PDF Time-delayed reaction-diffusion fronts

Time-delayed reaction-diffusion fronts

Neus Isern and Joaquim Fort

Departament de Física, Universitat de Girona, 17071 Girona, Catalonia, Spain 共Received 22 July 2009; published 20 November 2009兲

A time-delayed second-order approximation for the front speed in reaction-dispersion systems was obtained by Fort and Méndez关Phys. Rev. Lett. 82, 867共1999兲兴. Here we show that taking proper care of the effect of the time delay on the reactive process yields a different evolution equation and, therefore, an alternate equation for the front speed. We apply the new equation to the Neolithic transition. For this application the new equation yields speeds about 10% slower than the previous one.

DOI:10.1103/PhysRevE.80.057103 PACS number共s兲: 89.65.Ef, 87.23.Cc, 89.20.⫺a

I. INTRODUCTION

Reaction-diffusion systems have been applied to many complex biological and physical systems such as population dispersals 关1兴, viral infections 关2兴, chemical reaction pro- cesses 关3兴, combustion flames 关4兴, etc. In Ref. 关5兴 a time- delayed model for the front speed was presented including terms up to second order. However, here we will show that there was an error in the mathematical derivation, and we will derive and analyze the behavior of the correct time- delayed equation for the front speed.

In biological systems, variations in the population number density, p, are due to two processes: population growth共re- production minus deaths兲 and migration 共dispersion兲. The variation due to population growth can be expressed as a Taylor series,

关p共x,y,t+T兲−p共x,y,t兲兴g=T

冏

^⳵⳵^pt

冏

g

+T² 2

冏

^⳵⳵^2tp2

冏

g

+ ¯

=TF+T² 2

冏

^⳵⳵^Ft

冏

g

+ ¯, 共1兲

where the subindex g denotes growth, we have introduced the growth function as F共p兲=^⳵p_⳵t兩g, and T is the time delay 共one generation in most applications 关5兴兲. As usual, we as- sume thatF共p兲⬎0.

On the other hand, for the migration共dispersion兲we will define the dispersion kernel␾共⌬x,⌬y兲which gives the prob- ability per unit area that an individual initially placed at 共x +⌬x,y+⌬y兲has moved to共x,y兲after a time intervalT. Thus, the variation in population number density due to migration can be expressed as 关5兴

关p共x,y,t+T兲−p共x,y,t兲兴m=

冕冕

^p共x+⌬x,y

+⌬y,t兲␾共⌬x,⌬y兲d⌬xd⌬y

−p共x,y,t兲. 共2兲 In a system involving the two processes 共population growth and migration兲, the total variation in population den- sity during a time intervalTcan be expressed as the sum of both contributions,

p共x,y,t+T兲−p共x,y,t兲=

冕冕

^p共x+⌬x,y

+⌬y,t兲␾共⌬x,⌬y兲d⌬xd⌬y−p共x,y,t兲 +TF+T²

2

冏

^⳵⳵^Ft

冏

g

+ ¯. 共3兲

We assume that the kernel is isotropic, i.e., ␾共⌬x,⌬y兲

=␾共⌬兲, with⌬=

冑

⌬x2+⌬y2, and we Taylor expand Eq.共3兲 up to second order in time and space, thus, obtaining the follow- ing reaction-diffusion equation:

⳵^p

⳵^{t +} T 2

⳵^2p

⳵^{t2 =D}

冉

^⳵⳵^2xp2+

⳵^2p

⳵^y2

冊

^+F+T2

冏

^⳵⳵^Ft

冏

g

, 共4兲

whereDis the diffusion coefficientD=^具⌬_4T^2典=^具⌬x

2典 2T =^具⌬y

2典 2T. SinceF共p兲depends only on the population densityp, then the last term in Eq. 共4兲can be written as

T 2

冏

^⳵⳵^Ft

冏

g

=T 2

dF dp

冏

^⳵⳵^pt

冏

g

=T

2F

⬘

^F. 共5兲

In addition, as the density at the leading edge of the front is low, p⬇0, we have that F共p兲⬇pF

⬘

^{共0兲 and F}

⬘

^共p兲⬇F

⬘

^共0兲.

Therefore, for p⬇0 Eq.共4兲may be rewritten as

⳵^p

⳵^{t +} T 2

⳵^2p

⳵^{t2 =D}

冉

^⳵⳵^2xp2+

⳵^2p

⳵^y2

冊

^+pF

⬘

^共0兲+T2pF

⬘

^{共0兲F共0兲.}

共6兲 We now assume that fort→⬁andr→⬁the front can be considered locally planar. Thus, choosing thexaxis parallel to the local speed of the front, c⬅兩c_x兩, we can look for constant-shape solutions with the form p=¯pexp关␭共x−ct兲兴.

Applying thisansatzto Eq.共6兲we see that the value of␭can be obtained from

␭=

−c⫾

冑

^{c2− 4}

冉

^D−T2c2

冊

^F

⬘

^共0兲

冋

^{1 +T2F}

⬘

^共0兲

册

2

冉

^D−T2c2

冊

^{. 共7兲}

As␭has to be real, we obtain a lower bound for the front speed

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cⱖ

2

冑

^DF

⬘

^共0兲

冋

^{1 +T2F}

⬘

^共0兲

册

1 +TF

⬘

^{共0兲 . 共8兲}

However, the result obtained in Ref.关5兴was the so-called HRD speed, namely,

c_HRDⱖ 2

冑

DF

⬘

^共0兲 1 +T

2F

⬘

^共0兲

, 共9兲

which is different from Eq.共8兲.

The reason for this difference is the following. Here we have used Eq.共5兲, which allows us to rewrite Eq.共4兲as

⳵^p

⳵^{t +} T 2

⳵^2p

⳵^{t2 =D}

冉

^⳵⳵^2xp2+

⳵^2p

⳵^y2

冊

^{+F共p兲+T2dFdpF. 共10兲}

In contrast, in Ref.关5兴the following equation was used:

⳵^p

⳵^{t +} T 2

⳵^2p

⳵^{t2 =D}

冉

^⳵⳵^2xp2+

⳵^2p

⳵^y2

冊

^{+F共p兲+T2dFdp⳵}⳵^{pt. 共11兲} We can see that the last term is different. The reason is that in Ref.关5兴the subindexgwas omitted in the last term in Eq. 共4兲. Therefore, in Ref.关5兴, the last term in Eq.共4兲was not written as in Eq.共5兲but as follows:

T 2

⳵^F

⳵^{t =} T 2

dF dp

⳵^p

⳵^{t =} T 2F

⬘

^⳵_⳵^p

t, 共12兲 and thus leading to Eq.共11兲instead of Eq.共10兲. This is why in Ref.关5兴, speed共9兲was obtained instead of Eq.共8兲. How- ever, the derivation above clearly shows that Eq. 共8兲is the right result. In this Brief Report, we will apply variational analysis and show that Eq.共8兲is not only a lower bound but the exact speed共Sec.II兲. We will also analyze the difference between the new Eq.共8兲and the HRD speed共9兲by applying both equations to the Neolithic transition 共Sec. III兲. In Sec.

IVwe present our conclusions.

II. VARIATIONAL ANALYSIS: UPPER BOUND Equation共8兲is just a lower bound for the speed of front solutions to the new differential equation共4兲 关or Eq.共10兲兴. In order to find an upper bound, we apply variational analysis 关6兴 to Eq. 共10兲. As mentioned above, we assume that the fronts have a profile p共z兲=p共x−ct兲 traveling with a speed c⬎0, so all of the derivatives in Eq.共10兲can be expressed in terms ofz. We also assume that the population number den- sity p⬎0 cannot attain values above some value p_max, the so-called saturation density. Then, definingn共p兲= −p_zand as- suming thatn共0兲=n共pmax兲= 0 andn⬎0 in共0 ,pmax兲, differen- tial equation共10兲can be rewritten as

冉

^D−c2T2

冊

^n⳵⳵^np−cn+F

冉

^{1 +T2F}

⬘ 冊= 0. 共13兲

Now, introducing an arbitrary function g共p兲 such that g共p兲⬎0 and h共p兲= −g

⬘

共p兲⬎0, we multiply Eq. 共13兲 by g共p兲/n共p兲. Integrating the resulting expression by parts, we obtain

c

冕

0 p_max

gdp=

冕

0 p_max

冋 冉

^{D− T2c2}

冊

^hn+gnF

冉

^{1 +T2F}

⬘ 冊 册dp.

共14兲

Now, we can eliminaten共p兲from Eq. 共14兲applying that for any positive numbersrands, it follows from共r−s兲²ⱖ0 that 共r+s兲ⱖ2

冑

rs. Let us assume that the condition

1 +T

2F

⬘

^{共p兲⬎0 共15兲}

holds for all p苸共0 ,p_max兲. As g共p兲, h共p兲, n共p兲, F共p兲, and 共D−^T₂c²兲are positive关7兴, we may chooser⬅共D−^T₂c²兲hnand s⬅^g_nF共1 +^T₂F

⬘

^{兲into共r+s兲ⱖ2}

冑

rsand use Eq.共14兲to get the following restriction:

冑 冉

^Dc^−T2c2

冊

^ⱖ

2兰₀^pmax

冑

^hgF

冉

^{1 +T}2F

⬘ 冊dp 冕0

p_max

gdp

. 共16兲

Following the method in Ref. 关8兴, Sec. 3.3, it is easy to show that there is a functiongfor which the equality holds.

Then,

冑 冉

^Dc^−T2c2

冊

^{= maxg}

冢

^2兰0pmax

冑 冕

^hgF0pmax

冉

^{1 +gdpT2F}

⬘ 冊dp冣.

共17兲 In order to obtain the upper bound for the front speed we will use Jensen’s inequality关9兴

兰₀^pmax␮共p兲

冑

␣共p兲dp

兰₀^pmax␮共p兲dp ⱕ

冑

^兰0pmax兰₀^{pmax␮共p兲}␮共p兲dp^␣共p兲dp, 共18兲 where ␮共p兲⬎0 and ␣共p兲ⱖ0. We define ␮共p兲⬅g共p兲 and

␣共p兲⬅兵h共p兲F共p兲关1 +^T₂F

⬘

^{共p兲兴其/}g共p兲. Using these functions into Jensen’s inequality 共18兲, and applying the result to Eq.

共17兲, we obtain that

冑 冉

^Dc^−T2c²

冊

^ⱕ2maxg

冑

^兰0pmaxhF

冕

0

冉

^{1 +T2F}

⬘ 冊dp pmax

gdp .

共19兲 We want an upper bound independent ofg共p兲, so we will first find an expression in whichh共p兲= −g

⬘

共p兲no longer ap- pears by integrating by parts the numerator in the right-hand side of Eq.共19兲,

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冕

0 p_max

hF

冉

^{1 +T2F}

⬘ 冊dp=冕0 pmax

g

冋

^F

⬘ 冉1 +T2F⬘ 冊+ T2FF⬙ 册dp,

共20兲 where we have assumed that F共0兲=F共p_max兲= 0 共this holds, for example, for the logistic growth considered in Sec.III兲.

Moreover, from Eq.共20兲we obviously have

冕

0 p_max

hF

冉

^{1 +T}2F

⬘ 冊dpⱕp苸共0,psupmax兲冋F⬘ 冉1 +T2F⬘ 冊

+T

2FF

⬙ 册 冕0pmaxgdp, 共21兲

so now the upper bound in Eq.共19兲is independent ofg共p兲,

冑 冉

^Dc^−T2c2

冊

^ⱕ2

冑

^{p苸共sup0,pmax兲}

冋

^F

⬘ 冉1 +T2F⬘ 冊+T2FF⬙ 册.

共22兲 Let us assume that the population growth functionF共p兲is a continuous function with F

⬙

共p兲ⱕ0 and F共0兲= 0 共again these assumptions are true for the logistic growth considered in Sec.III兲. ThenF

⬘

^共p兲is a decreasing function for increas- ing values of p. Its maximum value is reached for p= 0.

Thus, using the value p= 0 in Eq. 共22兲 we obtain that the upper bound for the front speed is

cⱕ

2

冑

^DF

⬘

^共0兲

冋

^{1 +T2F}

⬘

^共0兲

册

1 +TF

⬘

^{共0兲 . 共23兲} As the lower bound given by Eq.共8兲is the same as upper bound共23兲, we can predict the speed of front solutions to Eq.

共10兲without any uncertainty,

c=

2

冑

^DF

⬘

^共0兲

冋

^{1 +T2F}

⬘

^共0兲

册

1 +TF

⬘

共0兲 . 共24兲 In contrast, for the HRD Eq. 共11兲, the exact speed was previously shown to be关5兴

cHRD= 2

冑

DF

⬘

共0兲 1 +T

2F

⬘

共0兲

. 共25兲

III. APPLICATION TO THE NEOLITHIC TRANSITION In order to compare the predictions from Eqs. 共24兲 and 共25兲, we will apply them to the spread of the Neolithic tran- sition in Europe, because this is the case to which Eq.共25兲 was initially applied 关5兴. The Neolithic transition is the change from hunter-gatherer to farming economics. In Eu- rope, it took place as an invasion of agricultural populations from the Southeast, which spread across Europe from 13 000 to 5000 years before present关10兴.

In order to make quantitative predictions we will use the logistic growth function, which has been widely applied to human populations关5,11兴:

F共p兲=ap

冉

^{1 −}p_max^p

冊

^{, 共26兲}

whereais called the initial growth rate andp_maxis the satu- ration density.

Using logistic function共26兲, Eq.共24兲can be rewritten as

c=

2

冑

^aD

冉

^{1 +aT2}

冊

1 +aT , 共27兲

whereas the HRD speed共25兲, used in Ref.关5兴, is

c_HRD= 2

冑

aD 1 +aT 2

. 共28兲

Both equations for the front speed depend on three param- eters: the initial growth rate, a, the diffusion coefficient, D

=^具⌬_4T^2典, and the generation time, T. We will use the ranges a

= 0.028⫾0.005 yr⁻¹ 关12兴, 具⌬²典= 900– 2200 km² 关10兴, and the characteristic valueT= 32 yr关13兴, which have been mea- sured for preindustrial farming populations. For these ranges, condition 共15兲 is fulfilled, so Eq. 共27兲 gives the speed of fronts.

Figure 1shows the front speeds obtained from Eqs. 共27兲 and 共28兲 for a characteristic mobility value 具⌬²典

= 1531 km². We can see that, for the range of values for the initial growth ratea appropriate to this application, the new Eq. 共27兲 yields slower speeds than Eq. 共28兲 共about 8%

slower兲. However, this is not the case for all values of a, as can be seen from the inset graph in Fig. 1.

In order to check the validity of Eq. 共27兲, we have also numerically integrated Eq.共10兲, withF共p兲given by Eq.共26兲, and initially p=p_max in a finite region and p= 0 elsewhere.

FIG. 1. Comparative plot between the front speed for Eq.共27兲 共solid line兲and Eq.共28兲 共dashed line兲. The symbols correspond to the speed obtained from numerically integrating Eq.共10兲, withF共p兲 given by Eq.共26兲. All results have been calculated for a character- istic mobility value具⌬²典= 1531 km².

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The speed obtained from the numerical integrations corre- sponds to the circles in Fig.1. They agree with the new Eq.

共27兲within less than 0.8%.

The range of speeds for the Neolithic transition front ob- tained from archeological data is 0.6–1.3 km/yr关10兴. We can see in Fig. 1 that the results from Eq. 共27兲 lie within this range.

To what extent does our result depend on the uncertainty in the value of the mobility? In Fig.2, we consider the front

speed values 0.6, 0.95, and 1.3 km/yr, corresponding to the range obtained from archeological data, for Eq. 共27兲 共full lines兲 and Eq. 共28兲 共dashed lines兲 关14兴. It is seen that the predictions of the new model共full lines兲are consistent with the observed front speed for most of the values of the mo- bility appropriate to this system.

IV. CONCLUDING REMARKS

In this Brief Report we have improved the derivation of the HRD speed in Ref. 关5兴. We have obtained the correct evolution Eq.共10兲and the new Eq.共24兲for the front speed.

We have applied the new Eq.共24兲to the Neolithic transi- tion. Using realistic parameters the front speeds are consis- tent with the observed range for the Neolithic transition in Europe 共0.6–1.3 km/yr 关10兴兲. Comparing these results with those from the HRD speed, we see that for the Neolithic transition our new equation leads to slower speeds.

In this case, the correction obtained is only about 10%, but it could be higher in other systems where generalizations of Eq.共24兲can be useful. For example, our framework could be applied in order to improve the predicted speeds of viral infection fronts关2兴.

ACKNOWLEDGMENTS

This work was funded by the European Commission 共Grant No. NEST-28192-FEPRE兲, the MICINN-FEDER 共Grant No. FIS2009-13050兲, and the Generalitat de Catalu- nya 共Grant No. SGR-2009-374兲. N.I. was supported by the MEC under the FPU program.

关1兴K. Davison, P. Dolukhanov, G. R. Sarson, and A. Shukurov, J.

Archaeol. Sci. 33, 641共2006兲.

关2兴J. Fort and V. Méndez, Phys. Rev. Lett. 89, 178101共2002兲. 关3兴L. Gálfi and Z. Rácz, Phys. Rev. A 38, 3151共1988兲. 关4兴J. Merikoski, J. Maunuksela, M. Myllys, J. Timonen, and M. J.

Alava, Phys. Rev. Lett. 90, 024501共2003兲.

关5兴J. Fort and V. Méndez, Phys. Rev. Lett. 82, 867共1999兲. 关6兴R. D. Benguria and M. C. Depassier, Phys. Rev. E 57, 6493

共1998兲.

关7兴The condition共D−Tc²/2兲⬎0 follows from␭⬍0 and Eq.共7兲. 关8兴J. Fort and V. Méndez, Rep. Prog. Phys. 65, 895共2002兲. 关9兴S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series and

Products 共Academic, San Diego, 1994兲; see p. 1133 formula HL151. The inequality is given for convex functions␾. If␾is a convex function, then −␾is concave and the inequality for

−␾holds with the sign reversed. Take the function −␾to be the square root.

关10兴R. Pinhasi, J. Fort, and A. J. Ammerman, PLoS Biol. 3, e410 共2005兲; for the estimation of mobility data see supporting text 3.

关11兴J. D. Murray,Mathematical Biology, 3rd ed.共Springer-Verlag, Berlin, 2002兲, Vol. 1.

关12兴N. Isern, J. Fort, and J. Pérez-Losada, J. Stat. Mech.: Theory Exp.共2008兲P10012.

关13兴J. Fort, D. Jana, and J. Humet, Phys. Rev. E 70, 031913 共2004兲; for the estimation of the generation timeT= 32 yr, see note关24兴in this reference.

关14兴The dashed lines in Fig.2are not exactly the same as the lines in Fig. 3 in Ref.关5兴because some parameter values have been estimated in a more precise way more recently especially the generation timeT关13兴.

FIG. 2. Predictions for the speed of the wave of advance in the Neolithic transition. The labeled curves correspond to the maxi- mum, minimum, and mean speeds from Neolithic data 共0.6–1.3 km/yr兲. The hatched regions correspond to realistic ranges of the initial growth rate and mobility for the Neolithic transition.

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