# A Multispecies Cross-Diffusion Model for Territorial Development

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## Abstract

**:**

## 1. Introduction

#### 1.1. Discrete Model

#### 1.2. Phases and an Order Parameter

#### 1.2.1. Expected Agent Density

#### 1.2.2. An Order Parameter

## 2. Simulations of the Discrete Model

#### 2.1. Well-Mixed State

#### 2.2. Segregated State

#### 2.3. System Parameters and the Discrete Phase Transition

#### 2.3.1. Effects of $\beta $

#### 2.3.2. Effects of Other Parameters

## 3. Deriving the Convection-Diffusion System

#### 3.1. Continuum Graffiti Density

#### 3.2. Continuum Agent Density

#### 3.2.1. Tools for the Derivation

#### 3.2.2. The Derivation

#### 3.3. Steady-State Solutions

## 4. Linear Stability Analysis

## 5. Variations of the Model: Varying $\mathbf{\beta}$ by Gang

#### 5.1. Timidity Model (Variation 1)

#### 5.2. Threat Level Model (Variation 2)

#### 5.3. Finding Critical ${\beta}_{i}$ for the Variations: Linear Stability Analysis

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Agent (

**left two**) and graffiti (

**right two**) densities’ temporal evolution for a well-mixed state. Here we have ${N}_{1}={N}_{2}={N}_{3}=$ 50,000, with $\lambda =\gamma =0.5$, $\beta =5\times {10}^{-6}$, $\delta t=1$ and the lattice size is $100\times 100$. Note that the initial graffiti lattice appears black because it is empty. The final graffiti lattice appears white because all sites have (almost) the same graffiti densities from all three gangs. It is clear from this figure that the agents remain well mixed over time.

**Figure 2.**Agent (

**top**) and graffiti (

**bottom**) densities temporal evolution for a segregated state. Here, we have ${N}_{1}={N}_{2}={N}_{3}=$ 50,000, with $\lambda =\gamma =0.5$, $\beta =3\times {10}^{-5}$, $\delta t=1$ and the lattice size is $100\times 100$. We see that the agents segregate into distinct territories, coarsening over time.

**Figure 3.**How changing the $\beta $ parameter affects the system. Here we have ${N}_{1}={N}_{2}={N}_{3}=$ 50,000, with $\lambda =\gamma =0.5$ and the lattice size is $100\times 100$. (

**Left**) It is seen that for a small $\beta $ value, the system remains well-mixed and the order parameter is almost zero over all time steps. For larger $\beta $ values, we see that the order parameter increases quickly as the system segregates and levels off to around one. (

**Right**) After 100,000 time steps, we take the order parameter value for different $\beta $ values. We clearly see that as the $\beta $ value increases, there is a critical $\beta $ value at which a phase transition occurs.

**Figure 4.**Agent density lattices taken after 100,000 time steps for different $\beta $ values for $\delta t=1,L=100,\frac{\gamma}{\lambda}=1,$ and ${N}_{1}={N}_{2}={N}_{3}=$ 50,000 agents. Starting from the top left, the resulting order parameter was around $0,0.20,0.39,0.69,0.83,$ and $0.96$, respectively. The $\beta $ values used were $5\times {10}^{-6},9.4\times {10}^{-6},1\times {10}^{-5},1.2\times {10}^{-5},1.4\times {10}^{-5}$ and $3\times {10}^{-5}$, respectively. The critical $\beta $ value is around $9.3\times {10}^{-6}$. We clearly see when the order parameter is between $0.1$ and $0.9$, then the system is in a partially segregated state. The territories formed also become more evident as $\beta $ increases, which also results in the order parameter value becoming larger.

**Figure 5.**The Effect of Parameters on the Phase Transition. Here we have ${N}_{1}={N}_{2}={N}_{3}=$ 50,000, with $\lambda =\gamma =0.5$ and $\delta t=1$. The order parameter values were computed after 100,000 time steps. On the top three plots, it is seen that for small $\beta $ values, the system remains well-mixed and the resulting order parameter values is approximately zero over time. However, for larger $\beta $ values we see that the order parameter increases quickly as the system segregates and levels off to around one. We clearly see that as the $\beta $ value increases there is a critical $\beta $ value in which a phase transition occurs. The plots also show the effects of changing the lattice grid, mass and the ratio $\gamma /\lambda $. The bottom three plots are magnified versions of the top three plots.

**Figure 6.**Temporal evolution of the graffiti densities lattice for a segregated state for different grid sizes. Here we have ${N}_{1}={N}_{2}={N}_{3}=$ 50,000, with $\lambda =\gamma =0.5$, $\beta =3\times {10}^{-5}$, $\delta t=1$. First row, $L=50$, Second row $L=75$, third row $L=100$, and fourth row $L=150$. We see that the graffiti density is similar as L increases.

**Figure 7.**The six eigenvalues versus the wave number k plotted for different $\beta $ values. Here we have $D=0.0001$, ${\rho}_{1}={\rho}_{2}={\rho}_{3}=\mathrm{50,000}$ with the $\frac{\gamma}{\lambda}$ ratio $=1$.

**Figure 8.**Critical $\beta $ against the mass and $\frac{\gamma}{\lambda}$ ratio. The red curves represent critical $\beta $ value from the linear stability analysis of the PDE system, and the blue curves represent the critical $\beta $ value from the discrete model. Here we have $\delta t=1$ and the lattice size is $100\times 100$. (

**Left**) Critical $\beta $ against the mass when the ratio is $\frac{\gamma}{\lambda}=1$. (

**Right**) Critical $\beta $ against the ratio $\frac{\gamma}{\lambda}$ when the mass is 50,000. We see that the phase transitions occurs at a smaller $\beta $ value whenever as the systems’ mass increase or when the $\frac{\gamma}{\lambda}$ ratio increase. We also see that the critical $\beta $ value of both the discrete model and the PDE system match, and this is a good indicator that our continuum equations replicate the behavior of the discrete model.

**Figure 9.**Critical $\beta $ against the total preserved system mass and $\frac{\gamma}{\lambda}$ ratio comparison for two and three gangs. Here, we have $\delta t=1$ and $L=100$. (

**Left**) Critical $\beta $ against the mass when the ratio $\frac{\gamma}{\lambda}=1$ for both 2 and 3 gangs. (

**Right**) Critical $\beta $ against the $\frac{\gamma}{\lambda}=1$ ration for both 2 and 3 gangs when the total mass of the system is preserved to 150,000. We see that increasing the number of gangs from 2 to 3 makes the critical $\beta $ and the phase transition larger.

**Figure 10.**Critical $\beta $ against the mass and $\frac{\gamma}{\lambda}$ ratio comparison for two and three gangs. Here, we have $\delta t=1$ and $L=100$. (

**Left**) Critical $\beta $ against the mass of each gang when the ratio $\frac{\gamma}{\lambda}=1$ for both 2 and 3 gangs. (

**Right**) Critical $\beta $ against the $\frac{\gamma}{\lambda}=1$ ratio for both 2 and 3 gangs when the mass of each is 50,000. We see that increasing the number of gangs from 2 to 3 does not affect the phase transition of the system, and that the critical $\beta $ is similar when the number of gangs is increased.

**Figure 11.**

**Top row**: Agent and graffiti densities’ temporal evolution for the Timidity Model. Here, ${\beta}_{1}=2\times {10}^{-5},\phantom{\rule{3.33333pt}{0ex}}{\beta}_{2}=3.5\times {10}^{-5}$ and ${\beta}_{3}=0.5\times {10}^{-5}$. We also have ${N}_{1}={N}_{2}={N}_{3}=$ 50,000, with $\lambda =\gamma =0.5$, $\delta t=1$ and the lattice size is $100\times 100$. It is clearly seen that the agents segregate over time.

**Bottom row**: The densities for gangs 1 (left), 2 (middle), and 3 (right) can be seen after 100,000 time steps.

**Figure 12.**Here, we plot graphs of the beta values ${\beta}_{j}$ against the percentage of the area dominated by gang j for the Timidity Model (variation 1). We use six sets of parameters, enumerated in Table 1.

**Figure 13.**Cross-sectional slices of the agent and graffiti densities for different $\beta $ extensions at the final time step for a segregated state. Here, we have ${N}_{1}={N}_{2}=N=3=\mathrm{50,000}$ with $\delta t=1$ and the lattice size is $100\times 100$; in both simulations, ${\beta}_{1}=2\times {10}^{-5},{\beta}_{2}=3.5\times {10}^{-5}$ and ${\beta}_{3}=0.5\times {10}^{-5}$. (

**Left**): Here, we consider the Timidity variation of the model. We observe that the territories range from small and very dense, with little incursion from the other gangs, to large and spread out, with other gang members encroaching on the territory, as the gangs’ ${\beta}_{j}$ value varies from high to low. (

**Right**): Here, we consider the Threat Level variation of the model. We see that the size of the territory here is correlated with the ${\beta}_{j}$ value for the gang, and that all of the territories here seem well-defined, with little of the territorial encroachment seen in the model pictured on the left.

**Figure 14.**

**Top row**: Temporal evolution of the agent and graffiti densities for the Threat Level Model. Here ${\beta}_{1}=2\times {10}^{-5},{\beta}_{2}=3.5\times {10}^{-5}$ and ${\beta}_{3}=0.5\times {10}^{-5}$. We also have ${N}_{1}={N}_{2}={N}_{3}=$ 50,000, with $\lambda =\gamma =0.5$, $\delta t=1$ and the lattice size is $100\times 100$. It is clearly seen that the agents segregate over time.

**Bottom row**: The agent and graffiti densities for gangs 1 (left), 2 (middle), and 3 (right) can be seen after 100,000 time steps.

**Figure 15.**Here, we plot the ${\beta}_{j}$ values against the percentage of the area dominated by gang j for the Threat Level variation of the model. We use the six sets of parameters enumerated in Table 1.

**Figure 16.**Agent density lattices for the first variation (Timidity Model) taken after 100,000 time steps for $\delta t=1,L=100,\frac{\gamma}{\lambda}=1,$ and ${N}_{1}={N}_{2}=$ 75,000 agents. On all lattices, the first $\beta $ value is fixed to ${\beta}_{1}=5\times {10}^{-6}$. Starting from the top left, the second $\beta $ value used was $1\times {10}^{-6},1\times {10}^{-5}$ and $2\times {10}^{-5}$, respectively. The color of gang 1 is red, while gang 2’s color is blue. In the first lattice, ${\beta}_{1}{\beta}_{2}<{\beta}_{*}$, and this resulted in the agents being well-mixed. In the second and third lattice, ${\beta}_{1}{\beta}_{2}>{\beta}_{*}$, and this resulted in the agents segregating.

**Figure 17.**Agent density lattices for the second variation (Threat Level) taken after 100,000 time steps for $\delta t=1,L=100,\frac{\gamma}{\lambda}=1,$ and ${N}_{1}={N}_{2}=$ 75,000 agents. On all lattices, the first $\beta $ value is fixed to ${\beta}_{1}=5\times {10}^{-6}$. Starting from the top left, the second $\beta $ value used was $1\times {10}^{-6},1\times {10}^{-5}$ and $2\times {10}^{-5}$, respectively. The color of gang 1 is red, while gang 2’s color is blue. In the first lattice, ${\beta}_{1}{\beta}_{2}<{\beta}_{*}$, and this resulted in the agents to be well-mixed. In the second and third lattice, ${\beta}_{1}{\beta}_{2}>{\beta}_{*}$, and this resulted in the agents to segregate.

**Figure 18.**Agent density lattices for the first variation (Timidity Model) taken after 100,000 time steps for $\delta t=1,L=100,\frac{\gamma}{\lambda}=1,$ and ${N}_{1}={N}_{2}={N}_{3}=$ 50,000 agents. On all lattices, the first and second $\beta $ values were fixed to ${\beta}_{1}=6\times {10}^{-6},{\beta}_{2}=8\times {10}^{-6}$. Starting from the top left, the third $\beta $ value used was $1\times {10}^{-5},2.2\times {10}^{-5}$ and ${\beta}_{3}=3\times {10}^{-5}$, respectively. The color of gang 1 is red, while gang 2 and 3’s colors are blue and green, respectively. In the first lattice, ${\beta}_{1}{\beta}_{2}{\beta}_{3}<{\beta}_{*}$, and this resulted in the agents being well-mixed. In the second and third lattice, ${\beta}_{1}{\beta}_{2}{\beta}_{3}>{\beta}_{*}$, and this resulted in the agents segregating.

**Figure 19.**Agent density lattices for the second variation (Threat Level) taken after 100,000 time steps for $\delta t=1,L=100,\frac{\gamma}{\lambda}=1,$ and ${N}_{1}={N}_{2}={N}_{3}=$ 50,000 agents. On all lattices, the first and second $\beta $ values were fixed to ${\beta}_{1}=6\times {10}^{-6},{\beta}_{2}=8\times {10}^{-6}$. Starting from the top left, the third $\beta $ value used was $1\times {10}^{-5},2.2\times {10}^{-5}$ and ${\beta}_{3}=3\times {10}^{-5}$, respectively. The color of gang 1 is red, while gang 2 and 3 colors are blue and green, respectively. In the first lattice, ${\beta}_{1}{\beta}_{2}{\beta}_{3}<{\beta}_{*}$, and this resulted in the agents being well-mixed. In the second and third lattice, ${\beta}_{1}{\beta}_{2}{\beta}_{3}>{\beta}_{*}$, and this resulted in the agents segregating.

**Table 1.**Here, we see the results of both variations of the original model for six different sets of ${\beta}_{j}$ in three-gang simulations. Here, Model 1 refers to the Timidity Model variation, while Model 2 refers to the Threat Level Model variation. The ${\beta}_{j}$ values are listed, along with the percentage of the lattice dominated by each gang at equilibrium. Note that the percentages do not add to 100% because in each simulation, a small percentage of the lattice is not clearly dominated by any one of the gangs.

Parameter Set | Gang | Value of ${\mathit{\beta}}_{\mathit{j}}$ | % Territory, Model 1 | % Territory, Model 2 |
---|---|---|---|---|

Set 1 | Gang 1 | ${\beta}_{1}=0.000005$ | $55.02$% | $11.27$% |

Gang 2 | ${\beta}_{2}=0.00002$ | $28.23$% | $32.48$% | |

Gang 3 | ${\beta}_{3}=0.000035$ | $10.20$% | $54.10$% | |

Set 2 | Gang 1 | ${\beta}_{1}=0.000015$ | $41.50$% | $25.95$% |

Gang 2 | ${\beta}_{2}=0.00002$ | $31.28$% | $32.62$% | |

Gang 3 | ${\beta}_{3}=0.000025$ | $25.19$% | $39.70$% | |

Set 3 | Gang 1 | ${\beta}_{1}=0.00001$ | $55.13$% | $16.27$% |

Gang 2 | ${\beta}_{2}=0.00002$ | $28.40$% | $28.63$% | |

Gang 3 | ${\beta}_{3}=0.00004$ | $14.15$% | $53.92$% | |

Set 4 | Gang 1 | ${\beta}_{1}=0.000012$ | $51.45$% | $19.39$% |

Gang 2 | ${\beta}_{2}=0.000024$ | $27.02$% | $34.85$% | |

Gang 3 | ${\beta}_{3}=0.000032$ | $19.99$% | $44.58$% | |

Set 5 | Gang 1 | ${\beta}_{1}=0.000022$ | $33.11$% | $32.72$% |

Gang 2 | ${\beta}_{2}=0.000022$ | $32.71$% | $32.84$% | |

Gang 3 | ${\beta}_{3}=0.000022$ | $32.92$% | $33.04$% | |

Set 6 | Gang 1 | ${\beta}_{1}=0.000018$ | $44.85$% | $23.66$% |

Gang 2 | ${\beta}_{2}=0.000028$ | $29.50$% | $34.32$% | |

Gang 3 | ${\beta}_{3}=0.000034$ | $24.89$% | $41.16$% |

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Alsenafi, A.; Barbaro, A.B.T.
A Multispecies Cross-Diffusion Model for Territorial Development. *Mathematics* **2021**, *9*, 1428.
https://doi.org/10.3390/math9121428

**AMA Style**

Alsenafi A, Barbaro ABT.
A Multispecies Cross-Diffusion Model for Territorial Development. *Mathematics*. 2021; 9(12):1428.
https://doi.org/10.3390/math9121428

**Chicago/Turabian Style**

Alsenafi, Abdulaziz, and Alethea B. T. Barbaro.
2021. "A Multispecies Cross-Diffusion Model for Territorial Development" *Mathematics* 9, no. 12: 1428.
https://doi.org/10.3390/math9121428