Journal of Applied Mathematics and Physics, 2015, 3, 1249-1255 Published Online October 2015 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2015.310153

How to cite this paper: Muhamediyeva, D.K. (2015) Qualitative Properties and Numerical Solution of the Kolmogorov- Fisher Type Biological Population Task with Double Nonlinear Diffusion. Journal of Applied Mathematics and Physics, 3, 1249- 1255. http://dx.doi.org/10.4236/jamp.2015.310153

Qualitative Properties and Numerical Solution of the Kolmogorov-Fisher Type Biological Population Task with Double Nonlinear Diffusion Dildora Kabulovna Muhamediyeva National University of Uzbekistan, Tashkent, Uzbekistan Email: matematichka@inbox.ru Received 24 July 2015; accepted 23 October 2015; published 26 October 2015

Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract In the present work we study the global solvability of the Kolmogorov-Fisher type biological pop-ulation task with double nonlinear diffusion and qualitative properties of the solution of the task based on the self-similar analysis. In additional, in this paper we consider the model of two com-peting population with dual nonlinear cross-diffusion.

Keywords Double Nonlinearity, Cross-Diffusion, Biological Population, A Parabolic System of Quasilinear Equations, Convective Heat Transfer, Numerical Solution, Iterative Process, Self-Similar Solutions

1. Introduction Let’s consider in the domain ( ){ }, : 0 ,Q t x t x R= < ∈ parabolic system of two quasilinear equations of reac-tion-diffusion with double nonlinear diffusion

( ) ( )

( ) ( )

1 1

2 2

211 1 1

1 2 1 1 1

212 2 2

2 1 2 2 2

1

1

pm

pm

u u uD u k t u ut x x x

u u uD u k t u ut x x x

β

β

−−

−−

∂ ∂ ∂∂= + − ∂ ∂ ∂ ∂

∂ ∂ ∂∂

= + − ∂ ∂ ∂ ∂

(1)

( ) ( )1 0 10 2 0 20, ,t tu u x u u x= == =

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which describes the process of biological populations of the Kolmogorov-Fisher in two-component nonlinear

medium, the diffusion coefficient which is equal to 1

21 1

1 2

pm uD u

x

−− ∂

∂, 2

21 2

2 1

pm uD u

x

−− ∂

∂, where 1 2 1 2, , , ,m m p β β

are positive real numbers, and ( )1 1 , 0u u t x= ≥ , ( )2 2 , 0u u t x= ≥ are sought solution . Below we investigate the qualitative properties of the considered problem by constructing self-similar system

of equations for (1).

2. Self-Similar System of Equations Self-similar system of equations we will construct by the method of nonlinear splitting [1]-[3].

Substitution in (1)

( ) ( ) ( ) ( )1 21 1 2 2, e , , , e , ,k t k tu t x v t x u t x v t x= =

lead (1) to the form:

( ) ( )

( ) ( )

1 1 1 21 1

2 2 2 12 2

22 11 11 1 1

1 2 1 11

22 11 12 2 2

2 1 2 22

e

e

pp k m k tm

pp k m k tm

v v vD v k vx x x

v v vD v k vx x x

β β

β β

τ

τ

−− + − − − +

−− + − − − +

∂ ∂ ∂∂= + ∂ ∂ ∂ ∂

∂ ∂ ∂∂

= + ∂ ∂ ∂ ∂

(2)

( ) ( )1 0 10 2 0 20, .t tv v v vη η= == =

Choosing ( ) ( )

( ) ( )( ) ( )

( ) ( )1 2 1 1 1 21 2 1 2

1 1 2 2 2 1

e e2 1 2 1

m k p k t m k p k t

p k m k p k m kτ

− + − − + −

= =− + − − + −

,

we get the following system of equations:

1 1 1

2 2 2

21 11 1 1

1 2 1 1

21 12 2 2

2 1 2 2

pm b

pm b

v v vD v a vx x x

v v vD v a vx x x

β

β

ττ

ττ

−− +

−− +

∂ ∂ ∂∂= − ∂ ∂ ∂ ∂

∂ ∂ ∂∂

= − ∂ ∂ ∂ ∂

(3)

where

( ) ( ) 11 1 1 1 22 1

ba k p k m k= ⋅ − + − ,

( ) ( )( ) ( )1 1 1 2

11 1 2

2 12 1

p k m kb

p k m kβ − + − −

=− + −

( ) ( ) 22 2 2 2 12 1

ba k p k m k= ⋅ − + − ,

( ) ( )( ) ( )2 2 2 1

22 2 1

2 12 1

p k m kb

p k m kβ − + − −

=− + −

For the purpose of obtaining self-similar system for the system of Equation (3) we find first the solution of a system of ordinary differential equations [4]-[7]

1 1

2 2

111 1

122 2

d,

dd

,d

b

b

a v

a v

β

β

ντ

τν

ττ

+

+

= − = −

in the form

( ) ( )1 21 2, ,α αν τ τ ν τ τ− −= =

where

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1 21 2

1 2

1 1, .

b bα α

β β+ +

= =

And then the solution of system (3) is sought in the form

( ) ( ) ( ) ( ) ( ) ( )1 1 1 2 2 2, , , , ,v t x v t w x v t x v t w xτ τ= = (4)

and ( )tτ τ= is selected so

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 2

1 2 1

2 1 2 11 2 2 1

0 0

2 1 11 2 1

1 2 1

1 2 1

d d

1 , if 1 2 1 0,1 2 1

ln , if 1 2 1 0,

p m p m

p m

v t v t t v t v t

T p mp m

T p m

τ τ

α α

τ τ

τ α αα α

τ α α

− − − −

− − + − +

= =

+ − − + − ≠ − − + − = + − − + − =

∫ ∫

if ( ) ( ) ( ) ( )1 2 1 2 1 22 1 2 1p m p mα α α α− + − = − + − . Then for ( ), , 1, 2iw x iτ = we get the system of equations

( )

( )

1 1

2 2

21 11 1 1

1 2 1 1 1

21 12 2 2

2 1 2 2 2

pm

pm

w w wD w w wx x x

w w wD w w wx x x

β

β

ψτ

ψτ

−− +

−− +

∂ ∂ ∂∂= + − ∂ ∂ ∂ ∂

∂ ∂ ∂∂

= + − ∂ ∂ ∂ ∂

(5)

where

( ) ( )( )3

1 , 1, 21 2 1i

i i i

ip m

ψα α τ−

= = − − + −

(6)

Consider the self-similar solution of system (5) of the form

( ) ( ) ( ) ( ) ( )11 1 2 2, , , , pw t x f w t x f x Tξ ξ ξ τ= = = + (7)

Then substituting (7) into (5) with respect to ( ) ( )1 2,f fξ ξ we get the following system of nonlinear dege-nerate self-similar equations:

( )

( )

1 1

2 2

21 11 1 1

2 1 1 1

21 12 2 2

1 2 2 2

d d dd 0,d d d d

d d dd 0,d d d d

pm

pm

f f ff f fp

f f ff f fp

β

β

ξ θξ ξ ξ ξ

ξ θξ ξ ξ ξ

−− +

−− +

+ + − =

+ + − =

(8)

where ( ) ( )( )3

11 2 1i

i i ip mθ

α α τ−

= − − + −

, 1, 2.i = Let’s build an upper solutions for system (8).

3. Construction an Upper Solution If

( ) ( )( ) ( ) ( )( )21 22 1 1 1 1i ip m m p p mβ = − − − − − − + , ( )( )1 22 1 1p m m> + − − , 1, 2i = ,

Equation (8) has a local solution of the form

( ) ( ) ( ) ( )1 2

1 2, ,n n

f A a f B aγ γξ ξ ξ ξ+ +

= − = −

where ( ) ( )max 0,b b+= , ( )1p pγ = − ,

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( ) ( )( )( ) ( ) ( )

( ) ( )( )( ) ( ) ( )

1 21 22 2

1 2 1 2

1 1 1 1; .

2 1 1 2 1 1

p p m p p mn n

p m m p m m

− − + − − += =

− − − − − − − −

Then in the domain Q according to the comparison principle of solutions [1] [8] we get Theorem 1. Let ( ) ( )0, 0, , .i iu x u x x R±≤ ∈ Then the solution of the task (1) in the domain Q takes place an

estimation

( ) ( ) ( )( ) ( ) ( )

1 1

2 2

1 1 1

2 2 2

, , e ,

, , e ,

k t

k t

u t x u t x f

u t x u t x f

α

α

τ ξ

τ ξ

−+

−+

≤ =

≤ = 1 pxξ τ=

where ( ) ( ) ( )1 2,f f и tξ ξ τ —above-defined functions. Note that the solution of system (1) when ( ) ( )( ) ( ) ( )( )2

1 22 1 1 1 1i ip m m p p mβ = − − − − − − + has the following representation in the

1 2

1 1 2 21 1,1 ,1

n na P B n P B n

γ γ

γ γγ γ

= + = +

.

where ( ),В a b —Beta Euler function [9]. It is proved that this view is self-similar asymptotics of solutions of systems (1).

( )

( )

11

22

1

1 1

1

2 2

d

d

n

n

a x P

a x P

µ γ

µ γ

τ ξ

τ ξ

∞−

+−∞

∞−

+−∞

− =

− =

∫

∫

Thence

( ) ( )

( ) ( )

1 11 11

2 22 22

1 11 1

1 1 10

1 11 1

2 2 20

1 1 1d 1 d ,1

1 1 1d 1 d ,1

n nn n

n nn n

a x a a B n P

a x a a B n P

µ γ γ γ γ

µ γ γ γ γ

τ ξ η η ηγ γ γ

τ ξ η η ηγ γ γ

∞− −

+−∞

∞− −

+−∞

− = − = + =

− = − = + =

∫ ∫

∫ ∫

1 2

1 1 2 21 1,1 ,1

n na P B n P B n

γ γ

γ γγ γ

= + = +

.

Carried out computational experiments and numerical results are obtained (see Table 1, Table 2).

4. Conclusion Thus, it is assumed that the possibility of adequate study of nonlinear equations, biological populations with double nonlinearity based on the method of nonlinear splitting and numerical study of nonlinear processes described by equations with double nonlinearity and analysis of results on the basis of the estimates of the solutions gives a comprehensive picture of the process of multicomponent competing systems of biological populations.

5. Results

( ) ( )

( ) ( )

1 1

2 2

21

1 1

21

2 2

1 ,

1 ,

pm

pm

u u uD v k t u ut x x x

v v vD u k t v vt x x x

β

β

−−

−−

∂ ∂ ∂ ∂= + − ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂

= + − ∂ ∂ ∂ ∂

D. K. Muhamediyeva

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Table 1. Numerical results in the case of fast diffusion.

Parameter values 1 21; 1;x x= =

2x = 1 22; 2;x x= =

2 2x = 1 23; 3;x x= =

3 2x = 1 24; 4;x x= =

4 2x = 1 25; 5;x x= =

5 2x =

1 2.5m = , 2 2.4m = , 3.1p = ,

310eps −= ,

1 0.944 0n = > ,

2 0.708 0n = > , 0.89 0q = − < ,

1 1β = , 2 1β =

1 3.7m = , 2 3.8m = , 4p = ,

310eps −=

1 0.59 0n = > ,

2 0.674 0n = > , 3.56 0q = − < ,

1 2β = , 2 2β =

1 4.1m = , 2 4.0m = , 4.4p = ,

310eps −= ,

1 0.672 0n = > ,

2 0.576 0n = > , 3.54 0q = − < ,

1 1β = , 2 1β =

1 3.7m = , 2 3.3m = , 4p = ,

310eps −= ,

1 0.95 0n = > ,

2 0.407 0n = > , 2.21 0q = − < ,

1 2β = , 2 2β =

1 7m = , 2 7.1m = , 4p = ,

310eps −= ,

1 0.368 0n = > ,

2 0.377 0n = > , 32.6 0q = − < ,

1 3β = , 2 4β =

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Table 2. Numerical results in the case of slow diffusion.

Parameter values 1 21; 1;x x= =

2x = 1 22; 2;x x= =

2 2x = 1 23; 3;x x= =

3 2x = 1 24; 4;x x= =

4 2x = 1 25; 5;x x= =

5 2x =

1 1.3m = , 2 1.2m = , 2.5p = ,

310eps −= ,

1 1.579 0n = > ,

2 2.368 0n = > 0.19 0q = > ,

1 1β = , 2 1β =

1 1.9m = , 2 1.7m = , 3p = ,

310eps −= ,

1 0.541 0n = > ,

2 1.622 0n = > , 0.37 0q = > ,

1 22, 2β β= =

1 3.7m = , 2 3.5m = , 5.4p = ,

310eps −= ,

1 0.64 0n = > ,

2 0.823 0n = > , 4.81 0q = > ,

1 3β = , 2 4β =

1) Fast diffusion is shown in Table 1. As an initial approximation it is necessary to take:

( ) ( ) ( ) 110 ,

nu x t T t aα γξ−= + + , ( ) ( ) ( ) 22

0 ,n

v x t T t aα γξ−= + + , 1p

xξ

τ

= , 1

pp

γ =−

,

( ) ( )1 1ii

p p mn

q − − + = , 1, 2i = , ( ) ( ) ( )2

1 22 1 1q p m m= − − − −

Parameter values must be 1 20, 0, 0.n n q> > < Constant a is determined from the condition ( )1 1,0 du x x P∞

−∞

=∫ ,

( )2 2,0 du x x P∞

−∞

=∫ : 1 2

1 1 2 21 1,1 ,1

n na P B n P B n

γ γ

γ γγ γ

= + = +

2) Slow diffusion is shown in Table 2. As an initial approximation it is necessary to take:

( ) ( ) ( ) 110 ,

nu x t T t aα γξ−

+= + − , ( ) ( ) ( ) 22

0 ,n

v x t T t aα γξ−

+= + − , 1

p

xξ

τ

= ,

1p

pγ =

−, т, 1,2i = , ( ) ( )( )2

1 22 1 1q p m m= − − − −

D. K. Muhamediyeva

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Parameter values must be 1 20, 0, 0in n q> > > . Constant a is determined from the condition

1 2

1 1 2 21 1,1 ,1 .

n na P B n P B n

γ γ

γ γγ γ

= + = +

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