^{1}

^{2}

^{*}

^{2}

^{2}

^{2}

^{2}

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 [

where

On the other hand concentration of the bacteria is described by the equation [

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

the presence of source term

The second tern on the right side of Equation (3) describes the chemotactic response of the species. In

Equation (3),

chemotaxis, and is termed as chemotactic coefficient. The function

Memory effect in DR equation arises when dispersal of the particle is not mutually independent [

is given by

At

Thus, at

where

the coefficient of nonlinear convective flux term

rate of change of concentration at time t and

Here

(6) reduces to Fisher-Burger equation with finite memory [

where

By taking

and

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

Simplifying Equations (5) and (10) one obtains

In Equation (11) we will take

Now using the transformation

For

From Equation (13) one can see that

In Equation (7) if we take

Now comparing Equations (17) and (18) one can see that

that

velocity term v. Similarly, by measuring

For Fisher type reaction term

where

equation without finite memory transport

Solutions of Equations (19) and (20) is already obtained by us in ref. [

and

where wave velocity w is given by

Equation (21) is a solitary wave solution of Equation (19) whereas solution (22) diverges. Since

and

and wave speed w as

Equation (24) is again a solitary wave solution of Equation (20) while Equation (25) is physically not acceptable. Now using Equation (13), we have

where C is constant of integration. Substituting the value of

and

where w is given by Equation (23). Similiarly, substituting the value of

and

where w is given by Equation (26). From Equations (28)-(31) one can see that concentration

Certain aspects ignored earlier [

that coefficient of non-linear convective flux,

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

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.

Bhupendra Singh,Loukrakpam Kennedy Meitei,Ranjit Kumar,Varun Malik,Yogesh Kumar,Nihal Kumar, (2016) Memory Effect in Chemotaxis Equation. Journal of Modern Physics,07,1105-1111. doi: 10.4236/jmp.2016.710099