In this chapter we consider several different parabolic-parabolic systems of chemotaxis which depend on time and one space coordinate. For these systems we obtain the exact analytical solutions in terms of traveling wave variables. Not all of these solutions are acceptable for biological interpretation, but there are solutions that require detailed analysis. We find this interesting, since chemotaxis is present in the continuous mathematical models of cancer growth and invasion (Anderson, Chaplain, Lolas, et al.) which are described by the systems of reaction–diffusion-taxis partial differential equations, and the obtaining of exact solutions to these systems seems to be a very interesting task, and a more detailed analysis is possible in a future study.
- parabolic-parabolic system
- exact solution
- soliton solution
- Patlak-Keller-Segel model
This chapter uses the publications of Shubina M.V.:
Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis, Russian J Math Phys., Maik Nauka/Interperiodica Publishing (Russian Federation), 25(3), 383–395, 2018.
The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: The particular integrable case and soliton solution, J Math Phys., 57(9), 091501, 2016.
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  and by Keller and Segel in the 1970s [2, 3, 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  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  and Perthame . In the review of Hillen and Painter , 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 . 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, 10, 11, 12] and the links below.
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 and a concentration of a chemical substance v . The general form of this model is:
where and are cell and chemical substance diffusion coefficients, respectively, and is a chemotaxis coefficient; when , this is an attractive chemotaxis (“positive taxis”), and when , this is a repulsive (“negative”) one [13, 14]. is the chemosensitivity function, and 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  and of Wang  and consider the models presented therein.
This paper is concerned with one-dimensional simplified models when the coefficients , , and are positive constants, , , and .
2. Signal-dependent sensitivity model
Let us start with a model that allows nonnegative bounded solutions that may be of interest from a biological point of view. Now consider the “logistic” model, one of versions of signal-dependent sensitivity model  with the chemosensitivity functions , where , and . In the review  one can see a mathematical analysis of this model. When and , the existence of traveling waves was established in [16, 17]. The replacements of and give , , , and . We also set , , as well as . It should be noted that a sign of may effect on the mathematical properties of the system. So, corresponds to an increase of a chemical substance, proportional to cell density, whereas corresponds to its decrease. And as we shall see later, various solutions correspond to these two cases.
After the above replacements, the model reads:
If we introduce the function , in terms of traveling wave variable , where , this system has the form:
where , , and is an integration constant.
In this chapter we will consider the case of . Then Eq. (2) gives:
is a constant and we will examine the following equation for :
Since is a positive constant, we consider two cases: [Eq. (4) is a linear nonhomogeneous equation] and .
Let us begin with . We introduce the new variable and the new function :
and Eq. (4) becomes:
where and . Eq. (6) is the Lommel differential equation [18, 19] with , and we consider . Since this is a linear inhomogeneous second-order differential equation, one can integrate it by the method of variation of parameters. We assume a solution in the form:
where and are Bessel functions and and are the functions of that satisfy the equations:
Considering that Wronskian , we obtain:
where and are constants. If both of the numbers are positive, the lower limits in the integrals may be taken to be zero. Then a particular integral of Lommel equation “proceeding in ascending powers of ” is ; if one considers a solution of Lommel equation “in the form of descending series,” one obtains the function  [see Eq. (8)]. Thus, quoting Watson  “...and so, of Lommel’s two functions and , it is frequently more convenient to use the latter.” Then the general solution of Eq. (6) has the form:
where and are constants,
We first consider the case . Then and . Eq. (6) becomes homogeneous, and for , its general solution is:
However one can check that the function diverges as for all .
Consider now . For to be real, let . Then Eq. (6) becomes the modified Bessel equation; the analysis of solution behavior at leads to suitable solutions for and :
with restrictions and . So one can see that as for all ; for and for as and as for all . The curves of these functions are presented in Figures 1 and 2, and the plots for are thicker than for . Thus, the solution obtained may be considered as a biologically appropriated one, and this requires further investigation.
with and . The latter condition leads to the requirement . The , as and and as . Thus, one can see that for , , and , is satisfied but . These functions are presented in Figures 3 and 4. It should be noted that or because of the pole in function.
Using the analysis of Eq. (11), one can see that the condition along with and () leads to the fact that the function has not changed, but becomes positive on all domains of definition. This function is presented in Figure 5.
Let us return to Eq. (4) and rewrite it in terms of the variable :
To integrate this equation, we use the Lie group method of infinitesimal transformations . We find a group invariant of a second prolongation of one-parameter symmetry group vector of (12), and with its help, we transform Eq. (12) into an equation of the first order. It turns out that nontrivial symmetry group requires some conditions:
and we consider the case . Thus, , and for:
we obtain the Abel equation of the second kind:
where we take
and Eq. (15) becomes an equation for the function . Solving it, for , we obtain:
in order for , and to be real. Then one can see that all functions in Eq. (19) are continuous bounded ones for and are positive. Hence, one may try to biologically interpret the functions and , and this requires further investigation. In Figure 6 one may see the different curves for and different . Figure 7 demonstrates and for two values and , see Figure 7. Further, for larger values of and , it seems more convenient to present the curves , , and to analyze them (see Figures 8–10). One can see from Eq. (13) that when , and the case of and is presented in Figure 11.
3. Logarithmic sensitivity
The model with logarithmic chemosensitivity function is also studied. For the case of , where , an extensive analysis is performed in . This survey is focused on different aspects of traveling wave solutions. When , this model coincides with Eq. (1) for . When and , the system was studied in [22, 23]. The complete analysis for is performed in . An existence of global solution is established in .
Now we consider the system with and . Similarly, a replacement of and gives , , , , and . Then the model has the form:
Let us rewrite the system (21) in terms of the function :
Then in terms of the traveling wave variable , where , Eq. (22) has the form:
where , , and is an integration constant. To integrate Eq. (23), we tested this system on the Painlevé ODE test. One can show that for , it passes this test only if with the additional condition . If we express as from Eq. (23), we obtain an equation only for ; for , it has the form:
For , this equation can be linearized. It becomes equivalent to the following linear equation for :
that gives the equation for :
Another possibility to solve this equation exactly is to put equal to zero. When , that means , for ; Eq. (27) can be linearized by . Its solution has three forms according to a sign of the expression . Since should be a nonnegative and bounded function as , the only suitable solution is:
where are positive constants and . Unfortunately, the corresponding solution for is alternating and has the form:
4. Linear sensitivity
Let us consider the system with linear function . When , the system is called the minimal chemotaxis model following the nomenclature of . This model is often considered with (and are constants), and it is studied in many papers. As was proved in [27, 28], the solutions of this system are global and bounded in time for one space dimension. The case of positive and nonnegative is studied in [29, 30, 31, 32, 33]. As we noted earlier, a sign of may effect on the mathematical properties of the system, which changes its solvability conditions .
Now we consider the linear chemosensitivity function and . The replacement of , , and leads to , , , and . Then the system has the form:
This system reduces to the system of ODEs in terms of traveling wave variable , where :
where , , and is an integration constant. As shown in , this system passes the Painlevé ODE test only if and . Let us focus on this case.
It is convenient to solve Eq. (32) in terms of variable:
where is an arbitrary constant. Then for and , we obtain the solutions in the form:
The function satisfies the modified Bessel’s equation and can be present as a linear combination of Infeld’s and Macdonald’s functions.
Using the series expansion of the Infeld’s function, as well as theirs asymptotic behavior , one may obtain the following asymptotic forms for and :
where the expression for agrees with Eq. (39).
So, the exact solution obtained has the form:
where , , and are arbitrary constants and the functions and are Infeld’s and Macdonald’s functions, respectively. This solution is not satisfactory from the biological point of view, since is an alternating function for any . However it seems interesting because of the following: in the case of and in terms of , its form coincides with the well-known Korteweg-de Vries soliton.
Consider now the class of solutions with half-integer index , when can be expressed in hyperbolic functions. The requirement of absence of divergence for finite leads to the following form for :
At first let us consider the solutions obtained for and as functions of . We begin with or . It is interesting to present the expressions for and :
where Eq. (40) appears the one-soliton solution exactly the same as the well-known one of the Korteweg-de Vries equation. Returning to the variable :
One can see that for (an increase of a chemical substance), the cell density for and that for is the solitary continuous solution vanishing as , whereas for has a point of discontinuity. One can say that when , we obtain “blow-up” solution in the sense that it goes to infinity for finite , and this is true for different .
The explicit form of our solution in terms of the variable can be obtained by direct substitution of Eq. (33) into Eq. (39), where . The resulting formulae are complicated and slightly difficult for analytic analysis; it seems to be more convenient to present the plots.
For in the function , we have the “step” whose altitude depends on the values of velocity and arbitrary constant . One may see that these curves become higher and shift to the right with different rates for the rising . The is the positive function whose altitude and sharpness of peak depend on c (see Figures 16 and 17).
For we can see that the solitonic behavior of is retained for different values of and ; the curves become higher and more tight, and they shift to the right also with an increase of and . For the cell density , the obtained solution has the negative section converging to zero for (Figures 18–21).
The curves for the concentration of the chemical substance are presented in Figure 22. Since has to be positive (nonnegative), we see that these functions do not satisfy this requirement in all domains of definition.
In conclusion it seems interesting to present the plots for and for different values of (Figures 23–25). It is interesting to see that there are irregular solutions for ; however, the corresponding solutions for are regular [see Eqs. (35)–(37)].
We investigate three different one-dimensional parabolic-parabolic Patlak-Keller-Segel models. For each of them, we obtain the exact solutions in terms of traveling wave variables. Not all of these solutions are acceptable for biological interpretation, but there are solutions that require detailed analysis. It seems interesting to consider the latter for the experimental values of the parameters and see their correspondence with experiment. This question requires further investigations.