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arxiv: 2605.23110 · v1 · pith:O6D5DQT3new · submitted 2026-05-22 · 🧮 math.DS

Delay-induced dynamics in a nonlinear crime interaction model with periodic forcing

Pablo Amster , Andr\'es Rivera , Sebasti\'an Pedersen This is my paper

Pith reviewed 2026-05-25 03:34 UTC · model grok-4.3

classification 🧮 math.DS
keywords crime dynamicsdelay differential equationsperiodic solutionstopological degreeHolling type II responselaw enforcementstability switchesnonlinear model
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The pith

A crime interaction model with delay and periodic enforcement admits positive periodic solutions determined by average law enforcement intensity.

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

The paper introduces a nonlinear delay differential equation model for criminal and non-criminal population interactions that includes a Holling type II functional response for saturation and periodic terms for time-dependent law enforcement. It establishes positivity, global existence, and invariance of the feasible region, derives explicit stability thresholds for the criminal-free equilibrium in the autonomous case, and shows that delays produce stability switches leading to oscillations. In the non-autonomous case, topological degree arguments establish the existence of strictly positive periodic solutions, implying that long-term behavior is governed by averaged enforcement measures instead of short-term fluctuations.

Core claim

In the non-autonomous case, topological degree arguments guarantee the existence of strictly positive periodic solutions, indicating that long-term dynamics depend primarily on the averaged law enforcement intensity measures rather than on short-term fluctuations.

What carries the argument

Topological degree arguments applied to the non-autonomous delay differential equation to establish existence of strictly positive periodic solutions.

If this is right

  • Threshold conditions determine stability of the criminal-free equilibrium and emergence of coexistence states in the autonomous setting.
  • The delay produces stability switches and oscillatory regimes via characteristic equation root crossings.
  • Long-term dynamics are shaped by averaged enforcement intensity rather than instantaneous variations.
  • Recurrent patterns in crime arise as a structural consequence of the delay mechanism.

Where Pith is reading between the lines

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

  • Policy emphasis on maintaining consistent average enforcement levels over time could be more effective than reacting to transient changes.
  • The averaging principle might apply to other delayed population models with periodic external forcing, such as in epidemiology or ecology.
  • Fitting the model to enforcement and crime time series could test whether observed oscillations align with predicted delay-induced switches.

Load-bearing premise

The model assumes the chosen delay term and Holling type II functional response accurately represent latency in behavioral response and saturation of criminal influence.

What would settle it

Numerical or data-driven observation that long-term crime patterns change substantially when short-term enforcement fluctuations vary while the time-average remains fixed would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.23110 by Andr\'es Rivera, Pablo Amster, Sebasti\'an Pedersen.

Figure 1
Figure 1. Figure 1: Diagram of the regions of the positive roots of F(s) according to the values of h and F(0) (see equation (8) and what follows). The blue curve corresponds to F(0) = h 2/4. Proof. Let x = s 2 , and consider the quadratic function G(x) = F(s(x)) given by G(x) = x 2 + hx + F(0), with h = ν 2 1 − 2ν2 − ς 2 2 . The roots of F(s) are s = ± √ x with 2x = −h ± p h 2 − 4F(0), In case I. the real part of x is non-po… view at source ↗
Figure 2
Figure 2. Figure 2: Simulation of the autonomous system (9) when τ = 2, with f(N) = 1 − N, φ = 1, ν = 0.9, σ = 0.4, γ = 1.4, le = 0.51. η and initial condition varies. We observe that criminal-free equilibrium (N†, 0) (green point), where N† = 1 (because f(1) = 0), switches stability depending on N†gγ(N) ≷ η + le, where gγ(N) = γ ν+N . Switching our attention to the co-existence equilibrium (N, ˆ Cˆ) (see section 3), here we … view at source ↗
Figure 3
Figure 3. Figure 3: Directions fields of the autonomous system (9) for the non-delayed case. The trivial equilibrium is not of much interest mainly because, as section 3.1 shows, it is unstable for all delays τ ≥ 0, but furthermore, it is not of much interest in the context of the problem that our system (1) is modeling, namely the dynamics between criminal and non-criminal populations [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Simulation of the autonomous system (9) with initial conditions (N0(t), C0(t)) = (1.5, 1.5), −τ ≤ t ≤ 0, and the rest same as figure 2b. We observe that co-existence equilibrium (N, ˆ Cˆ) ≃ (0.6949, 1.344) (green point) is asymptotically stable. 6.2. Numerical simulations in the non-autonomous case. We present two simulations of system (1): non-delay and delay cases in [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 5
Figure 5. Figure 5: Simulation of the non-autonomous system (1) when τ = 0 and τ = 2, with le(t) = 0.2 sin(t/4) + 0.5 and initial condition (N0(t), C0(t)) = (1.5, 1.5), −τ ≤ t ≤ 0 (rest same as figure 2b). We observe the asymptotic behavior toward a periodic solution. Conclusions A nonlinear time-delay model has been proposed and analyzed to describe the interaction be￾tween criminal and non-criminal populations, incorporatin… view at source ↗
read the original abstract

A nonlinear time-delay model is proposed to describe the interaction dynamics between criminal and non-criminal populations, combining social influence mechanisms, saturation effects represented by a Holling type II functional response, and time-dependent law-enforcement actions. The delay accounts for the latency between exposure to criminal behavior and behavioral response, introducing memory effects that naturally lead to a delay differential equations framework. Fundamental analytical properties, including positivity, global existence, and invariance of the feasible region, are established to ensure the mathematical consistency of the population interpretation. In the autonomous setting, explicit threshold conditions governing the stability of the criminal-free equilibrium and the emergence of coexistence states are derived, while the delay is shown to induce stability switches and oscillatory regimes through characteristic root crossings. In the non-autonomous case, topological degree arguments guaranty the existence of strictly positive periodic solutions, indicating that long-term dynamics depend primarily on the averaged law enforcement intensity measures rather than on short-term fluctuations. These results identify time delay as a key structural mechanism underlying recurrent patterns and complex temporal behavior in crime dynamics.

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 / 1 minor

Summary. The manuscript introduces a nonlinear delay differential equation model for interactions between criminal and non-criminal populations, incorporating a Holling type II functional response, a delay for behavioral latency, and periodic law-enforcement forcing. It establishes positivity, global existence, and invariance of the feasible region. In the autonomous case, explicit threshold conditions are derived for stability of the criminal-free equilibrium and emergence of coexistence equilibria, with the delay shown to induce stability switches via characteristic equation analysis. In the non-autonomous case, topological degree arguments are applied to prove existence of strictly positive periodic solutions, from which the authors conclude that long-term dynamics depend primarily on averaged enforcement intensity rather than short-term fluctuations.

Significance. If the existence result is rigorously established and the averaging interpretation is supported by additional analysis, the work would contribute to the mathematical understanding of delay-induced oscillations and robustness properties in periodically forced population models, with potential relevance to policy questions on enforcement timing.

major comments (2)
  1. [Abstract] Abstract: The claim that topological degree arguments guaranteeing existence of strictly positive periodic solutions 'indicate' that long-term dynamics depend primarily on averaged law enforcement intensity measures rather than short-term fluctuations is not entailed by the existence result. Standard applications of Mawhin's continuation theorem or similar yield existence under a priori bounds and non-resonance on the averaged nonlinearity but provide no information on uniqueness, stability, global attractivity, or invariance of solution properties under mean-preserving changes to the periodic forcing. An explicit comparison with the averaged autonomous system or an analysis showing independence from higher Fourier modes is required to support the inference.
  2. [Non-autonomous analysis section] Non-autonomous analysis section: The manuscript should explicitly state the a priori bounds derived for the periodic solutions and verify the non-resonance condition on the averaged vector field; without these details it is not possible to confirm that the topological degree is non-zero for the given periodic forcing term.
minor comments (1)
  1. [Abstract] Typo: 'guaranty' should read 'guarantee'.

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.

read point-by-point responses

  1. Referee: [Abstract] The claim that topological degree arguments guaranteeing existence of strictly positive periodic solutions 'indicate' that long-term dynamics depend primarily on averaged law enforcement intensity measures rather than short-term fluctuations is not entailed by the existence result. Standard applications yield existence under a priori bounds and non-resonance but provide no information on uniqueness, stability, global attractivity, or invariance under mean-preserving changes. An explicit comparison with the averaged autonomous system or analysis showing independence from higher Fourier modes is required.

    Authors: We agree that the interpretive statement in the abstract is not rigorously supported by the existence result alone. In the revised version we will remove this claim from the abstract and limit the statement to the proven existence of strictly positive periodic solutions. No additional comparison with the averaged system will be added, as that would require substantial new analysis beyond the scope of the current work. revision: yes

  2. Referee: [Non-autonomous analysis section] The manuscript should explicitly state the a priori bounds derived for the periodic solutions and verify the non-resonance condition on the averaged vector field; without these details it is not possible to confirm that the topological degree is non-zero for the given periodic forcing term.

    Authors: We will revise the non-autonomous section to explicitly derive and state the a priori bounds on candidate periodic solutions and to verify the non-resonance condition on the averaged vector field, thereby confirming that the topological degree is nonzero under the stated hypotheses on the periodic forcing. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities are detailed beyond standard modeling assumptions such as positivity and the form of the functional response.

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Abbas, J

    S. Abbas, J. P. Tripathi, and A. Neha. Dynamical analysis of a model of social behavior: Criminal vs non-criminal population.Chaos, Solitons & Fractals, 98:121–129, 2017

    work page 2017

  2. [2]

    Amster.Topological Methods in the study of boundary value problems

    P. Amster.Topological Methods in the study of boundary value problems. Universitext. Springer, 2014

    work page 2014

  3. [3]

    Amster, A

    P. Amster, A. Rivera, and J. A. Arredondo. Periodic solutions in a tumor-immune competition system with time-delay and chemotherapy effects, 2025

    work page 2025

  4. [4]

    Bezanson, A

    J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah. Julia: A fresh approach to numerical computing. SIAM Review, 59(1):65–98, 2017. 25 Delay-induced dynamics in a nonlinear crime interaction model with periodic forcing

    work page 2017

  5. [5]

    O. N. Bjørnstad, K. Shea, M. Krzywinski, and N. Altman. The seirs model for infectious disease dynamics. Nature methods, 17(6):557–559, 2020

    work page 2020

  6. [6]

    Brauer and C

    F. Brauer and C. Castillo-Chavez.Mathematical Models in Population Biology and Epidemiology. Springer New York, NY, 2nd edition, 2012

    work page 2012

  7. [7]

    T. Breloff. Plots.jl, Mar. 2025

    work page 2025

  8. [8]

    Calatayud, M

    J. Calatayud, M. Jornet, and J. Mateu. A dynamical mathematical model for crime evolution based on a compartmental system with interactions.International Journal of Computer Mathematics, 102(1):44–59, 2025

    work page 2025

  9. [9]

    Chen-Charpentier

    B. Chen-Charpentier. Delays and exposed populations in infection models.Mathematics, 11(8):1919, 2023

    work page 1919

  10. [10]

    Crokidakis

    N. Crokidakis. Dynamics of drug trafficking: Results from a simple compartmental model.ArXiv, abs/2409.02659, 2024

    work page arXiv 2024

  11. [11]

    Deutsch.Mathematical Modeling of Biological Systems, Volume II: Epidemiology, Evolution and Ecol- ogy, Immunology, Neural Systems and the Brain, and Innovative Mathematical Methods

    A. Deutsch.Mathematical Modeling of Biological Systems, Volume II: Epidemiology, Evolution and Ecol- ogy, Immunology, Neural Systems and the Brain, and Innovative Mathematical Methods. Mathematical Modeling of Biological Systems. Birkh¨ auser Boston, 2007

    work page 2007

  12. [12]

    Diekmann, H

    O. Diekmann, H. Heesterbeek, and T. Britton.Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press, 2013

    work page 2013

  13. [13]

    Diekmann, H

    O. Diekmann, H. Thieme, and H. Inaba. Mathematical epidemiology of infectious diseases: an ongoing challenge.Japan Journal of Industrial and Applied Mathematics, 42, 09 2025

    work page 2025

  14. [14]

    T. W. Gutema, A. G. Wedajo, and K. P. Rao. A mathematical analysis of the corruption dynamics model with optimal control strategy.Frontiers Appl. Math. Stat., 10, 2024

    work page 2024

  15. [15]

    Ibrahim, O

    O. Ibrahim, O. Daniel, and M. N. O. Ikhile. Mathematical modeling of the population dynamics of age- structured criminal gangs with correctional intervention measures.Applied Mathematical Modelling, 107, 02 2022

    work page 2022

  16. [16]

    B. S. Kotola and S. W. Teklu. A mathematical modeling analysis of racism and corruption codynamics with numerical simulation as infectious diseases.Computational and Mathematical Methods in Medicine, 2022, 2022

    work page 2022

  17. [17]

    Kuang.Delay Differential Equations with Applications in Population Dynamics

    Y. Kuang.Delay Differential Equations with Applications in Population Dynamics. Academic Press. Aca- demic Press, 1993

    work page 1993

  18. [18]

    Kwofie, M

    T. Kwofie, M. Dogbatsey, and S. E. Moore. Curtailing crime dynamics: A mathematical approach.Fron- tiers in Applied Mathematics and Statistics, 8:1086745, 2023

    work page 2023

  19. [19]

    D. Mamo, M. Kinyanjui, S. Teklu, and G. Hailu. Mathematical modeling and analysis of the co-dynamics of crime and drug abuse.Scientific Reports, 14, 11 2024

    work page 2024

  20. [20]

    McMillon, J

    D. McMillon, J. Morenoff, C. Simon, and E. Lane. A dynamical systems analysis of criminal behavior using national longitudinal survey of youth data.PLOS One, 20, 08 2025

    work page 2025

  21. [21]

    Mohammad and U

    F. Mohammad and U. M. Roslan. Analysis on the crime model using dynamical approach. InAIP confer- ence proceedings, volume 1870, page 040067. AIP Publishing LLC, 2017

    work page 2017

  22. [22]

    Murray.Mathematical Biology: I

    J. Murray.Mathematical Biology: I. An Introduction. Interdisciplinary Applied Mathematics. Springer New York, 2007

    work page 2007

  23. [23]

    Rackauckas and Q

    C. Rackauckas and Q. Nie. Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia.Journal of Open Research Software, 5(1):15, 2017

    work page 2017

  24. [24]

    Rihan.Delay Differential Equations and Applications to Biology

    F. Rihan.Delay Differential Equations and Applications to Biology. Forum for Interdisciplinary Mathe- matics. Springer Nature Singapore, 2021

    work page 2021

  25. [25]

    Rossini, N

    L. Rossini, N. B. Rossello, O. Benhamouche, M. Contarini, S. Speranza, and E. Garone. A general dde framework to describe insect populations: Why delays are so important?Ecological Modelling, 499:110937, 2025

    work page 2025

  26. [26]

    Saade, S

    M. Saade, S. Ghosh, M. Banerjee, and V. Volpert. Delay epidemic models determined by latency, infection, and immunity duration.Mathematical Biosciences, 370, 02 2024

    work page 2024

  27. [27]

    Sarmah, A

    A. Sarmah, A. Das, K. Dehingia, A. Phukan, H. kr. Sarmah, and E. Hıncal. Exploring the dynamics of a mathematical model of terrorism with time-delay.Iranian Journal of Science, 2025

    work page 2025

  28. [28]

    M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, and L. B. Chayes. A statistical model of criminal behavior.Mathematical Models and Methods in Applied Sciences, 18:1249–1267, 2008. 26 Pablo Amster, Andr´ es Rivera and G. Sebasti´ an Pedersen

    work page 2008

  29. [29]

    Smith.An Introduction to Delay Differential Equations with Applications to the Life Sciences

    H. Smith.An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics. Springer, 2011

    work page 2011

  30. [30]

    Sooknanan and T

    J. Sooknanan and T. A. Seemungal. Criminals and their models - a review of epidemiological models describing criminal behaviour.Applied Mathematics and Computation, 458:128212, 2023

    work page 2023

  31. [31]

    A. K. Srivastav, M. Ghosh, and P. Chandra. Modeling dynamics of the spread of crime in a society. Stochastic Analysis and Applications, 37(6):991–1011, 2019

    work page 2019

  32. [32]

    S. W. Teklu and B. B. Terefe. Mathematical modeling investigation of violence and racism coexistence as a contagious disease dynamics in a community.Computational and Mathematical Methods in Medicine, 2022, 2022

    work page 2022

  33. [33]

    J. P. Tripathi, S. Bugalia, K. Burdak, and S. Abbas. Dynamical analysis and effects of law enforcement in a social interaction model.Physica A: Statistical Mechanics and its Applications, 567:125725, 2021. 27

    work page 2021