**1. Introduction**

This chapter uses the publications of Shubina M.V.:


Chemotaxis, or the directed cell (bacteria or other organisms) movement up or down a chemical concentration gradient, plays an important role in many biological and medical fields such as embryogenesis, immunology, cancer growth, and invasion. The macroscopic classical model of chemotaxis was proposed by Patlak in 1953 [1] and by Keller and Segel in the 1970s [2–4]. Since then, the mathematical modeling of chemotaxis has been widely developed. This model is described by the system of coupled nonlinear partial differential equations. Proceeding from the study of the properties of these equations, it is concluded that the model demonstrates a deep mathematical structure. The survey of Horstmann [5] provides a detailed

introduction into the mathematics of the Patlak-Keller-Segel model and summarizes different mathematical results; the detailed reviews also can be found in the textbooks of Suzuki [6] and Perthame [7]. In the review of Hillen and Painter [8], a number of variations of the original Patlak-Keller-Segel model are explored in detail. The authors study their formulation from a biological perspective, summarize key results on their analytical properties, and classify their solution forms [8]. It should be noted that interest in the Patlak-Keller-Segel model does not weaken and new works appear devoted to the study of various properties of equations and their solutions [9–12] and the links below.

After the above replacements, the model reads:

*DOI: http://dx.doi.org/10.5772/intechopen.91214*

*y* ¼ *x* � *ct*, where *c* ¼ *const*, this system has the form:

nonhomogeneous equation] and *η* 6¼ 1.

and Eq. (4) becomes:

where *<sup>ν</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

**3**

*z*2

� � and <sup>Λ</sup> ¼ � <sup>4</sup>*β<sup>b</sup>*

*<sup>J</sup>ν*ð Þ*<sup>z</sup> CJ*ð Þ*<sup>z</sup>* � �

*<sup>J</sup><sup>ν</sup>* ð Þ ð Þ*<sup>z</sup> <sup>z</sup> CJ*ð Þ*<sup>z</sup>* � �

Considering that Wronskian *W J<sup>ν</sup>* ð Þ , *<sup>Y</sup><sup>ν</sup>* ð Þ¼ *<sup>z</sup>* <sup>2</sup>

*<sup>α</sup>*<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>*αβ c*2

equation [18, 19] with *<sup>μ</sup>* ¼ �<sup>1</sup> � <sup>1</sup>

of *z* that satisfy the equations:

A. *η* ¼ 1

(

where *u* ¼ *u y* ð Þ, *υ* ¼ *υ*ð Þ*y* , and *λ* is an integration constant.

In this chapter we will consider the case of *λ* ¼ 0. Then Eq. (2) gives:

*u* ¼ *Cue*

*Cu* is a constant and we will examine the following equation for *υ*:

*αυyy* þ *cυ<sup>y</sup>* � *βυ* þ *βb* þ *σCue*

Since *η* is a positive constant, we consider two cases: *η* ¼ 1 [Eq. (4) is a linear

Let us begin with *η* ¼ 1. We introduce the new variable *z* and the new function *w*:

*wzz* <sup>þ</sup> *zwz* <sup>þ</sup> *w z*<sup>2</sup> � *<sup>ν</sup>*<sup>2</sup> � � <sup>¼</sup> <sup>Λ</sup>*z*�<sup>1</sup>

4*σCu αc*<sup>2</sup> � �<sup>1</sup> 4

*αc*<sup>2</sup>

inhomogeneous second-order differential equation, one can integrate it by the

*w z*ð Þ¼ *CJ*ð Þ*z Jν*ð Þþ *z CY*ð Þ*z Yν*ð Þ*z* ,

where *Jν*ð Þ*z* and *Yν*ð Þ*z* are Bessel functions and *CJ*ð Þ*z* and *CY*ð Þ*z* are the functions

*<sup>z</sup>* þ *Yν*ð Þ*z* ð Þ *CY*ð Þ*z <sup>z</sup>* ¼ 0

*<sup>z</sup>* <sup>þ</sup> ð Þ *<sup>Y</sup>ν*ð Þ*<sup>z</sup> <sup>z</sup>* ð Þ *CY*ð Þ*<sup>z</sup> <sup>z</sup>* <sup>¼</sup> <sup>Λ</sup>*z*�<sup>1</sup>

*πz*

, we obtain:

method of variation of parameters. We assume a solution in the form:

*<sup>z</sup>* <sup>¼</sup> <sup>4</sup>*σCu αc*<sup>2</sup> � �<sup>1</sup> 2 *e* �*cy* 2

*<sup>w</sup>* <sup>¼</sup> <sup>4</sup>*σCu αc*<sup>2</sup> � �*<sup>α</sup>*�<sup>2</sup> 4*α υe cy* 2*α*

*ut* � *uxx* <sup>þ</sup> *<sup>η</sup> <sup>u</sup> vx*

*v*þ*b* � �

*vt* � *αvxx* � *σu* þ *βv* ¼ 0*:*

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis*

*uy* þ *cu* � *ηu*ð Þ ln ð Þ*υ <sup>y</sup>* þ *λ* ¼ 0 *αυyy* <sup>þ</sup> *<sup>c</sup>υ<sup>y</sup>* � *βυ* <sup>þ</sup> *<sup>β</sup><sup>b</sup>* <sup>þ</sup> *<sup>σ</sup><sup>u</sup>* <sup>¼</sup> 0, (

�*cyυη*

If we introduce the function *υ* ¼ *v* þ *b*, in terms of traveling wave variable

*<sup>x</sup>* <sup>¼</sup> <sup>0</sup>

(1)

(2)

(5)

, (3)

�*cyυ<sup>η</sup>* <sup>¼</sup> <sup>0</sup>*:* (4)

*<sup>α</sup>*, (6)

. Eq. (6) is the Lommel differential

*α:*

*<sup>α</sup>*, and we consider *σCu* > 0. Since this is a linear

In this chapter we investigate a number of different models describing chemotaxis. The aim of this paper is to obtain exact analytical solutions of these models. For one-dimensional parabolic-parabolic systems under consideration, we present these solutions in explicit form in terms of traveling wave variables. Of course, not all of the solutions obtained can have appropriate biological interpretation since the biological functions must be nonnegative in all domains of definition. However some of these solutions are positive and bounded, and their analysis requires further investigation. Despite the large number of works devoted to the systems under consideration and their properties, as well as the properties of their solutions, it seems to us that the solutions obtained in this paper are new.

The Patlak-Keller-Segel model describes the space–time evolution of a cell density *u*ð*t*, *r* !Þ and a concentration of a chemical substance *<sup>v</sup>*ð*t*, *<sup>r</sup>* !Þ. The general form of this model is:

$$\begin{cases} \boldsymbol{\mu}\_t - \nabla(\delta\_1 \nabla \boldsymbol{\mu} - \eta\_1 \boldsymbol{\mu} \nabla \phi(\boldsymbol{\nu})) = \mathbf{0}, \\ \boldsymbol{\nu}\_t - \delta\_2 \nabla^2 \boldsymbol{\nu} - \boldsymbol{f}(\boldsymbol{\nu}, \boldsymbol{\nu}) = \mathbf{0}, \end{cases}$$

where *δ*<sup>1</sup> >0 and *δ*<sup>2</sup> ≥0 are cell and chemical substance diffusion coefficients, respectively, and *η*<sup>1</sup> is a chemotaxis coefficient; when *η*<sup>1</sup> >0, this is an attractive chemotaxis ("positive taxis"), and when *η*<sup>1</sup> < 0, this is a repulsive ("negative") one [13, 14]. *ϕ*ð Þ*v* is the chemosensitivity function, and *f u*ð Þ , *v* characterizes the chemical growth and degradation. These functions are taken in different forms that correspond to some variations of the original Patlak-Keller-Segel model. We follow the reviews of Hillen and Painter [8] and of Wang [15] and consider the models presented therein.

This paper is concerned with one-dimensional simplified models when the coefficients *δ*1, *δ*2, and *η*<sup>1</sup> are positive constants, *x*∈ ℜ, *t* ≥0, *u* ¼ *u x*ð Þ , *t* , and *v* ¼ *v x*ð Þ , *t* .
