A dynamical systems model of unorganised segregation in two neighbourhoods
AA dynamical systems model of unorganised segregation intwo neighbourhoods
D. J. Haw and S. J. Hogan * Abstract
We present a complete analysis of the Schelling dynamical system [5] of two connected neighbourhoods, with orwithout 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 integrationand may destroy it.We conclude that a growing minority can only remain integrated if the majority increases its own tolerance. Ourresults show that an integrated single neighbourhood may not remain so when a connecting neighbourhood is created.
Segregation in society takes on many forms, occurring not only in ethnicity or religion, but also gender, forexample amongst toddlers [10] and in professional hierarchies [1]. Segregation is seen as so divisive thatsome have argued it “puts the whole idea of a peaceful society with its constitutional and civic liberties atrisk." (cited in [11, p.4]), with € segregation. These papers have been cited thousands of times and inspiredinnumerable other works. They are also the subject of controversy in terms of precedence [6] andapplicability [2].What appears less well known is that, in the same two papers, Schelling introduced another model, thebounded neighbourhood model (BNM) of unorganised segregation. In Schelling‘s BNM, a neighbourhoodis like a district within a city. Within the neighbourhood, every member is concerned about the overallpopulation mixture , not with any particular configuration. A member moves out if they are not happy withthe population mixture.Suppose that the population of a single neighbourhood is divided into two types and let X ( t ) , Y ( t ) ≥ t . In this neighbourhood, tolerance limits are allocated to a given population type via a tolerance schedule , as follows . The X -population toleranceschedule R X ( X ) describes the minimum ratio Y / X required in order for all of the X -population to remainin that neighbourhood. A similar function R Y ( Y ) denotes the tolerance schedule of the Y -population.Schelling [12, 13] made the following assumptions:S1. The neighbourhood is preferred over other locations: populations of either type will enter/remain/leaveunless tolerance conditions are violated. * D. J. Haw, School of Public Health, Imperial College London. S. J. Hogan: Department of Engineering Mathematics, University ofBristol, Bristol BS8 1UB The modern usage would be self-organised . Tolerance is a measure of how members of one population remain in an area where there is another population present. In contrast,homophily (or self-segregation) [1] is a measure of how much one population seeks out members of the same population. a r X i v : . [ n li n . AO ] J u l
2. The tolerance schedule is neighbourhood-specific.S3. Each member of the population is aware of the ratio of population types within the neighbourhood atthe moment the decision is made to enter/remain/leave (perfect information).S4. There is no lower bound on tolerance: no population insists on the presence of the other type.S5. Tolerance schedules are monotone decreasing: the more tolerant population members are the first toenter and the last to leave.We gave the first dynamical systems formulation of Schelling’s BNM in [5]. This work studies thecontinuous movement of two populations in and out of a single neighbourhood. We presented the firstcomplete quantitative analysis of the model for linear tolerance schedules. A fully predictive model wasderived and each term within the model was associated with a social meaning. Schelling‘s qualitativeresults were recovered and generalised.For the case of unlimited population movement, we derived exact formulae for regions in parameterspace where stable integrated populations can occur and showed how neighbourhood tipping can be explainedin terms of basins of attraction. When population numbers are limited, we derived exact criteria for theoccurrence of integrated populations.A natural extension of [5] is to consider multiple neighbourhoods, constructing sets of differentialequations that describe the flow of population within and between these neighbourhoods. In this paper,we focus on the case of two populations X ( t ) , Y ( t ) moving within and between two neighbourhoods. Ourwork is related to the "two-room model" of segregation studied in [15, 16]. Our approach differs in that wework in continuous time, whereas they work in discrete time.We structure the paper as follows. In Section 2, we consider the situation in which both populationsare contained solely within the two neighbourhoods, so that any population leaving one neighbourhoodmust necessarily relocate to the other. We consider linear tolerance schedules, examining cases when thetolerance schedules are the same or different in both neighbourhoods.We examine the case when one population has its numbers limited in one area (Section 3), and look athow nonlinearity in the tolerance schedules can change outcomes (Section 4).We discuss our results in Section 5. We consider the situation where there are reservoirs of bothpopulations, outside the two neighbourhoods. Whilst the total population of each type in the wholesystem is conserved, populations can enter or exit either neighbourhood without recourse to the otherneighbourhood. We also examine similarities between the one- and the two-neighbourhood problems andconsider what happens when a second neighbourhood is added to the single neighbourhood problem.Our conclusions are presented in Section 6. Recall that for one neighbourhood with two populations X , Y , where population members can come andgo depending on their tolerance, the dynamical system derived in [5, Eq. (27)] is given by dXdt = X [ XR X ( X ) − Y ] , (1) dYdt = Y [ YR Y ( Y ) − X ] for general tolerance schedules R X ( X ) , R Y ( Y ) . This model satisfies the constraints on the X -population that dXdt ≷ R X ( X ) ≷ YX : in other words, the X -population grows (decays) when the tolerance schedule R X ( X ) exceeds (falls short of) Y / X and that dXdt = X =
0. The model satisfies similar constraints onthe Y -population. Schelling’s initial example of a tolerance schedule is linear, as shown in Figure 1. We set R X ( X ) = a ( − X ) , (2) R Y ( Y ) = b ( − kY ) ,here we scale the size of the X -population to 1 and the minority Y -population to 1 / k , where k ≥
1. Withour scalings, the most tolerant member of the X -population can abide a YX ratio of a > X -population can not abide any members of the Y -population. Likewise, the mosttolerant member of the Y -population can abide a XY ratio of b > Y -population can not abide any members of the X -population. Figure 1:
Linear tolerance schedules R X ( X ) , R Y ( Y ) , as defined in (2) . In the one-neighbourhood problem, it is assumed that outside the neighbourhood, there is a “place wherecolour 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 leavingone neighbourhood must necessarily enter the other. Let ( X i , Y i ) , i =
1, 2 denote the ( X , Y ) -populations inneighbourhood i . So X + X = X total and Y + Y = Y total where X total , Y total are both constant. As above, wescale populations such that X total = Y total = k . Hence X = − X , (3) Y = k − Y . (4)If the neighbourhoods were completely independent of one another, then ( X i , Y i ) , i =
1, 2 wouldseparately satisfy (1). But the population dynamics in one neighbourhood is affected by movement ofpopulation to and from the other neighbourhood. So we have to take that into account in our modelling.To do this we make the additional mild assumption that each member in both neighbourhoods only caresabout the population mixture of the neighbourhood that they are in.Let us consider neighbourhood 1. The dynamics is governed by the following equations. dX dt = X [ X R X ( X ) − Y ] − X [ X R X ( X ) − Y ] (5) dY dt = Y [ Y R Y ( Y ) − X ] − Y [ Y R Y ( Y ) − X ] (6)The first two terms on the right hand side of (5) are the same as those in (1) and correspond to themovement of X -population, subject to the presence of the Y -population, in and out of neighbourhood 1.The third and fourth terms account for the movement of the X -population, subject to the presence of the Y -population, in and out of neighbourhood 2. The terms on the right hand side of (6) can be considered inan analogous way.For linear tolerance schedules (2) in both neighbourhoods, for i =
1, 2 we set R X i ( X i ) = a i ( − X i ) , (7) R Y i ( Y i ) = b i ( − kY i ) .here a ≠ a , b ≠ b in general. From (3) and (4), the dynamics in neighbourhood 2 is given simply by dX dt = − dX dt , (8) dY dt = − dY dt . (9)So we have no need to consider these dynamics explicitly, since they can be obtained directly from thedynamics of neighbourhood 1.Substituting (3), (4) into (5) and (6), with linear tolerance schedules (7), we obtain the following equationsfor ( X , Y ) : dX dt = a X ( − X ) − X Y − a X ( − X ) + ( − X )( k − Y ) , (10) dY dt = b Y ( − kY ) − X Y − kb Y ( k − Y ) + ( − X )( k − Y ) .Note that the coordinate axes X = Y = t , ˆ Y given by ˆ t = tk , ˆ Y = Y a (11)and then set α = ka , β = a b , β = a b , γ = a a . (12)Now we drop the hats and simplify (10) to find: dX dt = ( − X )[ − αγ X + α ( + γ ) X ] − α Y , (13) a dY dt = ( − α Y )[ − β Y + α ( β + β ) Y ] − X .Equilibria (steady states) in neighbourhood 1 are given by ( X , Y ) = ( X e , Y e ) where dX e dt = dY e dt =
0. Ifboth X e ≠ Y e ≠
0, these equilibria correspond to integration. We find ( X e , Y e ) by considering theintersection in ( X , Y ) -space of the nullclines of (13), namely solutions of α Y = ( − X )[ − αγ X + α ( + γ ) X ] , (14) X = ( − α Y )[ − β Y + α ( β + β ) Y ] . (15)We will establish conditions under which real positive solutions of (14) and (15) can exist. Then we willfind further conditions under which such solutions are stable. We consider the case when the ( X , Y ) -population linear tolerance schedules are identical in both neigh-bourhoods. So in (7), we set a = a = a , b = b = b ,Hence from (12) we have α = ka , β = β = β , γ =
1. (16)arameters α and β are key to what follows. With scaling (11), both refer to the minority Y -population:large/small α corresponds to a small/large minority and large/small β refers to a tolerant/intolerantminority. From (13), the governing equations become dX dt = ( − X )[ − α X + α X ] − α Y , (17) a dY dt = ( − α Y )[ − β Y + αβ Y ] − X Our aim is to find how many possible equilibria ( X e , Y e ) exist and then to examine their stability. Thepoints ( X e , Y e ) correspond to the intersection of the nullclines α Y = ( − X )[ − α X + α X ] , (18) X = ( − α Y )[ − β Y + αβ Y ] . (19)Substitution of (18) into (19) gives an equation of the form p ( X ) = p ( X ) is a real polynomial of degree 9 in X . So there are at most 9 equilibria of (17).By inspection, we have ( X e , Y e ) = (
1, 0 ) , ( α ) , ( , α ) for any values of the parameters α , β . Thesesolutions correspond to: ( X e , Y e ) = (
1, 0 ) : all the X -population and none of the Y -population in neighbourhood 1; none of the X -population and all the Y -population in neighbourhood 2. ( X e , Y e ) = ( α ) : none of the X -population and all the Y -population in neighbourhood 1; all the X -population and none of the Y -population in neighbourhood 2. ( X e , Y e ) = ( , α ) : both X , Y -populations evenly split between neighbourhoods 1 and 2.So we can write p ( X ) = X ( − X )( − X ) p ( X ) (20)where p ( X ) ≡ ∑ i = a i X i (21)is a real polynomial of degree 6. It can be shown that p ( X ) = p ( X ) = p ( − X ) , from (3). This gives a = − a , a = a − a , a = − a + a − a and so the odd coefficients of p ( X ) can be written in terms ofthe even coefficients. We find that a = α + β + βα (22) a = − β ( α + ) a = β ( α + α + ) a = − αβ ( α + ) a = αβ ( α + ) a = − α β a = α β .Since both α , β >
0, the signs of a i in p ( X ) alternate. Hence by Descartes’ rule of signs, p ( X ) canhave 0, 2, 4 or 6 positive real roots ( X = X e > ) , depending on parameter values, and no negative real roots.y the symmetry of p ( X ) , any roots X e ∈ [
0, 1 ] . Hence p ( X ) can have 3, 5, 7 or 9 positive real roots, nonegative real roots and any roots must lie in the interval [
0, 1 ] .Note that we could have chosen to eliminate X from (18) and (19) to give a ninth order polynomial q ( Y ) in Y . Similar considerations would then apply to the roots of q ( Y ) with any roots Y e ∈ [ α ] .We show examples of three qualitatively different cases in figure 2 for ( α , β ) = (
9, 16 ) , (
9, 40 ) , (
9, 80 ) . Onthe left hand side, we show the nullclines (18), (19) in ( X , Y ) -space for each case. We colour the basins ofattraction in our plots according to the stable equilibrium which it contains . On the right hand side offigure 2, we plot the corresponding p ( X ) .Figure 2a shows the case when α = β =
16. Here p ( X ) has only three real roots corresponding to ( X e , Y e ) = (
1, 0 ) , ( α ) , ( , α ) . The basin of attraction of ( X e , Y e ) = ( α ) is shown in pink and the basin ofattraction of ( X e , Y e ) = (
1, 0 ) is shown in blue. The equilibrium ( X e , Y e ) = ( , α ) has no basin of attraction.Figure 2b shows that the corresponding p ( X ) has no real zeros.In Figure 2c, we take α = β =
40. Now nullclines (18), (19) have five intersections, corresponding tofive real roots of p ( X ) . Three of these roots correspond to ( X e , Y e ) = (
1, 0 ) , ( α ) , ( , α ) , as before. Sothe two new roots correspond to zeros of p ( X ) . Figure 2d shows that p ( X ) has indeed developed twozeros. Neither of these two new equilibria has a basin of attraction .In Figure 2e, we take α = β =
80. Nullclines (18), (19) have nine intersections, corresponding to ninereal roots of p ( X ) and p ( X ) has six zeros, seen in figure 2f. Two of these six roots correspond tointegrated populations with a basin of attraction, shown in white in figure 2e.In this case, there is no example of p ( X ) having seven roots, corresponding to p ( X ) having fourzeros.From a societal perspective, figure 2 paints a rather gloomy picture. Figures 2a and 2c both imply thatpopulations will be segregated at these values of ( α , β ) , since both figures show only basins of attractionof segregated populations. The central (integrated) equilibrium ( X e , Y e ) = ( , α ) never has a basin ofattraction and so appears unstable. Stable integration seems possible only in figure 2e, corresponding toa small, highly tolerant, minority. As in the single-neighbourhood case [5], tipping points of the systemcorrespond to the boundaries of the basins of attraction.We will now establish exact criteria for the existence and stability of roots of p ( X ) as ( α , β ) vary. (17)Since we always have roots X e = , 1 of p ( X ) , we need only consider the existence of roots of p ( X ) .From the right hand side of figure 2, there will be at least two real roots of p ( X ) when p ( ) < p ( ) = α − αβ + β , then p ( ) < β > β c ≡ α α − β >
0, we must have α >
2. At β = β c , the two nullclines (18), (19) have a cubic tangency. Note theminimum value of β c =
16, which occurs when α =
4. The next step is to observe that if X = + η is a rootof p ( X ) =
0, then by symmetry, so too is X = − η , where η ∈ [ ] . So we have two equations for η : ∑ i = a i ( ± η ) i =
0, (24) In fact, the colour scheme is based on the index of dissimilarity [9] evaluated at the corresponding stable equilibrium state; seesection 5. Note that the central equilibrium ( X e , Y e ) = ( , α ) has changed its character from a saddle in figure 2a to an unstable node infigure 2c. We discuss the nature of these equilibria in section 2.1.2 below a) α = β = : ( X , Y ) phase space (b) α = β = : p ( X ) vs X (c) α = β = : ( X , Y ) phase space (d) α = β = : p ( X ) vs X (e) α = β = : ( X , Y ) phase space (f) α = β = : p ( X ) vs X 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 , Y e ) = ( α ) and the blue regionis the basin of attraction of ( X e , Y e ) = (
1, 0 ) . The white regions in figure 2e correspond to the basin of attraction of twoof the new equilibria in that figure. Stable nodes are denoted by (cid:32) , unstable nodes by (cid:35) and saddle points by ⊗ . Theright hand figures show p ( X ) in each case. here a i , i = A η + B η + C η + D = A = α β , (26) B = αβ ( − α ) , (27) C = β ( α − α + ) , (28) D = α − β + βα . (29)So η has either one or three real values, depending on the sign of the discriminant ∆ ≡ B C − AC − B D − A D + ABCD , because (25) is a cubic in η . Hence p ( X ) will have either two real roots orsix real roots; it can not have four real roots. Hence p ( X ) = D =
0, we have β = β c and (25) has solution η =
0. Hence p ( ) =
0, whichcorresponds exactly to (23).A lengthy calculation shows that ∆ = − α β [ α β + α ( − β − β ) + β ] . (30)Hence ∆ = β = β ± where β ± ≡ − α [ α − α ± √ α ( α − ) ] . (31)So η has three real values, and η has six real values, for β ∈ [ β − , β + ] . At β = β ± , the two nullclines (18),(19) have quadratic tangencies. We must have α ≥ β ± .The existence of equilibria of (17) is summarised in figure 3. Solid lines correspond to β = β c ≡ α /( α − ) ,from (23) and β = β ± ≡ − α [ α − α ± √ α ( α − ) ] , from (31). The apex of the dark shaded region is thepoint P = ( α , β ) = (
6, 36 ) . Since integrated equilibria occur only inside this region, we can deduce that thisversion of the two-room problem requires a very tolerant (large β ), small (large α ) minority to produce anintegrated population (which may or may not be stable).The above analysis also allows us to obtain an exact expansion of p ( X ) as a cubic in ( X − ) . Since η = ±( X − ) , we have from (24) and (25) that p ( X ) = ∑ i = a i X i (32) = A ( X − ) + B ( X − ) + C ( X − ) + D . (33)So p ( X ) = X ( − X )( − X )⎡⎢⎢⎢⎢⎣ α β ( X − ) + αβ ( − α )( X − ) + β ( α − α + )( X − ) + ( α − β + βα )⎤⎥⎥⎥⎥⎦ . (34)Hence we can give exact expressions for all the roots X = X e of p ( X ) =
0. We already have X e = , 1.Now we can solve p ( X e ) = ( X e − ) from (34) using the standard formula for roots of a cubic, takethe square root and obtain X e and then use (18) to obtain the corresponding value of Y e . These unwieldyexpressions, not given here, can then be used to check the numerical results in figure 2. igure 3: Number of equilibria of (17) , corresponding to the roots of p ( X ) . Solid lines correspond to β = β c ≡ α /( α − ) ,from (23) and β = β ± ≡ − α [ α − α ± √ α ( α − ) ] , from (31) . Integrated equilibria occur only inside the dark shadedregion, for β ∈ [ β − , β + ] , when (17) has nine equilibria. The polynomial p ( X ) has five roots in the light shaded regionand only three roots in the white shaded region. In both these cases, only segregated equilibria are possible. The apex P of the dark shaded region is the point ( α , β ) = (
6, 36 ) . (17)It is clear from figure 2 that not all equilbria of (17) are stable, since they have no basin of attraction. In thissection, we determine stability criteria for equilibria of (17). Let us write (17) in the form dX dt = P ( X , Y ) , (35) a dY dt = Q ( X , Y ) .where P ( X , Y ) = ( − X )[ − α X + α X ] − α Y , (36) Q ( X , Y ) = ( − α Y )[ − β Y + αβ Y ] − X .To establish stability criteria, we must calculate the eigenvalues of the Jacobian of (35), given by J ( X , Y ) ≡ ⎛⎝ ∂ P ∂ X ∂ P ∂ Y ∂ Q ∂ X ∂ Q ∂ Y ⎞⎠ = ( −( + α ) + α X ( − X ) − α − −( α + β ) + αβ Y ( − α Y ) ) , (37)evaluated at the various equilibria ( X , Y ) = ( X e , Y e ) .For the equilibrium ( X e , Y e ) = (
1, 0 ) , the eigenvalues of J ( X , Y ) are given by λ ± = [−( + α + β ) ± √ α + ( − β ) ] . (38)It is straightforward to show that both λ ± < α > β >
0. Hence the equilibrium ( X e , Y e ) = (
1, 0 ) ,corresponding to all of the X -population in neighbourhood 1 and all of the Y -population in neighbourhood, is a stable node, shown by a solid circle ( (cid:32) ) in figure 2. Unless the system is modified in some way,this means that there will always be a non-empty set of initial conditions that will lead to this segregatedoutcome.For the equilibrium ( X e , Y e ) = ( α ) , the eigenvalues are also given by (38). Similar considerationsapply to this (stable) segregated outcome.The equilibrium ( X e , Y e ) = ( , α ) has a more subtle behaviour. Its eigenvalues are given by λ ± = [ β − ( α + ) ± √( β + ) + α + α − αβ ] . (39)We can show that λ + > λ − ≷ β ≷ β c where β c ≡ α /( α − ) , from (23). Hence ( X e , Y e ) = ( , α ) is a saddle for β < β c , shown by a crossedcircle ( ⊗ ) in figure 2, and an unstable node for β > β c , shown by an open circle ( (cid:35) ). At β = β c , we have asupercritical pitchfork bifurcation, where the saddle at ( X e , Y e ) = ( , α ) becomes an unstable node andtwo saddles. This explains why no new stable equilibria are created at β = β c . This bifurcation occursprecisely at the boundary between three and five equilibria shown in figure 3.The three equilibria considered so far are zeros of p ( X ) ; they always exist for any values of α , β . Theremaining zeros of p ( X ) come from p ( X ) . They can be found analytically, as explained above. But theexpressions for the equilibria are unwieldy and the ensuing stability calculations are extremely lengthy. Sowe will simply summarise the results.In figure 4a, we set α = X e as a function of β (the values of Y e areomitted, for convenience). Since α <
2, none of the quantities β c , β ± is defined and p ( X ) only has threereal zeros: X e =
0, 1 (both stable nodes, shown in green) and X e ! = (a saddle, shown in black).In figure 4b, we take α =
5. Since 2 < α <
6, we have β c = ≈ β ± undefined. The saddleexisting for β < β c becomes an unstable node (shown in red) and the new solutions for β > β c are saddles.In figure 4c, we set α = β c ≈ β − ≈ β + ≈ β = β c is followed by fold bifurcations at β = β ± . For β ∈ [ β − , β + ] , we see two stable nodes, corresponding tostable integration. These two solution branches have different basins of attraction (see figure 2e).Finally in figure 4d, α =
10 and now the pitchfork bifurcation is at β = β c =
25. We can always find anintegrated population whenever β > β − ≈ ( α , β ) to lie in the dark shaded region of figure 3, corresponding to a small, highlytolerant, minority. For 6 < α <
8, we can only find an integrated solution when β − < β < β + . For α >
8, we canalways find an integrated population whenever β > β − , provided we start with the right initial conditions. In the previous section, the linear tolerance schedules (7) were the same in both neighbourhoods ( γ = β = β ) for both majority X - and minority Y -populations. Let us now consider the case in which the lineartolerance schedules are different. The types of people remain the same, but the two neighbourhoods inducea different tolerance in each population (this may be due to other factors such as urban environment,educational provision, etc). We revert to the original dynamical system (13) and nullclines (14) and (15).We distinguish three different cases:Case II; γ = β ≠ β : the majority population have the same tolerance in both neighbourhoods; theminority have different tolerances in both neighbourhoods.Case III; γ ≠ β = β : the majority population have different tolerances in both neighbourhoods; theminority have the same tolerance in both neighbourhoods.Case IV; γ ≠ β ≠ β : both populations have different tolerances in both neighbourhoods. a) α = : β c , β ± undefined. (b) α = β c = ≈ : β ± undefined. (c) α = : β c ≈ and β − ≈ β + ≈ . (d) α = : β c = and β − ≈ . Figure 4:
Case I: equilibrium values X e as a function of β for (a) α = , (b) α = , (c) α = and (d) α = . Substituting (14) into (15), we obtain a ninth order polynomial ˜ p ( X ) of possible equilibria. We knowthat X e =
0, 1, by inspection of (14), (15). But X e = is no longer a guaranteed equilibrium. Hence we write˜ p ( X ) = X ( − X ) ˜ p ( X ) ,where ˜ p ( X ) ≡ ∑ i = b i X i . (40)The real coefficients b i are given by b = −( α + β ) γ − β α (41) b = ( + α ) β + α ( + γ ) + β α + ( β + β ) γ + α ( β + β ) γ b = − ( β + β ) − [( + α ) β + ( + α ) β ] γ − α ( β + β ) γ − α ( β + β ) γ b = ( + α ) β + ( + α ) β + [( + α ) β + ( + α ) β ] γ + [ α ( + α ) β + α ( + α ) β ] γ + α ( β + β ) γ b = − α ( β + β ) − α [( + α ) β + ( + α ) β ] γ − α [( + α ) β + ( + α ) β ] γ − α ( β + β ) γ b = α ( β + β )[ + α + ( + α ) γ + ( + α ) γ + αγ ] b = − α ( β + β )( + γ + γ + γ ) b = α ( β + β )( + γ ) ote that ˜ p ( X ) evaluated at γ = β = β can be shown to equal ( − X ) p ( X ) , as expected. Equilibria of the governing equations (13), segregated or integrated, correspond to real zeros of ˜ p ( X ) .We know by inspection that X e =
0, 1 are equilibria. But owing to the lack of symmetry between the twoneighbourhoods, we can say very little analytically about any other equilibria, corresponding to real zerosof ˜ p ( X ) in (40).The signs of the coefficients b i of ˜ p ( X ) alternate for allowed values of the parameters. So Descartes’rule of signs tells us that ˜ p ( X ) has 1,3,5 or 7 real roots for X > X <
0. Hence˜ p ( X ) has 3,5,7 or 9 real roots for X > X <
0, as in case I.We can also show that there has to be at least one real root of ˜ p ( X ) for X ∈ (
0, 1 ) , for any allowedparameter values. Simple calculation shows that ˜ p ( ) = −( α + β ) γ − β α < p ( ) = ( α + β ) + β α > p ( X ) is continuous in X , the Intermediate Value Theorem tells us that there always has to be atleast one zero of ˜ p ( X ) between X = X =
1. Since neither ˜ p ( ) nor ˜ p ( ) is identically zero, theroot must lie strictly between X = X = dX dt = K ( X , Y ) , (42) a dY dt = L ( X , Y ) .where K ( X , Y ) = ( − X )[ − αγ X + α ( + γ ) X ] − α Y , (43) L ( X , Y ) = ( − α Y )[ − β Y + α ( β + β ) Y ] − X .The Jacobian of (42) is given by J ( X , Y ) ≡ ⎛⎝ ∂ K ∂ X ∂ K ∂ Y ∂ L ∂ X ∂ L ∂ Y ⎞⎠= ( −( + αγ ) + α ( + γ ) X − α ( + γ ) X − α − −( α + β ) + α ( β + β ) Y − α ( β + β ) Y ) ) . (44)For the equilibrium ( X e , Y e ) = (
1, 0 ) , the eigenvalues of J ( X , Y ) are given by λ ± = [−( + α + β ) ± √ α + ( − β ) ] . (45)Both λ ± < β . Hence ( X e , Y e ) = (
1, 0 ) is always a stable node. When γ = β = β = β , (45) reduces to (38).For the equilibrium ( X e , Y e ) = ( α ) , the eigenvalues of J ( X , Y ) are given by λ ± = [−( + β ) − α ( + γ ) ± √ α ( γ − ) + ( β − ) + α [( γ + ) + β ( − γ )]] . (46)These eigenvalues are both negative for all allowed values of α , β and are independent of β . Hence ( X e , Y e ) = ( α ) is always a stable node. Equation (46) reduce to (38) when γ = β = β = β .Our considerations above show that there is always at least one other equilibrium of (42) in additionto ( X e , Y e ) = (
1, 0 ) , ( α ) . In the case when there is exactly one additional equilibrium, the fact that both ( X e , Y e ) = (
1, 0 ) , ( α ) are always stable nodes means that this third equilibrium must be a saddle.n general, owing to the lack of symmetry, further equilibria must be created by fold bifurcations. Thesehappen when turning points of ˜ p ( X ) (locally quadratic maxima or minima) cross the X axis. We do notexpect pitchfork bifurcations in case II, III or IV, which occur in systems with an inversion or reflectionsymmetry [4], for example case I.Fold bifurcations produce either a stable node (a desirable outcome because X e ≠ Y e ≠
0) and anew saddle or an unstable node and a new saddle. We then face the following possibilities: (i) furtherequilibria (stable or unstable) can be produced by additional new fold bifurcations, (ii) the original saddlecan disappear in a fold bifurcation with the newly created node or (iii) the original fold bifurcation can bereversed.Finally in this section, we show some bifurcation diagrams to illustrate the wealth of possible behaviourthat can occur. For case II, γ = β ≠ β , we take α = β =
40 and vary β , shown in figure 5a. The phasespace diagram for β =
56 is shown in figure 5b. For case III, γ ≠ β = β , we take γ = β = β =
60 andvary α , shown in figure 5c. The phase space diagram for α = γ ≠ β ≠ β , we take γ = β = β =
40 and vary α , as shown in figure 5e. Note the presence of atranscritical bifurcation near α = α = For the case of a single neighbourhood, when parameters α , β are such that only segregation is possible,Schelling [12, 13] proposed that an integrated population could be obtained by limiting numbers of oneor both populations. In [5], we gave exact conditions under which this could occur. We also analysed thestability of the resulting equilibria and showed that the removal of the most intolerant individuals can leadto integration for most values of α , β . For some parameter values, it is possible to create up to seven newequilibria. But there are some values of α , β where limitation of the population can not produce integration.In this section, we consider how limiting numbers might affect the population mixture for two neighbour-hoods in case I (cases II, III and IV can be treated similarly). The picture is considerably more complicatedthan the one neighbourhood case. There are at least two ways to limit the population when there is morethan one neighbourhood. We can restrict the overall number of one population. So for example, we couldtake X + X = u for u ∈ (
0, 1 ) . This is equivalent to the solution proposed by Schelling [12, 13] in the case ofone neighbourhood. Note that this is not the same as a simple rescaling the X -population, since the leasttolerant member of the X -population can now abide a YX ratio of ( − u ) ≠ X = u <
1, but keep the overall population unchanged. Consequentlyin this case X ∈ [ − u , 1 ] , since X + X = X -population (and hence the X -population), as illustrated in figure 6.The Y -population is not restricted. Integration will correspond to intersections of the line X = u with thecubic nullcline (19) given by X = ( − α Y )[ − β Y + αβ Y ] . In figure 6, we show the case β ∈ [ α , 8 α ] ,when nullcline (19) has a middle branch, which lies completely within the feasible ( X , Y ) phase plane.In the absence of any population restriction, there will be three integrated equilibria, given by ( X , Y ) =( X e , Y e ) = ( X a , Y a ) , ( X c , Y c ) , ( X b , Y b ) .The turning points ( X ± , Y ± ) of nullcline (19) are important. If the X -population is restricted at X = u ,we exclude the most intolerant people. But for u ∈ [ X + , 1 ) , we do not gain any extra equilibrium. Instead thesegregated equilbrium at ( X , Y ) = (
0, 1 ) becomes the (slightly) integrated equilbrium ( X , Y ) = ( u , Y ul ) .Figure 6 shows the case when the upper unrestricted equilibrium Y a > Y + with u ∈ [ X a , X + ] . Thereare three new equilbria, denoted by ( X , Y ) = ( u , Y ul ) , ( u , Y um ) , ( u , Y uu ) on the lower, middle and upperbranches, respectively, where Y = Y u ( l , m , u ) are the real roots of the cubic equation u = ( − α Y )[ − β Y + αβ Y ] (47) a) Case II: γ = β ≠ β ; α = β = . (b) Case II with α = β = β = . (c) Case III: γ ≠ β = β ; γ = β = β = . (d) Case III with γ = β = β = α = . (e) Case IV: γ ≠ β ≠ β ; γ = β = β = . (f) Case IV with γ = β = β = α = . Figure 5:
Bifurcation diagrams and example phase portraits for case II, III and IV. Stable equilibria are denoted by (cid:32) , unstablenodes by (cid:35) and saddle points by ⊗ . when the discriminant of (47) is positive. Note that if Y a < Y + , we can only get two new equilibria if u ∈ [ X a , X + ] (not shown).So to proceed we must first find out when the turning points ( X ± , Y ± ) of nullcline (19) exist. Thensince the case Y a = Y + (and hence by symmetry Y b = Y − ) separates different types of behaviour, we must In cases II, III and IV, there will be two different intersections, due to the lack of symmetry. igure 6:
Limiting the X population: X = u: β ∈ [ α , 8 α ] , Y a > Y + . investigate when the two nullclines (18), (19) intersect there.Turning points ( X ± , Y ± ) of nullcline (19) exist when it has a vertical tangent. From (19), we find dY dX = [−( α + β ) + αβ Y − α β Y ] . (48)It is straightforward to show that, when β > α , the nullcline (19) has vertical tangents at ( X ± , Y ± ) = ( ± αβ √ β ( β − α ) , 12 α ± αβ √ β ( β − α )) . (49)When β ∈ [ α , 8 α ] , these vertical tangents lie within the feasible ( X , Y ) -plane. When β = α , the twonullclines (18), (19), have a cubic tangency at the central equilibrium ( X , Y ) = ( X c , Y c ) = ( X + , Y + ) =( X − , Y − ) = ( , α ) .Let us now investigate ways in which equilibria can occur on the middle branch of nullcline (19), when β ∈ [ α , 8 α ] . Our aim is to find a curve Γ u ( α , β ) = X ± ≡ ± αβ √ β ( β − α ) = u , (50) α Y ± ≡ ± β √ β ( β − α ) = ( − u )( − α u + α u ) . (51)Equation (50) is the statement that the lines X = u and X = − u intersect, due to symmetry, the verticaltangents of nullcline (19). Equation (51) is the statement that points ( u , Y ± ) lie on nullcline (19).Substituting (50) into (51), it can be shown that Γ u ( α , β ) ≡ β − αβ + α ( α + ) β + α . (52) igure 7: Limiting the X -population. Γ u ( α , β ) = (equation (52) , shown by black dashed line); β = β c (equation (23) , shown inred); β = β ± (equation (31) , shown in green), together with the lines α = α = and β = α , β = α . The curve Γ u ( α , β ) = β = β c and β = β − , as α → ∞ . Also shown in the same figure are β = β c (equation (23), shown in red); β = β ± (equation (31), shown in green), together with the lines α = α = β = α , β = α (shown in blue). Wecan see that the ( α , β ) parameter space is then divided up into 19 regions (some of which are extremelysmall). In each of these regions, the effect of restricting the X -population is slightly different. We shalldiscuss below the behaviour in two of these regions.Let us now consider the case when we limit the Y -population (and hence the Y -population), but notthe X -population. New integrated equilibria will correspond to intersections of the line Y = v with thecubic nullcline (18), given by solutions Y = Y v of α Y = ( − X )[ − α X + α X ] (not shown). In a similarmanner to above, it can be shown that turning points ( X ± , Y ± ) are given by ( X ± , Y ± ) = ( ± α √ α ( α − ) , 12 α ± α √ α ( α − ) ) , (53)which exist whenever α >
2. When α =
2, the two nullclines (18), (19), have a cubic tangency at the centralequilibrium ( X , Y ) = ( X c , Y c ) = ( X + , Y + ) = ( X − , Y − ) = ( , α ) . It can be shown that the middle branch ofnullcline (19) exists wholly with the permitted phase plane region when α ∈ [
2, 8 ] .Our aim is to find a curve Γ v ( α , β ) = ( α , β ) plane that separates regions where three integratedequilibria are possible from regions where two integrated equilibria are possible. Points on this curve mustsatisfy: Y ± ≡ ± α √ α ( α − ) = v , (54) X ± ≡ ± α √ α ( α − ) = ( − α v )( − β v + αβ v ) . (55)Equation (54) is the statement that the line Y = v intersects both horizontal tangents of nullcline (18).Equation (55) is the statement that points ( X ± , v ) lie on nullcline (18). We retain the same notation for these turning points. No confusion should arise. t can be shown that Γ v ( α , β ) ≡ β − α ( − α )( α − )( α + ) . (56)We do not show the curve Γ v ( α , β ) = α =
2, 8.The effect of restricting the Y -population can be seen in ( α , β ) parameter space, by replacing the curve Γ u ( α , β ) = Γ v ( α , β ) = X - and Y -populations can be seenby adding the curve Γ v ( α , β ) = Γ u , v ( α , β ) = β . The minimum of Γ u occurs at the point ( α u , β ) ≈ ( ) , where nullcline (15) passes through both the maximum and minimum of nullcline(14). Similarly, the minimum of Γ v occurs at the point ( α v , β ) ≈ ( ) , where nullcline (14) passesthrough both the maximum and minimum of nullcline (15).So far in this section, we have only considered the existence of new equilibria when a population islimited. Let us now consider the stability of these new solutions. When the X -population is restricted to X = u , the dynamics on this line is governed by the second equation in each of (42) and (43): a dY dt = L ( u , Y ) = ( − α Y )[ − β Y + αβ Y ] − u . (57)The equilibrium Y = Y u of (57) is the solution of (47). There are either one or three real values of Y u ,depending on whether the line X = u crosses the nullcline once or three times. Stability is governed by theeigenvalue λ u = −( α + β ) + αβ Y u − α β ( Y u ) .Analytical progress can be made in finding λ u , but in general it has to be evaluated numerically. System(57) can never undergo a Hopf bifurcation to a periodic solution since it is only one-dimensional. Similarconclusions apply when the Y -population is restricted to Y = v . The dynamics on this line are governedby the first equation in each of (42) and (43): dX dt = K ( X , v ) = ( − X )[ − α X + α X ] − α v , (58)and the stability of the equilibrium X = X v is governed by the eigenvalue λ v − ( + α ) + α X v − α ( X v ) ,which can be evaluated numerically.We demonstrate the effects of restricting the X -population in two different areas of ( α , β ) parameterspace shown in figure 7. First we take ( α , β ) = (
6, 30 ) , so we are in the light shaded region of figure 3.When u =
1, shown in figure 8a, we have no restriction on population. We have 5 equilibria: two stablesegregated equilibria with their basins of attraction, plus three unstable integrated equilibria (a centralunstable node, flanked by two saddles). When u = ( X , Y ) = (
1, 0 ) is no longer accessible, being replaced by a (slightly) integrated stable equilibrium.The three unstable integrated equilibria still survive. A further decrease to u = X = u ∈ [ X a , X + ] .When u = X = u = u = ( X , Y ) = ( α ) .In our second case, we take ( α , β ) = (
7, 49 ) , so we are in the dark shaded region of figure 3. Now wehave 9 equilibria, two of which correspond to stable integration. The effect of restricting the X -populationis shown in figure 9, for u =
1, 0.89, 0.75, 0.25, 0.11, 0.09.Hence it can be seen that in the two neighbourhood case, restriction of population does not always leadto new integrated equilibria, and those that are produced may only have small basins of attraction. Ofpossible greater concern is that population restriction can also eliminate integration. a) u = (b) u = (c) u = (d) u = (e) u = (f) u = Figure 8:
Limiting the X population: ( α , β ) = (
6, 30 ) . Stable equilibria are denoted by (cid:32) , unstable nodes by (cid:35) and saddle pointsby ⊗ . In this section, we consider other ways in which the linear tolerance schedule can be modified to produceintegrated populations. We illustrate phenomena that can occur when the tolerance schedules are nonlinear,using the original equations (1) in their unscaled form, without restricting the population. We focus on the a) u = (b) u = (c) u = (d) u = (e) u = (f) u = Figure 9:
Limiting the X population: ( α , β ) = (
7, 49 ) . Stable equilibria are denoted by (cid:32) , unstable nodes by (cid:35) and saddle pointsby ⊗ . case a = b = k =
1, that is ( α , β ) = (
2, 20 ) . This is the simplest case when the tolerance schedules arelinear, corresponding to dynamics in the white region of figure 3, where we have two stable segregatedequilibria and one unstable integrated equilibria.In our first example, let us replace the linear X -population tolerance schedule (2) with an exponentialolerance schedule [5], of the form R X ( X ) = RE X ( X ) ≡ − e − [ e − X − e − ] , (59)where RE X ( ) = RE X ( ) =
0. The Y -population has the linear tolerance schedule R Y ( Y ) ≡ ( − Y ) .The results shown in figure 10a are similar to the linear case.Keeping R X ( X ) = RE X ( X ) , we now replace the linear tolerance schedule of the Y -population withan exponential tolerance schedule of the form R Y ( Y ) = RE Y ( Y ) ≡ − e − [ e − Y − e − ] , (60)where RE Y ( ) = RE Y ( ) =
0. In this case, the central fixed point - a fully integrated population in bothneighbourhoods - is now stable, with a large basin of attraction (figure 10b). However, near the saddlepoints, a small change in initial conditions can lead to a stable segregated population.Next, consider polynomial tolerance schedules [5], of the form R X ( X ) = RQ pX ( X ) ≡ a ( − X ) p (61) R Y ( Y ) = RQ pY ( Y ) ≡ b ( − kY ) p , (62)where p ∈ Z + and a = b = k =
1, that is ( α , β ) = (
2, 20 ) , as before. When p =
1, we have the lineartolerance schedule case. When p =
2, the corresponding nullcline is a straight line in ( X , Y ) phase space.In figure 10c, we set R X = RQ X , given by (61) and R Y ( Y ) = ( − Y ) , that is, p = X and p = Y .The results are similar to both the linear case and to figure 10a. However when we set we set R X = RQ X ,and R Y = RQ Y , we have an open set of equilibria, given by the line α Y = − X , shown in figure 10d. Anydesired population mixture can be obtained simply by the correct choice of initial conditions. However it isclear that this outcome is not robust - any small change in p will lead to a different outcome.Finally, in figure 10e, we set R X = RQ X , and R Y ( Y ) = ( − Y ) . The results are similar to both thelinear case and to figures 10a and 10c. However when we set we set R X = RQ X , and R Y = RQ Y , thecentral fixed point is stable and its basin of attraction covers the whole of phase space, shown in figure 10f. RQ X and RQ Y represent globally less tolerant populations than the equivalent linear cases. We caninterpret the stable fixed point in figure 10f phenomenon as follows: low tolerance constraints mean thatboth types are trying to leave both neighbourhoods simultaneously. The stable equilibrium thereforerepresents a best case scenario, rather than a state in which all tolerance demands are satisfied.So overall we can conclude that modification of the linear tolerance schedule of only one populationmay not be enough to induce integration. Instead, both populations may need to modify their toleranceschedules to become integrated. Schelling [14] states that “. . . [in the BNM] . . . an important phenomenon can be that a too-tolerant majoritycan overwhelm a minority and bring about segregation". We have that β b = a = α k .So β = bk α . Hence lines through the origin of ( α , β ) parameter space correspond to increasing a , theupper tolerance limit of the majority X -population. We show two of these lines in Figure 7, for bk =
2, 8corresponding to the case when middle branch of nullcline (19) both exists and lies wholly within theallowed phase space. As we can see, asymptotic to that range is the area of existence of stable integration.Outside that range, even if a is very big, we can not get stable integration, thus quantifying Schelling’s [14]statement. a) R X = RE X ; R Y = ( − Y ) . (b) R X = RE X ; R Y = RE Y . (c) R X = RQ X ; R Y = ( − Y ) . (d) R X = RQ X ; R Y = RQ Y . (e) R X = RQ X ; R Y = ( − Y ) . (f) R X = RQ X ; R Y = RQ Y . Figure 10:
Nullclines, fixed points and basins of attraction for some candidate nonlinear tolerance schedules. Stable equilibria aredenoted by (cid:32) , unstable nodes by (cid:35) and saddle points by ⊗ . So far, we have only considered equilibria of our governing Schelling dynamical system (10). Do theseequations have periodic solutions? They have been observed in discrete time “two rooms" models ofsegregation. But these oscillations appear to be neutrally stable and consist of population swings in bothrooms. Periodic solutions have also been observed [3, 7] in Lotka-Volterra predator-prey models in twohabitats (or patches). But there the dynamics is substantially different.n our case, we have not found any Hopf bifurcations in our calculations and extensive numericalsimulations have not produced any limit cycles. So if they exist, they are most likely unstable (or have avery small basin of attraction). Since (10) is a planar system, Dulac’s criterion [8] could be used to showthat (10) does not have limit cycles. But up to now, we have been unable to find the correct Dulac function.So the existence and stability of periodic solutions to (10) must remain an open question.We can use our results to consider the effects of variation in parameters α and β . In particular weare interested in what might happen as the minority Y -population grows. Provided the variation is slowenough to be considered quasi-static, we can simply move around parameter space. A key parameter is α ≡ ak . If we fix a , the maximum tolerance of the X -population, and then decrease k , the Y -populationgrows as α decreases. Then for fixed combined tolerance parameter β ≡ ab >
36 in figure 3, we see thatintegration is only a transient phase as we enter and then leave the dark shaded region. So as k decreases,we need to ensure that α stays fixed, and that can only happen if a increases . In other words, when aminority grows, stable integration is only possible if the majority population increases its own toleranceas well. This runs counter to the populist idea that a growing minority should integrate more into themajority to be accepted.In [5], we considered the case where the two populations could live either in one neighbourhood orremove themselves to a place “where colour does not matter" [13]. Suppose now that this place changes insuch a way that colour does matter. This could happen for example by the creation or removal of bordersor as the result of a change in government. If there is an integrated population in the single neighbourhood,will it remain integrated after the change to two neighbourhoods?If we overlay part of figure 3 with part of figure 2 of [5], we obtain figure 11. We can now make thefollowing observation: in the light shaded region of figure 11 with apex P , a single neighbourhood canhave a stable, integrated population. But, for the two neighbourhood case, a stable integrated population ispossible only in the dark shaded region with apex P .We then arrive at the remarkable conclusion that, if the minority population is such that its values of α , β lie in the light shaded region, a re-organisation of neighbourhoods can lead to the loss of the stableintegrated population, without any change in the numbers or attitudes of either population. Put anotherway, some types of minority population in two neighbourhoods can only achieve integration by creatingone neighbourhood and a place where type does not matter, or after a perturbation of the system (tipping)into the basin of attraction of an integrated equilibrium.The two curves shown in figure 11 appear similar. In fact, a simple mapping takes one curveinto the other. Apex P has coordinates ( α , β ) = (
6, 36 ) , whereas at apex P , ( α , β ) = (
3, 9 ) . If we set ( α , β ) − neighbourhood = ( α , 4 β ) − neighbourhood in (30), then we recover equation (9) of [5]. These equations givecurves in parameter space where the discriminant of cubic is zero. But it can be shown that the cubicequations from which they originate are completely different and that the governing dynamical systemscannot be mapped to one another.We can extend the two-neighbourhood problem, by introducing an option to be in neither neighbourhood(figure 12). Let X and Y denote the X - and Y -populations present in neither neighbourhood. Thesereservoirs are segregated. Thus X = − X − X and Y = α − Y − Y . Neighbourhoods 1 and 2 are occupiedupon demand as follows. People enter a neighbourhood when the tolerance in that neighbourhood meansthat they would stay (and of course people leave a neighbourhood when the tolerance there means theyshould leave). To move between neighbourhoods, population members must go through either X or Y (inthis version of the problem, there is no direct movement between neighbourhoods 1 and 2).If f ( X i , Y i ) = X i ( X i R X i ( X i ) − Y i ) , i =
1, 2 and g ( X i , Y i ) = Y i ( Y i R Y i ( Y i ) − Y i ) , i =
1, 2, then dX dt = ⎧⎪⎪⎨⎪⎪⎩ f ( X , Y ) f ( X , Y ) ≤ X > ( − f ) f ( X , Y ) > X = dY dt = ⎧⎪⎪⎨⎪⎪⎩ g ( X , Y ) g ( X , Y ) ≤ Y > ( − g ) g ( X , Y ) > Y =
0, (64)which is equivalent to the single-neighbourhood case with limiting numbers and u = X + X , v = Y + Y . igure 11: Comparison of integrated population parameters for two-neighbourhoods with the single-neighbourhood problem [5].Curves with apex P = (
3, 9 ) are β = β ± from [5]. Curves with apex P = (
6, 36 ) are β = β ± from (31) above. neighbourhood ( X , Y ) neighbourhood ( X , Y ) X Y − ˙ X ˙ X − ˙ Y ˙ Y − ˙ X ˙ X − ˙ Y ˙ Y Figure 12:
Dynamics of the two neighbourhoods problem with reservoirs of population. All arrows represent population flowwhen the corresponding quantities are positive. Dotted arrows have the additional constraint that X > (top part ofschematic) or Y > (bottom part of schematic). We summarise the dynamics in table 1.Figure 13 shows the results of simulating this system. For each ( α , β ) , we simulate 20 randomly selectedinitial conditions, and compute the index of dissimilarity D at equilibrium. The conventional definition ofdissimilarity [9] is D ∶= [∣ X − α Y ∣ + ∣ X − α Y ∣] ∈ [
0, 1 ] , and corresponds to the proportion of the minoritypopulation that would have to relocate in order to yield a uniform distribution of both types across allneighbourhoods. One typical criticism of the index of dissimilarity is its sensitivity to neighbourhoodboundaries. Since boundaries are an inherent property of any BNM, this index is a natural tool to employin our study.Our colour scale for basins of attraction in cases I-IV uses the modified measure ¯ D ∶= X − α Y ∈ [−
1, 1 ] applied to the corresponding stable equilibrium, in order to distinguish between X - and Y -dominance. Awhite basin of attraction denotes an even distribution of types in both neighbourhoods. In figure 13, wereturn to the conventional D ∈ [
0, 1 ] as this yields a single measure for the whole system, rather than just > X = Y > Y = f ≤ f f g ≤ g g f > f g > g f ≤ f f g ≤ g g f > f g > g Table 1:
Summary of dynamics for two neighbourhoods with population reservoirs, given in (63) and (64) . First two columns: dX dt . Second two columns: dY dt . one neighbourhoods.We repeat this procedure for systems with two neighbourhoods without conservation of population,namely two decoupled, single-neighbourhood models, as in [5]. In the former case, the constraint imposedby conservation typically results in stable integrated states. In the latter case, the colour scale clearly showsthe relative area of basin of attraction of stable segregated states ( D =
We have considered Schelling’s BNM for the case of two neighbourhoods, with and without populationreservoirs. In the latter case, we presented the governing dynamical systems in (5) and (6). For the case ofidentical ( X , Y ) linear tolerance schedules (case I), we have carried out an extensive analysis, showing boththe existence and stability of integration. We have shown that such an outcome requires a small minority,in the presence of a highly tolerant majority. Similar results were obtained when the linear tolerances differbetween neighbourhoods (cases II, III, IV).If one or the other population is restricted, by removing the most intolerant individuals, we have shownthat new stable population mixtures can be created. But they may only exist in narrow regions of parameterspace, with small basins of attraction. In some cases, existing stable integrated populations can even bedestroyed by this process. We also considered how different nonlinear tolerance schedules can affect ourresults.Our results shed light on some popular notions of integration. So if a minority grows slowly, existingstable integration will be destroyed unless the majority population becomes more tolerant (and yet theymay feel less tolerant as a result of the increase in the minority). Any increase in tolerance by the minoritywill have little effect.Similarly the transition from one neighbourhood to two neighbourhoods (or vice versa) is not necessarilystraightforward. A well-integrated single neighbourhood can become segregated after such a transition,without any change in the tolerance of either population.Finally, when considering the case of 2 neighbourhoods connected by population reserves, we seethat integrated stable states exist for low values of tolerance parameters as a result of competition forfinite resources. The notion of trade-off between tolerance demands and external constraints, namelyneighbourhood structures and finite population reserves, is crucial to understanding the dynamics ofsegregation. We invite readers from socio-economic disciplines to offer insight into interpretation of ourresults, and to aid the construction of more complex model variants that better describe the flows ofpopulation in real urban areas. a) Two-neighbourhood model with reservoirs. (b) independent single neighbourhood models without conser-vation of population. (c) Example trajectories (two-neighbourhood model). (d)
Example trajectories (single neighbourhood models).
Figure 13:
For each ( α , β ) , we simulated sets of initial conditions for t = to t = , calculated dissimilarity D for each andplotted the mean D in parameter space ( D = indicates complete segregation, D = indicates an even distributionbetween neighbourhoods). Note that two empty neighbourhoods yields D = . Results for two neighbourhoods withreservoirs are shown in (a), and two independent simulations without conservation of total population are shown in (b).Example trajectories from (a) and (b) are given in (c) and (d) respectively, with α = β = X ( ) = X ( ) = Y ( ) = Y ( ) = . References [1] S. M. Clifton, K. Hill, A. J. Karamchandani, E. A. Autry, P. McMahon, and G. Sun. Mathematical modelof gender bias and homophily in professional hierarchies.
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