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arxiv: 1907.01941 · v1 · pith:ATUZQEI3new · submitted 2019-07-03 · 🌊 nlin.AO · math.DS

A dynamical systems model of unorganised segregation in two neighbourhoods

D. J. Haw , S. J. Hogan This is my paper

Pith reviewed 2026-05-25 09:44 UTC · model grok-4.3

classification 🌊 nlin.AO math.DS
keywords Schelling modeldynamical systemssegregationtwo neighbourhoodstolerance schedulesstable integrationpopulation dynamics
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The pith

Stable integration in two connected neighborhoods occurs only when the minority is small and combined tolerance is large.

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

The paper extends a dynamical systems model of population movement to two connected neighborhoods, with or without reservoirs. It demonstrates that stable integration requires both a small minority population and large combined tolerance from both groups. Unlike the single-neighborhood case, restricting one population does not guarantee or produce stable integration and can instead cause segregation. The work concludes that a growing minority stays integrated only if the majority raises its tolerance, and that an already integrated single neighborhood can lose stability when a connecting neighborhood is added.

Core claim

Complete analysis of the Schelling dynamical system for two neighborhoods shows stable integration is possible exclusively when the minority is small and combined tolerance is large. Limiting one population does not necessarily produce stable integration and may destroy it. A growing minority can remain integrated only if the majority increases its own tolerance. An integrated single neighbourhood may not remain so when a connecting neighbourhood is created.

What carries the argument

The Schelling dynamical system extended to two connected neighbourhoods with linear and nonlinear tolerance schedules that determine population movement.

If this is right

  • Stable integration requires a small minority population together with large combined tolerance.
  • Limiting one population may destroy stable integration rather than create it.
  • A growing minority population remains integrated only when the majority increases its tolerance.
  • Integration achieved in a single neighbourhood can be lost when a second connected neighbourhood is introduced.

Where Pith is reading between the lines

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

  • Policies that restrict population sizes in multi-neighbourhood settings might increase segregation instead of reducing it.
  • Stability of integration could depend on the specific pattern of connections among multiple neighbourhoods.
  • The model could be tested by measuring real tolerance levels against the combined tolerance threshold for given minority sizes.

Load-bearing premise

The dynamical system and tolerance schedules are taken directly from the prior single-neighborhood model, inheriting its movement rules without other social or economic factors.

What would settle it

Observe whether populations in a two-neighbourhood setup with small minority and high combined tolerance remain mixed over time, or whether reducing one population size produces segregation rather than integration.

Figures

Figures reproduced from arXiv: 1907.01941 by D. J. Haw, S. J. Hogan.

Figure 1
Figure 1. Figure 1: Linear tolerance schedules RX(X), RY(Y), as defined in (2). In the one-neighbourhood problem, it is assumed that outside the neighbourhood, there is a “place where colour does not matter" [13]. Population members can move between this place and the neighbourhood. We now consider the case of two neighbourhoods and two populations, where any population leaving one neighbourhood must necessarily enter the oth… view at source ↗
Figure 2
Figure 2. Figure 2: Three qualitatively different possibilities for Case I. In the left hand figures, nullcline (18) is shown in blue and nullcline (19) is shown in red. The pink region is the basin of attraction of the equilibrium (X e 1 ,Y e 1 ) = (0, 1 α ) and the blue region is the basin of attraction of (X e 1 ,Y e 1 ) = (1, 0). The white regions in figure 2e correspond to the basin of attraction of two of the new equili… view at source ↗
Figure 3
Figure 3. Figure 3: Number of equilibria of (17), corresponding to the roots of p9(X1). Solid lines correspond to β = βc ≡ 2α 2 /(α − 2), from (23) and β = β± ≡ 4 8−α [9α − α 2 ± √ α(α − 6) 3], from (31). Integrated equilibria occur only inside the dark shaded region, for β ∈ [β−, β+], when (17) has nine equilibria. The polynomial p9(X1) has five roots in the light shaded region and only three roots in the white shaded region… view at source ↗
Figure 4
Figure 4. Figure 4: Case I: equilibrium values Xe 1 as a function of β for (a) α = 1.9, (b) α = 5, (c) α = 7.1 and (d) α = 10. Substituting (14) into (15), we obtain a ninth order polynomial p˜9(X1) of possible equilibria. We know that X e 1 = 0, 1, by inspection of (14), (15). But X e 1 = 1 2 is no longer a guaranteed equilibrium. Hence we write p˜9(X1) = X1(1 − X1)p˜7(X1), where p˜7(X1) ≡ 7 ∑ i=0 biX i 1 . (40) The real coe… view at source ↗
Figure 5
Figure 5. Figure 5: Bifurcation diagrams and example phase portraits for case II, III and IV. Stable equilibria are denoted by , unstable nodes by # and saddle points by ⊗. when the discriminant of (47) is positive. Note that if Y a 1 < Y + 1 , we can only get two new equilibria if u ∈ [X a 1 , X + 1 ] (not shown). So to proceed we must first find out when the turning points (X ± 1 ,Y ± 1 ) of nullcline (19) exist. Then since… view at source ↗
Figure 6
Figure 6. Figure 6: Limiting the X1 population: X1 = u: β ∈ [2α, 8α] , Ya 1 > Y + 1 . investigate when the two nullclines (18), (19) intersect there. Turning points (X ± 1 ,Y ± 1 ) of nullcline (19) exist when it has a vertical tangent. From (19), we find dY1 dX1 = 1 [−(α + β) + 6αβY1 − 6α 2βY 2 1 ] . (48) It is straightforward to show that, when β > 2α, the nullcline (19) has vertical tangents at (X ± 1 ,Y ± 1 ) = ( 1 2 ± 1 … view at source ↗
Figure 7
Figure 7. Figure 7: Limiting the X1 -population. Γu(α, β) = 0 (equation (52), shown by black dashed line); β = βc (equation (23), shown in red); β = β± (equation (31), shown in green), together with the lines α = 2, α = 8 and β = 2α, β = 8α. The curve Γu(α, β) = 0 is shown by the black dashed line in figure 7. Note that it is asymptotic to β = βc and β = β−, as α → ∞. Also shown in the same figure are β = βc (equation (23), s… view at source ↗
Figure 8
Figure 8. Figure 8: Limiting the X1 population: (α, β) = (6, 30). Stable equilibria are denoted by , unstable nodes by # and saddle points by ⊗. 4 Nonlinear tolerance schedules In this section, we consider other ways in which the linear tolerance schedule can be modified to produce integrated populations. We illustrate phenomena that can occur when the tolerance schedules are nonlinear, using the original equations (1) in the… view at source ↗
Figure 9
Figure 9. Figure 9: Limiting the X1 population: (α, β) = (7, 49). Stable equilibria are denoted by , unstable nodes by # and saddle points by ⊗. case a = 2, b = 10, k = 1, that is (α, β) = (2, 20). This is the simplest case when the tolerance schedules are linear, corresponding to dynamics in the white region of figure 3, where we have two stable segregated equilibria and one unstable integrated equilibria. In our first examp… view at source ↗
Figure 10
Figure 10. Figure 10: Nullclines, fixed points and basins of attraction for some candidate nonlinear tolerance schedules. Stable equilibria are denoted by , unstable nodes by # and saddle points by ⊗. So far, we have only considered equilibria of our governing Schelling dynamical system (10). Do these equations have periodic solutions? They have been observed in discrete time “two rooms" models of segregation. But these oscill… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of integrated population parameters for two-neighbourhoods with the single-neighbourhood problem [5]. Curves with apex P1 = (3, 9) are β = β± from [5]. Curves with apex P2 = (6, 36) are β = β± from (31) above. neighbourhood 1 (X1,Y1) neighbourhood 2 (X2,Y2) X3 Y3 −X˙ 1 X˙ 1 −Y˙ 1 Y˙ 1 −X˙ 2 X˙ 2 −Y˙ 2 Y˙ 2 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Dynamics of the two neighbourhoods problem with reservoirs of population. All arrows represent population flow when the corresponding quantities are positive. Dotted arrows have the additional constraint that X3 > 0 (top part of schematic) or Y3 > 0 (bottom part of schematic). We summarise the dynamics in table 1 [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: For each (α, β), we simulated 20 sets of initial conditions for t = 0 to t = 200, calculated dissimilarity D for each and plotted the mean D in parameter space (D = 1 indicates complete segregation, D = 0 indicates an even distribution between neighbourhoods). Note that two empty neighbourhoods yields D = 0. Results for two neighbourhoods with reservoirs are shown in (a), and two independent simulations w… view at source ↗
read the original abstract

We present a complete analysis of the Schelling dynamical system [Haw2018] of two connected neighbourhoods, with or without population reservoirs, for different types of linear and nonlinear tolerance schedules. We show that stable integration is only possible when the minority is small and combined tolerance is large. Unlike the case of the single neighbourhood, limiting one population does not necessarily produce stable integration and may destroy it. We conclude that a growing minority can only remain integrated if the majority increases its own tolerance. Our results show that an integrated single neighbourhood may not remain so when a connecting neighbourhood is created.

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 manuscript extends the Schelling dynamical system of Haw2018 to two connected neighbourhoods (with or without population reservoirs) and performs a complete analysis for linear and nonlinear tolerance schedules. It concludes that stable integration is possible only when the minority population is small and combined tolerance is large; unlike the single-neighbourhood case, limiting one population does not necessarily produce stable integration and may destroy it. The authors further conclude that a growing minority can remain integrated only if the majority increases its tolerance, and that an integrated single neighbourhood may lose integration when a connecting neighbourhood is added.

Significance. If the stability conclusions hold, the work supplies a parameter-free dynamical-systems account of how neighbourhood connectivity alters integration thresholds relative to the isolated case. The explicit treatment of multiple tolerance schedules and the demonstration that single-neighbourhood stability does not automatically carry over to the coupled system are concrete strengths that could generate falsifiable predictions for multi-neighbourhood urban dynamics.

major comments (2)
  1. [Model definition and extension from Haw2018] The central claim that limiting one population can destroy integration (rather than produce it) is obtained by substituting the Haw2018 vector field into a two-neighbourhood coupling. The manuscript provides no independent derivation or justification for why the per-neighbourhood tolerance function and relocation probability continue to govern choice between the two connected sites; if the coupling term modifies the effective tolerance experienced across the boundary, the reported bifurcation structure and the “minority small + combined tolerance large” region can shift.
  2. [Stability analysis and bifurcation results] The stability conclusions for the two-neighbourhood system are stated to be independent of fitted parameters, yet the analysis inherits all movement rules and tolerance schedules from the single-neighbourhood model without a sensitivity check on the coupling strength. A concrete test (e.g., varying the inter-neighbourhood relocation weight while holding tolerance schedules fixed) is needed to confirm that the qualitative change from the single-neighbourhood case survives.
minor comments (2)
  1. [Model section] Notation for the two-neighbourhood state variables and the coupling term should be introduced explicitly before the stability theorems are stated.
  2. [Abstract and introduction] The abstract asserts a “complete analysis”; the main text should include a brief statement of which cases (linear vs. nonlinear schedules, with vs. without reservoirs) have been exhaustively classified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our extension of the Schelling model. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Model definition and extension from Haw2018] The central claim that limiting one population can destroy integration (rather than produce it) is obtained by substituting the Haw2018 vector field into a two-neighbourhood coupling. The manuscript provides no independent derivation or justification for why the per-neighbourhood tolerance function and relocation probability continue to govern choice between the two connected sites; if the coupling term modifies the effective tolerance experienced across the boundary, the reported bifurcation structure and the “minority small + combined tolerance large” region can shift.

    Authors: The coupled system is obtained by applying the original local tolerance schedules and relocation probabilities to each neighbourhood separately, with inter-neighbourhood flows determined by the same rules. This is a direct and natural extension that preserves the single-site dynamics while introducing coupling through population movement. We agree that an explicit step-by-step derivation of the two-neighbourhood vector field would improve clarity and will add this to the revised manuscript. revision: yes

  2. Referee: [Stability analysis and bifurcation results] The stability conclusions for the two-neighbourhood system are stated to be independent of fitted parameters, yet the analysis inherits all movement rules and tolerance schedules from the single-neighbourhood model without a sensitivity check on the coupling strength. A concrete test (e.g., varying the inter-neighbourhood relocation weight while holding tolerance schedules fixed) is needed to confirm that the qualitative change from the single-neighbourhood case survives.

    Authors: The bifurcation results are derived analytically for general linear and nonlinear tolerance schedules and do not rely on fitted numerical values. Nevertheless, we acknowledge the utility of checking robustness to the relative strength of inter-neighbourhood relocation. In the revision we will include numerical explorations that vary this coupling weight while holding tolerance functions fixed, confirming that the reported qualitative distinctions from the single-neighbourhood case remain intact. revision: yes

Circularity Check

0 steps flagged

Two-neighborhood extension derives stability results from coupled dynamical system; self-citation to base model is not load-bearing

full rationale

The paper extends the single-neighborhood Schelling system defined in [Haw2018] by coupling two neighborhoods and analyzes the resulting ODEs for equilibria and stability under various tolerance schedules. The central claims (stable integration only when minority is small and combined tolerance large; limiting one population may destroy integration) follow from this mathematical analysis of the extended vector field rather than from any re-fitting, self-definition, or uniqueness theorem imported from the authors' prior work. The citation to [Haw2018] supplies the base movement rules and tolerance functions as modeling assumptions; the two-neighborhood conclusions are obtained by direct substitution and bifurcation analysis, which constitutes independent content. No equations in the provided abstract or description reduce a prediction to a fitted input or rename a known result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5620 in / 1054 out tokens · 38560 ms · 2026-05-25T09:44:41.396491+00:00 · methodology

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Reference graph

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