1 Introduction

Lane Formation by side-stepping

Abstract.

In this paper we study a system of nonlinear partial differential equations, which describes the evolution of two pedestrian groups moving in opposite direction. The pedestrian dynamics are driven by aversion and cohesion, i.e. the tendency to follow individuals from the own group and step aside in the case of contraflow. We start with a 2D lattice based approach, in which the transition rates reflect the described dynamics, and derive the corresponding PDE system by formally passing to the limit in the spatial and temporal discretization. We discuss the existence of special stationary solutions, which correspond to the formation of directional lanes and prove existence of global in time bounded weak solutions. The proof is based on an approximation argument and entropy inequalities. Furthermore we illustrate the behavior of the system with numerical simulations.

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Keywords: diffusion, size exclusion, cross-diffusion, global existence of solutions.

1. Introduction

In the last decades demographics, urbanization and changes in our society resulted in an increased emergence of large pedestrian crowds, for example the commuter traffic in urban underground stations, political demonstrations or the evacuation of large buildings. Understanding the dynamics of these crowds has become a fast growing and important field of research. The first research activities started in the field of transportation research, physics and social sciences, but the ongoing development of mathematical models initiated a lot of research also in the applied mathematics community. Nowadays mathematical tools to analyze and investigate the derived models provide useful new insights into the dynamics of pedestrian crowds.
A variety of different mathematical models has been proposed in the past which can be generally classified into microscopic and macroscopic approaches. In the microscopic framework the dynamics of each individual is modeled taking into account social interactions with all others as well as interactions with the physical surrounding. This approach results in high dimensional and very complex systems of equations. Examples include the social force model by Helbing (cf. [8], [17], [16]), cellular automata (cf. [22], [3], [15], [1]) or stochastic optimal control approaches, cf. [19].
Macroscopic models, where the crowd is treated as a density, can be derived by coarse graining procedures from microscopic equations (see e.g. [6]), leading to nonlinear conservation laws or coupled systems of such (see e.g. [20], [10], [9]). Other approaches heuristically motivating macroscopic models are based upon optimal transportation theory, cf. [26], mean field games (cf. [25], [24], [5]) or optimal control cf. [14]. Piccoli and co-workers (cf. [28] and [11]) proposed a measure based approach capable to describe pedestrian dynamics on both microscopic and macroscopic scale - hence bridging the gap between the two description levels. Recently there has been an increasing interest in kinetic models and their respective hydrodynamic limits in pedestrian dynamics, see for example [27] and [12].
For an extensive review on the mathematical literature concerning crowd dynamics and the closely related field of traffic dynamics we refer to [2].

In this paper we (formally) derive and rigorously analyze a PDE system describing the evolution of two pedestrian groups moving in opposite direction. The individual dynamics are driven by two forces, cohesion and aversion. We show that this minimal dynamics already result in complex macroscopic features, namely the formation of directional lanes. We start with a 2D lattice model, in which the transition rates, i.e. the rate at which a particle jumps from one site to the next, express the tendency of individuals to stay within their own group (i.e. follow individuals moving in the same direction) while stepping aside when individuals from the other group approach. The corresponding mean-field PDE model can be derived by a Taylor expansion (up to second order) and is a nonlinear cross diffusion system with degenerate mobilities.
Similar models have been proposed in the literature, for example in the context of ion transport, cf. [4] or population dynamics, cf. [30]. The coherent difference of our model to these works are additional challenging features, namely a perturbed gradient flow structure as well as an anisotropic degenerate diffusion matrix. Although the system lacks the classical gradient flow structure, we can show that the entropy grows at most linearly in time. The corresponding entropy estimates are a crucial ingredient for deriving the global existence result for bounded weak solutions. The existence proof is based on an implicit time discretization and an -regularization of the time-discrete problem. Note that we follow a different approach than Jüngel in [21], which has the advantage that the method is based on an -regularization only and does not require the additional Bilaplace operator. We define a fixed point operator in and use Schauder’s fixed point theorem to deduce the existence of a solution to the regularized problem. The derived entropy estimates as well as a generalized version of the Aubin-Lions lemma justify the limit in the regularization parameter.

This paper is organized as follows: In Section 2 we present the 2D lattice based model and derive its (formal) mean-field limit via Taylor expansion up to second order. Furthermore we discuss the existence of special stationary solutions in Section 2.3. Section 3 focuses on structural features of the resulting PDE system, such as the corresponding entropy functional and the related dissipation inequality. Furthermore we study the boundedness of the densities, which is an essential prerequisite for the global existence proof outlined in Section 4. Finally we illustrate the behavior of the model with various numerical experiments, which reproduce well known phenomena such as lane formation in Section 5.

2. Mathematical modelling

In this section we present the formal derivation of the proposed PDE model from a microscopic discrete lattice approach. We consider two groups of individuals moving in opposite direction, i.e. one group is moving to the right, the other to the left. The individual dynamics are driven by two basic objectives: first individuals try to stay within or close to their own group, i.e. pedestrians walking in the same direction. Moreover they step aside when being approached by an individual moving in the opposite direction. Based on this minimal interaction rules we derive the corresponding PDE model by Taylor expansion up to second order in the following.

2.1. The microscopic model

Throughout this paper we refer to the groups of individuals moving to the right and left as red and blue individuals respectively. Their dynamics are driven by the objectives described above and correspond to cohesion and aversion. Let us consider a domain , partitioned into an equidistant grid of mesh size . Each grid point , and can be occupied by either a red or a blue individual. The probability to find a red individual at time at location is given by:

with an analogous definition for . We set for and use the abbreviation if the time step is obvious. The dynamics of the individuals are driven by the evolution of the probabilities and . These probabilities depend on the transition rates of individuals. Let denote the transition rate of an individual to move from the discrete point to . We define the transition probabilities for the reds as:

(1)

where , and . The factor corresponds to size exclusion, i.e. an individual cannot jump into the neighboring cell if it is occupied. Note that we assume that individuals only anticipate the dynamics in their direction of movement, i.e. they do not look backwards, which is reasonable when modeling the movement of pedestrians. The second factor in the transition probabilities (1) corresponds to cohesion and aversion. If the probability of moving in the walking direction is increased if the individual in front, i.e. at position , is moving in the same direction (assuming that the cell is not occupied).
Aversion corresponds to sidestepping. If , an individual steps aside if another individual, in (1) a blue particle located at , is approaching. If , there is a preference to make a step to the right hand side with respect to their direction of movement, if , to the left. From the perspective of an observer red individuals prefer to make a jump down if a blue individual is ahead of them in the case . The parameter includes diffusion in the -direction. In the case of no diffusion, i.e. , individuals only step aside when being approached by an individual moving in opposite direction.
The master equation for the red particles then reads as

(2)

The probability to find a red particle at location in space corresponds to the probability that a particle located at jumps forward (first term), particles located above or below, i.e. at jump up or down (second line), minus the probability that a particle located at moves forward or steps aside (third line). The corresponding transition rates for the blue particles are defined accordingly to (1) by:

(3)

The master equation for the blue particles has the same structure as (2.1), i.e.:

(4)

2.2. Derivation of the macroscopic model

In the following we shall consider the formal limit in equations (2.1) and (2.1) to derive the corresponding PDE system. After performing a Taylor expansion up to second order, we obtain

(5)

where

and

denote the fluxes for and respectively. The first order terms correspond to the movement of the reds and blues to the right and left in -direction respectively as well as to the preference of either stepping to the right or left in -direction (depending on the difference ). The second order terms correspond to the cross diffusion terms where the prefactor denotes the lattice size. We consider system (5) on , where is a bounded domain. In our computational examples, see Section 5, the domain corresponds to a corridor, i.e. with . As individuals cannot penetrate the walls, we set no flux boundary conditions on the top and bottom, i.e.

At the entrance and exit of the corridor, i.e. at , we assume periodic boundary conditions. Note that Robin type boundary conditions, where the in- and outfluxes at the entrance and exits are directly proportional to the local density, would be more realistic. The boundary conditions set above correspond to the simplest choice and shall serve as a starting point for the investigation of more realistic and complex models in the near future, cf. [7].
We would like to remark that the lengthy Taylor expansion and formal limiting procedure can be accomplished automatically using computer algebra techniques, even for more general classes of models, see [23].

2.3. Stationary Solutions

In this last part of the modelling section we study the existence of specific stationary solutions, which correspond to the formation of lanes. These segregation phenomena can be observed in crowded streets with pedestrians as well as in experiments. Lane formation is a rather intuitive phenomenon, but a strict mathematical definition is less obvious. In the following we shall distinguish between strict segregation and the case when still some pedestrians might get into the counterflow, leading to the definition:

Definition 1.

Let denote a stationary solution to system (5) for , which is x-independent, i.e. for all and any we have . Considering therefore as a function of only, we call

  • a solution with strong lane formation, if the functions and have a compact support in y-direction with

  • a solution with weak lane formation, if the sufficiently smooth solution satisfies

    and there exists a point , such that

Note that the definition of weak lane formation has to be changed accordingly if individuals have the preference to step to the left instead of right, i.e. . We expect that the side-stepping initiates the formation of directional lanes, an assumption that has also been confirmed by the numerical experiments in Section 5 for specific ranges of parameters. In particular we consider system (5) in the case and , i.e.

(6)

where we set without loss of generality . Note that the second order terms in are dropped out, i.e. the terms including - and -derivatives are neglected. In this case we can prove weak lane formation for and postulate the formation of strong lanes as .

Therefore, we consider system (6) and analyze its equilibrium solutions which are constant in -direction. In this case system (6) reduces to

(7a)
(7b)

Note that we have assumed a preference for stepping to the right in (6), which corresponds to the different sign in the first terms of (7). If , we can rewrite (7) as

(8a)
(8b)

Summation of (8a) and (8b) and subsequent integration gives

(9)

for some constant . Equation (8) allows us to study the behavior of stationary solution curves with respect to the densities and . Figure 1 illustrates these stationary solutions in the case for different values of .

Figure 1. Stationary solution curves for , and

If or then or respectively. Hence solution curves can get arbitrarily close to the - and -axes, but they can only reach them in the case of a trivial solution curve, i.e. consisting only of one stationary point lying on one of the axes. The actual starting and end points of the solution curves as well as the corresponding constants depend the chosen parameters and on the initial masses of the system, i.e. on

In the case of small values of we observe a quick change of the densities and from high to low values and the other way around. For larger values the densities increase or respectively decrease slower along the solution curves.
The following additional solution properties can be deduced from equations (8) and (9).

Lemma 1.

Let denote solutions to system (8) and let be a constant with .

  1. There exists no solution with and .

  2. There exists no solution with .

  3. Any solution is monotone with and .

Proof.

To show (i) we assume to the contrary that there exists a solution with and . Then equation (9) implies that is a positive constant and therefore the same holds true for and individually. This is a contradiction to (8) as is only true if .

To see (ii) we again argue by contradiction. If we immediately deduce from system (8) that .

To prove the monotinicity properties in (iii) we first observe that (8a) and (8b) imply

(10)

for some constant . This allows to exclude the existence of a with , since in this case equations (10) would yield as well as . Therefore has to be monotone and due to symmetry is also monotone with the opposite sign.
To show the stated signs of the derivatives we assume and . Subtracting the equations in (10) then leads to and thus to a contradiction, since for equation (10) as well as the assumptions imply , whereas for the second equation in (10) gives . We therefore obtain the desired monotonicity properties and .  ∎

Lemma 1 indicates the existence of weak lane formation. From (iii) we know that and are monotone functions which are strictly positive. Hence there exists a single point where . Due to the side-stepping tendency the reds will move to the bottom, while the blues move up. In the case of equal masses it is impossible that one density is larger than the other on the whole domain, which implies the formation of weak lanes in the sense of Definition 1 in this case:

Theorem 1.

Let and . Then system (8) has non-trivial stationary states, and any stationary solution constant in the -direction exhibits weak lane formation.

Further properties of solutions to (7) and (8) can be observed for different asymptotic parameter regimes:

  • : We deduce from (9) and (8) that , i.e the smaller , the sharper the separation of and .

  • : In this case for some constant and for some constant which corresponds to lane formation.

Remark 2.

If pedestrians have the preference to step to the left instead of to the right the monotonicity behavior of and is reversed.

3. Basic properties

In this section we discuss basic properties of system (5). In the following we set , and . Then system (5) reads as

(11)

We shall prove global existence of weak solutions of system (11) in Section 4, a result which can be extended to the case and . The proof uses several structural features of system (11), such as the corresponding entropy functional and the boundedness of solutions, which we discuss in this section.

3.1. Entropy functional

A key point in the existence analysis are estimates based on the corresponding entropy functional

(12)

where the potentials and correspond to the motion of the red and blue individuals to the right and left respectively. Note the difference in the prefactor of the entropy term compared to other entropies used in the literature for similar PDE models, cf. [4], [30]. This prefactor results from the anisotropic diffusion as we shall explain in the following.
Introducing the entropy variables and

we can rewrite (11) as follows

(13)

where

We observe from equation (13) that we do not have a gradient flow structure. The additional terms result from the different structure of the second order terms in (11). They are either of the form , or , which correspond to different entropies. This lack of structure results in the different prefactor in (12).
Note that the entropy functional (12) is also not an entropy in the classical sense as we cannot ensure that it is non-increasing. Nevertheless the entropy grows at most linearly in time, which is sufficient for proving existence of global weak solutions.

Lemma 2.

Let be a sufficiently smooth solution to system (11) satisfying

Then there exists a constant such that

(14)

where

for some constant .

Proof.

System (13) enables us to deduce the entropy dissipation relation:

(15)

where we have used integration by parts. For the non-quadratic term in -direction, we use the fact that

and deduce that

(16)

The first term on the right hand side is negative, for the second we derive that

Therefore we obtain

As and the integration is over a bounded domain, there is a positive constant such that

Applying Young’s inequality, we get

and therefore

Altogether we deduce the following estimate from (16):

We use the same arguments for the term and obtain the following entropy dissipation from (15):

For the analysis it will be sufficient to use a reduced version of the entropy inequality, given by

where . Using the definitions of and , applying Young’s inequality to estimate the mixed terms as well as the fact that

we obtain

Since and we get the estimate

(17)

for some constant , which concludes the proof. ∎

3.2. Positivity

We want the global weak solution of system (11) to satisfy for all , if the latter condition is prescribed for the initial data. System (11) can be written in the form

where is the diffusion matrix given by

Note that the diffusion matrix is neither symmetric nor positive definite in general. Hence, we cannot use the maximum principle to prove nonnegativity and boundedness of , and . However, the system allows to use a more direct approach to deduce upper and lower bounds for the variables , and . We therefore consider the entropy density

where

(18)
Lemma 3.

The function is strictly convex and belongs to . Its gradient is invertible and the inverse of the Hessian is uniformly bounded.

Proof.

The invertibility of can be shown directly. Using the definitions of the entropy variables and , we get

Solving these relations for gives

which leads to a quadratic equation in with exactly one positive solution

and therefore

Simple calculations ensure that .
To show the uniform boundedness of the inverse of , we observe that

Since , we can deduce that the inverse of exists and is bounded in . ∎

Hence, Lemma 3 ensures that if there exists a weak solution to (21), the original variables satisfy for almost everywhere. This gives us -bounds, necessary for the global in time existence proof in the following section.

4. Main result

We start this section by stating the notion of weak solutions to system (11).

Definition 2.

A function is called a weak solution to system (11) if it satisfies the formulation