Diffusion-Reaction (DR) equation has been used to model a large number of phenomena in nature. It may be mentioned that a linear diffusion equation does not exhibit any traveling wave solution. But there are a vast number of phenomena in different branches not only of science but also of social sciences where diffusion plays an important role and the underlying dynamical system exhibits traveling wave features. In contrast to the simple diffusion when the reaction kinetics is combined with diffusion, traveling waves of chemical concentration are found to exist. This can affect a biochemical change, very much faster than straight diffusional processes. This kind of coupling results into a nonlinear (NL) DR equation. In recent years, memory effect in DR equation has been found to play an important role in many branches of science. The effect of memory enters into the dynamics of NL DR equation through its influence on the speed of the travelling wavefront. In the present work, chemotaxis equation with source term is studied in the presence of finite memory and its solution is compared with the corresponding chemotaxis equation without finite memory. Also, a comparison is made between Fisher-Burger equation and chemotaxis equation in the presence of finite memory. We have shown that nonlinear diffusion-reaction-convection equation is equivalent to chemotaxis equation.
Diffusion-Reaction (DR) equation has been used to explain many phenomena in nature [
In addition to diffusion, there are many phenomena in nature in which convection velocity term also becomes important [
Nonlinear convective flux term arises naturally in the study of chemotaxis equation [
For example, the female silk moth Bombyx mori exudes a pheromone, called bombykol, as a sex attractant for the male, which has a remarkably efficient antenna filter to measure the bombykol concentration, and it moves in the direction of increasing concentration. The acute sense of smell of many deep sea fish is particularly important for communication and predation [
It has been shown that certain species of bacteria and insects can move toward higher concentrations of nutrients [
∂ s ∂ t = − G ( s ) u + D 0 ∂ 2 s ∂ x 2 , (1)
where G ( s ) is the concentration of the attractant per cell, u ( x , t ) is the density of the bacteria and D 0 is the diffusion constant of the attractant. For most of the practical purpose, G ( s ) is taken as constant and diffusion constant D 0 is also neglected [
∂ s ∂ t = − G u . (2)
On the other hand concentration of the bacteria is described by the equation [
∂ u ∂ t = ∂ ∂ x [ μ ( s ) ∂ u ∂ x ] − ∂ ∂ x [ u χ ( s ) ∂ s ∂ x ] + f ( u ) , (3)
where the first tern on the right side represent the motion of the bacteria in the absence of chemotaxis. In the
absence of chemical gradient ( ∂ s ∂ x = 0 ) , Equation (3) becomes identical to the diffusion-reaction equation in
the presence of source term f ( u ) . Here motility factor, μ , takes the place of the diffusion coefficient D. In Equation (3) μ is taken as function of substrate concentration s. In principle μ could also vary with bacterial concentration u and space variable x. But the effect of substrate concentration is not known at present [
The second tern on the right side of Equation (3) describes the chemotactic response of the species. In
Equation (3), u χ ( s ) [ ∂ s ∂ x ] is flux of species due to chemotaxis where χ ( s ) is a measure of strength of
chemotaxis, and is termed as chemotactic coefficient. The function χ is also called the chemotactic sensitivity function. In the next section we will discuss memory effect in DR equation.
Memory effect in DR equation arises when dispersal of the particle is not mutually independent [
∂ u ∂ t = D ∂ 2 u ∂ x 2
is given by
u ( t , x ) = 1 4 D π t exp ( − x 2 4 D t )
At t = 0 , the solution of the equation is Dirac delta function, i.e.,
u ( x , t = 0 ) = δ ( x )
Thus, at t = 0 all the particles are sitting at x = 0 . For t > 0 , the solution of the equation is non zero for all x. If we take a value of x such that | x | > c t , where c is the speed of light, we see that there is a finite probability, however small, for particles to diffuse at superluminal speeds. The error lies in the diffusion equation itself, which does not recognize any limiting propagation speed. Thus it becomes necessary to include memory effect, which takes care of the finite speed. When memory effect is taken into account then we have the following modification of Fick’s law [
J ( x , t + τ ) = − D ∂ u ∂ x + g ( u ) v , (4)
∂ u ∂ t = − ∂ J ∂ x + f ( u ) , (5)
where u = u ( x , t ) , is the concentration or the density variable depending on the phenomenon under study; v is
the coefficient of nonlinear convective flux term g ( u ) and D is the diffusion coefficient. Here ∂ u ∂ t is the time
rate of change of concentration at time t and J ( x , t + τ ) is the flux at a later time t + τ , while f ( u ) is the source term. Here τ is delay time and its value depends on the system under study [
u t t − β D u x x − f ′ ( u ) u t + β ( u t − f ( u ) ) + β v g ′ ( u ) u x = 0. (6)
Here β ≡ 1 τ , f ′ ( u ) = d f d u and g ′ ( u ) = d g d u . In particular for f ( u ) = α u − γ u 2 and g ( u ) = u 2 , Equation
(6) reduces to Fisher-Burger equation with finite memory [
u t t − β D u x x − f ′ ( u ) u t + β ( u t − f ( u ) ) + k β u u x = 0
where k ≡ 2 v . Note that Equation (6) describe a transport phenomenon in which both diffusion and convection processes are of equal importance. After using the transformation ξ = x − w t in Equation (6), we get
( w 2 − β D ) u ″ + ( f ′ ( u ) − β ) w u ′ − β f ( u ) + β v g ′ ( u ) u ′ = 0. (7)
By taking τ = 1 / β = 0 in Equations (6) and (7) one obtain the corresponding DR equation without finite memory transport
u t + v g ′ ( u ) u x = D u x x + f ( u ) , (8)
and
D u ″ + w u ′ − v g ′ ( u ) u ′ + f ( u ) = 0. (9)
One can see from above that Equations (6) and (7) are hyperbolic nonlinear DR equation while Equations (8) and (9) are parabolic nonlinear DR equation.
In the presence of finite memory, Equation (5) remains unchanged while Equation (4) gets modified to
J ( x , t + τ ) = − [ μ ( s ) ∂ u ∂ x − u χ ( s ) ∂ s ∂ x ] . (10)
Simplifying Equations (5) and (10) one obtains
u t t + ( β − f ′ ( u ) ) u t − β f ( u ) − β ∂ ∂ x [ μ ( s ) ∂ u ∂ x ] + β ∂ ∂ x [ u χ ( s ) ∂ s ∂ x ] = 0. (11)
In Equation (11) we will take μ ( s ) = μ 0 , χ ( s ) = χ 0 . Afetr this substitution Equation (11) becomes
u t t + ( β − f ′ ( u ) ) u t − β f ( u ) − β μ 0 u x x + β χ 0 s x u x + β χ 0 u s x x = 0. (12)
Now using the transformation ξ = x − w t in Equations (2) and (12) we get
s ′ = G w u , (13)
( w 2 − β μ 0 ) u ″ + ( f ′ ( u ) − β ) w u ′ − β f ( u ) + β χ 0 s ′ u ′ + β χ 0 u s ″ = 0. (14)
For τ = 1 / β = 0 , Equations (12) and (14) becomes
u t − μ 0 u x x + χ 0 s x u x + χ 0 u s x x = f ( u ) , (15)
μ 0 u ″ + w u ′ − χ 0 s ′ u ′ − χ 0 u s ″ + f ( u ) = 0. (16)
From Equation (13) one can see that s ″ = G u ′ / w . Substituting the value of s ′ and s ″ from Equation (13) into Equation (14) one obtains
( w 2 − β μ 0 ) u ″ + ( f ′ ( u ) − β ) w u ′ − β f ( u ) + 2 β χ 0 G w u u ′ = 0. (17)
In Equation (7) if we take g ( u ) = u 2 , then we obtain the following equation
( w 2 − β D ) u ″ + ( f ′ ( u ) − β ) w u ′ − β f ( u ) + 2 β v u u ′ = 0. (18)
Now comparing Equations (17) and (18) one can see that D = μ 0 and v = χ 0 G / w . Under this condition Equations (7) and (17) becomes identical. Note that here v is coefficient of nonlinear convection term and from Equation (17) one can see that this coefficient depends on χ 0 , G and wave velocity w. Thus, we have mapped the chemotaxis equation to nonlinear diffusion-reaction-convection equation. In this mapping we have assumed
that v = χ 0 G w and hence by measuring χ 0 , G and w, experimentally, we can find the velue of convective
velocity term v. Similarly, by measuring μ 0 experimentally, we can determine the diffusion coefficient D.
For Fisher type reaction term f ( u ) = α u − γ u 2 , Equation (17) or (18) takes the form
( w 2 − β D ) u ″ + w ( α − β ) u ′ + ( k β − 2 γ w ) u u ′ − β α u + β γ u 2 = 0 , (19)
where k ≡ 2 v = 2 χ 0 G w . By taking τ = 1 / β = 0 in Equation (19) one obtain the corresponding nonlinear DR
equation without finite memory transport
D u ″ + w u ′ − k u u ′ + α u − γ u 2 = 0. (20)
Solutions of Equations (19) and (20) is already obtained by us in ref. [
u ( ξ ) = α 2 γ [ 1 − tanh ( γ k α ( α + β ) ( 4 D β γ 2 − α 2 k 2 ) ξ ) ] , (21)
and
u ( ξ ) = α 2 γ [ 1 − coth ( γ k α ( α + β ) ( 4 D β γ 2 − α 2 k 2 ) ξ ) ] , (22)
where wave velocity w is given by
w = β ( α k 2 + 4 D γ 2 ) 2 γ k ( α + β ) . (23)
Equation (21) is a solitary wave solution of Equation (19) whereas solution (22) diverges. Since u ( ξ ) represent the concentration of certain species which cannot go to infinity hence solution (22) is physically not acceptable. On the other hand solutions of Equation (20) is given by
u ( ξ ) = α 2 γ [ 1 − tanh ( α k 4 γ D ξ ) ] , (24)
and
u ( ξ ) = α 2 γ [ 1 − coth ( α k 4 γ D ξ ) ] , (25)
and wave speed w as
w = α k 2 + 4 D γ 2 2 γ k . (26)
Equation (24) is again a solitary wave solution of Equation (20) while Equation (25) is physically not acceptable. Now using Equation (13), we have
s ( ξ ) = ∫ G w u ( ξ ) d ξ + C , (27)
where C is constant of integration. Substituting the value of u ( ξ ) from Equations (21) and (22) in Equation (27), we obtain the following value of s ( ξ ) of Equation (13) as
s ( ξ ) = G α 2 γ w [ ξ − ( 4 D β γ 2 − α 2 k 2 ) γ k α ( α + β ) ln [ cosh ( γ k α ( α + β ) ( 4 D β γ 2 − α 2 k 2 ) ξ ) ] ] + C , (28)
and
s ( ξ ) = G α 2 γ w [ ξ − ( 4 D β γ 2 − α 2 k 2 ) γ k α ( α + β ) ln [ sinh ( γ k α ( α + β ) ( 4 D β γ 2 − α 2 k 2 ) ξ ) ] ] + C , (29)
where w is given by Equation (23). Similiarly, substituting the value of u ( ξ ) from Equations (24) and (25) in Equation (27), we obtain the following value of s ( ξ ) of Equation (13) as
s ( ξ ) = G α 2 γ w [ ξ − 4 γ D α k l n [ c o s h ( α k 4 γ D ξ ) ] ] + C , (30)
and
s ( ξ ) = G α 2 γ w [ ξ − 4 γ D α k l n [ s i n h ( α k 4 γ D ξ ) ] ] + C , (31)
where w is given by Equation (26). From Equations (28)-(31) one can see that concentration s ( x , t ) of attractant increases with G and decreases as wave speed w increases. Also, it is directly proportional to coef- ficient of linear term α and inversely proportional to coefficient of nonlinear term γ .
Certain aspects ignored earlier [
that coefficient of non-linear convective flux, v = χ 0 G w , decreases as wave speed w increases. Thus, using the
nonlinear Diffusion-Reaction-Convection equation one can find the coefficient of nonlinear convective flux v, of chemotaxis equation.
Although result obtained in this paper is highly simplified, the solutions obtained here can explain such physical phenomena which is governed by chemotaxis equation. The case when μ and χ depend on bacteria concentration u and space variable x will be discussed in a separate paper. Such studies are in progress.
We would like to thank R. S. Kaushal, Awadhesh Prasad and Ram Ramaswamy for helpful discussion. We would also like to thank Dyal Singh College for providing us the computational facility during the course of this work.