Proliferating L\'evy Walkers and Front Propagation

arXiv:1512.08559v1 [cond-mat.stat-mech] 28 Dec 2015

Proliferating Lévy Walkers and Front Propagation H. Stage a , S. Fedotov a1 and V. Méndezb a

b

School of Mathematics, The University of Manchester, M13 9PL Manchester, UK Universitat Autónoma de Barcelona, Departament de F´ısica, Facultat de Ciències. Edifici Cc 08193-Cerdanyola del Valls (Barcelona) Spain

Abstract. We develop non-linear integro-differential kinetic equations for proliferating Lévy walkers with birth and death processes. A hyperbolic scaling is applied to them from the onset without further long-time large scale (diffusion) approximations. The resulting Hamilton-Jacobi equations are used to determine the rate of front propagation. We found the conditions on switching, birth and death rates under which the propagation velocity is equal to the walker’s speed. In the strong anomalous case the death rate was found to influence the velocity of propagation to fall below the walker’s maximum speed. Key words: anomalous diffusion AMS subject classification:

1.

Introduction

Lévy walks have become of increasing interest in the past years, both mathematically and in the context of their many applications [13, 23, 24, 27]. They are typical in describing anomalous superdiffusion and other transport phenomena. A variety of experimental results indicate superdiffusive motion which is consistent with Lévy walks, including the spreading of cancer cells [18, 19]. Lévy walks with appropriate proliferation kinetics could therefore be used to model cancer propagation. So far modelling of Lévy walks has mainly been concerned with passive movements, and not proliferation. Some attempts have been made taking into account models with finite and random velocities [15, 16]. The purpose of our paper is to develop a theory of proliferating Lévy walkers with birth and 1

Corresponding author. E-mail: [email protected]

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Proliferating Lévy Walkers and Front Propagation

death kinetics, starting from a mesoscopic level. A Lévy walker has non-Markovian motion, which poses a challenge as we do not already have a non-linear integro-differential equation for proliferating walkers. We are further interested in the front propagation in such systems. Our main technique is hyperbolic scaling applied from the onset without long-time large scale approximation. This is a known method previously applied to problems of front propagation involving non-Markovian transport [17]. Our reasoning for this is as follows. The standard method to treat kinetic equations is to apply parabolic scaling and obtain a diffusion equation with diffusion coefficient D = v 2 T where v is the velocity and T mean running time [12, 22]. If we na¨ıvely take into account birth and death processes, we obtain the well-known Fisher equation [17, 20] ∂ 2u ∂u = D 2 + αu(1 − u), ∂t ∂x

(1.1)

where the propagation speed of the travelling wave emerging from an initial condition with compact support is √ √ (1.2) u = 2 Dα = 2v T α. It is clear that at αT = 1, the propagation speed is 2v, which contradicts the maximum possible speed v of the walker and violates the causality principle. This difficulty can be resolved using hyperbolic corrections to (1.1). The corresponding equation of motion [17] (p.38) is T

∂ρ ∂ 2ρ ∂ 2ρ 0 + [1 − T F (ρ)] = D + F (ρ), ∂t2 ∂t ∂x2

(1.3)

which is the reaction-telegraph equation with non-linear kinetic term F (ρ) = αρ(1 − ρ). This has propagation speed √ √ 4Dα 2v αT u= = , (1.4) 1 + αT 1 + αT which has the right limit as αT → 1. The simpler form of (1.2) is recovered for αT 1 [10]. For this reason, our intention in this paper is to study the kinetic equation for proliferating Lévy walkers without applying diffusion approximations [3, 4]. As we do not have a general equation for the density of proliferating walkers with arbitrary running time distribution, we cannot simply introduce a modification in the form of an additional reaction term as in (1.1). Similar problems have been solved recently for subdiffusive proliferating walkers [6, 8, 21, 25, 26]. In this paper we focus on superdiffusive kinetic models. Our challenge is to implement the birth-death kinetic terms into non-Markovian Lévy models. Our main objective is to find the equation for the density of proliferating Lévy walkers. Our specific goals are then to use this equation to find the resulting travelling wave and rate of front propagation. It should then be possible to find conditions under which the propagation velocity is upper bounded. We intend to show that this mesoscopic outset of the model does not necessarily lead to the same macroscopic velocity as if we were to use standard diffusion methods. Our method has the benefit of applying to various forms of the running time distribution, which is of interest given recent evidence that this may not be exponential [2]. Our definition of Lévy walks consequently considers distributions that are not exponential as these further reduce the 2

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problems to be Markovian in nature. We make a distinction between Lévy flights where individuals make the displacement instantaneously and rest in that position for some time, and walks where the displacement is continuous. The former again reduces the problem to be Markovian.

2.

Proliferating Lévy Walkers

For simplicity we will treat one-dimensional Lévy walks with walkers moving in the x-direction with constant velocity v. We assume that walkers change their direction with rate γ(τ ) depending on the running time τ : ψ(τ ) , (2.1) γ(τ ) = Ψ(τ ) where ψ(τ ) is the probability density R ∞function (PDF) of running times and Ψ(τ ) the corresponding survival probability of the walker t ψ(τ )dτ [9]. To model persistent movement, we assume that γ(τ ) is a decreasing function with running time. Hence, the longer a walker has been moving in a certain direction, the less likely it is to change. For constant γ(τ ), we obtain an exponential distribution of running times [1]. Our intention is to derive a kinetic equation for the mean total density ρ(x, t) = ρ+ (x, t) + ρ− (x, t),

(2.2)

where ρ± (x, t) are the densities of individuals moving to the right (+) and left (-). Our challenge now is to implement the birth and death rates into this non-Markovian transport process. We assume that the production or birth term for walkers moving to the right is f + (ρ)(κρ+ + (1 − κ)ρ− ),

(2.3)

f + (ρ)(κρ− + (1 − κ)ρ+ ).

(2.4)

and to the left We assume death terms of the form θρ± . The equations account for correlations in the direction of motion of the ’mother’ individual and corresponding ’daughter’ individuals. The degree of correlation, κ, takes three notable values. At κ = 1/2, there is no correlation. At κ = 1 we have complete correlation whereas κ = 0 corresponds to complete anticorrelation [17] (p.43). In what follows we consider three different models for this. Model A We first consider the case κ = 1, corresponding to complete correlation such that (newborn) daughter individuals move in the same direction as the (producing) mother individual. We assume the death rate θ is constant and the birth rate f + (ρ) is dependent on the mean total density. This can describe both competition in highly dense areas or freely available resources in less occupied areas should such systems be of interest. The governing equation for the densities ρ± can be written as ∂ρ+ ∂ρ+ +v = −i+ + i− − θρ+ + f + (ρ)ρ+ , ∂t ∂x 3

(2.5)

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∂ρ− ∂ρ− −v = i+ − i− − θρ− + f + (ρ)ρ− , (2.6) ∂t ∂x where we note the birth terms only contribute to the respective direction of movement. See Appendix A for the details of the derivation. In order to sustain the travelling wave solution, we assume f + (ρ) > θ and f + (ρ), θ > 0. Furthermore, the newborn walkers are initialised with a running time of τ = 0. The interaction/exchange terms accounting for the changes in direction [9] Z t i± (x, t) = K(t − τ )ρ± (x ∓ v(t − τ ), τ )e−θ(t−τ ) dτ (2.7) 0

depend on the PDF of walkers moving in a particular direction for all running times between the beginning t = 0 and t = τ , i.e. all walkers which have travelled from other positions for any amount of time to now be at (x, t). This clearly demonstrates the non-Markovian nature of the model. As the walkers may die at any point throughout their walk the term in θ arises naturally, but f + does not appear as walkers are produced with zero running time. Finally, the exchange term also depends on the distribution of running times ψ(τ ) via the kernel K in Laplace space: b b sψ(s) ψ(s) b = . K(s) = b b Ψ(s) 1 − ψ(s)

(2.8)

It is important to note that (2.7) was obtained by integration to remove the dependency of the equations on τ . This involves convolutions in Fourier-Laplace space via the transformation Z Z ∞ FL[f (x, t)] = f˜(k, s) = eikx−st f (x, t)dtdx. (2.9) R

0

Consequently, seemingly small changes to the production terms in the equations of motion may lead to structurally different solutions for ρ(x, t). Model B In contrast to our previous model, we now consider the uncorrelated case κ = 1/2 where the direction of movement mother and daughter walkers is not correlated at all. In this case, half of the produced individuals are sent in each direction (still with zero running time). This changes the equations for the densities to ∂ρ+ ρ+ + ρ− ∂ρ+ +v = −i+ + i− − θρ+ + f + (ρ) , ∂t ∂x 2

(2.10)

ρ+ + ρ− ∂ρ− ∂ρ− −v = i+ − i− − θρ− + f + (ρ) , (2.11) ∂t ∂x 2 and an exchange term as in (2.7). The derivation is similar to that of Appendix A. Although these are deceptively similar to (2.5), (2.6) the equations are now further coupled. The consequences of this are explored later in this paper. 4

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Model C Finally, we consider the case of a constant birth rate f + with a running time equal to that of the walker which produced it in addition to κ = 1. This means that there is a correlation not only between the direction of movement of the mother and daughter walkers, but also their running times. The governing equations are now ∂ρ+ ∂ρ+ +v = −i+ + i− − θρ+ + f + ρ+ , ∂t ∂x

(2.12)

∂ρ− ∂ρ− −v = i+ − i− − θρ− + f + ρ− , (2.13) ∂t ∂x where the derivation is again similar to that of Appendix A, but the interaction terms are necessarily different. It may be possible to make simplifications to a net growth rate for the system. The interaction term becomes Z t + K(t − τ )ρ± (x ∓ v(t − τ ), τ )e(f −θ)(t−τ ) dτ, (2.14) i± (x, t) = 0

so that we can write δ = f + − θ as the net constant growth rate. Clearly in this case there is a correlation between the transport and production, which does not appear in the previously introduced models.

2.1.

Markovian Model

In the particular case when the switching rate γ is constant, ψ(τ ) = γe−γτ the distribution of running times is exponential. We then recover the Markovian model ∂ρ+ ∂ρ+ +v = −γ(ρ+ − ρ− ) + (f + − θ)ρ+ , ∂t ∂x

(2.15)

∂ρ− ∂ρ− −v = γ(ρ+ − ρ− ) + (f + − θ)ρ− . (2.16) ∂t ∂x In this case, one can consider a net growth term δ = f + − θ > 0 without needing to worry about the subtleties in the running time dependencies. These correspond with a telegraph equation for f + , θ = 0 and is well-documented [11, 14].

2.2.

Non-Markovian Model

If we instead of an exponential distribution consider ψ(τ ) = λ2 te−λt corresponding to a gammadistribution, equations (2.15), (2.16) become ∂ρ+ ∂ρ+ +v = −i+ + i− + (f + − θ)ρ+ , ∂t ∂x 5

(2.17)

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∂ρ− ∂ρ− −v = i+ − i− + (f + − θ)ρ− , ∂t ∂x where the interaction terms (using (2.8)) are now written as Z t i± (x, t) = λ2 e−2λτ e−θτ ρ± (x ∓ vτ, t − τ )dτ.

(2.18)

(2.19)

0

This case is now clearly non-Markovian. Furthermore, it is trivially seen that the death rate couples with the memory kernel by virtue of both being expressed in exponential form, thus changing its effective form.

3.

Single equation for the total density

We now proceed to obtaining a single equation for the total density in each of the introduced models. Our method is similar to ones used in previous work [7]. Defining a flux term J(x, t) = vρ+ (x, t) − vρ− (x, t) of the directionality of movement and using (2.2), we obtain J(x, t) 1 ρ(x, t) ± . ρ± (x, t) = 2 v

(3.1)

(3.2)

In what follows our intention is to consider the propagation of the front into the unstable state. It is well known that in such cases we need only consider the linear terms in the birth rates as additional terms vanish under our intended hyperbolic scaling. Consequently we can conveniently write f + (ρ) = r = const.

(3.3)

This holds even if the original expressions for f + (ρ) are highly non-linear. Model A By expression of (2.5), (2.6) in terms of ρ, J and cross-differentiation to remove the flux, we get 0 two different expressions for the differential equation (see Appendix B for details). If f + (ρ) = 0, i.e. for a constant birth rate r, we get the expression 2 ∂ ∂ρ ∂ 2ρ 2∂ ρ − v − 2v (i+ − i− ) + (r − θ)2 ρ − 2(r − θ) = 0. 2 2 ∂t ∂x ∂x ∂t Using (2.7), one obtains the final integro-differential equation for the constant birth rate 2 ∂ 2ρ ∂ρ 2∂ ρ − v + 2(θ − r) + (θ − r)2 ρ = 2 2 ∂t ∂x ∂t Z t ∂ ∂ −θz − K(z)e +θ−r−v ρ(x − vz, t − z)dz ∂t ∂x 0 Z t ∂ ∂ −θz − K(z)e +θ−r+v ρ(x + vz, t − z)dz. ∂t ∂x 0

6

(3.4)

(3.5)

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This is one of our main results which will be used to find the equation for the front propagation of the system. Model B Using the same method as above, we obtain the non-linear equation 2 ∂ρ ∂ 2ρ ∂ 0 2∂ ρ = 0. − v − 2v (i+ − i− ) − θ(f + (ρ) − θ)ρ + (2θ − f + (ρ) − ρf + (ρ)) 2 2 ∂t ∂x ∂x ∂t

(3.6)

Using (2.7), this has full integro-differential form 2 ∂ 2ρ ∂ρ 0 2∂ ρ − v + (2θ − f + (ρ) − ρf + (ρ)) − θ(f + (ρ) − θ)ρ 2 2 ∂t ∂x ∂t Z t ∂ ∂ −θz + =− K(z)e + θ − f (ρ) − v ρ(x − vz, t − z)dz ∂t ∂x 0 Z t ∂ ∂ −θz + K(z)e − + θ − f (ρ) + v ρ(x + vz, t − z)dz. ∂t ∂x 0 (3.7)

We note that depending on both the forms of f + (ρ) and K(z), these equations may be highly nonlinear. However, as discussed above, the influential terms will be constant, simplifying (3.7). Model C By a similar method to before where we assume a constant production rate r from the onset, 2 ∂ ∂ρ ∂ 2ρ 2∂ ρ 2 − v − 2v (i − i ) + (r − θ) ρ − 2(r − θ) = 0, + − ∂t2 ∂x2 ∂x ∂t

(3.8)

where by (2.14), we obtain a final equation of the form 2 ∂ 2ρ ∂ρ 2∂ ρ − v + 2(θ − r) + (θ − r)2 ρ = 2 2 ∂t ∂x ∂t Z t ∂ ∂ (r−θ)z − K(z)e +θ−r−v ρ(x − vz, t − z)dz ∂t ∂x 0 Z t ∂ ∂ (r−θ)z − K(z)e +θ−r+v ρ(x + vz, t − z)dz. ∂t ∂x 0

(3.9)

While the correlations are seemingly trivial changes to the equations of motion, they lead to structurally different solutions for the front propagation.

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Before Scaling

After Rescaling

ρϵ (x,t)=ρ(x/ϵ,t/ϵ)

ρ(x,t) 1

1

x

x

Figure 1: Graphical illustration of the effect of hyperbolic scaling on the propagation front.

4.

Front Propagation

We are now in a position to find the rate of the travelling wave solutions. To ensure a minimum propagation speed we will study the equations with a front-like initial condition ( 1, if x < x0 ρ0 (x) = (4.1) 0, if x > x0 . For the models we are considering, the speed of propagation is independent of the profile. Consequently, we choose to apply a hyperbolic scaling in order to reduce the front to a geometric structure from which the velocity can easily be found [17]. We now proceed to apply a hyperbolic scaling x → x , t → t and introduce a rescaled density ρ (x, t) = ρ( x , t ) which becomes the reaction front in the limit → 0 (see Figure 1). We then apply an exponential transformation to ρ x G(x, t) t x→ , ρ (x, t) = exp − t→ , , (4.2) where the full form or any non-linearities in ρ would be preserved for more involved models. It follows from (4.2) that the position for the reaction front x(t) can be found from G[t, x(t)] = 0. It is clear that provided lim→0 G(x, t) > 0 we obtain lim→0 ρ (x, t) = 0. Otherwise, when G(x, t) = 0, lim→0 ρ (x, t) = 1. [5] Our reasoning for ignoring non-linear terms now becomes clear. Under the aforementioned limits as → 0, the birth terms simplify to linear terms r = f + (0). Model A Applying this to Model A (3.5), we obtain 2 ρ ∂ρ 2 2∂ ρ − v + 2(θ − r) + (θ − r)2 ρ = 2 2 ∂t ∂x ∂t Z t/ ∂ ∂ x t −θτ ρ − K(τ )e + θ − r − v − vτ, − τ dτ ∂t ∂x 0 Z t/ ∂ ∂ x t −θτ − K(τ )e + θ − r + v ρ + vτ, − τ dτ. ∂t ∂x 0 2∂

2

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(4.3)

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Substitution of (4.2) into (4.3) and expanding to first order in , give an equation for G(x, t) 2 2 ∂G ∂G ∂G 2 + (θ − r)2 = −v − 2(θ − r) ∂t ∂x ∂t Z ∞ ∂G ∂G ∂G τ ( ∂G −θτ (4.4) − K(τ )e − +θ−r+v e ∂t +v ∂x ) dτ ∂t ∂x 0 Z ∞ ∂G ∂G ∂G τ ( ∂G −θτ − K(τ )e − +θ−r−v e ∂t −v ∂x ) dτ. ∂t ∂x 0 The advantage of this equation is that it can be expressed using the Laplace transformation of the memory kernel. This naturally leads to the formulation 2 2 ∂G ∂G ∂G 2 + (θ − r)2 = −v − 2(θ − r) ∂t ∂x ∂t ∂G ∂G b ∂G ∂G (4.5) − − K θ− +θ−r+v −v ∂t ∂x ∂t ∂x ∂G ∂G b ∂G ∂G − − +θ−r−v K θ− +v . ∂t ∂x ∂t ∂x This is the generalised, relativistic Hamilton-Jacobi equation corresponding to (3.5). An alternative method for finding this equation is demonstrated in Appendix C. Model B Applying the same method as before to (3.7), we obtain the expression 2 2 ∂G ∂G ∂G 2 + θ(θ − r) = −v − (2θ − r) ∂t ∂x ∂t ∂G ∂G b − − +θ−r+v K θ− ∂t ∂x ∂G b ∂G K θ− +θ−r−v − − ∂t ∂x

∂G ∂G −v ∂t ∂x ∂G ∂G +v . ∂t ∂x

(4.6)

We note that in this model, the birth rate features less heavily than in the previous case. Model C We now use (3.9) to find the last Hamilton-Jacobi equation. 2 2 ∂G ∂G ∂G 2 −v − 2(θ − r) + (θ − r)2 = ∂t ∂x ∂t ∂G ∂G b − − K θ−r− +θ−r+v ∂t ∂x ∂G ∂G b − − +θ−r−v K θ−r− ∂t ∂x 9

∂G ∂G −v ∂t ∂x ∂G ∂G +v . ∂t ∂x

(4.7)

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This can be expressed in terms of an effective growth rate δ = r − θ, as these two rates now always appear together. 2 2 ∂G ∂G b ∂G ∂G ∂G ∂G 2 +δ −v +δ−v −v = K −δ − ∂t ∂x ∂t ∂x ∂t ∂x (4.8) ∂G b ∂G ∂G ∂G +δ+v +v . + K −δ − ∂t ∂x ∂t ∂x The Hamilton-Jacobi equations can be expressed in terms of a Hamiltonian H and generalised momentum p ∂G ∂G H=− , p= (4.9) ∂t ∂x so that Model A (4.5) becomes b b (H + θ − r)2 − v 2 p2 = − (H + θ − r + vp) K(H + θ − vp) − (H + θ − r − vp) K(H + θ + vp). (4.10) b though the appropriate This form allows for immediate application for a multitude of kernels K, continuation from here depends on its structure. This is of particular note as one often does not know the equation for the kernel, but its Laplace transform is readily available. (4.10) is employed to derive the front propagation velocity u by requiring the functional G[t, x(t)] = 0 so that the total + u ∂G = 0. Consequently, H(p) = up by using (4.9). From Hamilton’s equations differential ∂G ∂t ∂x ∂H we know that ∂p = q˙ = xt where x(t) = ut so we can set u=

H ∂H = . p ∂p

(4.11)

This is equivalent to [5] u = min p

4.1.

H p

.

(4.12)

Forms of the Kernel

b = λ (see (2.8)) Exponential running time PDF: ψ(τ ) = λe−λτ . In this case, the memory kernel K leads to a Hamiltonian (H + θ − r)2 − v 2 p2 = −2λ (H + θ − r) ,

(4.13)

which one can differentiate with respect to p employing (4.11) to find H 2 + H(θ − r + λ) − v 2 p2 = 0.

(4.14)

This corresponds to the velocity function u(p) = p

2v 2 p (λ + θ − r)2 + 4v 2 p2 10

,

(4.15)

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from which is trivially follows that if θ − r + λ = 0, the front propagation speed will be u = v. This corresponds to a system where the various rates affecting to the motion of the walker nullify each other, leaving the front to travel at the undisturbed individual velocity. Gamma-distributed running time PDF: ψ(τ ) = λ2 τ e−λτ . The Laplace transform of this PDF is λ2 λ2 b ψb = (s+λ) 2 . This leads to a memory kernel K(s) = 2λ+s which now will depend on both H, p. Explicitly, the Hamilton-Jacobi equation is (H + θ − r)2 − v 2 p2 = −

λ2 (H + θ − r + vp) λ2 (H + θ − r − vp) − . 2λ + H + θ − vp 2λ + H + θ + vp

By the same method as before, one obtains the front velocity p u = v provided λ = r − θ + r(r − θ).

(4.16)

(4.17)

This result is obtained by iterativelyp applying (4.11). However, it does not preclude the existence still v, but the of other minima. If λ < r − θ + r(r − θ), the upper bound on the system is p front velocity no longer sharply tends to this limits. Alternatively, if λ > r − θ + r(r − θ) the propagation velocity may be less than the upper bound. This assumes a reasonable value of λ such that propagation may occur and we do not limit ourselves to a model which is akin to vibration about a point. Traditionally, the front velocity estimate from diffusion would be derived from a Hamiltonian of the form H = Dp2 + r − θ (4.18) p with consequent velocity of u = 2 D(r − θ). This is an overestimate wrt. the result from (4.17) when 49 r > θ, (4.19) 48 indicating that the growth term need not be much greater than the death rate before issues are encountered. In the case of (4.14), the diffusion estimate is twice as large (see (1.2)) regardless of the values of the reaction parameters. Therefore, in cases where growth rates are relatively larger than death rates, a more careful estimate of the front velocity should be considered. We compare the velocities of this kernel for various λ in Figure 2. b = s1−µ , the Strong anomalous case: ψ(τ ) ∼ τ 1−µ , 0 < µ < 1. For the corresponding kernel K τ0µ Hamilton-Jacobi equation becomes (H + θ + vp)1−µ (H + θ − vp)1−µ −(H + θ − r − vp) . τ0µ τ0µ (4.20) Explicitly finding the propagation velocity in this case is more difficult but can be illustrated graphically in Figure 3 where for ease of comparison we consider similar values of the model rates. We observe that the death rate not only modifies the propagation of the system, but effectively kills the anomalous effects of the transport. We can see from Figure 3 that the death rate has a large implication for the velocity profile. If the death rate is zero, the only minimum velocity is v. However, for non-zero death rates, a minimum appears below this limit. This is a result of the (H + θ − r)2 −v 2 p2 = − (H + θ − r + vp)

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Velocity for r=0.5, λ=r-θ+

r (r - θ)

Velocity for r=0.5, λ=2.2

3.0

3.0

θ=0

θ=0 2.5

θ=0.4

2.5

θ=0.35

θ=0.35

2.0

θ=0.3

θ=0.3 p

p

H

2.0 H

θ=0.4

1.5

1.0

1.5

0.5 1.0 0.0 0.0

0.5

1.0

1.5

2.0

2.5

0.0

3.0

0.5

1.0

Momentum p

b (a) K(s) =

λ2 2λ+s

1.5

2.0

2.5

3.0

Momentum p

b (b) K(s) =

for r = 0.5, v = 1.

λ2 2λ+s

for r = 0.5, v = 1.

Figure 2: u(p) = Hp for a gamma-distributed kernel with various production-to-turning ratios. In the case of λ obeying (4.17), the death rate has little effect on the propagation velocity and minp (u) = v. The curves tend to u = v increasingly quickly as θ grows. However, in the case where the switching rate is larger all velocities have a minimum, even in the case of no death rate θ = 0. Now minp (u) < v where the velocity is dependent on θ. The details of this are defined by (4.17). Velocity for r=0.5, μ=0.75 3.0

θ=0 2.5

θ=0.4 θ=0.35

2.0

p

H

θ=0.3 1.5

1.0

0.5

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Momentum p

b Figure 3: Anomalous kernel K(s) = the values of r.

s1−µ τ0µ

for r = 0.5, v = 1. The behaviour is independent of

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coupling of the death rate with the turning kernel. This phenomenon is unchanged by variations in the growth rate r for the same model. However, in the gamma-distributed kernel, the behaviour of the propagation is dependent upon the relative values of r, λ.

5.

Conclusion

We have formulated three different models for superdiffusive transport (Lévy walkers) with birth and death processes and found the corresponding integro-differential equations for the mean density for all three cases. We have applied a hyperbolic transformation from the onset without further diffusion approximations. We have determined the Hamilton-Jacobi equations for these models and consequently equations for the fronts of the travelling waves. Our technique is applicable to non-linear forms of f + (ρ), though these higher order terms vanish in the transformation as imposed by the conditions on ρ . We found the conditions under which the parameters of the models led to a front propagation velocity equal to the speed of the walkers. These conditions depend on the equation for the memory kernel. Consequently, we found conditions under which standard diffusion methods produce an overestimate of the propagation velocity. In the strong anomalous case we found that the death rate influenced the velocity of propagation, making it less than the maximum speed of the walkers. We interpret this as the death rate tempering the anomalous effects in the transport. We also demonstrated for two separate kernels that in the superdiffusive case u < v, where the velocity was dependent upon the death rate θ but held even for the case θ = 0. In this case we again observed a tempering in the anomalous behaviour, effected by the death process. This is in contrast to the slowing down one would expect from simpler models using an approach of the form (1.2).

1.

Appendix A

In this appendix we derive the non-Markovian master equations for the densities ρ± and determine the structure of the exchange terms which describe the changes in direction of the individuals. If we simply denote the number of walkers moving in either direction as n± (x, t, τ ), we can write the equations of motion ∂n+ ∂n+ ∂n+ + +v = −γ(τ )n+ − θn+ + f + n+ , ∂t ∂τ ∂x

(A.1)

∂n− ∂n− ∂n− + −v = −γ(τ )n− − θn− + f + n− , (A.2) ∂t ∂τ ∂x where γ(τ ) is the switching rate dependent on the running time so that our equations depend on τ . We want to eliminate this variable and do so by defining the density Z t ρ± (x, t) = n± (x, t, τ )dτ. (A.3) 0

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We will need the conditions n± (x, 0, τ ) =

t

Z

ρ0± (x)δ(τ ),

γ(τ )n∓ (x, t, τ )dτ

n± (x, t, 0) =

(A.4)

0

where ρ0± (x) are theR initial distributions. Using the method of characteristics, n± (x, t, τ ) = n± (x∓ τ vt, t − τ, 0)e−θτ e− 0 γ(u)du . Choosing to write Ψ(τ ) = e−

Rτ 0

γ(u)du

=−

Ψ0 (τ ) , γ(τ )

(A.5)

Rt and noting that by definition, i± = 0 γ(τ )n± (x, t, τ )dτ = n∓ (x, t, 0) we can find write n± (x, t, τ ) = i∓ (x − vτ, t − τ )Ψ(τ )e−θτ . By substitution, this yields Z t ρ± (x, t) = i∓ (x ∓ vτ, t − τ )Ψ(τ )e−θτ dτ, (A.6) 0

Z

t

i± (x, t) =

γ(τ )i∓ (x ∓ vτ, t − τ )Ψ(τ )e−θτ dτ

Z0 t

(A.7) −θτ

i∓ (x ∓ vτ, t − τ )ψ(τ )e

=

dτ,

0

where in on the last line we have used the result Ψ0 (τ ) = −ψ(τ ). Applying the Fourier-Laplace transform to these equations, we get ˆ ∓ ikv + θ), ˜ı± (k, s) = [˜ı∓ (k, s) + ρ0± (k)]ψ(s

(A.8)

ˆ ∓ ikv + θ), ρ˜± (k, s) = [˜ı∓ (k, s) + ρ0± (k)]Ψ(s

(A.9)

from which it follows ˜ı(k, s) =

ˆ ∓ ikv + θ) ψ(s ˆ ∓ ikv + θ)˜ ρ˜(k, s) = K(s ρ(k, s) ˆ ∓ ikv + θ) Ψ(s

(A.10)

from (2.8). Taking the inverse Laplace transform, we obtain (2.7) using the shift theorem. We note that under the assumption that new individuals are produced with zero running time, the form of the production term is irrelevant and so the derivation also applies for sufficiently well-behaved f + = f + (ρ).

2.

Appendix B

This appendix finds the full integro-differential equation of the total density of walkers, using methods previously considered in [12]. From (2.5), (2.6) we add and subtract the terms, ∂(ρ+ + ρ− ) ∂(ρ+ − ρ− ) ∂ρ ∂J +v = + = (f + − θ)(ρ+ + ρ− ) = (f + − θ)ρ ∂t ∂x ∂t ∂x 14

(B.1)

H. Stage et al.


∂(ρ+ − ρ− ) ∂(ρ+ + ρ− ) 1 ∂J ∂ρ +v = +v = −2(i+ − i− ) + (f + − θ)(ρ+ − ρ− ) ∂t ∂x v ∂t ∂x (B.2) J + = −2(i+ − i− ) + (f − θ) , v where the second forms arise from the definitions of J = vρ+ − vρ− , ρ = ρ+ + ρ− . By crossdifferentiation wrt. space and time, ∂ 2ρ ∂ρ ∂ 2J = (f + − θ) + 2 ∂t ∂x∂t ∂t v2

(B.3)

∂ 2ρ ∂ ∂J ∂ 2J + = −2v (i − i ) + (f − θ) + + − ∂x2 ∂x∂t ∂x ∂x ∂ρ ∂ + 2 = −2v (i+ − i− ) + (f − θ) ρ − (f + − θ) ∂x ∂t

(B.4) 2

∂ J where we have used (B.1) in the last step. Hence we can eliminate the remaining flux term ∂x∂t to obtain (3.4). Using (2.7), and our expression for ρ± (x, t), we can write Z t ∂ ∂ρ ∂J −2v (i+ − i− ) = − K(t − τ ) v + (x − v(t − τ ), τ )e−θ(t−τ ) dτ ∂x ∂x ∂x 0 Z t ∂ρ ∂J K(t − τ ) v + − (x + v(t − τ ), τ )e−θ(t−τ ) dτ, ∂x ∂x 0 (B.5) Z t ∂ ∂ + −θ(t−τ ) +θ−f −v =+ K(t − τ )e ρ(x − v(t − τ ), τ )dτ ∂t ∂x 0 Z t ∂ ∂ −θ(t−τ ) + K(t − τ )e +θ−f +v + ρ(x + v(t − τ ), τ )dτ, ∂t ∂x 0

so that (3.5) is no longer flux dependent.

3.

Appendix C

This appendix discusses an alternative method by which the Hamilton-Jacobi equation of our twostate systems may be found. By the same exponential transformations as before we write G

G

ρ− (x, t) = Be− ,

ρ+ (x, t) = Ae− ,

(C.1)

which we substitute directly into (2.5), (2.6). Together with a hyperbolic transformation, this yields the governing equations −

∂G ∂G ρ+ − v ρ = −θρ+ + f + (ρ )ρ+ − i+ + i− , ∂t ∂x +

(C.2)

−

∂G ∂G ρ− + v ρ− = −θρ− + f + (ρ )ρ− + i+ − i− , ∂t ∂x

(C.3)

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with the interaction terms i±

Z =

t/

K(τ )e−θτ ρ± (x ∓ vτ, t − τ )dτ.

(C.4)

0

By expansion to first order in , these equations take the form ∂G ∂G ∂G ∂G ∂G ∂G b b − ρ −v ρ = −θρ+ + rρ+ − ρ+ K − −v + θ + ρ− K − +v +θ , ∂t + ∂x + ∂t ∂x ∂t ∂x (C.5) ∂G ∂G ∂G ∂G ∂G ∂G b b − ρ +v ρ = −θρ− + rρ− + ρ+ K − −v + θ − ρ− K − +v +θ . ∂t − ∂x − ∂t ∂x ∂t ∂x (C.6) These can be expressed in matrix form b − ∂G − v ∂G + θ − ∂G − v ∂G +θ−r+K ∂t ∂x ∂t ∂x M= b − ∂G − v ∂G + θ −K ∂t ∂x A 0 M = 0 B

− ∂G ∂t

! b − ∂G + v ∂G + θ −K ∂t ∂x b − ∂G + v ∂G + θ , + v ∂G + θ − r + K ∂x ∂t ∂x

(C.7) where non-trivial solutions arise from the requirements that the determinant of the matrix be zero. By direct evaluation, ∂G ∂G ∂G ∂G b − −v +θ−r+K − −v +θ × ∂t ∂x ∂t ∂x ∂G ∂G ∂G ∂G b +v +θ−r+K − +v +θ − (C.8) − ∂t ∂x ∂t ∂x ∂G ∂G ∂G ∂G b − b − −K +v +θ K −v + θ = 0. ∂t ∂x ∂t ∂x This simplifies to (4.5).

Acknowledgements S.F. acknowledges the support of the EPSRC Grant No. EP/J019526/1 ”Anomalous reactiontransport equations”.

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