pith. sign in

arxiv: 2509.18146 · v1 · submitted 2025-09-15 · 🌊 nlin.PS · math.DS

Global Bifurcations and Pattern Formation in Target-Offender-Guardian Crime Models

Madi Yerlanov , Qi Wang , Nancy Rodriguez This is my paper

Pith reviewed 2026-05-18 16:21 UTC · model grok-4.3

classification 🌊 nlin.PS math.DS
keywords crime modelingreaction-diffusion systemsbifurcation theorypattern formationpolicing dynamicsurban crimeHopf bifurcationadvection-diffusion
0
0 comments X

The pith

Critical policing intensity thresholds trigger either persistent crime hotspots or oscillatory cycles in a three-agent urban model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a reaction-advection-diffusion system with three interacting components: targets of crime, offenders, and guardians who represent mobile police. It applies Crandall-Rabinowitz bifurcation theory together with spectral analysis to locate precise values of policing intensity at which a spatially uniform equilibrium loses stability. Past those thresholds the system settles into either stationary heterogeneous patterns that correspond to fixed crime hotspots or time-periodic oscillations that appear as recurring waves of criminal activity. A reader cares because the thresholds mark tipping points where modest changes in guardian behavior can produce large, lasting shifts in neighborhood crime levels. The analysis extends earlier two-equation crime models by treating law enforcement itself as a dynamic participant whose movement feeds back into the pattern-forming process.

Core claim

Using Crandall-Rabinowitz bifurcation theory and spectral analysis, the authors derive rigorous conditions for both steady-state and Hopf bifurcations in the target-offender-guardian system. These conditions identify critical thresholds of policing intensity at which spatially uniform equilibria lose stability, producing either persistent heterogeneous hotspots or oscillatory crime-policing cycles. Numerical simulations illustrate the predicted stationary patterns, periodic oscillations, and chaotic dynamics that emerge beyond the thresholds.

What carries the argument

Crandall-Rabinowitz bifurcation theory and spectral analysis applied to the three-component reaction-advection-diffusion equations.

If this is right

  • Above a critical policing intensity, neighborhoods become locked into stable clusters of criminal activity.
  • Guardian mobility can produce recurrent waves of hotspot formation and dissipation.
  • The system can exhibit stationary patterns, periodic oscillations, and chaotic dynamics depending on parameter values.
  • Treating law enforcement as an explicit interacting component reveals nonlinear interactions absent from two-equation models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The identified thresholds could guide simulation tests of alternative patrol strategies to determine which ones avoid crossing into unwanted pattern regimes.
  • Analogous three-agent bifurcation analyses might apply to other social systems in which one population (such as health workers or regulators) actively responds to the actions of two others.
  • The presence of chaotic regimes suggests that even when thresholds are known, long-term forecasts of exact hotspot locations remain limited.

Load-bearing premise

The target-offender-guardian interactions are accurately captured by a reaction-advection-diffusion system to which standard bifurcation theory applies directly, without dominant unmodeled factors such as external economic shocks or data limitations.

What would settle it

Direct observation of whether crime distributions in a monitored urban area switch from uniform to clustered or oscillatory states exactly when measured policing intensity crosses the mathematically predicted thresholds.

Figures

Figures reproduced from arXiv: 2509.18146 by Madi Yerlanov, Nancy Rodriguez, Qi Wang.

Figure 1
Figure 1. Figure 1: Illustration of the effect of diffusion rates on the ordering of χ, χ −, and χ + obtained from linear stability analysis. Roman numerals indicate the corresponding inequality regimes, while letters denote parameter combinations selected for further simulations. Region I corresponds to the linearly stable regime. The remaining parameters are fixed at α = 1.0, β = 1.0, λ = 1.5, χ = 2, L = π. Each region repr… view at source ↗
Figure 2
Figure 2. Figure 2: Examples of amplitude evolutions from [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of patterns corresponding to the points in [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representation of a one-dimensional simulation of (E) with the same parameters as in 2c. (a) Emergence and stabilization of periodic behavior in the amplitude of the A component, Aamp = RMS[A(x, t) − λ]. Data are sampled every 3 time units to provide a clearer visualization of the oscillations. The figure shows that, after an initial transient, the solution does not converge to a steady state but instead s… view at source ↗
Figure 5
Figure 5. Figure 5: Bifurcation diagram of the midpoint value A(π/2, t∗ ) in the one-dimensional simulation of (E) in (0, π) with initial perturbation of the form 0.05 cos(x). The y-axis records all extreme values of A(π/2, t) with t ∈ [800, 1000], after transients have decayed. t ∗ denotes a maximizer or a minimizer. Simulations are run up to Tmax = 1000, treating t ∈ [0, 800] as transient. Parameters are fixed at DA = 0.1, … view at source ↗
Figure 6
Figure 6. Figure 6: Representations of the chaotic solution to (E) with the same parameters as in [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Representations of a periodic solution to (E). The parameters are the same as in [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: As expected, smaller diffusion rates lead to chaoti [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Solutions to (E) for various Dρ (columns) and Du (rows). The non-transient ρ(π/2, t) and u(π/2, t) are plotted against each other. The remaining parameters are: DA = 0.1, α = β = 1, λ = (1 + √ 5)/2, χ = 2. The colors represent the time: dark blue – t = 800 and yellow – t = 1000. The star represents the constant-in-time solutions. Whenever ρ(π/2, t) = 0.382 and u(π/2, t) = 1, this indicates constant-in-spac… view at source ↗
Figure 9
Figure 9. Figure 9: Solutions to (E) for various Du (columns) and χ (rows). The non-transient ρ(π/2, t) and u(π/2, t) are plotted against each other. The remaining parameters are: DA = 0.1, Dρ = 0.01, α = β = 1, λ = (1 + √ 5)/2. We represent two snapshots using dark blue (t = 800) and yellow (t = 1000). The star represents the constant-in-time solutions. Here, we note that not only diffusion rates, but other parameters as wel… view at source ↗
read the original abstract

We study a reaction-advection-diffusion model of a target-offender-guardian system designed to capture interactions between urban crime and policing. Using Crandall-Rabinowitz bifurcation theory and spectral analysis, we establish rigorous conditions for both steady-state and Hopf bifurcations. These results identify critical thresholds of policing intensity at which spatially uniform equilibria lose stability, leading either to persistent heterogeneous hotspots or oscillatory crime-policing cycles. From a criminological perspective, such thresholds represent tipping points in guardian mobility: once crossed, they can lock neighborhoods into stable clusters of criminal activity or trigger recurrent waves of hotspot formation. Numerical simulations complement the theory, exhibiting stationary patterns, periodic oscillations, and chaotic dynamics. By explicitly incorporating law enforcement as a third interacting component, our framework extends classical two-equation models. It offers new tools for analyzing nonlinear interactions, bifurcations, and pattern formation in multi-agent social systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies a reaction-advection-diffusion model for a target-offender-guardian system in urban crime. It applies Crandall-Rabinowitz bifurcation theory and spectral analysis to derive rigorous conditions for steady-state and Hopf bifurcations from spatially uniform equilibria, identifying critical thresholds in policing intensity that lead to heterogeneous hotspots or oscillatory cycles. Numerical simulations illustrate stationary patterns, periodic oscillations, and chaotic dynamics. The framework extends classical two-component crime models by treating law enforcement as an explicit interacting component.

Significance. If the derivations hold, the work offers a mathematically grounded extension of pattern-formation models in criminology by incorporating guardian dynamics, potentially identifying tipping points in policing intensity with implications for understanding stable crime clusters or recurrent waves. The explicit use of established bifurcation tools on a three-component system is a constructive step beyond prior two-equation models.

major comments (2)
  1. The abstract asserts that Crandall-Rabinowitz theory yields 'rigorous conditions' for both steady-state and Hopf bifurcations, yet the provided text contains no explicit model equations, linearization, or verification of the transversality and crossing conditions required by the theorem. Without these details it is not possible to confirm that the claimed thresholds are correctly derived.
  2. The modeling assumption that target-offender-guardian interactions are fully captured by a reaction-advection-diffusion system (without dominant unmodeled factors such as economic shocks) is load-bearing for the interpretation of the bifurcation thresholds as criminological tipping points; this choice requires explicit justification and sensitivity analysis in the model section.
minor comments (2)
  1. The title refers to 'Global Bifurcations' while the abstract and claimed results focus on local bifurcations from uniform states; clarify whether global continuation or other global techniques are actually employed.
  2. Numerical methods, parameter values, and initial conditions used in the simulations should be stated explicitly to allow reproduction of the reported stationary patterns, oscillations, and chaotic regimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to improve clarity and completeness.

read point-by-point responses

  1. Referee: The abstract asserts that Crandall-Rabinowitz theory yields 'rigorous conditions' for both steady-state and Hopf bifurcations, yet the provided text contains no explicit model equations, linearization, or verification of the transversality and crossing conditions required by the theorem. Without these details it is not possible to confirm that the claimed thresholds are correctly derived.

    Authors: We agree that the current presentation does not make the full verification steps sufficiently explicit for independent confirmation. In the revised manuscript we will add the complete model equations, the linearization around the spatially uniform equilibrium, the explicit computation of the bifurcation parameter values, and direct verification of the transversality condition for the steady-state bifurcation together with the crossing condition for the Hopf bifurcation, following the standard requirements of Crandall-Rabinowitz theory. revision: yes

  2. Referee: The modeling assumption that target-offender-guardian interactions are fully captured by a reaction-advection-diffusion system (without dominant unmodeled factors such as economic shocks) is load-bearing for the interpretation of the bifurcation thresholds as criminological tipping points; this choice requires explicit justification and sensitivity analysis in the model section.

    Authors: We accept that the modeling assumptions need more explicit defense. The revised model section will include a dedicated paragraph justifying the reaction-advection-diffusion framework by reference to prior two-component crime models and empirical literature on guardian mobility, followed by a sensitivity analysis that perturbs parameters representing possible unmodeled influences (including proxies for economic shocks) and reports the resulting shifts in the computed bifurcation thresholds. revision: yes

Circularity Check

0 steps flagged

No significant circularity: standard bifurcation theory applied to new model

full rationale

The paper formulates a three-component reaction-advection-diffusion system for target-offender-guardian interactions and applies the external Crandall-Rabinowitz bifurcation theorem plus spectral analysis to derive conditions for steady-state and Hopf bifurcations. These tools are independent mathematical results not derived from the present model's outputs or fitted parameters. No predictions reduce to inputs by construction, no self-citation chains justify the core premises, and the extension from classical two-equation models is presented as a modeling choice rather than a tautology. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the modeling choice that crime-policing dynamics obey reaction-advection-diffusion equations and on the applicability of Crandall-Rabinowitz theory; specific reaction rates, diffusion coefficients, and advection speeds are not detailed in the abstract.

axioms (1)
  • domain assumption Target-offender-guardian interactions are governed by a system of reaction-advection-diffusion equations
    This is the foundational modeling assumption that allows spatial pattern formation to be analyzed via bifurcation theory.

pith-pipeline@v0.9.0 · 5685 in / 1278 out tokens · 56482 ms · 2026-05-18T16:21:01.273752+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages

  1. [1]

    Hopf bifurcation in quasilinear reactio n-diffusion systems

    Herbert Amann. Hopf bifurcation in quasilinear reactio n-diffusion systems. In Stavros Busen- berg and Mario Martelli, editors, Delay Differential Equations and Dynamical Systems , pages 53–63, Berlin, Heidelberg, 1991. Springer Berlin Heidelbe rg

    work page 1991

  2. [2]

    Self-organise d critical hot spots of criminal activity

    Henri Berestycki and Jean-Pierre Nadal. Self-organise d critical hot spots of criminal activity. European Journal of Applied Mathematics , 21(4-5):371–399, 2010

    work page 2010

  3. [3]

    Ex istence of symmetric and asymmetric spikes for a crime hotspot model

    Henri Berestycki, Juncheng Wei, and Matthias Winter. Ex istence of symmetric and asymmetric spikes for a crime hotspot model. SIAM Journal on Mathematical Analysis , 46(1):691–719, 2014

    work page 2014

  4. [4]

    The effects of hot spots policing on crime

    Anthony A Braga. The effects of hot spots policing on crime . Annals of the American Academy of Political and Social Science , 578(1):104–125, 2001

    work page 2001

  5. [5]

    The effects of hot spots polic- ing on crime: An updated systematic review and meta-analysi s

    Anthony A Braga, Andrew V Papachristos, and David M Hurea u. The effects of hot spots polic- ing on crime: An updated systematic review and meta-analysi s. Justice Quarterly, 31(4):633– 663, 2014. 29

    work page 2014

  6. [6]

    Cops- on-the-dots: The linear stability of crime hotspots for a 1- D reaction-diffusion model of urban crime

    Andreas Buttenschoen, Theodore Kolokolnikov, Michael J Ward, and Juncheng Wei. Cops- on-the-dots: The linear stability of crime hotspots for a 1- D reaction-diffusion model of urban crime. European Journal of Applied Mathematics , 31(5):871–917, 2020

    work page 2020

  7. [7]

    Global bifurcation of solutions for crime modeling equations

    Robert Stephen Cantrell, Chris Cosner, and Ra´ ul Man´ as evich. Global bifurcation of solutions for crime modeling equations. SIAM Journal on Mathematical Analysis , 44(3):1340–1358, 2012

    work page 2012

  8. [8]

    Crime modeling with L´ evy flights

    Sorathan Chaturapruek, Jonah Breslau, Daniel Yazdi, Th eodore Kolokolnikov, and Scott G McCalla. Crime modeling with L´ evy flights. SIAM Journal on Applied Mathematics , 73(4):1703–1720, 2013

    work page 2013

  9. [9]

    Bifurcation fr om simple eigenvalues

    Michael G Crandall and Paul H Rabinowitz. Bifurcation fr om simple eigenvalues. Journal of Functional Analysis, 8(2):321–340, 1971

    work page 1971

  10. [10]

    Bifurcation, perturbation of simple eigenvalues and linearized stability

    Michael G Crandall and Paul H Rabinowitz. Bifurcation, perturbation of simple eigenvalues and linearized stability. Archive for Rational Mechanics and Analysis , 52:161–180, 1973

    work page 1973

  11. [11]

    Statistical physic s of crime: A review

    Maria R D’Orsogna and Matjaˇ z Perc. Statistical physic s of crime: A review. Physics of Life Reviews, 12:1–21, 2015

    work page 2015

  12. [12]

    Preventing crime at places

    John E Eck. Preventing crime at places. In David P. Farri ngton, Doris Layton MacKenzie, Lawrence Sherman, and Brandon C. Welsh, editors, Evidence-based crime prevention , pages 241–294. Routledge, 2003

    work page 2003

  13. [13]

    Global solutions to a higher-dimensio nal system related to crime modeling

    Marcel Freitag. Global solutions to a higher-dimensio nal system related to crime modeling. Mathematical Methods in the Applied Sciences , 41(16):6326–6335, 2018

    work page 2018

  14. [14]

    Fairness and effectiveness in policing: The evidence

    Kathleen Frydl and Wesley Skogan. Fairness and effectiveness in policing: The evidence . National Academies Press, 2004

    work page 2004

  15. [15]

    The nucleation-annihila tion dynamics of hotspot patterns for a reaction-diffusion system of urban crime with police deplo yment

    Chunyi Gai and Michael J Ward. The nucleation-annihila tion dynamics of hotspot patterns for a reaction-diffusion system of urban crime with police deplo yment. SIAM Journal on Applied Dynamical Systems, 23(3):2018–2060, 2024

    work page 2018

  16. [16]

    E xistence of solutions for a 1-D boundary value problem coming from a model for burglary

    M Garcia-Huidobro, Raul Manasevich, and Jean Mawhin. E xistence of solutions for a 1-D boundary value problem coming from a model for burglary. Nonlinear Analysis: Real World Applications, 14(5):1939–1946, 2013

    work page 1939

  17. [17]

    Sta te of the art in agent-based modeling of urban crime: An overview

    Elizabeth R Groff, Shane D Johnson, and Amy Thornton. Sta te of the art in agent-based modeling of urban crime: An overview. Journal of Quantitative Criminology , 35:155–193, 2019

    work page 2019

  18. [18]

    Stationary patterns an d their selection mechanism of urban crime models with heterogeneous near-repeat victimi zation effect

    Yu Gu, Qi Wang, and Guangzeng Yi. Stationary patterns an d their selection mechanism of urban crime models with heterogeneous near-repeat victimi zation effect. European Journal of Applied Mathematics, 28(1):141–178, 2017

    work page 2017

  19. [19]

    Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source term

    Frederic Heihoff. Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source term. Zeitschrift f¨ ur angewandte Mathematik und Physik, 71(3):80, 2020

    work page 2020

  20. [20]

    Global well-posedness and un iform boundedness of 2D urban crime models with nonlinear advection enhancement

    Wenjing Jiang and Qi Wang. Global well-posedness and un iform boundedness of 2D urban crime models with nonlinear advection enhancement. European Journal of Applied Mathemat- ics, pages 1–13, 2024. 30

    work page 2024

  21. [21]

    Statistical models of criminal behavior: The effects of law enforcement actions

    Paul A Jones, P Jeffrey Brantingham, and Lincoln R Chayes . Statistical models of criminal behavior: The effects of law enforcement actions. Mathematical Models and Methods in Applied Sciences, 20(supp01):1397–1423, 2010

    work page 2010

  22. [22]

    Do more police lead to more crime deterrence? Crime & Delinquency, 60(5):716–738, 2014

    Gary Kleck and James C Barnes. Do more police lead to more crime deterrence? Crime & Delinquency, 60(5):716–738, 2014

    work page 2014

  23. [23]

    The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban cr ime

    Theodore Kolokolnikov, Michael J Ward, and Juncheng We i. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban cr ime. Discrete and Continuous Dynamical Systems-B, 19(5):1373–1410, 2014

    work page 2014

  24. [24]

    Modelling and prevention of crime using age-structure and law enforcement

    Manoj Kumar and Syed Abbas. Modelling and prevention of crime using age-structure and law enforcement. Journal of Mathematical Analysis and Applications , 519(2):126849, 2023

    work page 2023

  25. [25]

    Linear and Quasilinear Elliptic Equations , volume 46 of Mathematics in Science and Engineering

    Olga A Ladyzhenskaia and Nina N Ural’tseva. Linear and Quasilinear Elliptic Equations , volume 46 of Mathematics in Science and Engineering . Academic Press, 1968

    work page 1968

  26. [26]

    L´ ev y processes, subordinators and crime modelling

    Tijana Levajkovic, Hermann Mena, and Martin Zarfl. L´ ev y processes, subordinators and crime modelling. Novi Sad Journal of Mathematics , 46(2):65–86, 2016

    work page 2016

  27. [27]

    On localized hotsp ots of an urban crime model

    David JB Lloyd and Hayley O’Farrell. On localized hotsp ots of an urban crime model. Physica D: Nonlinear Phenomena , 253:23–39, 2013

    work page 2013

  28. [28]

    Exploring data assimilation and forecasting issues for an urban crime model

    David JB Lloyd, Naratip Santitissadeekorn, and Martin B Short. Exploring data assimilation and forecasting issues for an urban crime model. European Journal of Applied Mathematics , 27(3):451–478, 2016

    work page 2016

  29. [29]

    Global existence of solutions for a chemotaxis-type system arising in crime modeling

    Raul Manasevich, Quoc Hung Phan, and Philippe Souplet. Global existence of solutions for a chemotaxis-type system arising in crime modeling. European Journal of Applied Mathematics , 24(2):273–296, 2013

    work page 2013

  30. [30]

    Bi-dimensional crime model b ased on anomalous diffusion with law enforcement effect

    Francisco Javier Mart ´ ınez-Far ´ ıas, Anah ´ ı Alvarado-S´ anchez, Eduardo Rangel-Cortes, and Ar- turo Hern´ andez-Hern´ andez. Bi-dimensional crime model b ased on anomalous diffusion with law enforcement effect. Mathematical Modelling and Numerical Simulation with Appl ications, 2(1):26–40, 2022

    work page 2022

  31. [31]

    The existence and stabili ty of spike solutions for a chemo- taxis system modeling crime pattern formation

    Linfeng Mei and Juncheng Wei. The existence and stabili ty of spike solutions for a chemo- taxis system modeling crime pattern formation. Mathematical Models and Methods in Applied Sciences, 30(09):1727–1764, 2020

    work page 2020

  32. [32]

    Geographic profiling from kinetic models of criminal behavior

    George O Mohler and Martin B Short. Geographic profiling from kinetic models of criminal behavior. SIAM Journal on Applied Mathematics , 72(1):163–180, 2012

    work page 2012

  33. [33]

    Self-exciting point process modeling of crim e

    George O Mohler, Martin B Short, P Jeffrey Brantingham, F rederic Paik Schoenberg, and George E Tita. Self-exciting point process modeling of crim e. Journal of the American Statis- tical Association, 106(493):100–108, 2011

    work page 2011

  34. [34]

    Crime modeling with truncated L´ evy flights for re sidential burglary models

    Chaohao Pan, Bo Li, Chuntian Wang, Yuqi Zhang, Nathan Ge ldner, Li Wang, and Andrea L Bertozzi. Crime modeling with truncated L´ evy flights for re sidential burglary models. Math- ematical Models and Methods in Applied Sciences , 28(09):1857–1880, 2018

    work page 2018

  35. [35]

    Adding police to a mathematical model of burglary

    Ashley B Pitcher. Adding police to a mathematical model of burglary. European Journal of Applied Mathematics, 21(4-5):401–419, 2010. 31

    work page 2010

  36. [36]

    A continuum model of residential bur- glary incorporating law enforcement

    Lee Ricketson. A continuum model of residential bur- glary incorporating law enforcement. preprint on webpage a t www.academia.edu/570840/A_Continuum_Model_of_Residential_Burglary_Incorporating_Law_Enforceme 2010

    work page 2010

  37. [37]

    On the global well-posedness theory for a class of PDE models for criminal activity

    Nancy Rodr ´ ıguez. On the global well-posedness theory for a class of PDE models for criminal activity. Physica D: Nonlinear Phenomena , 260:191–200, 2013

    work page 2013

  38. [38]

    Understanding the effects of on-and off-hotspot policing: Evidence of hotspot, oscillating, and chaotic ac tivities

    Nancy Rodr ´ ıguez, Qi Wang, and Lu Zhang. Understanding the effects of on-and off-hotspot policing: Evidence of hotspot, oscillating, and chaotic ac tivities. SIAM Journal On Applied Dynamical Systems, 20(4):1882–1916, 2021

    work page 1916

  39. [39]

    On the global existence and qualitative behavior of one- dimensional solutions to a model for urban crime

    Nancy Rodr ´ ıguez and Michael Winkler. On the global existence and qualitative behavior of one- dimensional solutions to a model for urban crime. European Journal of Applied Mathematics , 33(5):919–959, 2022

    work page 2022

  40. [40]

    The limits of hot spots policing

    Dennis P Rosenbaum. The limits of hot spots policing. In David Weisburd and Anthony A. Braga, editors, Police innovation: Contrasting perspectives , pages 245–263. Cambridge Uni- versity Press, 2006

    work page 2006

  41. [41]

    An age- structured population approach for the mathematical model ing of urban burglaries

    Joan Saldana, Maria Aguareles, Albert A viny´ o, Marta P ellicer, and Jordi Ripoll. An age- structured population approach for the mathematical model ing of urban burglaries. SIAM Journal on Applied Dynamical Systems , 17(4):2733–2760, 2018

    work page 2018

  42. [42]

    broken windows

    Robert J Sampson and Stephen W Raudenbush. Seeing disor der: Neighborhood stigma and the social construction of “broken windows” . Social Psychology Quarterly, 67(4):319–342, 2004

    work page 2004

  43. [43]

    hot spots

    Lawrence W Sherman and David Weisburd. General deterre nt effects of police patrol in crime “hot spots”: A randomized, controlled trial. Justice Quarterly, 12(4):625–648, 1995

    work page 1995

  44. [44]

    On global bifurcation for quasilinear elliptic systems on bounded domains

    Junping Shi and Xuefeng Wang. On global bifurcation for quasilinear elliptic systems on bounded domains. Journal of Differential Equations , 246(7):2788–2812, 2009

    work page 2009

  45. [45]

    Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression

    Martin B Short, Andrea L Bertozzi, and P Jeffrey Branting ham. Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression. SIAM Journal on Applied Dynamical Systems , 9(2):462–483, 2010

    work page 2010

  46. [46]

    Dissipation and displacement of hotspots in reaction-diffusion models o f crime

    Martin B Short, P Jeffrey Brantingham, Andrea L Bertozzi , and George E Tita. Dissipation and displacement of hotspots in reaction-diffusion models o f crime. Proceedings of the National Academy of Sciences , 107(9):3961–3965, 2010

    work page 2010

  47. [47]

    A statistical m odel of criminal behavior

    Martin B Short, Maria R D’orsogna, Virginia B Pasour, Ge orge E Tita, Paul J Branting- ham, Andrea L Bertozzi, and Lincoln B Chayes. A statistical m odel of criminal behavior. Mathematical Models and Methods in Applied Sciences , 18(supp01):1249–1267, 2008

    work page 2008

  48. [48]

    Community policing, Chicago style

    Wesley G Skogan and Susan M Hartnett. Community policing, Chicago style . Oxford Univer- sity Press, 1999

    work page 1999

  49. [49]

    The distribution of police protection

    David Thacher. The distribution of police protection. Journal of Quantitative Criminology , 27:275–298, 2011

    work page 2011

  50. [50]

    MATLAB version: 24.2.0.2740171 (r2 024b) update 1, 2024

    The MathWorks Inc. MATLAB version: 24.2.0.2740171 (r2 024b) update 1, 2024. 32

    work page 2024

  51. [51]

    Partial differential equation toolb ox version: 24.2 (r2024b), 2024

    The MathWorks Inc. Partial differential equation toolb ox version: 24.2 (r2024b), 2024

    work page 2024

  52. [52]

    Hotspot formation and d ynamics for a continuum model of urban crime

    Wang Hung Tse and Michael J Ward. Hotspot formation and d ynamics for a continuum model of urban crime. European Journal of Applied Mathematics , 27(3):583–624, 2016

    work page 2016

  53. [53]

    Asynchronous instabil ities of crime hotspots for a 1-D reaction-diffusion model of urban crime with focused police patrol

    Wang Hung Tse and Michael J Ward. Asynchronous instabil ities of crime hotspots for a 1-D reaction-diffusion model of urban crime with focused police patrol. SIAM Journal on Applied Dynamical Systems, 17(3):2018–2075, 2018

    work page 2018

  54. [54]

    Global well-posednes s and uniform boundedness of urban crime models: One-dimensional case

    Qi Wang, Deqi Wang, and Yani Feng. Global well-posednes s and uniform boundedness of urban crime models: One-dimensional case. Journal of Differential Equations , 269(7):6216– 6235, 2020

    work page 2020

  55. [55]

    Time-periodic and s table patterns of a two-competing- species Keller-Segel chemotaxis model: Effect of cellular g rowth

    Qi Wang, Jingyue Yang, and Lu Zhang. Time-periodic and s table patterns of a two-competing- species Keller-Segel chemotaxis model: Effect of cellular g rowth. Discrete and Continuous Dynamical Systems-B, 22(9):3547–3574, 2017

    work page 2017

  56. [56]

    What can police do to reduc e crime, disorder, and fear? The annals of the American academy of political and social sc ience, 593(1):42–65, 2004

    David Weisburd and John E Eck. What can police do to reduc e crime, disorder, and fear? The annals of the American academy of political and social sc ience, 593(1):42–65, 2004

    work page 2004

  57. [57]

    Policing drug hot sp ots: The Jersey City drug market analysis experiment

    David Weisburd and Lorraine Green. Policing drug hot sp ots: The Jersey City drug market analysis experiment. Justice Quarterly, 12(4):711–735, 1995

    work page 1995

  58. [58]

    Is problem-oriented policing effective in reducing crime and disorder? Findings from a Campbell systematic review

    David Weisburd, Cody W Telep, Joshua C Hinkle, and John E Eck. Is problem-oriented policing effective in reducing crime and disorder? Findings from a Campbell systematic review. Criminology & Public Policy , 9(1):139–172, 2010

    work page 2010

  59. [59]

    Wilson and George L

    James Q. Wilson and George L. Kelling. Broken windows: T he police and neighborhood safety. Atlantic Monthly, 249(3):29–38, 1982

    work page 1982

  60. [60]

    Global solvability and stabilizatio n in a two-dimensional cross-diffusion system modeling urban crime propagation

    Michael Winkler. Global solvability and stabilizatio n in a two-dimensional cross-diffusion system modeling urban crime propagation. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 36(6):1747–1790, 2019

    work page 2019

  61. [61]

    Formatio n and suppression of hotspots in urban crime models with law enforcement

    Madi Yerlanov, Qi Wang, and Nancy Rodr ´ ıguez. Formatio n and suppression of hotspots in urban crime models with law enforcement. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(6), 2025

    work page 2025

  62. [62]

    C ops on the dots in a mathematical model of urban crime and police response

    Joseph R Zipkin, Martin B Short, and Andrea L Bertozzi. C ops on the dots in a mathematical model of urban crime and police response. Discrete and Continuous Dynamical Systems-B , 19(5):1479–1506, 2014. 33

    work page 2014