The source term in a reaction-diffusion system, in general, does not involve
explicit time dependence. A class of self-limiting growth models dealing with
animal and tumor growth and bacterial population in a culture, on the other
hand are described by kinetics with explicit functions of time. We analyze a
reaction-diffusion system to study the propagation of spatial front for these
models.Comment: RevTex, 13 pages, 5 figures. To appear in Physical Review 

A. K. Laird

C. Cattaneo

Deb Shankar Ray

G. Abramson

G. Klein

H. Malchow

J. Folkman

J. M. Sancho

K. K. Manne

M. Mimura

M. R. Roussel

N. Shigesada

S. Fedotov

Sandip Kar

Suman Kumar Banik

T. Hillen

V. Méndez

W. Horsthemke

English

arXiv

The source term in a reaction-diffusion system, in general, does not involve explicit time dependence. A class of self-limiting growth models dealing with animal and tumor growth and bacterial population in a culture, on the other hand, are described by kinetics with explicit functions of time. We analyze a reaction-diffusion system to study the propagation of spatial front for these models

Kar, Sandip

Banik, Suman Kumar

Ray, Deb Shankar

Indian Academy of Sciences

Class of self-limiting growth models in the presence of nonlinear diffusion

Crossref

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Class of self-limiting growth models in the presence of nonlinear
diffusion
Sandip Kar, Suman Kumar Banik and Deb Shankar Ray∗
Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India
(February 2, 2008)
Abstract
The source term in a reaction-diffusion system, in general, does not involve
explicit time dependence. A class of self-limiting growth models dealing with
animal and tumor growth and bacterial population in a culture, on the other
hand are described by kinetics with explicit functions of time. We analyze a
reaction-diffusion system to study the propagation of spatial front for these
models.
PACS number(s): 87.10.+e, 87.15.Vv, 05.45.-a
Typeset using REVTEX
∗Electronic-mail: pcdsr@mahendra.iacs.res.in
1
I. INTRODUCTION
Reaction-diffusion systems are ubiquitous in almost all branches of physics [1], chemistry
[2] and biology [3–5] dealing with population growth, fluid dynamics, pulse propagation in
nerves, chemical reactions, optical and other processes. The basic equation describes the
dynamics of a field variable n(x, t), a function of space and time in terms of a source term
(also known as reaction term) and a diffusion term. An important early endeavor in this
direction is the study of self-limiting growth models of which the most well-known is the
Fisher equation [6,7] which takes into account of a linear growth and a nonlinear decay.
The model and many of its variants have found wide applications both from theoretical
and experimental point of view [4]. A notable feature of these models is that the source or
the reaction terms do not involve any explicit time dependence. On the other hand there
are situations [8–10] where the source terms contain explicit functions of time which put a
constraint on the growth process in the long time limit. For example, the Gompertz growth
[8,9] is a model used for study of growth of animals and tumors, where the growth rate is
proportional to the current value, but the proportionality factor decreases exponentially in
time so that
dn
dt
= rn exp(−αt) , (1a)
where r and α are positive experimentally determined constants. Similarly another type of
model proposed to analyze the growth of bacterial population in culture [10] is described by
dn
dt
= knt exp(−βt2) . (1b)
Again k and β are positive constants required to fit the experimental data. An important
feature of these models is that unlike the logistic growth process the asymptotic value of the
density function n depends on its initial population.
Keeping in view of these experimental observations it is therefore worthwhile to generalize
the specific cases in terms of an explicit function of time φ(t) such that we write
2
dn
dt
= rnφ(t) (2)
where r is a constant for the growth process and φ(t) may of the type (i) φ(t) = 1 for
exponential growth (ii) φ(t) = exp(−αt) for Gompertz growth (iii) φ(t) = t exp(−βt2) for
bacterial growth, etc.
The object of the present paper is to study a reaction-diffusion system with a reaction
term describing a class of self-limiting growth processes (2). Since in many living organisms
concentration dependent diffusivity [4,5,11–15] has been found to be essential to the modeling
of reaction-diffusion systems we investigate the interplay of this nonlinear diffusion and self-
limiting growth process in the dynamics. We show that the model and its variant with a
finite memory transport [16–25] admit of exact solutions. The dependence of the rate of
spread of the wave front on various parameters is explored.
II. THE REACTION-DIFFUSION SYSTEM
We consider a reaction-diffusion system with a source term describing self-limiting growth
and with a nonlinear diffusion term in the following form:
∂n(x, t)
∂t
= rnφ(t) +
∂
∂x
Dn
∂n
∂x
(3)
where D is the diffusion coefficient. Our primary aim in this section is to provide an exact
solution of Eq.(3). To this end we first make use of the following transformation
n(x, t) = ũ(x, t) exp
(
r
∫ t
0
φ(t′)dt′
)
(4)
in Eq.(3) to obtain
∂ũ(x, t)
∂t
= D exp
(
r
∫ t
0
φ(t′)dt′
)
∂
∂x
{
ũ
∂ũ
∂x
}
. (5)
We now introduce the scaled time variable τ as
τ = D
∫ t
0
f(t′)dt′ ≡ G(t) (say) (6a)
3
where
f(t) = exp[r
∫ t
0
φ(t′)dt′] . (6b)
This reduces Eq.(5) to the following form
∂u(x, τ)
∂τ
=
∂
∂x
{
u(x, τ)
∂u(x, τ)
∂x
}
(7)
with ũ(x, t) ≡ ũ(x, G−1(τ)) = u(x, τ) where time t has been expressed as an inverse function
G−1(τ) according to Eq.(6a-6b).
Eq.(7) is the well-known Boltzmann nonlinear diffusion equation [1,26]. Now subject to
the initial condition of a unit point source at the origin,
n(x, 0) = δ(x) = ũ(x, 0) = u(x, 0) (8)
we solve Eq.(7) under the following boundary conditions
lim
x→±∞
u(x, τ) = 0 ∀τ > 0 (9)
and
∫ +∞
−∞
u(x, τ)dx = 1 ∀τ > 0 . (10)
Next we seek the similarity solution of the nonlinear diffusion equation (7). We make
use of the well-known similarity transformation [1,5,26,27]
u = τ−1/3v(z) and z = xτ−1/3 (11)
in Eq.(7) to obtain
3
d
dz
(
v
dv
dz
)
+ v + z
dv
dz
= 0 (12)
On integration Eq.(12) yields
3
(
v
dv
dz
)
+ zv = 0 (13)
4
Since we are interested in the symmetric solutions with v′(0) = 0 we have put the integration
constant zero in going from Eq.(12) to (13). On further integration Eq.(13) results in the
solution
v(z) = (A2 − z2)/6 |z| < A
= 0 |z| > A
(14a)
where A is a constant which can be determined from the condition (10) to obtain
A = (9/2)1/3 (14b)
Therefore the solution of Eq.(7) in x and τ is given by
u(x, τ) =
1
6τ
[
A2τ 2/3 − x2
]
|x| < Aτ 1/3
= 0 |x| > Aτ 1/3 (15)
It is interesting to note that by virtue of the relations (6a-6b) τ is dependent on r and
φ(t) which control the growth and self-limiting factors, respectively of the source term.
This implies that the shock-wave like behaviour with propagating wave-front at x = xf =
Aτ 1/3 as evident from the similarity solutions (15) critically depends on the reaction terms.
Specifically, the wave front propagates in the medium with speed
dxf
dt
=
1
3
(
9D
2
)1/3
f(t)
[
∫ t
0
f(t′)dt′
]−2/3
(16)
where f(t) is given by (6a) and in turn depends on the functional form of φ(t).
We now consider two specific cases to illustrate the spatial propagation patterns.
(i) φ(t) = 1
For a constant value of φ the model suggests an exponential growth. The relation (6a)
in this case can then be utilized to obtain f(t) = exp(rt) so that τ = (D/r)[exp(rt) − 1].
Putting this expression for τ in the solution (15) we have after using Eq.(4)
n(x, t) =
[
A2 {(D/r) (exp(rt) − 1)}2/3
]
− x2
(6D/r)[exp(rt) − 1] exp(−rt)
(17)
This solution clearly has a sharp wave-front at xf = Aτ
1/3 which propagates at a speed
5
dxf
dt
=
1
3
A
(
Dr2
)1/3
exp(rt) (exp(rt) − 1)−2/3 (18)
To illustrate the spatial propagation of the population n(x, t) in time we plot in Fig-1
the spatial shock-wave like patterns for r = 1.0 and D = 1.0. It is apparent that the sharply
peaked distribution at t = 0 starts spreading relatively slowly with peak at x = 0 diminishing
with time upto a period t = 0.1 . Beyond this time the spatial growth of population becomes
comparatively large and it diverges due to the combined effect of exponential growth and
nonlinear diffusion. For a much lower growth rate (r = 0.001), however, the population
spreads monotonically due to the nonlinear diffusion which overwhelms the effect of growth
process. This is evident in Fig-1(b).
(ii) φ(t) = t exp(−βt2)
With the above expression for φ(t) for bacterial self-limiting growth we obtain from
(6a-6b)
f(t) = exp[(−r/2β)(exp(−βt) − 1)] (19)
and
τ = D exp(r/2β)
∫ t
0
exp[(−r/2β) exp(−βt′)]dt′ (20)
By defining z = (r/2β) exp(−βt) the above expression can be reduced to the following form
τ = −D
exp(r/2β)
β
∫ (r/2β) exp(−βt)
(r/2β)
exp(−z)
z
dz (21)
The integral in (21) can be put into a standard form with the help of Ei-function [28] so
that τ can be expressed as
τ = D
exp(r/2β)
β
[Ei (−r/2β) − Ei ((−r/2β) exp(−βt))] (22)
The corresponding density n(x, t) and the speed of the wave front dxf/dt at xf are given by
n(x, t) =
A2 (D exp(r/2β)/β)2/3 [Ei (−r/2β) − Ei (−(r/2β) exp(−βt))]2/3 − x2
(6D/β) [Ei(−r/2β) − Ei((−r/2β) exp(−βt))] exp[(r/2β) exp(−βt)]
(23)
and
6
dxf
dt
= A
(
D
exp(r/2β)
β
)1/3
d
dt
[Ei(−r/2β) − Ei((−r/2β) exp(−βt))]1/3 (24)
respectively.
In Fig-2(a,b) we show the shock-wave like spread of population by plotting n(x, t) vs
x for several values of time for D = 1 and r = 1 . Since β puts a limit to the growth at
large time the peak of n(x, t) at x = 0 as shown in Fig-2(a) (β = 0.1) does not increase too
much as compared to the earlier case considered in Fig-1(a). It has been observed that for
a unique value of β ≥ 1.0 there is a monotonic decrease in the peak population n(x, t) at
x = 0. For smaller values of β (Fig-2(b)) the spread is similar to that in Fig-1(a). In Fig-2(c)
we exhibit the spatial front propagation for several values of growth rate r at a time t = 1.0
keeping D = 1 and β = 0.01. It is apparent that with increase of r the reaction dominates
over diffusion so that the peak population at x = 0 increases compared to spreading.
III. EFFECT OF FINITE MEMORY TRANSPORT
We now generalize the proposed reaction-diffusion model to include the effect of finite
memory transport. It has been observed that an animal’s movement at a particular instant of
time often depends on its motion in the immediate past. This results in a delay in population
flux, or a memory in the diffusion coefficient. A number of attempts have been made in
the recent literature [16–25] to analyze the delayed population growth in several models and
related context in heat conduction and transport processes. To consider a finite memory in
the present model we modify the nonlinear diffusion term in Eq.(3) to the following form:
∂n(x, t)
∂t
= rn(x, t)φ(t) +
∂
∂x
[
Dγ
∫ t
0
exp[−γ(t − τ)]n(x, τ)
∂n(x, τ)
∂x
dτ
]
(25)
where γ refers to the inverse of relaxation time. The population flux takes into account
of the relaxation effect due to the delay of the particles in adopting a definite direction of
propagation. Differentiating both sides of the above equation with respect to t and using it
again we obtain
7
∂2n
∂t2
= (rφ − γ)
∂n
∂t
+ (rφ̇ + rφγ)n +
∂
∂x
[
Dγn
∂n
∂x
]
(26)
In the limit of vanishing relaxation time i.e, 1/γ →0 Eq.(26) reduces to Eq.(3). When
memory effects are taken into account, the dispersal of the organisms are not mutually inde-
pendent. Hence the correlation between the successive movement of the diffusing particles
results in a delay in the transport. Thus Eq.(26) is a typical form of a delayed transport
equation.
We now consider a specific case φ(t) = 1. Substitution of the traveling wave form N(z)
(= n(x, t)) with z = x + ct satisfies
c2
∂2N
∂z2
= c(r − γ)
∂N
∂z
+ rγN + Dγ
∂
∂z
(
N
∂N
∂z
)
(27)
where c is the speed of the traveling wave to be determined.
We now consider the trial solution of Eq.(27) of the form N(z) = N0 exp(sz
b) subject
to the initial condition that at z = 0, N = N0, where s and b are positive constants to be
determined. Substitution of this solution in Eq.(27) yields the following relation
[c2s2b2z2(b−1) + c2sb(b − 1)z(b−2) − csb(r − γ)z(b−1)
−rγ] exp(szb) − DγN0sb[2sbz
2(b−1) + (b − 1)z(b−2)]
× exp(2szb) ≡ L(z) = 0 (28)
For L(z) = 0, for all z, the coefficients of exp(szb) and exp(2szb) within the square brackets
must vanish identically. For this the only acceptable solution for b is b = 1. We obtain
2s2DγN0 = 0 (29a)
and
c2s2 − cs(r − γ) − rγ = 0 (29b)
From the above two equations the solution for s is given by
s =
c[(1/γ) − (1/r)] + [c2 ((1/γ) − (1/r))2 + 4/r ((c2/γ) + 2DN0)]
1/2
2 [(c2/γr) + (2DN0/r)]
(30)
8
In the limit of instantaneous relaxation, i.e, 1/γ →0 Eq.(30) yields
s =
c
[
−1 + (1 + (DN0r/c
2))
1/2
]
4DN0
(31)
Furthermore the above expression in the limit of weak diffusion D →0 we obtain from
Eq.(31) after a Taylor expansion
s =
r
c
(32)
To determine the speed of the propagation of the wave front we now rearrange the
solution for s in (30) to obtain
c =
(r − γ) + [(r − γ)2 + 4(rγ − 2s2DγN0)]
1/2
2s
(33)
For real values of c, the quantity inside the square root must be positive, which determines
the minimum value of c for s = r/c [ Eq.(32) ] as
cmin =
2r2DγN0
(r + γ)2
(34)
Eq.(27) therefore admits of an exact traveling wave like solution
N(z) = N0 exp


c(r − γ) + (c2(r − γ)2 + 4rγ(c2 + 2DγN0))
1/2
2(c2 + 2DγN0)

 z (35)
It is interesting to observe that the speed of the traveling wave front not only depends
on nonlinear diffusion and growth rate but also on the initial concentration and memory. A
comparison of the solutions in this section and in the previous one shows that (35) does not
reduces to Eq.(17) in the limit of vanishing relaxation time (1/γ → 0) although Eq.(26) goes
over to Eq.(3) under this condition. This is because of the fact that the nature of the partial
differential equation changes due to the inclusion of relaxation terms and also the boundary
conditions for the shock wave like ‘diffusing solutions’ (17) are different for the travelling
wave front solution (35). The nature of the two solutions are thus generically different. We
point out in passing that the dependence on initial concentration on speed as shown in (34)
is rather an unusual feature in reaction-diffusion system.
9
IV. CONCLUSIONS
In this paper we have analyzed a class of reaction-diffusion systems in which the kinetic
term describes the self-limiting growth processes of the Gompertz type and is an explicit
function of time. We have shown that the model can be solved exactly to analyze the spatial
front propagation problem. To make the model more realistic we have included the effect
of finite relaxation to concentration-dependent diffusive processes. In view of the fact that
the source terms have their direct relevance on experimental measurement on animal and
tumor growth or bacterial culture we think that the solutions discussed in this paper will
be pertinent in the context of reaction-diffusion systems, in general.
ACKNOWLEDGMENTS
The authors are indebted to C.S.I.R. (Council of Scientific and Industrial Research),
Government of India, for financial support.
10
REFERENCES
[1] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers
(Birkhäuser, Boston, 1997).
[2] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics:
Oscillations, Waves, Patterns and Chaos (Oxford University Press, New York, 1998).
[3] N. F. Britton, Reaction-Diffusion Equations and their applications to Biology (Aca-
demic, New York, 1986).
[4] J. D. Murray, Mathematical Biology, Second Corrected Edition (Springer, Berlin, 1993).
[5] M. Kot, Elements of Mathematical Ecology (Cambridge University Press, Cambridge,
2001).
[6] R. A. Fisher, Ann. Eugenics. 7, 355 (1937).
[7] A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, in Selected Works of A. N.
Kolmogorov, edited by V. M. Tikhomirov (Kluwer, Dordrecht, 1991).
[8] J. Folkman and M. Hochberg, J. Exp. Med. 4, 745 (1973).
[9] A. K. Laird, Brit. J. Cancer 18, 490 (1964); G. Klein and L. Révész, J. Nat. Cancer
Inst. 14, 229 (1953).
[10] D. Kaplan and L. Glass, Understanding Nonlinear Dynamics (Springer-Verlag, New
York, 1995).
[11] N. Shigesada, J. Math. Bio. 9, 85 (1980).
[12] A. Okubo, Diffusion and Ecological Problems: Mathematical Models (Springer, Berlin,
1980).
[13] M. R. Roussel and J. Wang, Phys. Rev. Lett. 87, 188302 (2001).
[14] H. Malchow, J. Theor. Biol. 135, 371 (1988).
11
[15] M. Mimura and K. Kawasaki, J. Math. Bio. 9, 49 (1980).
[16] C. Cattaneo, C. R. Acad. Sci. 247, 431 (1958).
[17] K. P. Hadeler, in Reaction Transport Systems in Mathematics Inspired by Biology, edited
by V. Capasso and O. Diekmann, CIME Lectures, Florence (Springer-Verlag, Berlin,
1998).
[18] T. Hillen, Math. Models Methods Appl. Sci. 8, 507 (1998).
[19] V. Méndez, J. Fort and J. Farjas, Phys. Rev. E 60, 5231 (1999).
[20] W. Horsthemke, Phys. Rev. E 60, 2651 (1999).
[21] K. K. Manne, A. J. Hurd and V. M. Kenkre, Phys. Rev. E 61, 4177 (2000).
[22] G. Abramson, A. R. Bishop and V. M. Kenkre, Phys. Rev. E 64, 066615 (2001).
[23] S. Fedotov, Phys. Rev. Lett. 86, 926 (2001).
[24] J. M. Sancho and A. Sánchez, Phys. Rev. E 63, 056608 (2001).
[25] R. D. Benguria and M. C. Depassier, nlin.PS/0202003.
[26] J. Crank, The Mathematics of Diffusion, Second Edition (Oxford University Press,
Oxford, 1975).
[27] G. Birkhoff, Hydrodynamics (Princeton University Press, Princeton, 1950).
[28] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic,
New York, 1980).
12
FIGURES
FIG. 1. Evolution of spatial front in time for the model with φ(t) = 1. (a) The population
n(x, t) is plotted against x for different times using r = 1.0 and D = 1.0. (b) Same as in Fig.(1a)
but for r = 0.001 (units arbitrary).
FIG. 2. Evolution of spatial front in time for the model with φ(t) = t exp(−βt2). (a) The
population n(x, t) is plotted against x for different times using r = 1.0, D = 1.0 and β = 0.1. (b)
Same as in Fig.(2a) but for β = 0.01. (c) The population n(x, t) is plotted against x at t = 1.0 for
different r using D = 1.0 and β = 0.01 (units arbitrary).
13
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0
1
2
3
4
5
(1a)
Fig.(1a)
 t = 0.001
 t = 0.01
 t = 0.1
 t = 1.0
 t = 2.0
 t = 3.0
n(
x,
t)
x
-3 -2 -1 0 1 2 3
0
1
2
3
4
5
(1b)
Fig.(1b)
 t = 0.001
 t = 0.01
 t = 0.1
 t = 1.0
n(
x,
t)
x
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
1
2
3
4
5
Fig.(2a)
(2a)  t = 0.001
 t = 0.01
 t = 0.1
 t = 1.0
 t = 2.0
 t = 3.0
 t = 5.0
n(
x,
t)
x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0
1
2
3
4
5
Fig.(2b)
(2b)
 t = 0.001
 t = 0.01
 t = 0.1
 t = 1.0
 t = 2.0
 t = 3.0
 t = 5.0
n(
x,
t)
x
-4 -3 -2 -1 0 1 2 3 4
0
1
2
3
4
5
Fig.(2c)
(2c)  r = 0.01
 r = 0.1
 r = 1.0
 r = 2.0
 r = 3.0
 r = 5.0
n(
x,
t)
x


Publications of the IAS Fellows

ar
X
iv
:p
hy
sic
s/0
20
30
92
v1
  [
ph
ys
ics
.bi
o-
ph
]  
29
 M
ar 
20
02
Class of self-limiting growth models in the presence of nonlinear
diffusion
Sandip Kar, Suman Kumar Banik and Deb Shankar Ray∗
Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India
(February 2, 2008)
Abstract
The source term in a reaction-diffusion system, in general, does not involve
explicit time dependence. A class of self-limiting growth models dealing with
animal and tumor growth and bacterial population in a culture, on the other
hand are described by kinetics with explicit functions of time. We analyze a
reaction-diffusion system to study the propagation of spatial front for these
models.
PACS number(s): 87.10.+e, 87.15.Vv, 05.45.-a
Typeset using REVTEX
∗Electronic-mail: pcdsr@mahendra.iacs.res.in
1
I. INTRODUCTION
Reaction-diffusion systems are ubiquitous in almost all branches of physics [1], chemistry
[2] and biology [3–5] dealing with population growth, fluid dynamics, pulse propagation in
nerves, chemical reactions, optical and other processes. The basic equation describes the
dynamics of a field variable n(x, t), a function of space and time in terms of a source term
(also known as reaction term) and a diffusion term. An important early endeavor in this
direction is the study of self-limiting growth models of which the most well-known is the
Fisher equation [6,7] which takes into account of a linear growth and a nonlinear decay.
The model and many of its variants have found wide applications both from theoretical
and experimental point of view [4]. A notable feature of these models is that the source or
the reaction terms do not involve any explicit time dependence. On the other hand there
are situations [8–10] where the source terms contain explicit functions of time which put a
constraint on the growth process in the long time limit. For example, the Gompertz growth
[8,9] is a model used for study of growth of animals and tumors, where the growth rate is
proportional to the current value, but the proportionality factor decreases exponentially in
time so that
dn
dt
= rn exp(−αt) , (1a)
where r and α are positive experimentally determined constants. Similarly another type of
model proposed to analyze the growth of bacterial population in culture [10] is described by
dn
dt
= knt exp(−βt2) . (1b)
Again k and β are positive constants required to fit the experimental data. An important
feature of these models is that unlike the logistic growth process the asymptotic value of the
density function n depends on its initial population.
Keeping in view of these experimental observations it is therefore worthwhile to generalize
the specific cases in terms of an explicit function of time φ(t) such that we write
2
dn
dt
= rnφ(t) (2)
where r is a constant for the growth process and φ(t) may of the type (i) φ(t) = 1 for
exponential growth (ii) φ(t) = exp(−αt) for Gompertz growth (iii) φ(t) = t exp(−βt2) for
bacterial growth, etc.
The object of the present paper is to study a reaction-diffusion system with a reaction
term describing a class of self-limiting growth processes (2). Since in many living organisms
concentration dependent diffusivity [4,5,11–15] has been found to be essential to the modeling
of reaction-diffusion systems we investigate the interplay of this nonlinear diffusion and self-
limiting growth process in the dynamics. We show that the model and its variant with a
finite memory transport [16–25] admit of exact solutions. The dependence of the rate of
spread of the wave front on various parameters is explored.
II. THE REACTION-DIFFUSION SYSTEM
We consider a reaction-diffusion system with a source term describing self-limiting growth
and with a nonlinear diffusion term in the following form:
∂n(x, t)
∂t
= rnφ(t) +
∂
∂x
Dn
∂n
∂x
(3)
where D is the diffusion coefficient. Our primary aim in this section is to provide an exact
solution of Eq.(3). To this end we first make use of the following transformation
n(x, t) = u˜(x, t) exp
(
r
∫ t
0
φ(t′)dt′
)
(4)
in Eq.(3) to obtain
∂u˜(x, t)
∂t
= D exp
(
r
∫ t
0
φ(t′)dt′
)
∂
∂x
{
u˜
∂u˜
∂x
}
. (5)
We now introduce the scaled time variable τ as
τ = D
∫ t
0
f(t′)dt′ ≡ G(t) (say) (6a)
3
where
f(t) = exp[r
∫ t
0
φ(t′)dt′] . (6b)
This reduces Eq.(5) to the following form
∂u(x, τ)
∂τ
=
∂
∂x
{
u(x, τ)
∂u(x, τ)
∂x
}
(7)
with u˜(x, t) ≡ u˜(x,G−1(τ)) = u(x, τ) where time t has been expressed as an inverse function
G−1(τ) according to Eq.(6a-6b).
Eq.(7) is the well-known Boltzmann nonlinear diffusion equation [1,26]. Now subject to
the initial condition of a unit point source at the origin,
n(x, 0) = δ(x) = u˜(x, 0) = u(x, 0) (8)
we solve Eq.(7) under the following boundary conditions
lim
x→±∞
u(x, τ) = 0 ∀τ > 0 (9)
and
∫ +∞
−∞
u(x, τ)dx = 1 ∀τ > 0 . (10)
Next we seek the similarity solution of the nonlinear diffusion equation (7). We make
use of the well-known similarity transformation [1,5,26,27]
u = τ−1/3v(z) and z = xτ−1/3 (11)
in Eq.(7) to obtain
3
d
dz
(
v
dv
dz
)
+ v + z
dv
dz
= 0 (12)
On integration Eq.(12) yields
3
(
v
dv
dz
)
+ zv = 0 (13)
4
Since we are interested in the symmetric solutions with v′(0) = 0 we have put the integration
constant zero in going from Eq.(12) to (13). On further integration Eq.(13) results in the
solution
v(z) = (A2 − z2)/6 |z| < A
= 0 |z| > A
(14a)
where A is a constant which can be determined from the condition (10) to obtain
A = (9/2)1/3 (14b)
Therefore the solution of Eq.(7) in x and τ is given by
u(x, τ) =
1
6τ
[
A2τ 2/3 − x2
]
|x| < Aτ 1/3
= 0 |x| > Aτ 1/3 (15)
It is interesting to note that by virtue of the relations (6a-6b) τ is dependent on r and
φ(t) which control the growth and self-limiting factors, respectively of the source term.
This implies that the shock-wave like behaviour with propagating wave-front at x = xf =
Aτ 1/3 as evident from the similarity solutions (15) critically depends on the reaction terms.
Specifically, the wave front propagates in the medium with speed
dxf
dt
=
1
3
(
9D
2
)1/3
f(t)
[∫ t
0
f(t′)dt′
]−2/3
(16)
where f(t) is given by (6a) and in turn depends on the functional form of φ(t).
We now consider two specific cases to illustrate the spatial propagation patterns.
(i) φ(t) = 1
For a constant value of φ the model suggests an exponential growth. The relation (6a)
in this case can then be utilized to obtain f(t) = exp(rt) so that τ = (D/r)[exp(rt) − 1].
Putting this expression for τ in the solution (15) we have after using Eq.(4)
n(x, t) =
[
A2 {(D/r) (exp(rt)− 1)}2/3
]
− x2
(6D/r)[exp(rt)− 1] exp(−rt)
(17)
This solution clearly has a sharp wave-front at xf = Aτ
1/3 which propagates at a speed
5
dxf
dt
=
1
3
A
(
Dr2
)1/3
exp(rt) (exp(rt)− 1)−2/3 (18)
To illustrate the spatial propagation of the population n(x, t) in time we plot in Fig-1
the spatial shock-wave like patterns for r = 1.0 and D = 1.0. It is apparent that the sharply
peaked distribution at t = 0 starts spreading relatively slowly with peak at x = 0 diminishing
with time upto a period t = 0.1 . Beyond this time the spatial growth of population becomes
comparatively large and it diverges due to the combined effect of exponential growth and
nonlinear diffusion. For a much lower growth rate (r = 0.001), however, the population
spreads monotonically due to the nonlinear diffusion which overwhelms the effect of growth
process. This is evident in Fig-1(b).
(ii) φ(t) = t exp(−βt2)
With the above expression for φ(t) for bacterial self-limiting growth we obtain from
(6a-6b)
f(t) = exp[(−r/2β)(exp(−βt)− 1)] (19)
and
τ = D exp(r/2β)
∫ t
0
exp[(−r/2β) exp(−βt′)]dt′ (20)
By defining z = (r/2β) exp(−βt) the above expression can be reduced to the following form
τ = −D
exp(r/2β)
β
∫ (r/2β) exp(−βt)
(r/2β)
exp(−z)
z
dz (21)
The integral in (21) can be put into a standard form with the help of Ei-function [28] so
that τ can be expressed as
τ = D
exp(r/2β)
β
[Ei (−r/2β)− Ei ((−r/2β) exp(−βt))] (22)
The corresponding density n(x, t) and the speed of the wave front dxf/dt at xf are given by
n(x, t) =
A2 (D exp(r/2β)/β)2/3 [Ei (−r/2β)− Ei (−(r/2β) exp(−βt))]2/3 − x2
(6D/β) [Ei(−r/2β)− Ei((−r/2β) exp(−βt))] exp[(r/2β) exp(−βt)]
(23)
and
6
dxf
dt
= A
(
D
exp(r/2β)
β
)1/3
d
dt
[Ei(−r/2β)− Ei((−r/2β) exp(−βt))]1/3 (24)
respectively.
In Fig-2(a,b) we show the shock-wave like spread of population by plotting n(x, t) vs
x for several values of time for D = 1 and r = 1 . Since β puts a limit to the growth at
large time the peak of n(x, t) at x = 0 as shown in Fig-2(a) (β = 0.1) does not increase too
much as compared to the earlier case considered in Fig-1(a). It has been observed that for
a unique value of β ≥ 1.0 there is a monotonic decrease in the peak population n(x, t) at
x = 0. For smaller values of β (Fig-2(b)) the spread is similar to that in Fig-1(a). In Fig-2(c)
we exhibit the spatial front propagation for several values of growth rate r at a time t = 1.0
keeping D = 1 and β = 0.01. It is apparent that with increase of r the reaction dominates
over diffusion so that the peak population at x = 0 increases compared to spreading.
III. EFFECT OF FINITE MEMORY TRANSPORT
We now generalize the proposed reaction-diffusion model to include the effect of finite
memory transport. It has been observed that an animal’s movement at a particular instant of
time often depends on its motion in the immediate past. This results in a delay in population
flux, or a memory in the diffusion coefficient. A number of attempts have been made in
the recent literature [16–25] to analyze the delayed population growth in several models and
related context in heat conduction and transport processes. To consider a finite memory in
the present model we modify the nonlinear diffusion term in Eq.(3) to the following form:
∂n(x, t)
∂t
= rn(x, t)φ(t) +
∂
∂x
[
Dγ
∫ t
0
exp[−γ(t− τ)]n(x, τ)
∂n(x, τ)
∂x
dτ
]
(25)
where γ refers to the inverse of relaxation time. The population flux takes into account
of the relaxation effect due to the delay of the particles in adopting a definite direction of
propagation. Differentiating both sides of the above equation with respect to t and using it
again we obtain
7
∂2n
∂t2
= (rφ− γ)
∂n
∂t
+ (rφ˙+ rφγ)n+
∂
∂x
[
Dγn
∂n
∂x
]
(26)
In the limit of vanishing relaxation time i.e, 1/γ →0 Eq.(26) reduces to Eq.(3). When
memory effects are taken into account, the dispersal of the organisms are not mutually inde-
pendent. Hence the correlation between the successive movement of the diffusing particles
results in a delay in the transport. Thus Eq.(26) is a typical form of a delayed transport
equation.
We now consider a specific case φ(t) = 1. Substitution of the traveling wave form N(z)
(= n(x, t)) with z = x+ ct satisfies
c2
∂2N
∂z2
= c(r − γ)
∂N
∂z
+ rγN +Dγ
∂
∂z
(
N
∂N
∂z
)
(27)
where c is the speed of the traveling wave to be determined.
We now consider the trial solution of Eq.(27) of the form N(z) = N0 exp(sz
b) subject
to the initial condition that at z = 0, N = N0, where s and b are positive constants to be
determined. Substitution of this solution in Eq.(27) yields the following relation
[c2s2b2z2(b−1) + c2sb(b− 1)z(b−2) − csb(r − γ)z(b−1)
−rγ] exp(szb)−DγN0sb[2sbz
2(b−1) + (b− 1)z(b−2)]
× exp(2szb) ≡ L(z) = 0 (28)
For L(z) = 0, for all z, the coefficients of exp(szb) and exp(2szb) within the square brackets
must vanish identically. For this the only acceptable solution for b is b = 1. We obtain
2s2DγN0 = 0 (29a)
and
c2s2 − cs(r − γ)− rγ = 0 (29b)
From the above two equations the solution for s is given by
s =
c[(1/γ)− (1/r)] + [c2 ((1/γ)− (1/r))2 + 4/r ((c2/γ) + 2DN0)]
1/2
2 [(c2/γr) + (2DN0/r)]
(30)
8
In the limit of instantaneous relaxation, i.e, 1/γ →0 Eq.(30) yields
s =
c
[
−1 + (1 + (DN0r/c
2))
1/2
]
4DN0
(31)
Furthermore the above expression in the limit of weak diffusion D →0 we obtain from
Eq.(31) after a Taylor expansion
s =
r
c
(32)
To determine the speed of the propagation of the wave front we now rearrange the
solution for s in (30) to obtain
c =
(r − γ) + [(r − γ)2 + 4(rγ − 2s2DγN0)]
1/2
2s
(33)
For real values of c, the quantity inside the square root must be positive, which determines
the minimum value of c for s = r/c [ Eq.(32) ] as
cmin =
2r2DγN0
(r + γ)2
(34)
Eq.(27) therefore admits of an exact traveling wave like solution
N(z) = N0 exp

c(r − γ) + (c2(r − γ)2 + 4rγ(c2 + 2DγN0))1/2
2(c2 + 2DγN0)

 z (35)
It is interesting to observe that the speed of the traveling wave front not only depends
on nonlinear diffusion and growth rate but also on the initial concentration and memory. A
comparison of the solutions in this section and in the previous one shows that (35) does not
reduces to Eq.(17) in the limit of vanishing relaxation time (1/γ → 0) although Eq.(26) goes
over to Eq.(3) under this condition. This is because of the fact that the nature of the partial
differential equation changes due to the inclusion of relaxation terms and also the boundary
conditions for the shock wave like ‘diffusing solutions’ (17) are different for the travelling
wave front solution (35). The nature of the two solutions are thus generically different. We
point out in passing that the dependence on initial concentration on speed as shown in (34)
is rather an unusual feature in reaction-diffusion system.
9
IV. CONCLUSIONS
In this paper we have analyzed a class of reaction-diffusion systems in which the kinetic
term describes the self-limiting growth processes of the Gompertz type and is an explicit
function of time. We have shown that the model can be solved exactly to analyze the spatial
front propagation problem. To make the model more realistic we have included the effect
of finite relaxation to concentration-dependent diffusive processes. In view of the fact that
the source terms have their direct relevance on experimental measurement on animal and
tumor growth or bacterial culture we think that the solutions discussed in this paper will
be pertinent in the context of reaction-diffusion systems, in general.
ACKNOWLEDGMENTS
The authors are indebted to C.S.I.R. (Council of Scientific and Industrial Research),
Government of India, for financial support.
10
REFERENCES
[1] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers
(Birkha¨user, Boston, 1997).
[2] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics:
Oscillations, Waves, Patterns and Chaos (Oxford University Press, New York, 1998).
[3] N. F. Britton, Reaction-Diffusion Equations and their applications to Biology (Aca-
demic, New York, 1986).
[4] J. D. Murray, Mathematical Biology, Second Corrected Edition (Springer, Berlin, 1993).
[5] M. Kot, Elements of Mathematical Ecology (Cambridge University Press, Cambridge,
2001).
[6] R. A. Fisher, Ann. Eugenics. 7, 355 (1937).
[7] A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, in Selected Works of A. N.
Kolmogorov, edited by V. M. Tikhomirov (Kluwer, Dordrecht, 1991).
[8] J. Folkman and M. Hochberg, J. Exp. Med. 4, 745 (1973).
[9] A. K. Laird, Brit. J. Cancer 18, 490 (1964); G. Klein and L. Re´ve´sz, J. Nat. Cancer
Inst. 14, 229 (1953).
[10] D. Kaplan and L. Glass, Understanding Nonlinear Dynamics (Springer-Verlag, New
York, 1995).
[11] N. Shigesada, J. Math. Bio. 9, 85 (1980).
[12] A. Okubo, Diffusion and Ecological Problems: Mathematical Models (Springer, Berlin,
1980).
[13] M. R. Roussel and J. Wang, Phys. Rev. Lett. 87, 188302 (2001).
[14] H. Malchow, J. Theor. Biol. 135, 371 (1988).
11
[15] M. Mimura and K. Kawasaki, J. Math. Bio. 9, 49 (1980).
[16] C. Cattaneo, C. R. Acad. Sci. 247, 431 (1958).
[17] K. P. Hadeler, in Reaction Transport Systems in Mathematics Inspired by Biology, edited
by V. Capasso and O. Diekmann, CIME Lectures, Florence (Springer-Verlag, Berlin,
1998).
[18] T. Hillen, Math. Models Methods Appl. Sci. 8, 507 (1998).
[19] V. Me´ndez, J. Fort and J. Farjas, Phys. Rev. E 60, 5231 (1999).
[20] W. Horsthemke, Phys. Rev. E 60, 2651 (1999).
[21] K. K. Manne, A. J. Hurd and V. M. Kenkre, Phys. Rev. E 61, 4177 (2000).
[22] G. Abramson, A. R. Bishop and V. M. Kenkre, Phys. Rev. E 64, 066615 (2001).
[23] S. Fedotov, Phys. Rev. Lett. 86, 926 (2001).
[24] J. M. Sancho and A. Sa´nchez, Phys. Rev. E 63, 056608 (2001).
[25] R. D. Benguria and M. C. Depassier, nlin.PS/0202003.
[26] J. Crank, The Mathematics of Diffusion, Second Edition (Oxford University Press,
Oxford, 1975).
[27] G. Birkhoff, Hydrodynamics (Princeton University Press, Princeton, 1950).
[28] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic,
New York, 1980).
12
FIGURES
FIG. 1. Evolution of spatial front in time for the model with φ(t) = 1. (a) The population
n(x, t) is plotted against x for different times using r = 1.0 and D = 1.0. (b) Same as in Fig.(1a)
but for r = 0.001 (units arbitrary).
FIG. 2. Evolution of spatial front in time for the model with φ(t) = t exp(−βt2). (a) The
population n(x, t) is plotted against x for different times using r = 1.0, D = 1.0 and β = 0.1. (b)
Same as in Fig.(2a) but for β = 0.01. (c) The population n(x, t) is plotted against x at t = 1.0 for
different r using D = 1.0 and β = 0.01 (units arbitrary).
13
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0
1
2
3
4
5
(1a)
Fig.(1a)
 t = 0.001
 t = 0.01
 t = 0.1
 t = 1.0
 t = 2.0
 t = 3.0
n
(
x
,
t
)
x
-3 -2 -1 0 1 2 3
0
1
2
3
4
5
(1b)
Fig.(1b)
 t = 0.001
 t = 0.01
 t = 0.1
 t = 1.0
n
(
x
,
t
)
x
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
1
2
3
4
5
Fig.(2a)
(2a)
 t = 0.001
 t = 0.01
 t = 0.1
 t = 1.0
 t = 2.0
 t = 3.0
 t = 5.0
n
(
x
,
t
)
x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0
1
2
3
4
5
Fig.(2b)
(2b)
 t = 0.001
 t = 0.01
 t = 0.1
 t = 1.0
 t = 2.0
 t = 3.0
 t = 5.0
n
(
x
,
t
)
x
-4 -3 -2 -1 0 1 2 3 4
0
1
2
3
4
5
Fig.(2c)
(2c)
 r = 0.01
 r = 0.1
 r = 1.0
 r = 2.0
 r = 3.0
 r = 5.0
n
(
x
,
t
)
x


Class of self-limiting growth models in the presence of nonlinear
  diffusion

Class of self-limiting growth models in the presence of nonlinear diffusion

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