Spatially-driven Instability in Lane-changing Models
R. Eddie Wilsona,∗ , Tianyu Fan (樊天宇)aa Department of Engineering Mathematics, University of Bristol, Bristol, U.K.

RE.Wilson@bristol.ac.uk, tianyu.fan@bristol.ac.uk∗ Corresponding author

Extended abstract submitted for presentation at the 11th Triennial Symposium onTransportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

1st April 2022
Keywords: Turing instability; Macroscopic traffic models; Lane-changing

1

INTRODUCTION

Our scope is macroscopic models of n-lane highways that generalise the kinematic wave equation(Lighthill & Whitham, 1955) in the form
∂ρi ∂qi+= Fi−1 − Fi ,∂t∂x(1)where ρi and qi are respectively the density and flow in lane i = 1, 2, . . . , n, and Fi is the net rate
at which traffic in lane i moves to lane i + 1. We take Fi := Fi (ρi , ρi+1 ) for i = 1, 2, . . . , n − 1, sothat the rate of lane-changing between adjacent lanes is a function of their respective densities,and F0 = Fn = 0 to model the left and right boundaries of the highway.
The simplest case is n = 2, where Munjal & Pipes (1971) chose F1 = k(ρ2 −ρ1 ) generalised toF1 = k(ρ2 − λρ1 ) by Holland & Woods (1997), who demonstrated how this apparently wave-likesystem can display a diffusive effect known as Taylor dispersion, that is well known in the fluid
mechanics literature.
In the mathematical biology literature, there is an effect known as the Turing Instability(Turing, 1952) (or Diffusion Driven Instability), in which reaction-diffusion models can display
spatiotemporal patterns, even when their spatially-independent dynamics (analogous to the RHS
source terms in (1)) are stable. Since (1) can display diffusion-like properties, our goal here is tostudy whether spatiotemporal instabilities can be driven by lane-changing effects.

2LINEAR STABILITY AND THE DISPERSION RELATION

We consider the linear stability of a spatially-independent equilibrium (known as ‘uniform flows’)ρ ∗ = (ρ∗1 , ρ∗2 , . . . , ρ∗n )T , for which the lane-changing terms cancel out, so that Fi (ρ∗i , ρ∗i+1 ) =Fi−1 (ρ∗i−1 , ρ∗i ) for i = 1, 2, . . . , n. Following e.g. Wilson (2008), we examine linear stability bythe substitution ρ = ρ ∗ + ρ̃ρ(x, t), where ρ̃ρ is small. We try the ansatz ρ̃ρ(x, t) = Re(ceikx eλt ),which for non-zero solutions c yields|M − ikQ − λI| = 0,(2)where I is the n × n identity matrix, Q is the diagonal matrix diag(q10 (ρ∗1 ), q20 (ρ∗2 ), . . . , qn0 (ρ∗n ), andM is a tridiagonal matrix with non-zero entries mi,i−1 = (D1 Fi−1 )(ρ∗i−1 , ρ∗i ), mi,i = (D2 Fi−1 )1(ρ∗i−1 , ρ∗i ) − (D1 Fi )(ρ∗i , ρ∗i+1 ), and mi,i+1 = −(D2 Fi )(ρ∗i , ρ∗i+1 ), where (DF ) terms denote partial
derivatives evaluated at the equilibrium. The determinant equation (2) can be used to derive
the dispersion relation, which gives growth rates Re(λ) as a function of the wavenumber k.
Motivated by our search for Turing-like instabilities, we assume that the lane-changing source
terms are ‘ODE stable’ — i.e., that the eigenvalues of M have negative (or zero) real parts —corresponding to k = 0 in (2). The question is then whether there is non-zero k for which
Re(λ(k)) > 0, which would indicate a spatially driven instability.
The simplest case for illustration is n = 2, where there is a single source term F1 . We thenfind−D1 F1 −D2 F1
M=(3)+D1 F1 +D2 F1
which has eigenvalues 0 and D2 F1 −D1 F1 . The 0 eigenvalue has eigenvector (D2 F1 , −D1 F1 )T best
explained by reference to the Munjal and Pipes model, where it becomes (1, 1)T . This null-vector
thus corresponds to translation along the continuous curve of equilibria ρ1 = ρ2 . More broadly,for n = 2 systems, we expect the equilibria to trace out curves in the (ρ1 , ρ2 ) plane. Generally
speaking, we might expect that if the density of one lane were to increase, then for equilibrium,
the density of the other lane should also increase, which implies that D1 F1 and D2 F1 must have
opposite signs. If the equilibria are also ODE stable, it follows that
(4)

(D2 F1 ) < 0 < (D1 F1 ).
Unfortunately, it can be shown by analysing the quadratic(5)

λ2 + (A + Bi)λ + C + Di = 0whereB = k(q10 (ρ∗1 ) + q20 (ρ∗2 ))(6)

D = k(q20 (ρ∗2 )D1 F1 − q10 (ρ∗1 )D2 F1 )(7)

A = D1 F1 − D2 F1 ,C = −k 2 q10 (ρ∗1 )q20 (ρ∗2 ),that arises from (2) that condition (4) implies that there is no spatial instability, see Figure 1.
More complicated analyses for n > 2 suggest that spatial instability is only possible if theequilibrium relationships between lane densities ρi have negative slope — which does not seemrealistic.
01-0.5

0.5-10-1.5-0.5-2-1-2.5-1.5-3-50(a) D1 F1 = 1-2-55D2 F1 = −20(b) D1 F1 = −15D2 F1 = −2

Figure 1: Examples of the dispersion relation. For illustration, we set q10 (ρ∗1 ) = 1 and q20 (ρ∗2 ) = 2for both examples. (a) Stable example (branches take negative values). (b) Unstable example (onebranch is positive).

23CONTRIBUTION-BASED LANE-CHANGING MODELS

We propose a new approach, first introduced by Noble (2019), to macroscopic lane-changing
models which extends beyond the state-of-the-art in two ways:• Similar to Subraveti et al. (2019), it explicitly considers the contributions of flows of vehicles
from lane i to i + 1 and from i + 1 to i, and not just their net outcome, by writing
Fi = fi,i+1 − fi+1,i .
• It explicitly distinguishes between the rate of lane-changing, interpreted at the level of atypical individual vehicle, from the size of the population that the rate applies to.
A natural choice for the contribution functions is fi,j = ρi g(vj − vi )h(ρj ), describing a lanechanging rate from lane i to lane j which depends on the velocity advantage vj − vi of lane j,
and is multiplied by ρi (representing the population that the rate applies to). The function h(ρj )should be decreasing and models how lane-changing into a dense lane becomes increasingly moredifficult due to lack of space: henceforth, the choice h(ρ) = ρmax − ρ suffices for our purposes
where ρmax is the jam density.
In the ordinary way, we might expect the function g to be increasing, to represent an increased
incentive for changing to a lane that is much faster than our current one. However, we argue thatin fact, a large speed difference cuts off lane changing altogether, since gap acceptance makes
it impossible. Note that the domain of function g is [−vmax , vmax ] where vmax is the maximum
velocity. In principle, the g functions could also be chosen differently to model asymmetry in thelane-changing process.

4COMPUTATIONAL EXAMPLE OF INSTABILITY

In the simplest n = 2 case, we note that (D1 F1 + D2 F1 )|ρ1 =ρ2 = 0. This makes D1 F1 and D2 F1have opposite signs, which implies by (4) that the natural equilibrium ρ1 = ρ2 does not havespatially driven instabilities. However, it can be shown, using asymptotic analysis, that for agiven velocity-density relation, some contribution functions give rise to extra equilibria wherethe lane densities are very different and the lanes are almost decoupled. See Figure 2. In thesesituations, we can exhibit spatially driven instability. In the TRISTAN talk we will discuss what
this implies for the solutions of the associated initial value problems.

References
Holland, E. N., & Woods, A. W. 1997. A continuum model for the dispersion of traffic on two-lane roads.
Transportation Research Part B Methodological, 31(6), 473–485.
Lighthill, M. J., & Whitham, G. B. 1955. On kinematic waves II. A theory of traffic flow on long crowdedroads. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences,229(1178), 317–345.
Munjal, P. K., & Pipes, L. A. 1971. Propagation of on-ramp density perturbations on unidirectional
two-and three-lane freeways. Transportation Research, 5(4), 241–255.
Noble, P. 2019. Multi-lane Multi-class Highway Traffic Modelling in Space and Time. MSc thesis, University of Bristol, UK.
Subraveti, H., Knoop, V., & Arem, B. 2019. First order multi-lane traffic flow model — an incentive based
macroscopic model to represent lane change dynamics. Transportmetrica B: Transport Dynamics, 7(1),1758–1779.
Turing, A. M. 1952. The chemical basis of morphogenesis. Bulletin of Mathematical Biology, 237(641),37–72.
Wilson, R. E. 2008. Mechanisms for spatio-temporal pattern formation in highway traffic models.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences,366(1872), 2017–2032.

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04-2

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0-1

-0.5

00.5

X 0.838384
Y 0.00171002

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1-0.5

(a) Contribution function.
0.01

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-0.01

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-0.02

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1(b) Net rate of lane-changing.

1000-0.04-11-0.500.5

1(d) Dispersion relation of point A.

(c) Branches of equilibria.

Figure 2: Example of spatially driven instability. (a) Contribution function that models a cut-off
in lane-changing when the speed difference is too high. Our construction uses linear (red) andcubic (green / blue) splines. (b) Plot of F1 when ρ̂1 + ρ̂2 is fixed, which suggests extra equilibria
exist with ρ̂1 6= ρ̂2 . Here we use non-dimensional densities ρ̂1 := ρ1 /ρmax and ρ̂2 := ρ2 /ρmax .
(c) Plot of equilibria in the (ρ̂1 , ρ̂2 ) plane. (d) Dispersion relation of equilibrium point A (atwhich the equilibrium relationship slopes downwards). One solution branch is positive, indicating
spatially driven instability.

4