Traveling fronts in systems of particles with random

PHYSICAL REVIEW E 82, 041119 共2010兲

Traveling fronts in systems of particles with random velocities Vicenç Méndez, Daniel Campos, and Ignacio Gómez-Portillo Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, Bellaterra, Barcelona 08193, Spain 共Received 28 July 2010; revised manuscript received 1 October 2010; published 21 October 2010兲 We explore a reaction-dispersal mechanism for the generation of wave fronts which consists of a set of particles traveling with random velocities 共chosen from arbitrary distributions兲 which experience an autocatalytic reaction. The differences found between this mechanism and approaches based on the continuous-time random walks, where the particles are assumed to perform discrete jumps from one position to another, are analyzed and discussed. A complete analytical treatment of our velocity model is achieved, which allows us to predict the constant speed of traveling fronts or their time dependence in case they are accelerated. Also, a general criterion to distinguish the situations of fronts with constant speed from those accelerated is provided. From it, we conclude that accelerated fronts should be expected in different situations easily achievable in nature. DOI: 10.1103/PhysRevE.82.041119

PACS number共s兲: 05.40.Fb, 02.50.Ey, 05.45.⫺a

I. INTRODUCTION

Reaction-dispersal models represent an archetypical framework to understand and implement spread phenomena and traveling solutions, in general, e.g., fronts, pulses, or waves. This has found application in a wide variety of fields like in chemical waves and patterns 关1,2兴, biological 关3,4兴 and human 关5兴 invasions, epidemics propagation 关6兴, tumor growth 关7兴, and so on. Classical reaction-dispersal equations are built on the hypothesis that individual particles move according to Fickian diffusion, so one typically refers then to “reaction-diffusion” systems. Such diffusive transport can be implemented by considering that particles either “jump” from one position to another or move with constant velocity, provided in both cases that the characteristic time scale of the transport process is very small compared to the observation time or the other time scales in the system. Thus, the potential effects of long-range dispersal 共plus other microscopic details of the transport pattern兲 are lost within that approach. This restriction can be relaxed by introducing the well-known idea of continuous-time random walks 共CTRWs兲. Within the CTRW framework, particles can jump from one position to another with an arbitrary distribution of jump lengths, these jumps being separated by waiting times also randomly distributed. This generalized mechanism allows one to implement, among others, transport patterns that exhibit anomalous diffusion. These are characterized by the power-law scaling t␣ for the mean-square displacement, with ␣ ⫽ 1 共␣ = 1 for Fickian diffusion兲. Actually, several authors have used this idea in the past years to explore the properties of traveling wave solutions and wave fronts for reaction-dispersal systems under the condition that the dispersal process on its own corresponds to subdiffusion 共␣ ⬍ 1兲 关8–11兴. However, the idea that particles travel through discrete jumps from one position to another is only realistic for some specific situations, like in compartmentalized media 共where particles move effectively just from one compartment to another兲 or patchy environments 关12兴. On the contrary, many systems in nature 共for instance, cell cultures or chemical re1539-3755/2010/82共4兲/041119共9兲

actions mediated by transport in a sparse medium兲 fit much better a velocity-based approach. For those cases, velocity models, where individual particles move continuously with a given velocity instead of making jumps, become extremely useful. The simplest versions of such velocity models assume that the speed of a particle takes a fixed value v0, while only the direction of movement can change in time. For the one-dimensional 共1D兲 case where the particle can choose only between values v0 or −v0 with probability 1/2, and the change from positive velocity to negative 共and vice versa兲 is governed by a Poisson process with characteristic rate ␤, one obtains the so-called hyperbolic reaction-diffusion equation 关1,13–15兴. The generalization to the case where the times between consecutive changes in the direction follow an arbitrary probability distribution 共that is, they are not simply determined by a constant rate ␤兲 was provided by Zumofen and Klafter 关16兴. The composite case where the particles can travel with a fixed speed while at the same time they experience Fickian diffusion 共which holds, for example, in a system of self-propelled particles influenced by thermal noise兲 has been studied in 关17–19兴. Finally, the situation where sojourns with a fixed velocity v0 are separated by waiting periods 共which has special interest in the field of cellular migration兲 has been explored in recent works 关20兴. An important advance within this context has been provided by Zaburdaev et al. 关21兴. They have shown how to extend the idea of the standard CTRW to the case where the individual particles can travel with an arbitrary distribution of velocities instead of making jumps. Their original work and a recently published continuation 关22兴, however, just focused on the properties of the transport process, while the effects resulting from a reaction 共that is, a birth-death兲 mechanism have not been considered yet. The present work tries to fill that gap by exploring the characteristic traveling fronts arising from the random velocity model presented in 关21兴 when a reaction term is considered. As we shall see, the properties of these fronts show some interesting differences if compared with those models based on the idea of “jumps.” For example, we find that accelerated fronts emerge for some velocity distributions which are not heavy tailed. This is in contrast with the CTRW model and all reaction-dispersal

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©2010 The American Physical Society

PHYSICAL REVIEW E 82, 041119 共2010兲

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models based on jumps, where a very slowly decaying distribution function of jump lengths 共typically slower than exponential兲 is necessary to obtain such accelerated solutions 关23,24兴. Also, we show in this work that the asymptotic equivalence found by Zaburdaev et al. between their model and the CTRW breaks down when the reaction process is implemented; this will become evident by proving that the characteristic speed of the fronts found in the two cases is different. This paper is organized as follows. The CTRW framework and its random-velocity counterpart are presented in Sec. II. In Sec. III we derive a criterion for distinguishing the cases where reaction-dispersal fronts with constant speed or accelerated are obtained. Different analytical results for the characteristic speed and the asymptotic behavior of these fronts are provided for both cases 共Secs. III and VI兲. These results will also be compared to numerical simulations for different cases of special interest. Finally, the conclusions of our work will be left for Sec. V.

II. CTRW AND MODEL WITH RANDOM VELOCITIES

Let us consider for simplicity an infinite 1D continuous medium in which a set of noninteracting particles can jump from one position to another according to the jump length probability distribution ⌽共x兲. Before they jump, particles have to wait at their current position for a random time distributed according to ␸共t兲. Then, if we define J j共x , t兲 as the density of particles arriving to the position x at time t, we have

冕 ⬘冕 t

J j共x,t兲 =

⬁

dt

0

dx⬘J j共x − x⬘,t − t⬘兲␸共t⬘兲⌽共x⬘兲

−⬁

+ J共x,0兲␦共t兲,

P j共x,t兲 =

t

dt⬘J j共x,t − t⬘兲␾共t⬘兲,

共2兲

0

where ␾共t兲 = 兰⬁t dt⬘␸共t⬘兲 is the “survival” probability distribution of ␸共t兲; i.e., it gives us the probability that the particle has not jumped yet after a waiting period of duration t standing at its current position. Equations 共1兲 and 共2兲 represent the cornerstone of the CTRW model. By inserting the solution of Eq. 共1兲 into Eq. 共2兲 and differentiating with respect to time one finds

⳵ P j共x,t兲 = ⳵t

冕 ⬘冕 冕 ⬘ t

⬁

dt

0

dx⬘ P j共x − x⬘,t − t⬘兲M共t⬘兲⌽共x⬘兲

−⬁

t

dt P j共x,t − t⬘兲M共t⬘兲,

−

共3兲

0

which is commonly known as the CTRW master equation. The memory kernel M共t兲 is defined through its Laplace transform in time 共denoted by ˆ 兲, ˆ 共s兲 = s␸ˆ 共s兲 , M 1 − ␸ˆ 共s兲

共4兲

where s is the corresponding argument. Now we will introduce a reaction process into this framework. Since we are interested here in the analysis of wave fronts propagating into unstable states, we will consider an autocatalytic process of the type A + 쏗 → 2A, which is one of the simplest mechanisms able to introduce such selfsimilar solutions. Defining r as the characteristic parameter of the reaction process, we have that the density R共x , t兲 of particles appearing from the reaction process can be written as a logistic growth term

共1兲

where the subindex j stands for jumps and the second term in the right-hand side represents the contribution from the initial condition. For the sake of clarity, let us mention that sometimes these initial conditions are introduced, ⬁ dx⬘ P共x alternatively, through a term of the type 兰−⬁ − x⬘ , 0兲␸共t兲⌽共x⬘兲, so defining an initial density of particles P共x , 0兲 共see, for example, Section 3.4 in Ref. 关1兴兲. The difference between this term and the one used in Eq. 共1兲 reflects just the technical question of whether the particles are “put at position x and then the time is reset” at t = 0 关then it is more natural to define an initial density P共x , 0兲兴 or “first the clock is reset at t = 0 and then the particles are put there” 关leading to J共x , 0兲兴. In any case, this is just a minor point which does not change at all the form of the differential master equation 关see Eq. 共3兲 below兴 or the global dynamics of the system, and obviously it does not affect the results for traveling fronts presented here. According to expression 共1兲, the density of particles standing at the position x at the time t will be computed through

冕

R共x,t兲 = rP共x,t兲关1 − P共x,t兲兴,

共5兲

where rP共x , t兲 and −rP共x , t兲2 stand for the birth and mortality terms, respectively 共where the latter term has been introduced to avoid unbounded growth of the particle density兲. In Eq. 共5兲 we have not specified the subindex for P because this relationship holds both for jump and velocity models. We can implement reaction term 共5兲 into the CTRW if we assume that the new particles created with the rate rP共x , t兲 have zero age 共see Section 3.4.2 in Ref. 关1兴 for further details兲. Then, Eqs. 共1兲 and 共2兲 turn into

冕 ⬘冕 t

J j共x,t兲 =

⬁

dx⬘J j共x − x⬘,t − t⬘兲␸共t⬘兲⌽共x⬘兲⍀ j共x

dt

0

−⬁

− x⬘,t,t⬘兲 + J j共x,0兲␦共t兲 + rP共x,t兲,

P j共x,t兲 =

冕

t

0

where

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共6兲

dt⬘J j共x,t − t⬘兲␾共t⬘兲⍀ j共x,t,t⬘兲,

共7兲

PHYSICAL REVIEW E 82, 041119 共2010兲

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discussed above. Following the prescriptions in the original reference, we can define h共v兲 as the probability distribution that the particle performs a single “flight” with velocity v. Then the velocity model 共with reaction兲 will be governed by reaction 共5兲 together with

















冕 ⬘冕 t











Jv共x,t兲 =

0

FIG. 1. Schematic representation of the trajectories of particles in the reaction-dispersal mechanism for the CTRW 共left兲 and its velocity counterpart presented here 共right兲. The balls with the number inside represent individual particles at the moment when they appear in the system, and so their dynamics starts. The x-t trajectories of the particles are represented by solid lines 共particle 0兲, dashed lines 共particle 1兲, and dashed-dotted lines 共particle 2兲. The values of ␶01 , ␶02 , . . . and z01 , z02 , . . . correspond to the waiting 共flight兲 times and the jump 共flight兲 lengths of the particle 0, respectively.

冉 冕

⍀ j共x,t,t⬘兲 ⬅ exp − r

t

t−t⬘

关P j共x,u兲兴2du

冊

is a survival probability term which takes into account the effects of the nonlinear 共mortality兲 term in Eq. 共5兲. Note that alternative forms to Eqs. 共6兲 and 共7兲 have also been proposed in the literature to implement nonlinear kinetic terms 共see 关9兴 and Sections 3.4.1 and 3.4.3 in 关1兴兲. Specifically, careful choice of this equation is necessary 共since it can lead to dramatically different results兲 in the case of heavy-tailed waiting time distributions since the system shows effects such as aging or ergodicity breaking which strongly affect its general dynamics 关8,9,11兴. The trajectory scheme of the particles experiencing a CTRW plus reaction as that described above is depicted in Fig. 1 共left兲. The figure depicts only the linear part of the reaction term 共birth process兲, as mortality processes underlying the nonlinear term can be argued to be irrelevant for the speed of the traveling front 共see Secs. IV and V兲. The solid lines there represent the trajectory of the particle “0,” initially standing at position x0, whose waiting times between jumps are denoted by ␶01 , ␶02 , . . ., while z01 , z02 , . . . represent its corresponding jump lengths. Also, the particle ‘‘0’’ is seen to generate, at a given time t1, a new particle “1” which then starts its independent dynamics and so on. The trajectory followed by each particle is drawn in a different line style 共see the figure caption兲. Figure 1 共right兲, on the contrary, represents the equivalent situation for the case in which the particles do not wait and jump from one position to another but move continuously with a certain random velocity. The velocity of the particle remains constant during a certain “flight time” and then it is given a new value randomly. Note that, in order to facilitate the comparison, the jump lengths and waiting times in the CTRW scheme 共left兲 are chosen equal to the distances covered by the particles and the corresponding “flight times” in the velocity scheme 共right兲. The scheme in the right of Fig. 1 corresponds to the velocity counterpart of the CTRW introduced by Zaburdaev et al. 关21兴, plus the reaction 共birth兲 process implemented as

⬁

dt

dvJv共x − vt⬘,t − t⬘兲␸共t⬘兲h共v兲⍀v共x

−⬁

− vt⬘,t,t⬘兲 + ␦共x兲␦共t兲 + rPv共x,t兲,

冕 ⬘冕 t

Pv共x,t兲 =

⬁

dt

0

共8兲

dvJv共x − vt⬘,t − t⬘兲␾共t⬘兲h共v兲⍀v共x,t,t⬘兲,

−⬁

共9兲 where the subindex v stands for “velocity.” Note that Eqs. 共8兲 and 共9兲 are formally very similar to Eqs. 共6兲 and 共7兲; however, it must be stressed that the specific meaning of the densities is different in both cases. Here, Jv represents the density of particles that finish a single flight 共that is, they change the value of their speed兲 at time t at position x. Likewise, Pv is the density of particles passing through position x at time t. If we compare the two schemes in Fig. 1, it can be seen that the position of the particle 0 coincides at times ␶01 , ␶01 + ␶02 , . . . albeit the trajectory of the particle is different in each case. This implies that the asymptotic transport properties 共without considering reaction兲 of the CTRW model and its velocity counterpart will be equivalent. Actually, Zaburdaev et al. gave the relation 关21兴 ⌽共x兲 =

冕 冕 ⬁

⬁

dt

0

dv␦共x − vt兲␸共t兲h共v兲

共10兲

−⬁

between the jump length distribution in the CTRW and the velocity distribution in their model, which ensures the asymptotic equivalence between both approaches. We will show that this equivalence between jumps and velocities, which is valid when only transport is being considered, breaks down when the reaction is introduced. Actually, this is not an effect of the reaction process on its own. Equation 共10兲 guarantees that the asymptotic statistical properties of both models coincide, but it is also true that the density profiles are different in jump and velocity models, as has been emphasized previously 关22兴. What the reaction process does is to strengthen these differences, which makes the macroscopic dynamics of the system different too, as we shall see. From Fig. 1 it is also easy to understand intuitively how these differences between jump and velocity models emerge. While the behavior of the particle ‘‘0’’ will be the same asymptotically in both cases, the particle ‘‘1’’ is created at position x0 + z01 in the CTRW scheme, but it appears forward in the velocity scheme 共and the same will happen, in general. for particles ‘‘2,’’ ‘‘3,’’ etc.兲. As a result, one must expect that the characteristic speed of traveling solutions obtained from this reaction-dispersal framework will be always faster in the

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velocity scheme than in the CTRW scheme, provided equivalence condition 共10兲 is fulfilled. This is exactly what one finds 共see Sec. IV兲.

具xn典 =

III. CRITERION FOR THE SPEED OF TRAVELING FRONTS

冕

xn⌽共x兲dx,

−⬁

冕

具tn典 =

⬁

冕

⬁

tn␸共t兲dt,

vnh共v兲dv ,

−⬁

respectively. From Eq. 共10兲, it is not difficult to show that the moments of the distribution read

J␧v共x,t兲 =

=

xndx

⬁

dt

dv␦共x − vt兲␸共t兲h共v兲

dt

0

⬁

0

冕 冕 冕 冉冊 冕 ⬁

⬁

−⬁

⬁

␸共t兲 t

x nh

−⬁

x dx = t

⬁

dt␸共t兲tn

0

冕

⬁

znh共z兲dz

−⬁

= 具tn典具vn典.

共11兲

⬁ The moment-generating function ⌽共p兲 = 兰−⬁ ⌽共x兲e pxdx can be expressed as the sum ⬁

⌽共p兲 = 兺

n=0

pn n n 具t 典具v 典. n!

As a result, if this series does not converge then the momentgenerating function does not exist; so, if 具tn典具vn典 grows with n faster than n ! an then the front will accelerate, with a as a certain constant. This requirement is a general condition to have accelerated fronts. In the following, we will show how to compute the front speed when it is constant or accelerated. IV. FRONTS PROPAGATING WITH CONSTANT SPEED

0

具 v n典 =

xn⌽共x兲dx

−⬁

The jump length distribution 共known as dispersal kernel in ecological literature兲 given in Eq. 共10兲 allows us to know whether the front will propagate with constant speed or will accelerate. Given ␸共t兲 and h共v兲, if the resulting ⌽共x兲 has finite moments and its moment-generating function is also finite then the front will propagate with constant speed. This criterion has been developed and discussed in detail for the case of jumps in 关25兴. Note that according to this, if the moment-generating function diverges 共even if the moments are finite兲 then the front will accelerate; some examples of this situation are provided below. Let us define the moments of the PDFs, ⌽共x兲, ␸共t兲, and h共v兲, as ⬁

⬁

−⬁

=

具xn典 =

冕 冕 冕

冕 ⬘冕 t/␧

⬁

dvJ␧v共x − ␧vt⬘,t − ␧t⬘兲␸共t⬘兲h共v兲⍀v

dt

0

To observe traveling fronts in reaction-dispersal systems it is necessary to choose appropriate initial conditions 共conditions with compact support are often required 关26兴兲; for the sake of simplicity we will consider here always Jv共x , 0兲 = ␦共x兲. When the criterion provided in Sec. III predicts front traveling with constant speed, this speed can be computed with the Hamilton-Jacobi formalism 关1兴. This method consists, first of all, in inserting the hyperbolic scaling x → x / ␧ and t → t / ␧ with ␧ Ⰶ 1 into Eqs. 共8兲 and 共9兲 to get

−⬁

P␧v共x,t兲 =

冕 ⬘冕 t/␧

⬁

dt

0

冉

冊

x t − vt⬘, ,t⬘ + ␦共x/␧兲␦共t/␧兲 + rP␧v共x,t兲, ␧ ␧

dvJ␧v共x − ␧vt⬘,t − ␧t⬘兲␾共t⬘兲h共v兲⍀v

−⬁

冉 冊

x t , ,t⬘ , ␧ ␧

共12兲

where we have defined the new fields J␧v共x , t兲 = Jv共x / ␧ , t / ␧兲, P␧v共x , t兲 = Pv共x / ␧ , t / ␧兲, and ⍀v

冉 冊 冉 冕 x t r , ,t⬘ ⬅ exp − ␧ ␧ ␧

t

t−␧t⬘

冊

关P␧v共x,u兲兴2du .

共13兲

Since J␧v共x , t兲 and P␧v共x , t兲 are positive we can make use of the transformations J␧v共x , t兲 = A1e−G which when inserted into Eq. 共12兲 give

冕 ⬘冕 t/␧

A1 = A1

⬁

dt

0

dv exp关vt⬘⳵xG␧ + t⬘⳵tG␧ + O共␧兲兴␸共t⬘兲h共v兲⍀v

−⬁

冕 冕 t/␧

A2 = A1

dt⬘

0

⬁

冉

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共x,t兲/␧

and P␧v共x , t兲 = A2e−G

␧

共x,t兲/␧

,

冊

x t ␧ − vt⬘, ,t⬘ + ␦共x/␧兲␦共t/␧兲eG 共x,t兲/␧ + rA2 , ␧ ␧

dv exp关vt⬘⳵xG␧ + t⬘⳵tG␧ + O共␧兲兴␾共t⬘兲h共v兲⍀v

−⬁

␧

冉 冊

x t , ,t⬘ . ␧ ␧

共14兲

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As ␧ → 0 the system evolves toward the asymptotic regime where the front is completely developed from its initial condition and travels at constant speed. It is expected then that Pv共x , t兲 converges to the indicator function of the set whose boundary is regarded as the front position that connects the unstable state Pv共x , t兲 = 0 to the stable state Pv共x , t兲 = 1. Any initial condition will ensure a front propagating with minimal speed if it has compact support, i.e., has an exponentially bounded tail 关26兴. We suppose that Jv共x , 0兲 and Pv共x , 0兲 are A1 or A2 if x ⬎ 0 and zero otherwise, respectively. The condition G␧共x , t兲 ⱖ 0 determines the front position when ␧ → 0. ␧ Since e−G 共x,t兲/␧ → 0 as ␧ → 0 for G␧共x , t兲 ⬎ 0, we conclude that the condition G共x , t兲 = 0 provides the front position, where G共x , t兲 = lim␧→0 G␧共x , t兲. The effect of the nonlinear kinetics disappears because ⍀v → 1 as ␧ → 0 关27兴. So that, taking the limit ␧ → 0 in Eq. 共14兲 we find

冢冕 1−

冕

⬁

␸共t兲e

−Ht

0

⬁

−

␾共t兲e

−Ht

dt

dt

冕

冕

⬁

h共v兲e

v pt

−r

−⬁

⬁

h共v兲e dv v pt

1

−⬁

0

冣

冕

⬁

h共v兲␸ˆ 共H − v p兲dv + r

冕

⬁

冉 冊

A1 = 0, A2

ˆ 共H − v p兲dv , h共v兲␾

which provides the Hamilton-Jacobi equation. In Eq. 共16兲 we have introduced the notation ␸ˆ 共s兲 = 兰⬁0 e−st␸共t兲dt. Finally, the front speed v f is obtained by solving the set of equations 关1兴, H p

and

dH H = , dp p

e−Ht+p

2 2t2/4 v0

dt,

共19兲

and the front speed can be computed by solving numerically the set of equations 1 = e−yv f /v0ey

2/4

+a

冕

1

e−yv f ␰/v0e␰

2 y 2/4

dv f = 0 共20兲 dy

d␰,

0

for v f / v0 共y = pv0␶ , a = r␶兲. On the other hand, it is interesting to compare this result with the equivalent CTRW model. According to Eq. 共10兲, the CTRW counterpart of the example considered here will correspond to a PDF of jump distances ⌽共x兲 2 −1 −x2/␶2v0 . Since the waiting time distribution is = ␲−1/2v−1 0 ␶ e ␸共t兲 = ␦共t − ␶兲 the Hamilton-Jacobi equation 关Eq. 共18兲兴 reads

vf 1 = min v0 2 z⬎0

2 2␶2/4 v0

+r

e H␶ − 1 . H

共21兲

1=

e−H␶ +r 1 − 共v0 p␶兲2

e−yv f /v0 +a 1 − y2

共22兲

,

冕

␶

0

e−Ht dt, 1 − 共v0 pt兲2

共23兲

冕

1

0

e−yv f ␰/v0 d␰, 1 − y 2␰ 2

dv f = 0. dy

共24兲

For the CTRW model the Hamilton-Jacobi equation reads e H␶ =

共18兲

e H␶ − 1 1 , + r H 1 − 共v0 p␶兲2

共25兲

and the front speed is then given by vf v0

Examples

共i兲 Consider first that particles perform flights of duration ␶ but their velocities are distributed according to a Gaussian 2 −v2/v0 and ␸共t兲 = ␦共t − ␶兲. As the PDF; hence, h共v兲 = ␲−1/2v−1 0 e moment-generating function of a Gaussian exists, the front will travel with constant speed. In this case the HamiltonJacobi equation 关Eq. 共16兲兴 turns into

冑冋冉 冊 册 a a ln e 1 − + z z z

and the speed is given by solving numerically the set of equations, 1=

and so it means that the front speed will be computed from the solution of Eqs. 共17兲 and 共18兲. Next, we will deal with some specific cases in order to compare these theoretical predictions with numerical simulations.

z

where z = H␶. 共ii兲 Let us consider now that particles perform again flights of duration ␶ but their velocities are now distributed according to the exponential PDF h共v兲 = 共2v0兲−1e−兩v兩/v0. The dispersal kernel reads ⌽共x兲 = 共2␶v0兲−1 exp共−兩x兩 / ␶v0兲, its moment-generating function exists, the Hamilton-Jacobi equation is, from Eq. 共16兲,

共17兲

together with Eq. 共16兲. For the CTRW model, the Hamilton-Jacobi equation led to 关1,12兴 ˆ 共H兲, 1 = ␸ˆ 共H兲⌽共p兲 + r␾

␶

0

共15兲

共16兲

vf =

冕

+r

From Eqs. 共17兲 and 共21兲 we get

−⬁

−⬁

2 2␶2/4 v0

e H␶ = e p

where p = ⳵xG and H = −⳵tG are the momentum and Hamiltonian, respectively. The system of algebraic equations for A1 and A2 has a nontrivial solution if the determinant of the matrix in Eq. 共15兲 is zero, i.e., 1=

1 = e−H␶+p

= min z⬎0

z

冑 冋冉 冊 册 1 − ez 1 −

a z

+

a

−1

.

z

The comparison of the theoretical expressions for the speed above with numerical results is provided in Fig. 2. In order to check the validity of all the results obtained throughout the paper, we performed a double numerical analysis: 共i兲 we solved numerically the original integral equations 关i.e., Eqs. 共8兲 and 共9兲 for the velocity model兴 and 共ii兲 we performed Monte Carlo simulations that reproduce the processes de-

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MÉNDEZ, CAMPOS, AND GÓMEZ-PORTILLO

3

Velocity model CTRW model

h共v兲 =

1 −兩v兩/v 0 e 2v0

and

␸共t兲 = ␶−1e−t/␶ .

共26兲

From Eq. 共10兲 the dispersal kernel is

冉冑 冊

v/v0

h(v) exponential 2 h(v) Gaussian 1

0 0.0

1 K0 2 v 0␶

⌽共x兲 =

0.5

1.0

1.5

2.0

a FIG. 2. Comparison for the front speed as a function of a obtained from the velocity model and the CTRW 共see Legend兲 for ␸共t兲 = ␦共t − ␶兲 with Gaussian and exponential speed distribution h共v兲. The jump kernel corresponding to the CTRW approach is that given by Eq. 共10兲. For simplicity, arbitrary values ␶ = 1, v0 = 1 are chosen. The corresponding numerical results obtained from Monte Carlo simulations are denoted by circles 共velocity model兲 and diamonds 共CTRW兲.

兩x兩 , v 0␶

共27兲

where K0 is the modified Bessel function of order 0. Making use of the asymptotic properties of the Bessel function for large x, the kernel can be expressed as ⌽共x兲 ⬃ 兩x兩−1/4 exp共−2冑兩x兩 / v0␶兲 that has finite moments but diverging moment-generating function. To obtain the speed of the front we linearize Eqs. 共8兲 and 共9兲 around the unstable state because the front speed is determined by the behavior of the front at the leading edge due to the fact that the reaction is monostable 关1,26兴. In consequence, the effects of the nonlinear part of the growth term vanish, as happened in the Hamilton-Jacobi formalism in Sec. IV 共we typically use the term pulled front for those cases when this happens兲. Transforming on space and time to the Fourier-Laplace space 共k , s兲 one gets Jv共k,s兲 = Jv共k,s兲

冕

⬁

␸ˆ 共s + ikv兲h共v兲dv + 1 + rPv共k,s兲,

−⬁

picted in Fig. 1. In all the cases we observed that the agreement between numerical integration and Monte Carlo simulations was excellent; basically the only differences we found were in the length and the shape of the transitory regime before the wave fronts are formed. So, in the present paper we just show for simplicity the results obtained from Monte Carlo simulations. The plot in Fig. 2 shows that the agreement between analytical predictions and simulations is excellent for both cases studied here. By comparing the results from the velocity model with those from the CTRW, it must be noted that the speed of fronts obtained from the latter is always smaller, as expected. Also, the differences found between the two models confirm that, even if condition 共10兲 is fulfilled, the equivalence between the jump and velocity approaches breaks down as a result of the interplay between the reaction mechanism and the differences in the density profiles 共according to our discussion in Sec. II兲. As can be seen from the plots, the differences between the models become more important as the reaction parameter a grows so the effect of the reaction process on the system dynamics increases.

Pv共k,s兲 = Jv共k,s兲

冕

⬁

␾ˆ 共s + ikv兲h共v兲dv .

共28兲

−⬁

By combining the two parts in Eq. 共28兲 one finds

Pv共k,s兲 = 1−

冕

冕

⬁

␾ˆ 共s + ikv兲h共v兲dv

−⬁

⬁

␸ˆ 共s + ikv兲h共v兲dv − r

冕

⬁

.

␾ˆ 共s + ikv兲h共v兲dv

−⬁

−⬁

共29兲 As we are interested in the asymptotic behavior of the system we consider s → 0 in Eq. 共29兲. Then, we can expand terms as follows:

冕

⬁

−⬁

␸ˆ 共s + ikv兲h共v兲dv ⯝ ⌽共k兲 + M共k兲s + ¯ ,

with V. ACCELERATED FRONTS

M共k兲 = −

In this section we want to compute the dependence of the front speed with time when there is acceleration. As explained above this occurs when the dispersal kernel ⌽共x兲 in Eq. 共10兲 decays asymptotically slower than an exponential function, i.e., its moment-generating function diverges. We center our attention to the situation where both h共v兲 and ␸共t兲 have finite moments and, as a consequence, ⌽共x兲 also has finite moments 关though ⌽共p兲 diverges兴. Consider for example the case where

冕

⬁

h共v兲t
␸共ikv兲dv ,

−⬁

where 共ˆikv兲 means Laplace transform where the argument is replaced by ikv. Analogously,

冕

⬁

−⬁

with

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␾ˆ 共s + ikv兲h共v兲dv ⯝ N1共k兲 + N2共k兲s + ¯ ,

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TRAVELING FRONTS IN SYSTEMS OF PARTICLES WITH…

N1共k兲 =

冕

⬁

−⬁

N2共k兲 = −

ˆ 共ikv兲dv h共v兲␾

冕

⬁

A共k兲 =

and

− M共k兲N1共k兲 − N2共k兲关⌽共k兲 − 1兴 , 关M共k兲 + rN2共k兲兴2

h共v兲t
␾共ikv兲dv .

−⬁

B共k兲 = −

Introducing these expansions into Eq. 共29兲 and inverting by Laplace we obtain, as t → ⬁, Pv共k,t兲 ⯝ A共k兲exp关B共k兲t兴,

共30兲

再

冋

⬁

exp关B共k兲t兴 exp关B共0兲t兴 exp关B共k兲t兴 kn dn = ⌽共k兲 A共0兲 +兺 n A共k兲 ⌽共k兲 ⌽共0兲 ⌽共k兲 n=1 n! dk

再

Inverting by Fourier, it turns into

冋

⬁

Pv共x,t兲 ⬃ ⌽共x兲exp关B共0兲t兴 1 +

共31兲

As ⌽共x兲 has finite moments we expand Eq. 共30兲 in Maclaurin’s series for k, following 关25兴

with

Pv共k,t兲 ⯝ A共k兲⌽共k兲

⌽共k兲 − 1 + rN1共k兲 . M共k兲 + rN2共k兲

1 dn exp关− B共0兲t兴 exp关B共k兲t兴 兺 n A共k兲 A共0兲 ⌽共k兲 n=1 n! dk

册

册

k=0

冎 冎

.

1 dn n ⌽共x兲 . k=0 ⌽共x兲 dx

v f ⬃ t.

If for all n = 1 , 2 , . . . the condition 1 dn n ⌽共x兲 = 0 兩x兩→⬁ ⌽共x兲 dx

共32兲

lim

Second, if we consider h共v兲 =

holds; then,

1

e v 0 冑␲

共33兲

the PDF of jumps is

To obtain the front speed we fix Pv共x , t兲 to a constant value of the front tail. The resulting equation Pv共x , t兲 = const relates the front position with time. Differentiating we find

⌽共x兲 =

Pv共x,t兲 ⬃ ⌽共x兲exp关B共0兲t兴

as 兩x兩 → ⬁.

冉 冊 冉 冊

⳵ Pv ⳵ Pv dx + 0= ⳵ x t dt ⳵t

, x

冕

共34兲

and

冑 冕

1

v 0␶ ␲

⬁

␸共t兲 = ␶−1e−t/␶ ,

t−1e−t/␶e−x

冑

Pv共x,t兲 ⬃ x−1/4e−␣ xert/共1+r␶兲 , which can be used to get, from Eq. 共34兲,

⬁

冋

f共␭, ␩兲e␭S共␩兲d␩ ⯝ −

0

2␲ ␭S⬙共␩0兲

册

1/2

f共␭, ␩0兲e␭S共␩0兲

as ␭ → ⬁

共38兲

tends to 0 for ␩ → 0 and ␩ → ⬁, ␭ ⬎ 0, and the if f共␭ , ␩兲e function S共␩兲 has a unique maximum at ␩ = ␩0. The maximum is movable when its position depends on x. The maximum of −t / ␶ − x2 / v20t2 is attained at t = 共x / v0兲2/3共2␶兲1/3. So that, introducing the change of variables t = ␩x2/3 into Eq. 共37兲 we find

as x → ⬁,

so that, integrating dx / 冑x ⬃ dt, we have finally x ⬃ t2 and

共37兲

dt

␭S共␩兲

⌽共x兲 = 共35兲

2/ 2t2 v0

0

+ O共1/␭兲

As an example, let us consider three cases. First of all, we take Eq. 共26兲. Clearly, Eq. 共27兲 satisfies condition 共32兲 and for x → ⬁ the dispersal kernel is ⌽共x兲 ⬃ x−1/4e−␣冑x, where ␣ is a certain constant and x ⬎ 0. From Eq. 共33兲 one has

4rx ⳵t Pv ⬃ 冑x = vf = − ⳵x Pv 共1 + r␶兲共1 + 2␣冑x兲

−v2/v20

but cannot be calculated explicitly. However, we can apply the Laplace method for integrals with movable maxima 关28兴 to evaluate the asymptotic behavior of Eq. 共37兲. This method is summarized by the following result

and the front speed is given by

⳵t Pv dx =− . vf = dt ⳵x Pv

共36兲

冑 冕

1

v 0␶ ␲

⬁

冋 冉

␩−1 exp − x2/3

0

1 ␩ + 2 2 ␶ v 0␩

冊册

d␩ . 共39兲

S共␩兲 = ␩ / ␶ + 1 / v20␩2

Now, and ␭ = −x2/3. Finally, from Eq. 共38兲 kernel 共39兲 turns into 共x ⬎ 0兲

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MÉNDEZ, CAMPOS, AND GÓMEZ-PORTILLO 100000

5

10

30000

5

10

4

80000

2

10

1

10

0

10

15000

-1

10

0

1

10

2

10 time

10000

10

60000

4

front position

20000

t

3

10

1.5

front position

front position

25000

front position

10

10

t

3

10

2

2

10

1

10

0

10

-1

10

40000

0

1

10

2

10 time

10

20000

5000 0 0

100

200

300

400

0

500

0

100

200

time FIG. 3. Time dependence of the front position 共in arbitrary units兲 for the case of h共v兲 Gaussian and ␸共t兲 = ␶−1e−t/␶. Arbitrary values ␶ = 0.3, v0 = 1, and a = 3 have been chosen. The log-log plot is given in the inset to observe the corresponding scaling, where the dotted line represents the theoretical scaling in Eq. 共35兲.

⌽共x兲 ⯝

1 共v20␶2x兲

e−3共x/2v0␶兲 1/3

2/3

as x → ⬁,

共40兲

which has also finite moments and satisfies Eq. 共32兲. From Eqs. 共33兲 and 共40兲 we obtain Pv共x,t兲 ⬃ x

−1/3 −␣x2/3 rt/共1+r␶兲

e

e

where ␣ is a certain constant and x ⬎ 0. From Eqs. 共34兲 and 共41兲 one finally obtains

gates with constant speed. Now we show how the method employed in this section also predicts the constant speed of the front in this limit ␴ → 0. From Eqs. 共33兲 and 共34兲, we obtain vf = −

x ⬃ t3/2 → v f ⬃ t1/2 .

and lim␴→0 ⌽共x兲 ⬃ e

x⬃t

→ vf ⬃ t

⬁

.

In Figs. 3 and 4 we show the behavior of wave fronts obtained from our Monte Carlo simulations for velocity distributions h共v兲 Gaussian and exponential, respectively. The accelerated nature of the fronts is evident from the plots. The scaling of the front position with time obtained theoretically in Eqs. 共35兲 and 共42兲 is confirmed using a log-log scale 共see insets兲. There one observes that, after a transient regime necessary for the formation of fronts, a power-law scaling for the front position is reached. To finish, let us consider the third example where the jump velocities are exponentially distributed and ␸共t兲 ⬃ ␴−1 exp关−共t − ␶兲2 / ␴2兴. In the limit where ␴ → 0, the flight duration distribution becomes ␸共t兲 = ␦共t − ␶兲 and we recover the second example studied in Sec. IV where the front propa-

1 −共t − ␶兲2/␴2 −兩x兩/␶v 0dt e e ␴t

−兩x兩/␶v0

lim v f ⬃ −

n

m/n

共43兲

then from Eq. 共43兲 we have lim ⌽共x兲

共42兲

In general, if we consider the PDFs h共v兲 ⬃ e−兩v兩 and ␸共t兲 m ⬃ e−t with n and m higher than 1, the dispersal kernel is n/共n+m兲 共x ⬎ 0兲, where ␣ is a cergiven by ⌽共x兲 ⬃ x−n/2共n+m兲e−␣x tain constant. Repeating the same procedure as above we have 1+m/n

冕

0

so that

⳵t Pv ⌽共x兲 . ⬃− ⳵x Pv ⌽⬘共x兲

Since the dispersal kernel is now ⌽共x兲 ⬃

3rx ⳵t Pv ⬃ x1/3 , = vf = − ⳵x Pv 共1 + r␶兲共1 + 2␣x2/3兲

400

FIG. 4. Time dependence of the front position 共in arbitrary units兲 for the case of h共v兲 exponential and ␸共t兲 = ␶−1e−t/␶. Arbitrary values ␶ = 0.3, v0 = 1, and a = 3 have been chosen. The log-log plot is given in the inset to observe the corresponding scaling, where the dotted line represents the theoretical scaling in Eq. 共42兲.

共41兲

,

300

time

␴→0

␴→0

lim ⌽⬘共x兲

= const

␴→0

as expected. VI. CONCLUSIONS

We have presented here a kind of reaction-dispersal mechanisms based on the idea that individual particles travel continuously with a speed taken from an arbitrary distribution h共v兲 instead of making jumps, as is implicitly assumed in most approaches. This mechanism represents the velocity counterpart of the CTRW with reaction, which has already been explored by different authors in the past years 关8–11兴. Two main results ought to be emphasized from all our discussion and derivations above. First, we have shown that the macroscopic dynamics in a reaction-dispersal mechanism based on velocities and the one based on jumps 共CTRW兲 are different, despite an equivalence exists in the case without reaction 关21兴. This is a consequence of the differences in the density profiles in each transport model, which become accentuated through the reaction process. Also, a microscopic

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TRAVELING FRONTS IN SYSTEMS OF PARTICLES WITH…

ponential, which are very usual distributions obtained from animal or cell tracking experiments 关29–31兴. As a whole, it means that accelerated reaction-dispersal fronts should be observed easily if the appropriate experimental setting is available 共probably settings based on chemical reactions or cell culture growth would be the best candidates兲. On the contrary, transport models based on jumps can only lead to accelerated fronts if long-tailed jump length distributions 共like stretched exponentials or power-law functions兲 are considered, which makes experimental observation more difficult to achieve.

justification for these differences has been sketched in Sec. II. We want to stress that the interest of this result is not merely academic; if there was an equivalence between velocity-based and jump-based approaches, then one could use a tool 关Eq. 共10兲 in this case兴 to translate from one to the other, so that there is no need to study them separately. Since this equivalence breaks down for reaction dispersal, it becomes necessary to specify the microscopic details of the system in order to obtain accurate results for its global dynamics. Second, we have provided a general criterion 共actually adapted from that in 关25兴兲 to predict when fronts with constant speed or accelerated are to be expected. From this criterion, we have observed, for example, that a system of particles changing speed with a constant characteristic rate ␶−1 关that is, for ␸共t兲 = ␶−1e−t/␶兴 and with a speed distribution of n the type h共v兲 ⬃ e−兩v兩 , with n ⱖ 1, will exhibit accelerated fronts. This includes the case when h共v兲 is Gaussian or ex-

This research has been partially supported by the Ministerio de Ciencia e Innovación under Grant Nos. CGL 200760797 共D.C.兲, FIS 2009-13370-C02-01 共V.M.兲, and by Generalitat de Catalunya under Grant No. SGR 2009-00164.

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ACKNOWLEDGMENTS

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