied in Sec.III A 3.Before investigating how larger β values affect our system, let us examine the well-mixedphase more closely by taking a cross-sectional slice which shows the distribution of agentdensity on a lattice row. In these cross-sections, the effects of the stochasticity of thesimulations can be seen quite clearly. The cross-sectional slices over time for β = 1 × − are shown in Fig. 2; behind the cross-section for the agents, we also plot the expectedagent density for the well-mixed phase approximated in equation (6). Note that we cannotconsider an ensemble average for the cross-sectional slice because, due to the stochasticity10 IG. 1. Temporal evolution of the agent density on the left, and the temporal evolution for thegraffiti territory on the right for a well-mixed state. Here we have N A = N B = 100 , λ = γ = 0 . β = 1 × − , δt = 1 and the lattice size is 100 × in the model, each simulation could have a different territory evolution, leading to vastlydiverging agent and graffiti distributions. Taking an ensemble average is therefore likely toresult in the density distributions being uniformly distributed over time, obfuscating theterritorial development and leading to incorrect conclusions.From the cross-sectional slice in Fig. 2, we notice that the agent and graffiti densities areroughly uniformly distributed across the row, with noise inherent to the stochastic agent-based simulations. It is clear from the figure that the agent density is quite close to 1 × ,which is the expected agent density for a well-mixed phase.11 IG. 2. Temporal evolution of a cross-sectional slice of the agent and graffiti densities for a well-mixed state. Here we have N A = N B = 100 , λ = γ = 0 . β = 1 × − , δt = 1 andthe lattice size is 100 × th row. It is clearly seen that the agentsand the graffiti remain well-mixed over time for these parameters, and that our predicted agentexpectation is a good approximation for the agent density.
2. Well-Segregated State
We now increase the value of beta twenty-fold to 2 × − , while maintaining the latticesize and the number of agents as in the well-mixed state described above. Four time pointsof the simulations are shown in Fig. 3. We observe that the initial state of the system iswell-mixed; over time, we see that the red and blue agents start to cluster together andform all-red and all-blue areas, coarsening over time. Thus, β = 2 × − is large enough toproduce a segregated phase, changing the agents’ movement from an unbiased random walkto a biased random walk. This indicates that the critical β or the phase transition shouldoccur somewhere in the interval β ∈ (1 × − , × − ).In Fig. 4, we now consider cross-sectional slices for the segregated state in order tocompare them against Fig. 2. From equation (8), we know the expected agent densities forgang i ∈ { A, B } ; these are plotted in the agent density plots as the red and blue dotted lines.Upon examining the cross-sectional slices in Fig. 4, we observe that initially, the agents ofboth colors are uniformly distributed across the row. The uniform value of the agent densityis roughly 1 × for both gangs. As time progresses, the agents begin to segregate, and12 IG. 3. Temporal evolution of the agent density lattice (top) and territory dominated by the gang’sgraffiti (bottom) for a segregated state. Here we have N A = N B = 100 , λ = γ = 0 . β = 2 × − , δt = 1 and the lattice size is 100 × increasingly large pockets of all-red or all-blue agents appear. Once segregated, we noticethat the red agents and graffiti are nearly uniformly distributed in the all-red pockets witha value 2 × , and vice versa for the blue agents and graffiti. The reason for the agentdensity doubling in value is that initially the gangs had to cover the entire row, but oncesegregated, the gangs had to cover only their own territory, which is approximately half ofthe row. This agrees with the expected agent density for a segregated state from equation(8); the roughness of the distributions is due to the stochastic nature of the simulation.We end our discussion of the different states of the system by providing an illustrativeexample of how β can amplify or dampen the effect of variations of the graffiti density ξ . Letus assume that the left, right, up and down neighbors of site ( x, y ) have the following gang B graffiti densities: { × , 0 . × , 0 . × , 0 . × } . The graffiti density valuesused here are taken from the initial cross-sectional slice of the well-mixed phase in Fig. 2.Using equation (1), we check the probabilities of a red agent moving to a neighboring sitefor different β values. We use the same β values as the ones used in the well-mixed and well-13 IG. 4. Temporal evolution of a cross-sectional slice of the agent density (top) and the graffitidensity (bottom) for a segregated state. Here we have N A = N B = 100 , λ = γ = 0 . β = 2 × − , δt = 1 and the lattice size is 100 × th row. Itis clearly seen that both the agents and graffiti segregate over time for these parameters and thatour predicted agent expectation is good at approximating the agent density. segregated states, in addition to β = 6 . × − , which creates a partially segregated state.The resulting probabilities are summarized in Table I. We notice that for the well-mixed β value 1 × − , the probability of moving to any of the four neighboring sites is approximately0 .
25, although there is a large difference in the amount of graffiti on each of the sites. Fora larger β value, the probability of moving down is 0 . . β amplifies the effects of the variations in the graffiti density, instigating the phasetransition. 14 ξ B = 1 × , ξ B = 0 . × , ξ B = 0 . × , ξ B = 0 . × ,M left M right M up M down × − . × − × − A moving to a neighboring site for different β values.Clearly, a larger β has a greater amplification effect on the variations in the graffiti density ξ B .
3. Phase Transition in the Discrete Model
We are now in a position to identify the critical β at which the model undergoes aphase transition, using the order parameter defined in equation (9). In Fig. 5, we plot theensemble average of four simulations of the order parameter over the course of a simulationfor different values of β . We expect that system has a high order parameter value in asegregated state, and a low order parameter value in a well-mixed state. In Fig. 5, we seethat for β = 0 and β = 0 . β = 0 . β = 0 . ,
000 time steps; afterwardsthe order parameter values equilibrate to around 0 .
95 and remain there over time. Finally,for β = 0 . . β , β ∗ , which is the point at which the behavior of the system changes from well-mixed (disordered) to segregated (well-ordered). To locate β ∗ numerically, we notice thatthe order parameter values of the system will eventually equilibrate to some constant (seeFig. 5). We produce a phase transition plot by taking the final value of that order parameter15 IG. 5. Temporal evolution of the order parameter for different values of β . Here we have N A = N B = 100 , λ = γ = 0 . δt = 1 and the lattice size is 100 × β the system was well-mixed and the order parameter stays close to zero through time. Forsufficiently larger β values, we have a segregated state and the order parameter increases and levelsoff just below one. constant and plotting it against different values of β . The critical parameter β ∗ is the valueat which the order parameter becomes nonzero; here, we approximate that value by taking β ∗ be the point where the order parameter of the system surpasses 0 .
01. Several phasetransition plots for our system are shown in Fig. 7.The phase transition and β ∗ might depend on the other parameters in the system, namelythe total mass of the system ( N ), the graffiti production and decay rates ( γ and λ , respec-tively), the lattice dimension ( L ), and the time step ( δt ). To investigate how the phasetransition depends on each of the parameters, we keep the other parameters fixed and varyonly one. In the first two figures of Fig. 7, we can observe that the mass indeed affectsthe critical β . The left plot of the figure shows the phase transition for N = 200 , N A = N B , while the middle plot shows the phase transition for N = 100 , N A = N B . We can observe that for the smaller mass, β ∗ is almost double that of the simu-lations with the larger mass. Hence, if the system has more agents, then the required β forsegregation is smaller. This makes sense in terms of our model, because if there are moreagents in the system, then there will be more graffiti added, and each site will have a largeramount of graffiti. This implies that a smaller β will be needed for the agents to react tothat graffiti. 16 IG. 6. Temporal evolution of the territory dominated by the gang’s graffiti for a segregated statewith different lattice sizes. Here we have N A = N B = 100 , λ = γ = 0 . β = 2 × − and δt = 1. The lattice sizes for the figure were 50 ×
50 for the first row, 75 ×
75 for the second row and100 ×
100 for the last row. It is clearly seen that the territory dominated by graffiti segregation issimilar over time regardless of the lattice size.
4. The Role of Lattice Size and Time Step
To examine how the lattice size can affect the phase transition of our system, we keepthe number of agents, which is also referred as the system mass N , fixed while we changethe grid size. Note that in our system simulations the total area of the lattice always equals17ne, therefore by increasing the lattice size L , we make the grid finer and the sites smaller.For the visualization, we use the following lattice sizes: 50 ×
50, 75 ×
75, and 100 × N = 200 ,
000 with N A = N B , and β = 2 × − fixed. The graffititerritory lattices for varying values of L are shown in Fig. 6, while the order parameterevolution for the different lattice sizes can be seen in Fig. 7. For the order parameterevolution, we plotted the ensemble average of four simulations.From Fig. 7, we observe that by keeping the system parameters fixed and only changingthe grid size, the rate of segregation is unaffected, and that the coarsening rate does notdepend on the grid size. This is important, because in Sec. III B, we will be deriving thecontinuum equations, and it is only natural to wonder how a finer grid may affect the discretemodel. In Fig. 6, we see that a finer grid produces a ‘smoother’ lattice visualization, asthere are more sites, and they are smaller in size. Hence, the lattice for a finer grid is less‘pixelated’ and the territories have smoother boundaries.In Sec. III B, we will derive the continuum equations by taking the time step δt to zeroas we take the lattice spacing l = L to zero. Hence, it is important to understand the rolenot only of the grid spacing but also of the time step in the dynamics of the discrete model.To this end, we numerically observe the effects of a smaller time step in the discrete model.In Fig. 8, we visualize the temporal evolution of order parameter and the phase transitionplot. We notice that decreasing the time step does not alter the critical β for the phasetransition or alter the evolution of the order parameter, indicating that the size of the timestep in our discrete model has little effect.Comparing the first and last plots of Fig. 7, we can see that the ratio γλ also affects thecritical β . We see from the figure that as the ratio is halved from 1 on the left to onthe right, the shape of the phase transition is maintained but β ∗ roughly doubles. This,again, is unsurprising, since increasing the decay rate λ in the ratio implies that the graffitidecays more quickly and there is less graffiti at each site, thereby forcing a higher β valuefor segregation to occur. The same can be said about decreasing the graffiti rate γ .We discuss the phase transitions again in Sec. III C 2, where we examine how β ∗ changesas we vary the parameters. There, we use the critical β as a means to compare this discretemodel with its continuum system counterpart, which is derived in the next section.18 IG. 7. The order parameter at the final time step against β for different lattice sizes and numberof agents. Here, we have λ = γ = 0 . δt = 1. In the first figure, the number of agents N A = N B = 100 , N A = N B = 50 , β is also equal for all three lattice sizes. Comparing the first and middle figures, we notice that thecritical β increased as the mass decreased. Comparing the first and bottom third, we notice thatthe critical β increased as the ratio decreased.FIG. 8. The phase transition as the time step is changed. In both figures, we have N A = N B =100 , λ = γ = 0 . × β = 2 × − (a segregated state). The evolution of the orderparameter is observed to be the same for all three time steps shown. On the right, we see the orderparameter at the final time step against β for different time steps. It is clear to observe that theplots for the three different time steps chosen are almost identical, with the same critical β ∗ . . Derivation of the Continuum Model In order to better understand our system, in this section, we formally derive a system ofcorresponding continuum equations by taking the time step and the grid spacing to zero.
1. Continuum Graffiti Density
The evolution equations for the graffiti density are easily found. Recalling the discretemodel (5), and taking the limit δt →
0, while assuming that the graffiti density ξ A issufficiently smooth, the evolution equation for gang A ’s graffiti becomes ∂ξ A ∂t ( x, y, t ) = γρ A ( x, y, t ) − λξ A ( x, y, t ) . (12)The evolution equation for gang B ’s graffiti follows identically.
2. Continuum Agent Density
Before deriving the continuum equation for the agent density, we define several quantitiesthat will be of significant notational help. First, let us define T A : T A ( x, y, t ) := e βξ B ( x,y,t ) l (cid:16) ( β ∇ ξ B ( x, y, t )) − β ∆ ξ B ( x, y, t ) (cid:17) , (13) Hereafter, we will drop the ( x, y, t ) as it is notationally superfluous. We will also needapproximations to ∇ T A and ∆ T A . Recalling the Taylor expansion1 a + h = 1 a − ha + O ( h ) . and letting a = 4 and h = l (cid:0) ( β ∇ ξ B ) − β ∆ ξ B (cid:1) , we can approximate T A : T A = e βξ B (cid:18) − l (cid:0) ( β ∇ ξ B ) − β ∆ ξ B (cid:1)(cid:19) + O ( l ) . (14) Taking the gradient, we find: ∇ T A = e βξ B (cid:18) β ∇ ξ B − l (cid:0) ( β ∇ ξ B ) + β ∇ ξ B ∆ ξ B − β ∇ ξ B (cid:1)(cid:19) + O ( l ) . (15)
20y taking the divergence, we find: ∆ T A = ∇ · ( ∇ T A )= e βξ B (cid:18) (cid:0) ( β ∇ ξ B ) + β ∆ ξ B (cid:1) − l (cid:16) β ( ∇ ξ B ) ∆ ξ B + β (∆ ξ B ) + ( β ∇ ξ B ) − β ∇ ξ B (cid:17)(cid:19) + O ( l ) . (16) We derive T B , ∇ T B , and ∆ T B similarly.We can now derive approximations to the probabilities of an agent arriving to the site( x, y ) from a neighboring site. Recall equation (2), M A (˜ x → x, ˜ y → y, t ) = e − βξ B ( x,y,t ) (cid:80) (˜˜ x, ˜˜ y ) ∼ (˜ x, ˜ y ) e − βξ B (˜˜ x, ˜˜ y,t ) , (17)where (˜˜ x, ˜˜ y ) are the neighbors of site (˜ x, ˜ y ). We use the discrete Laplacian to removethe influence of the neighbors’ neighbors (˜˜ x, ˜˜ y ) in the denominator. Recalling the discreteLaplacian: (cid:88) (˜˜ x, ˜˜ y ) ∼ (˜ x, ˜ y ) e − βξ B (˜˜ x, ˜˜ y,t ) = 4 e − βξ B (˜ x, ˜ y,t ) + l ∆ (cid:16) e − βξ B (˜ x, ˜ y,t ) (cid:17) + O ( l ) , (18) and noting that ∆ e − βξ B (˜ x, ˜ y,t ) = ∇ · ∇ (cid:16) e − βξ B (˜ x, ˜ y,t ) (cid:17) = ∇ · (cid:16) − β ∇ ξ B (˜ x, ˜ y, t ) e − βξ B (˜ x, ˜ y,t ) (cid:17) = (cid:104) ( β ∇ ξ B (˜ x, ˜ y, t )) − β ∆ ξ B (˜ x, ˜ y, t ) (cid:105) e − βξ B (˜ x, ˜ y,t ) , (19) we combine (18) with (19) to give (cid:88) (˜˜ x, ˜˜ y ) ∼ (˜ x, ˜ y ) e − βξ B (˜˜ x, ˜˜ y,t ) = e − βξ B (˜ x, ˜ y,t ) (cid:16) l (cid:16) ( β ∇ ξ B (˜ x, ˜ y, t )) − β ∆ ξ B (˜ x, ˜ y, t ) (cid:17)(cid:17) + O ( l ) . Finally, we substitute it back into equation (17), replacing the denominator to give M A (˜ x → x, ˜ y → y, t ) = e − βξ B ( x,y,t ) e βξ B (˜ x, ˜ y,t ) l (cid:16) ( β ∇ ξ B (˜ x, ˜ y, t )) − β ∆ ξ B (˜ x, ˜ y, t ) (cid:17) + O ( l ) . However, we notice that the term in squared brackets takes the form of (13), thus givingus the final form: M A (˜ x → x, ˜ y → y, t ) = e − βξ B ( x,y,t ) T A (˜ x, ˜ y, t ) + O ( l ) . (20)21he result for gang B is similar, completing our notational toolbox.We now formally derive the continuum equations for the agent density. Our main toolsare the discrete Laplacian for approximating the influence of the neighbors of site ( x, y ), andthe approximations (20). Recalling the discrete model (3) for the agent density, dividingboth sides by δt , and noting that, at any time t , the movement probabilities away from asite sum to one gives us ρ A ( x, y, t + δt ) − ρ A ( x, y, t ) δt = 1 δt (cid:34) e − βξ B ( x,y,t ) (cid:88) (˜ x, ˜ y ) ∼ ( x,y ) ρ A (˜ x, ˜ y, t ) T A (˜ x, ˜ y, t ) − ρ A ( x, y, t ) + O ( l ) (cid:35) . (21) We use the discrete Laplacian (18) to approximate the summation on the right, replacingthe contribution from the neighboring sites with information from the current site: δt (cid:34) e − βξ B ( x,y,t ) (cid:18) ρ A ( x, y, t ) T A ( x, y, t ) + l ∆ (cid:16) ρ A ( x, y, t ) T A ( x, y, t ) (cid:17)(cid:19) − ρ A ( x, y, t ) + O ( l ) (cid:35) . Now that the equation is governed entirely by quantities at site ( x, y ) and time t , we candrop ( x, y, t ) from the notation. Using definition (13), we substitute the full expression for T A into the first term and simplify: δt (cid:34) ρ A
14 + l (cid:16) ( β ∇ ξ B ) − β ∆ ξ B (cid:17) − ρ A + l e − βξ B ∆ (cid:16) ρ A T A (cid:17) + O ( l ) (cid:35) . (22) Using a Taylor series expansion on the fractional term and substituting this back intoexpression (22) gives δt ρ A − l (cid:16) ( β ∇ ξ B ) − β ∆ ξ B (cid:17) − ρ A + l e − βξ B ∆ (cid:16) ρ A T A (cid:17) + O ( l ) . Simplifying the expression yields ρ A ( x, y, t + δt ) − ρ A ( x, y, t ) δt = l δt (cid:104) − ρ A (cid:16) ( β ∇ ξ B ) − β ∆ ξ B (cid:17) + e − βξ B ∆ (cid:16) ρ A T A (cid:17)(cid:105) + O (cid:18) l δt (cid:19) . (23) We can further simplify by noting that ∆ (cid:16) ρ A T A (cid:17) = (cid:16) T A ∆ ρ A + 2 ∇ T A ∇ ρ A + ρ A ∆ T A (cid:17) . T A , ∇ T A , and ∆ T A , we have ∆ (cid:16) ρ A T A (cid:17) = e βξ B ρ A + 2 βe βξ B ∇ ξ B ∇ ρ A + e βξ B ρ A (cid:0) β ∆ ξ B + ( β ∇ ξ B ) (cid:1) + O ( l )= e βξ B (cid:104) ∆ ρ A + 2 β ∇ ξ B ∇ ρ A + ρ A (cid:16) ( β ∇ ξ B ) + β ∆ ξ B (cid:17)(cid:105) + O ( l ) . (24) Substituting (24) back into (23) gives us ρ A ( x, y, t + δt ) − ρ A ( x, y, t ) δt = l δt (cid:34) − ρ A (cid:0) ( β ∇ ξ B ) − β ∆ ξ B (cid:1) + ∆ ρ A + 2 β ∇ ξ B ∇ ρ A + ρ A (cid:16) ( β ∇ ξ B ) + β ∆ ξ B (cid:17) (cid:35) + O (cid:18) l δt (cid:19) . Combining like terms, we find ρ A ( x, y, t + δt ) − ρ A ( x, y, t ) δt = l δt (cid:34) ∆ ρ A + 2 β ∇ · (cid:16) ρ A ∇ ξ B (cid:17)(cid:35) + O (cid:18) l δt (cid:19) = l δt ∇ · (cid:34) ∇ ρ A + 2 β (cid:16) ρ A ∇ ξ B (cid:17)(cid:35) + O (cid:18) l δt (cid:19) . Assuming that the agent density ρ A is smooth, and that the limits l → ,δt → ,l δt → D, (25) hold, we arrive at the final form of the evolution equation for the density of red agents: ∂ρ A ∂t = D ∇ · (cid:34) ∇ ρ A + 2 β (cid:16) ρ A ∇ ξ B (cid:17)(cid:35) . (26) An identical derivation holds for the density of blue agents. Hence, our full system ofcontinuum equations is ∂ξ A ∂t ( x, y, t ) = γρ A ( x, y, t ) − λξ A ( x, y, t ) ∂ξ B ∂t ( x, y, t ) = γρ B ( x, y, t ) − λξ B ( x, y, t ) ∂ρ A ∂t ( x, y, t ) = D ∇ · (cid:104) ∇ ρ A ( x, y, t ) + 2 β (cid:0) ρ A ( x, y, t ) ∇ ξ B ( x, y, t ) (cid:1)(cid:105) ∂ρ B ∂t ( x, y, t ) = D ∇ · (cid:104) ∇ ρ B ( x, y, t ) + 2 β (cid:0) ρ B ( x, y, t ) ∇ ξ A ( x, y, t ) (cid:1)(cid:105) , (27) with periodic boundary conditions. 23he dimensionless form of the continuum system is ∂ξ A ∂t = cρ A − ξ A∂ξ B ∂t = cρ B − ξ B∂ρ A ∂t = ∇ X · (cid:104) ∇ X ρ A + 2 β (cid:0) ρ A ∇ X ξ B (cid:1)(cid:105) ∂ρ B ∂ ˜ t = ∇ X · (cid:104) ∇ X ρ B + 2 β (cid:0) ρ B ∇ X ξ A (cid:1)(cid:105) . (28) Details of the nondimensionalization can be found in Appendix A.
C. Studying the Continuum Model
Now that a continuum version of the model has been derived, we have more tools withwhich to understand the model. We first verify that the continuum system and the discretesystem share the same uniform and segregated equilibrium solutions; then we perform alinear stability analysis around the uniform equilibrium solution to gain insight into thephase transition.
1. Steady-State Solutions
Identifying the steady-state for the graffiti is straightforward: setting ∂ξ i ∂t = 0 yields ξ i = γλ ρ i for i ∈ { A, B } . (29)Looking more closely at the equations for agent density, we note that steady-state solutionsfor the red gang must satisfy ∇ ρ A ( x, y, t ) + 2 β (cid:16) ρ A ( x, y, t ) ∇ ξ B ( x, y, t ) (cid:17) = c A for c A ∈ R . Using the equilibrium graffiti density (29), we see that ∇ ρ A ( x, y, t ) + 2 βγλ (cid:16) ρ A ( x, y, t ) ∇ ρ B ( x, y, t ) (cid:17) = c A . (30)24nd similarly for the blue gang. Thus, any form of ρ A ( x, y, t ) and ρ B ( x, y, t ) satisfying ξ A ( x, y, t ) = γλ ρ A ( x, y, t ) ξ B ( x, y, t ) = γλ ρ B ( x, y, t ) ∇ ρ A ( x, y, t ) + 2 β (cid:0) ρ A ( x, y, t ) ∇ ξ B ( x, y, t ) (cid:1) = c A ∇ ρ B ( x, y, t ) + 2 β (cid:0) ρ B ( x, y, t ) ∇ ξ A ( x, y, t ) (cid:1) = c B , (31)is a steady-state solution of our system.The well-mixed state, with all agent and graffiti densities uniformly distributed so that ξ i = γλ ρ i , is clearly a steady-state for our system. Another obvious steady-state solutiontakes the following form: ξ A = γλ ρ A ξ B = γλ ρ B ρ A = c A , < x < . , . < x < ρ B = , < x < . c B , . < x < . (32) The steady-state solution (32) can be used to compare the discrete model and the con-tinuum model. Starting our simulations with the agents completely segregated, we visualizethe temporal evolution of the agent densities over time in Fig. 9. This figure shows anensemble average of 40 cross-sectional slices for agent density, where we can see that theagent density remains constant over time and the density clearly follows the form of thesteady-state solutions of equation (32). Note that in this context, unlike in Sec. III A 2,taking an ensemble average is sensible since we know the expected the solution. In thefigure, we also notice that the graffiti is proportional to the agent density: ξ A,B = γλ ρ A,B .To test whether (29) holds more generally for the segregated state in the discrete model,we simulate our system with different values for the ratio γλ , then comparing the resultingdensities with that of equation (32). We use the same starting conditions as in Sec. II A,and at the final time step we take a cross-sectional slice over the entire lattice. These cross-sectional slices are visualised in Fig. 10 for the following ratio values: , 1, and 2. The figure25ndicates that the relationship (29) derived from the continuum equations holds generallyfor simulations of the segregated state in the discrete model. FIG. 9. Temporal evolution of an ensemble average of cross-sectional slices of the agent densityfor a steady-state solution. Here we have N A = N B = 100 , λ = γ = 0 . β = 2 × − , δt = 1 and the lattice size is 100 ×
2. Linear Stability Analysis
To help us better understand our system, we perform a linear stability analysis on theuniformly distributed equilibrium solution corresponding to the well-mixed state. Takingthe same approach as [5, 16, 38, 44], we consider perturbations of the form (cid:15) = δe αt e ikx ,with δ (cid:28)
1, so that our solution takes the form: ξ A = ¯ ξ A + δ ξ A e αt e ikx ξ B = ¯ ξ B + δ ξ B e αt e ikx ρ A = ¯ ρ A + δ ρ A e αt e ikx ρ B = ¯ ρ B + δ ρ B e αt e ikx . (33)26 IG. 10. The final time step of the agent and graffiti densities for a segregated state using different γλ ratios. Here we have N A = N B = 100 , β = 2 × − , δt = 1 and the lattice size is100. The ratios for the figure were γ = 0 .
25 and λ = 0 . γ = λ = 0 . γ = 0 . , λ = 0 .
25 for the last row. Looking at the cross-sectional slices, we see that ξ A ≈ ρ A , ξ A ≈ ρ A and ξ A ≈ ρ A for the top, middle and bottom rows, respectively. Hence, thediscrete model agrees with the steady-state solution of equation, ξ A = γλ ρ A . α must be negative, forcing the perturbations todecay over time.Substituting the perturbed steady-state (33) into the graffiti equation yields ∂∂t (cid:0) ¯ ξ A + δ ξ A e αt e ikx (cid:1) = γ ( ¯ ρ A + δ ρ A e αt e ikx ) − λ ( ¯ ξ A + δ ξ A e αt e ikx ) . Since ¯ ξ A is an equilibrium solution, ∂ ¯ ξ A ∂t = γ ¯ ρ A − λ ¯ ξ A = 0. Hence, αδ ξ i = ( γδ ρ i − λδ ξ i ) for i ∈ { A, B } . (34)Substituting(33) into the evolution equation for the agent density gives us ∂∂t (cid:0) ¯ ρ A + δ ρ A e αt e ikx (cid:1) = D (cid:0) ¯ ρ A + δ ρ A e αt e ikx (cid:1) + Dβ ∇ · (cid:16) ( ¯ ρ A + δ ρ A e αt e ikx ) ∇ ( ¯ ξ B + δ ξ B e αt e ikx ) (cid:17) . Since our equilibrium solution is constant in both time and space, αδ ρ A e αt e ikx = − D | k | δ ρ A e αt e ikx + Dβ ddx (cid:0) ( ¯ ρ A + δ ρ A e αt e ikx )( ikδ ξ B e αt e ikx ) (cid:1) = − D | k | δ ρ A + 2 β ¯ ρ A δ ξ B ) e αt e ikx + O ( δ ρ A δ ξ B ) . Neglecting the term O ( δ ρ A δ ξ B ), αδ ρ A = − D | k | δ ρ A + 2 β ¯ ρ A δ ξ B ) , (35)and similarly for δ ρ B .Writing the linearized equations (34) through (35) in systems form, − λ γ − λ γ − βD ¯ ρ A | k | − D | k | − βD ¯ ρ B | k | − D | k | δ ξ A δ ξ B δ ρ A δ ρ B = α δ ξ A δ ξ B δ ρ A δ ρ B . gives us an equation of the form ( M − αI ) (cid:126)δ = 0 . This is an eigenvalue equation for the ma-trix M , and for it to have non trivial solutions (i.e. solutions where (cid:126)δ (cid:54) = 0), the determinantof ( M − αI ) must be zero. Thus, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ( λ + α ) 0 γ − ( λ + α ) 0 γ − βD ¯ ρ A | k | − (cid:16) D | k | + α (cid:17) − βD ¯ ρ B | k | − (cid:16) D | k | + α (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , f ( α ) = 116 (cid:18) (cid:0) ( λ + α ) − β γ ¯ ρ A ¯ ρ B (cid:1) D | k | + 8 α ( λ + α ) (2 α + D | k | ) (cid:19) = 0 . Solving the characteristic polynomial gives us the following four eigenvalues: α , = − (cid:18) λ + D | k | ± (cid:113) λ − D ( λ + 4 βγ √ ¯ ρ A ¯ ρ B ) | k | + D | k | (cid:19) (36)and α , = − (cid:18) λ + D | k | ± (cid:113) λ − D ( λ − βγ √ ¯ ρ A ¯ ρ B ) | k | + D | k | (cid:19) . (37) Fig. 11 shows the four eigenvalues as functions of the wave number k . In the figure, weset γλ = 1 and densities ρ A = ρ B = 100 , k = 0. However, for k (cid:54) = 0, the condition β ≤ γλ ) √ ¯ ρ A ¯ ρ B (cid:18) Dλ | k | λD | k | − (cid:19) . (38) is needed in order for α , to be real, and β ≥ − γλ ) √ ¯ ρ A ¯ ρ B (cid:18) Dλ | k | λD | k | − (cid:19) . (39) is needed to ensure that α , is real. We note that the condition on α , is satisfied for all β ≥
0, and λ > γ ≥ D ≥ α , the eigenvalue has a negative real part if Re (cid:18) λ + D | k | + (cid:113) λ − D ( λ + 4 βγ √ ¯ ρ A ¯ ρ B ) | k | + D | k | (cid:19) > . For λ >
D >
0, this is always true, and thus α always has a negative real part forthe parameter values we consider. Looking at the second eigenvalue, α has a negative realpart if Re (cid:18) λ + D | k | − (cid:113) λ − D ( λ + 4 βγ √ ¯ ρ A ¯ ρ B ) | k | + D | k | (cid:19) > , or equivalently, λ + D | k | > Re (cid:18)(cid:113) λ − D ( λ + 4 βγ √ ¯ ρ A ¯ ρ B ) | k | + D | k | (cid:19) . IG. 11. Eigenvalues for different values of β , plotted against wave number k . Here we have D = 1 × − , the ratio γλ = 1, and the density ρ A = ρ B = 100 , Squaring both sides and simplifying, we find λ > − βγ √ ¯ ρ A ¯ ρ B ⇐⇒ β > − γλ ) √ ¯ ρ A ¯ ρ B . α also always has a negative real part for the parameter values that we consider.Continuing on to the third eigenvalue, we recall that α is always real, so α has a negativereal part if λ + D | k | + (cid:113) λ − D ( λ − βγ √ ¯ ρ A ¯ ρ B ) | k | + D | k | > . This always holds for our choice of parameters, thus α has a negative real part for all wavenumbers k . Finally, α is also always real for our parameter choices and has negative realpart when λ + D | k | > (cid:113) λ − D ( λ − βγ √ ¯ ρ A ¯ ρ B ) | k | + D | k | . Squaring both sides and simplifying, we find that α has negative real part exactly when β < γλ ) √ ¯ ρ A ¯ ρ B . (40) Hence, α is the only eigenvalue which can have a positive real part, and the uniformlydistributed solution becomes linearly unstable for β ≥ γλ ) √ ¯ ρ A ¯ ρ B . (41) This allows us to define a critical β value where the stationary solution changes stability.In Fig. 12, we plot the critical parameter value β ∗ that we found numerically for thediscrete model in red and the critical β that we found in the continuum system in blue. Inthe left figure, we plot them as a function of the mass N A + N B , where we fix N A = N B and γ = λ = .
5, and in the figure on the right, we plot them as a function of the ratio γλ ,with N A = N B = 100 , β ∗ from thediscrete model matches the linearized PDE system’s critical β values as the system massincreases. Thus, for sufficiently large mass, the continuum equations predict our discretemodel results. We also see that as the system mass increases, the phase transition occurs ata smaller β . The match between the blue and the red plots is very good except for massesbelow 200 , β values of the discrete model very closely matches that ofthe linearized continuum system for all plotted ratio values. Hence, the results from ourcontinuum equations predict our discrete model results. We also observe that as the γλ ratioincreases, the phase transition occurs at a smaller β .31 IG. 12. On the Left: Critical β against the system mass. Here we have that λ = γ = 0 .
5, andfor the discrete model we have δt = 1 and the lattice size is 50 ×
50. The red and blue curvesrepresent the discrete model and the linearized PDE system respectively. On the right: Critical β against the ratio γλ . Here we have N A = N B = 100 , δt = 1 and the lattice size is 50 ×
50. The red and blue curves represent the discrete model andthe linearized PDE system respectively.
IV. CONCLUSIONS
In this work, we have presented an agent-based model for gang territorial developmentmotivated by graffiti markings. The model undergoes a phase transition as the parametersare changed. By deriving a continuum version of the model and performing a linear stabilityanalysis on the well-mixed state, a bifurcation point is found which matches the precise valueof the critical parameter found via numerical simulations of the discrete model.The continuum version of the model resembles a two-species version of the Keller-Segelmodel [18], though the graffiti acts as a chemo-repellent rather than a chemo-attractant,and the graffiti does not diffuse in space. Another interesting thing to notice is that thecoarsening that we observe in the simulations of the model, see for example Fig. 3, closelyresembles the Cahn-Hilliard equation [8], though no connection has yet been found. In short,there is much which remains to be explored about the models presented here.32
CKNOWLEDGMENTS
This research was supported by the National Science Foundation under Grant No. DMS-1319462. The authors are grateful to Gil Ariel, Daniel Balagu´e Guardia, Jes´us RosadoLinares, Wanda Strychalski, and Marie-Therese Wolfram for their helpful discussions andcomments.
Appendix A: Non-Dimensionalization
In this appendix, we will derive the nondimensionalized system for the continuum equa-tions derived in Sec. III B. We start the non-dimensionalization by first defining the naturaltime scale and characteristic length to be ˜ t = λt, ˜ X = Xl c , where X = ( x, y ) and l c = (cid:113) Dλ . Hence, ˜ t and ˜ X are dimensionless quantities. Because weare interested in the effect of variations of β on the dynamics of the system, we also define anondimensional ˜ β by identifying β ∗ as the critical β and letting ˜ β = ββ ∗ . Note that β ∗ carriesthe dimension Space Number of Individuals .For the derivation we first start non-dimensionalizing the continuum equations for thegraffiti. Recall that the dimensional version of the evolution equation for red graffiti densityis ∂ξ A ∂t = γρ A − λξ A . Dividing both sides by λ , we find ∂ξ A λ∂t = γλ ρ A − ξ A . Note that ∂λ∂t = ∂∂ ˜ t , and multiplying both sides by β ∗ , we arrive at β ∗ ∂ξ A ∂ ˜ t = β ∗ γλ ρ A − β ∗ ξ A . β ∗ is space Number of Individuals . Hence, β ∗ ξ A = ˜ ξ A and β ∗ ρ A = ˜ ρ A aredimensionless quantities, giving us the final dimensionless form of the evolution equation: ∂ ˜ ξ A ∂ ˜ t = c ˜ ρ A − ˜ ξ A . (A1)Note that here, c = λγ is a dimensionless π number.Next, we nondimensionalize the continuum equations for the agent densities (26) and(26). Recall that the dimensional form of the evolution equations for red agent density is ∂ρ A ∂t = D ∇ X · (cid:34) ∇ X ρ A + 2 β (cid:16) ρ A ∇ X ξ B (cid:17)(cid:35) . Dividing both sides by λ and employing β = β ∗ ˜ β , we see that ∂ρ A λ∂t = D λ ∇ X · (cid:34) ∇ X ρ A + 2 ˜ ββ ∗ (cid:16) ρ A ∇ X ξ B (cid:17)(cid:35) . Note again that ∂λ∂t = ∂∂ ˜ t .Next, we rewrite the operator ∇ X in terms of ˜ X , i.e. ∇ X = (cid:113) λD ∇ ˜ X : ∂ρ A ∂ ˜ t = D λ λD ∇ ˜ X · (cid:34) ∇ ˜ X ρ A + 2 ˜ ββ ∗ (cid:16) ρ A ∇ ˜ X ξ B (cid:17)(cid:35) . Finally, we multiply both sides by β ∗ : β ∗ ∂ρ A ∂ ˜ t = β ∗ ∇ ˜ X · (cid:34) ∇ ˜ X ρ A + 2 ˜ β (cid:16) ρ A ∇ ˜ X ( β ∗ ξ B ) (cid:17)(cid:35) , giving us ∂ ˜ ρ A ∂ ˜ t = 14 ∇ ˜ X · (cid:34) ∇ ˜ X ˜ ρ A + 2 ˜ β (cid:16) ˜ ρ A ∇ ˜ X ˜ ξ B (cid:17)(cid:35) . Dropping the tilde notation from the equations for notational simplicity gives us thedimensionless form of the continuum system: ∂ξ A ∂t = cρ A − ξ A∂ξ B ∂t = cρ B − ξ B∂ρ A ∂t = ∇ X · (cid:104) ∇ X ρ A + 2 β (cid:0) ρ A ∇ X ξ B (cid:1)(cid:105) ∂ρ B ∂ ˜ t = ∇ X · (cid:104) ∇ X ρ B + 2 β (cid:0) ρ B ∇ X ξ A (cid:1)(cid:105) . (A2)34e see that the non-dimensional form of the continuum equations is similar to that ofthe dimensional form in equation (31). However, it is sometimes more convenient to analyzethe non-dimensional continuum equations, as it allows us to see how the model behaves asdifferent parameters are scaled in relation to one another. [1] Karen Adams and Anne Winter. Gang graffiti as a discourse genre. Journal of Sociolinguistics ,1(3):337–360, 1997.[2] Luiz GA Alves, Haroldo V Ribeiro, and Renio S Mendes. Scaling laws in the dynamics ofcrime growth rate.
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