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Russian Physics Journal, Vol. 57, No. 1, May, 2014 (Russian Original No. 1, January, 2014)

ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY

NONLINEAR DIFFUSION OF BIOLOGICAL SYSTEMS

V. V. Lasukov1 and T. V. Lasukova2 UDC 517.9

It is shown that a nonlinear differential equation of diffusion type can be used to describe spatial mosaic structures of biological systems.

Keywords: Burgers’ nonlinear diffusion, microorganisms.

INTRODUCTION

It is well known that the basic equation describing the dynamics of a colony of bacteria or cells is the Fisher–Kolmogorov–Petrovskii–Piskunov (FKPP) equation, in which the kinetics is determined by local, nonlinear terms while diffusion determines the spatial distribution of the population [1, 2]. The FKPP model does not take into account a number of factors influencing the dynamics of biological systems, an account of which leads to nonlocal generalizations of the FKPP equation. Different generalizations of the FKPP equation were considered using numerical methods in [3–6]. In [7–9] we investigated spontaneous emission of microorganisms in a Coulomb field, in an oscillating electric field, and in a constant and uniform magnetic field.

In the present work, we analytically investigate the dynamics of biological systems on the basis of Burgers’ differential equation of nonlinear diffusion [10].

NONLINEAR DIFFUSION OF BIOLOGICAL SYSTEMS

For simplicity, we investigate analytically the dynamics of biological systems in planar geometry. Toward this end, we use the following system of partial differential equations:

2 2

2 2x

x y x xV

V V V D Vt x y x y

, (1)

2 2

2 2y

x y y yV

V V V D Vt x y x y

,  (2)

0x yV Vt x y

,  (3)

, , , , ,x ydx dyV x y t V x y tdt dt

,  (4)

1Physical-Technical Institute at National Research Tomsk Polytechnic University, Tomsk, Russia; 2Tomsk State Pedagogical University, Tomsk, Russia, e-mail: lav_9@list.ru. Translated from Izvestiya Vysshikh UchebnykhZavedenii, Fizika, No. 1, pp. 3–6, January, 2014. Original article submitted August 2, 2012.

1064-8887/14/5701-0001 2014 Springer Science+Business Media New York

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where , , ,x yV x y t are the components of the velocity of the microorganisms, D is the diffusion coefficient, and is

the density. For simplicity, we consider the case when andx yV V V V , where and are constants.

We seek the solution of the equation

2 2

2 2

V V V D Vt x y x y

  (5)

in the form 0 1 2 0,V V f u u k x k y V t . Then the partial differential equation (5) reduces to an ordinary

differential equation

2

22

df d fA Bf Dkdu du

,

where

1 2 0 1 21 ,A k k V B k k , 2 2 21 2k k k .

Next, for definiteness we choose 1 22, 1, 1k k , as a result of which the solution of Eq. (5) takes the form

00 0 0tanh 2

10

CV V C x y V t

D

, (6)

where 0C is an integration constant. Solution of Eqs. (4) taking into account relation (6) gives

20

0 0 100 0 0 02 arcsinh sinh 2

10 10

C tDC C

x y V t x y V t eD D

, (7)

where 0 00 and 0x x y y are the initial conditions. Taking into account law of conservation of mass (3), it is not

hard to obtain an expression for the density from Eq. (7):

20

20

20

20

22 0 10

0 0 00

00

0 0 100 0 0

22 0 010

0 0 0

0 0

0 100 0 0

1 sinh 210

, ,tanh 2

sinh 21010

1 sinh 2 cosh 210 10

cosh 210

C tD

C tD

C tD

C tD

C x y V t eA Dx y t A

C x y V t C x y V t eDD

C Cx y V t e x y VD D

C x y V t eD

20

0

22 0 10

0

,

sinh 210

C tD

t

C x y V t eD

(8)

where 0 0 is the initial condition. A plot of constant density contours is presented in Fig. 1.

It is not hard to show that system of equations (1)–(4) has the solution

3

20

00

02

2 0100

cos 210

sin 210

C tD

C x y V tD

Ce x y V tD

. (9)

The mosaic spatial structure of the contours of solution (9) is shown in Fig. 2. From Figs. 1 and 2 it can be seen that the microorganisms randomly (or deliberately) falling in the range of one density level can, in the absence of external forces, form ordered structures due to self-action of Burgers’ nonlinear diffusion [10].

Note that solutions (6) and (8) are easily generalized to the three-dimensional case:

00 0 0tanh 3

22

CV V C x y z V t

D

, (10)

20

00

02

2 0 220

cosh 322

sinh 322

C tD

C x y z V tD

C x y z V t eD

. (11)

In turn, when the pressure P and the force f compensate one another grad P f , solutions (10) and (11) satisfy

the Navier–Stokes equations

Fig. 1. Constant density contours for fixed value of dimensionless time

20 ln 10

10 2

Ct

D : , , constt .

4

1grad ,

div 0.

V V V D V P ft

Vt

In this case, the unknowns are not the two functions andP V , but the three functions , , andP V . This means that the

Navier–Stokes equations have an exact solution in three-dimensional space which is always valid.

CONCLUSIONS

The study presented above allows us to conclude that the Burgers' nonlinear diffusion equation describing the dynamics of biological systems can have a solution possessing a mosaic spatial structure. The solution of the nonlinear diffusion equation can be used in quantum geometrodynamics to solve the problem of the temporal asymmetry of the Universe [11–33].

This work was supported in part by of the Russian Scientific Foundation.

REFERENCES

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Ct

D : , , constt .

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