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Estimation of the Solution of the

Kolmogorov-Fisher Type Biological Population Task by Taking

Into Account the Reaction-Diffusion

Muhamediyeva Dildora Kabulovna

Abstract: The spatial-temporal dynamics of the population is one of the most interesting aspects and problems for environmental modeling. In this article, we will consider some mathematical models based on one-dimensional reaction-diffusion-advection equations for population growth in a heterogeneous habitat. Considering a number of models of increasing complexity, we investigate often the opposite roles of advection and diffusion for the conservation of the population.

Whenever possible, we demonstrate basic mathematical methods and provide critical conditions that ensure the survival of the population, in simple systems and in more complex resource-consumer models.

Keywords: biological population, reaction-diffusion, cross-diffusion, parabolic system, quasilinear equations.

I. INTRODUCTION

In the 1930s Fisher and Kolmogorov, Petrovsky, Piskunov in population dynamics and Zel'dovich, Frank-Kamenetsky in combustion theory began to study problems with reaction conditions [1-2].

Population modeling is of great importance in ecology and

economics,T forT example,T forT describingT predator-preyT

interactionsT andT competition,T predictingT theT dynamicsT ofT

cellT divisionsT andT infectiousT diseases,T andT forT managingT

renewableT resourcesT (harvesting).T PopulationT modelsT

describeT theT changeT inT theT numberT ofT speciesT dueT toT

fertility,T mortalityT andT movementT fromT positionT toT

positionT (inT space)T orT fromT stageT toT stageT (age,T size,T

etc.)T [3-6].T InT thisT briefT review,T weT willT lookT atT someT

mathematicalT resultsT forT deterministicT andT continuousT

populationT models.T WeT concentrateT onT theT followingT

classesT ofT models:

spatiallyT homogeneousT populationT models;

T •T spatiallyT heterogeneousT populationT models;

T •T ageT andT sizeT demographicT models;T and

T •T modelsT ofT theT populationT withT aT delayT inT time.

TheT evolutionT ofT spatiallyT homogeneousT populationsT canT

beT modeledT byT ordinaryT differentialT equations,T forT

example,T aT logisticT growthT model.T

Revised Manuscript Received on July 25, 2019.

Muhamediyeva Dildora Kabulovna, Ph.D, Tashkent university of information technologies.

TheT interactionT ofT competingT populationsT canT beT

describedT byT aT systemT ofT coupledT equations,T oneT

ofT whichT isT theT well-knownT Volterra-TrayT systemT

[7].T ImportantT issues,T inT additionT toT theT

justificationT ofT theTrelevantT problems,T areT theT stabilityT

ofT stableT statesT andT biologicalT consequences.T InT spatiallyT

heterogeneousT conditions,T theT populationT sizeT variesT inT

spaceT andT canT spreadT inT theT environment.T ThisT givesT aT

classT ofT reaction-diffusionT equationsT andT theirT systemsT

[8-9].T TuringT foundT thatT aT stationaryT solutionT ofT theT

diffusionT systemT canT becomeT unstableT evenT ifT theT stableT

stateT ofT theT correspondingT systemT withoutT diffusionT isT

stable.T Thus,T stabilityT analysisT isT muchT moreT involvedT

thanT inT theT spatiallyT homogeneousT case.T RoughlyT

speaking,T inT theT longT run,T extinctionT orT coexistenceT ofT

speciesT mayT occur.

SolutionsT ofT theT diffusionT modelT ofT theT Voltaire-LotkaT

competitionT doT notT showT theT formationT ofT patterns.T

Consequently,T thisT modelT isT notT capableT ofT describingT

segregationT phenomena.T AnalysisT ofT theT existenceT ofT

cross-diffusionT systemsT isT difficultT dueT toT strongT

nonlinearT couplingT andT sinceT diffusionT matricesT canT notT

beT eitherT symmetricT orT positiveT definiteT [7-8].T Recently,T

analyticalT toolsT haveT beenT developedT thatT proveT theT

existenceT ofT time-varyingT weakT solutions.T WhenT theT

individualsT ofT theT populationT areT notT identical,T butT mayT

differT inT age,T size,T etc.,T weT needT toT introduceT structuredT

populationT models.

TheT fieldT ofT populationT modelingT hasT becomeT soT

largeT thatT inT thisT reviewT weT canT considerT onlyT aT smallT

partT ofT theT publishedT model-mathematicalT topics.T ManyT

classesT ofT modelsT andT importantT questionsT willT notT beT

discussed.T ForT example,T weT ignoreT differenceT andT matrixT

equations,T stochasticT approachesT andT models,T includingT

mutations,T maturationT structures,T metapopulations,T andT

demographicT orT biomedicalT applications.

II. FORMULATION OF THE PROBLEM

InT thisT paperT weT investigateT theT propertiesT ofT solutionsT

ofT theT problemT ofT aT biologicalT populationT ofT theT

Fisher-KolmogorovT typeT inT theT caseT ofT variableT density.T TheT

mainT methodT ofT

(2)

self-inT theT fieldT Q={(t,x):T 0<T tT <,T x

N

R

T }T parabolicT

systemT ofT twoT quasilinearT reaction-diffusionT equations

1 1 2 2 2 1 1

1 2 1 1 1 1 1

2 1

2

2 1 1 2 2 2 2

1 , 1 , p m k p m k u

D u u u k u u

t u

D u u u k u u

t                    (1)

)

(

10 0

1

u

x

u

t

,

u

2 t0

u

20

(

x

)

, (2)

which describes the process of a biological population of the Kolmogorov-Fisher type in a nonlinear two-component medium, the coefficients of mutual diffusion are

respectively equal 1

2 1

1 2 1 1

p

m k

D u

u

u

,

2 1 2

2 1 2 2

p

m k

D u

u

u

. Numeric parameters

2 1 2

1

,

m

,

n

,

p

,

,

m

,

D

1,

D

2- positive real numbers,

x

grad

(.)

(.)

,

 

1, 21,

x

R

N

l

0

;

0

)

,

(

1 1

u

t

x

u

,

u

2

u

2

(

t

,

x

)

0

- sought solutions.

WeT willT studyT theT propertiesT ofT solutionsT ofT

problemT (1),T (2)T onT theT basisT ofT aT self-similarT

analysisT ofT solutionsT ofT theT systemT ofT equationsT

constructedT byT theT methodT ofT nonlinearT splittingT andT

standardT equations.

We note that the substitution in (1)

1

1

( , )

1

( ( ), )

k t

u t x

e

v

t x

,

2

2

( , )

2

( ( ), )

k t

u t x

e

v

t x

will bring it to mind:

 

 

1 1 1 1 2

1 1

2 2 2 2 1

2 2

2 ( 2) ( 1)

1 1

1

1 2 1 1 1 1

2 ( 2) ( 1)

1 1

2

2 1 2 2 2 2

,

, p k p kk m k t m k

p k p kk m k t m k

v

D v v v k e v

v

D v v v k e v

   

                         (3)

)

(

10 0

1

v

x

v

t

,

v

2 t0

v

20

(

x

)

.

(4)

If

k

1

[(

p

2)

k

(

m

1

1)]

k

2

[(

p

2)

k

(

m

2

1)]

,

then choosing

1 1 2 2 2 1 1 1

1 2 1 1 1 1

2

1 1

2

2 1 2 2 2 2

( ) , ( ) , p m k p m k v

D v v v a t v

v

D v v v a t v

 

                  (5) Where

1

1 1 ( 2) 1 ( 1 1) 2

b

ak pkkmk ,

1 1 1 2

1

1 1 2

(

(

2) )

(

1)

,

(

2)

(

1)

p

k k

m

k

b

p

kk

m

k

2

2 2 ( 2 1) 1 ( 2) 2

b

ak mkpkk ,

2 2 2 1

2

2 1 2

(

(

2) )

(

1)

.

(

1)

(

2)

p

k k

m

k

b

m

k

p

kk

If

b

i

0

, and

a

i

(

t

)

const

,

i

1

,

2

, then the

system has the form:

1 1 2 2 2 1 1 1

1 2 1 1 1 1

2

1 1

2

2 1 2 2 2 2

, , p m k p m k v

D v v v a v

v

D v v v a v

 

                   (6)

Below,T weT describeT oneT ofT theT methodsT forT obtainingT aT

self-similarT systemT forT theT systemT ofT equationsT (5).T ItT

consistsT inT theT following.T WeT firstT findT aT solutionT ofT theT

systemT ofT ordinaryT differentialT equation

1 2 1 1 1 11 1 2 2 22 , , dv a v d dv a v d  

          

in the form

1

1 1

1

1 ( ) ( ( )) ,

v

t

  , 2

2 2

2

1 ( ) ( ( )) ,

v

t

  ,

for the case

b

i

0

, and

a

i

(

t

)

const

,

i

1

,

2

. And in

(3)

1 1

2 2

1 1

1 11

1 2

2 22

,

, b

b

dv

a v

d dv

a v

d

 

 

 

 



intheform

1 1

2 1

1

1

( )

( ( ))

,

b

v

t

,

2 2

2 2

2

1

( )

( ( ))

,

b

v

t

,

Then the solution of system (5) is sought in the form

1 1 1

2 2 2

( , ) ( ) ( ( ), ), ( , ) ( ) ( ( ), ),

v t x v w t x

v t x v w t x

(7)

And

 

( )

t

is chosen so

1 21

1

1 [ ( 2) ( 1)]

1 2 1 1 2 1

( 1) ( 2)

1 1 2 1 2 1

0

1 1

( ) , 1 [ ( 2) ( 1)] 0, 1 [ ( 2) ( 1)]

( ) ( ) ( ) ln( ), 1 [ ( 2) ( 1)] 0,

( ), 2 1,

p k m

m p k

T если p k m

p k m

v t v t dt T если p k m

T если p и m

 

  

 

    

   

 

   

     

       

   

 

if

1

(

p

2

)

2

(

m

1

1

)

2

(

p

2

)

1

(

m

2

1

)

.

Then for

z

i

( , ( )),

 

x

i

1, 2

we obtain a system of equations:

1 1

2 2

2

1 1

1

1 2 1 1 1 1 1

2

1 1

2

2 1 2 2 2 2 2

(

),

(

),

p

m k

p

m k

w

D w

w

w

w

w

w

D w

w

w

w

w

 

 

 

 



 

 

(8)

where

1 [1( 2 ) 2(11)]

1 2 1

1 2 1

1

( )

1 1 1 2 1

1

,

1 [ (

2)

(

1)

0,

(1 [ (

2)

(

1)])

,

1 [ (

2)

(

1)

0,

p k m

if

p

k

m

p

k

m

с

 

if

p

k

m

   

 

 

 

 

(9)

2 1 2

2 1 2

2 1 2

2

(1 [ ( 2) ( 1)])

2 1 2 1 2

1

,

1 [ (

2)

(

1)]

0,

(1 [ (

2)

(

1)])

,

1 [ (

2)

(

1)]

0.

p m

if

p

m

p

m

с

 

if

p

m

    

 

 

 

 

 

If

1 [ (

1

p

2)

k

2

(

m

1

 

1)

0

, self-similar solution of system (9) has the form

1/

( ( ), )

( ),

1, 2,

/ [ ( )]

p

i i

w

t x

f

i

x

t

, . (10)

(4)

1 1

2 2

2 1

1 1 1 1 1

2 1 1 1

2 1

1 1 2 2 2

1 2 2 2

(

)

(1

)

0,

(

)

(1

)

0.

p k m

N N

p k m

N N

df

df

df

d

f

f

f

d

d

d

p d

df

df

df

d

f

f

f

d

d

d

p d

 

 

 

 

(11)

where 1

1 2 1

1

(1 [

k p

(

2)

(

m

1)])

 

and 2

2 1 2

1

(1 [

k p

(

2)

(

m

1)])

 

.

The system (11) has an approximate solution of the form

1

1

(

) ,

/ (

1)

n

f

A a

p

p

, 2

2

(

)

n

f

B a

 ,

WhereА and В constant and

1

1 2 2

1 2

( (

2) 1)( (

2) (

1))

(

2)

(

1)(

1)

k p

k p

m

n

k

p

m

m

 

 

,

2

2 2 2

1 2

( (

2) 1)( (

2) (

1))

(

2)

(

1)(

1)

k p

k p

m

n

k

p

m

m

 

 

.

InT thisT paper,T basedT onT theT methodT describedT above,T

theT qualitativeT propertiesT ofT theT solutionsT ofT theT

systemT (1)T areT investigated,T andT onT thisT basisT theT

problemT ofT choosingT theT initialT approximationT forT theT

iterativeT solutionT isT solved,T leadingT toT rapidT

convergenceT toT theT solutionT ofT theT CauchyT problemT

(1),T (2),T dependingT onT theT valueT ofT theT numericalT

parametersT andT initialT data.T ForT thisT purpose,T theT

asymptoticT representationT ofT theT solutionT foundT byT usT

wasT usedT asT theT initialT approximation.T ThisT allowedT

usT toT performT aT numericalT experimentT andT

visualizationT ofT theT process,T describedT byT theT systemT

(1),T dependingT onT theT valuesT enteringT intoT theT systemT

ofT numericalT parameters.

III. BUILDING THE UPPER SOLUTION

We now consider the construction of the upper solution for the system (11).

We note that the functions

f

1

( ),

f

2

( )

have

properties

 

 

1 2

2 2

2

( 2) 1

1 1 1 ( 2) 1 1

2 1 1

2

( 2) 1

1 2 2 1 ( 2) 1

1 2 2

(0, )

(0, )

p k

k p

m k p m

p k

k p

m m k p

df

df

f

A

B

f

C

d

d

df

df

f

A

B

f

C

d

d





 

   

 

   

 

 

and

1 1

2 2

2

( 2) 1

1 1

1 1 1 1 ( 2) 1 1

2 1 1 1

2

( 2) 1

1 1

1 1 2 2 ( 2) 1 2

1 2 2 2

p k

k p

m m

N N k p

p k

k p

m m

N N k p

df

df

df

d

f

A

B

Nf

d

d

d

d

df

df

df

d

f

A

B

Nf









 

 

   

 

 

   

  

  

(5)

1

2

( 2) 1 ( 2) 1 1

1 1

( 2) 1 1 ( 2) 1

2 2

1/

1/

k p k p m

k p m k p

A

B

p

A

B

p









   

   

Then the functions

f

1

,

f

2are solutions of the Zel'dovich-Kompaneets type for the system (1) and in the domain

( 1)/

( )

a

p p

they satisfy the system of equations

1

2

2 1

1 1 1 1 1

2 1

2 1

1 1 2 2 2

1 2

0

0

p k m

N N

p k m

N N

df

df

df

d

N

f

f

d

d

d

p d

p

df

df

df

d

N

f

f

d

d

d

p d

p

 

 

 

 

 



 

in the classical sense.

Due to the fact that

1

2

2 1

1 1 1

2 1

2 1

1 2 2

1 2

p k m

N N

p k m

N N

df

df

f

f

d

d

df

df

f

f

d

d

 

 

function

f

1

( ),

f

2

( )

and the flows have the following smoothness property

1

2 1

1 1 1

1 2 1

0

( ),

0,

,

p k m

N

df

df

N

f

f

f

C

d

d

 

 

2

2 1

1 2 2

2 1 2

0

( ),

0,

p k m

N

df

df

N

f

f

f

C

d

d

 

 

.

We choose A and B so that the inequalities carried out

1

2

( 2) 1 ( 2) 1 1

1 1

( 2) 1 1 ( 2) 1

2 2

1/

1/

k p k p m

k p m k p

A

B

p

A

B

p









   

   

(12)

(6)

1

2

2 1

1 1 1 1 1

2 1

2 1

1 1 2 2 2

1 2

p k m

N N

p k m

N N

df

df

df

d

N

f

f

d

d

d

p d

p

df

df

df

d

N

f

f

d

d

d

p d

p

 

 

 

 

 

 



 

 

That by virtue of the fact that

1

0,

2

0

(0,

)

df

df

при

d

d

from (12) we have

1

1

2 1

1 1 1 1 1

2

2 1

1 1 2 2 2

1

0,

0 ,

(0, ).

p k m

N N

p k m

N N

df df df

d

f

d d d p d

df df df

d

f

d d d p d

 

   

 

  

 

 

 

   

 

 

 

   

 

 

 

Then in the region

Q

according to the principle of

comparing solutions we have

Theorem 1. Let

u

i

(0, )

x

u

i

(0, ),

x x

R

.

Then for the solution of the problem (1) in the domain Ω we have the estimate

1 1

2 2

1 1 1

2 2 2

( , )

( , )

( ),

( , )

( , )

( ),

k t k t

u t x

u

t x

e

f

u t x

u

t x

e

f

 

 

 

1/

/ [ ( )]

p

x

t

where

f

1

( ),

f

2

( )

и

( )

t

- the functions defined

above.

IV. CONCLUSION

Thus,T theT proposedT nonlinearT mathematicalT modelT ofT

aT biologicalT populationT withT aT doubleT nonlinearityT

correctlyT reflectsT theT processT underT study.T CarryingT

outT theT analysisT ofT theT resultsT onT theT basisT ofT theT

obtainedT estimatesT ofT theT solutionsT givesT anT

exhaustiveT pictureT ofT theT processT inT two-componentT

competingT systemsT ofT theT biologicalT populationT withT

REFERENCES

1. AripovT M.T MethodT ofT ReferenceT EquationsT forT

SolvingT NonlinearT Boundary-ValueT ProblemsT

Tashkent,T Fan,T 1988,T 137T p.

2. BelotelovT N.V,T LobanovT A.IT PopulationT modelsT

withT nonlinearT diffusion.T //T MathT modeling.T -MT .;T

1997,T No.T 12,T pp.T 43-56.

3. GauzeT G.F.T OnT theT processesT ofT destructionT ofT

oneT speciesT byT anotherT inT populationsT ofT

infusoriansT //T ZoologicalT Journal,T 1934,T Vol.T 13,T

No.T 1.

4. AripovT M.,T MuhammadievT J.T AsymptoticT behaviourT

ofT automodelT solutionsT forT oneT systemT ofT

quasilinearT equationsT ofT parabolicT type.T

BuletinStiintific-UniversitateadinPitesti,T SeriaT

MatematicaT siT Informatica.T NT 3.T 1999.T pg.T 19-40 5. AripovT M.M.T MuhamediyevaT D.K.T ToT theT

numericalT modelingT ofT self-similarT solutionsT ofT

reaction-diffusionT systemT ofT theT oneT taskT ofT

biologicalT populationT ofT Kolmogorov-FisherT type.T

InternationalT JournalT ofT EngineeringT andT

Technology.T Vol-02,T Iss-11,T Nov-2013.T 2013. 6. AripovT M.M.T MukhamedievaT D.KT ApproachesT

toT solvingT oneT taskT ofT aT biologicalT

population.T QuestionsT ofT computationalT andT

appliedT mathematics.T -Tashkent.T 2013.T IssueT

129.T -C.22-31.AripovT M.M.T MukhamedievaT DKT

ApproachesT toT solvingT oneT taskT ofT aT

biologicalT population.T QuestionsT ofT

computationalT andT appliedT mathematics.T -Tashkent.T 2013.T IssueT 129.T

(7)

9. J.T Huisman,T J.T Sharples,T J.T M.T Stroom,T P.T

M.T Visser,T W.T E.T A.T Kardinaal,T J.T M.T H.T

Verspagen,T andT B.T Sommeijer.T ChangesT inT

turbulentT mixingT shiftT competitionT forT lightT

betweenT phytoplanktonT species.T Ecology,T

References

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