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
self-inT theT fieldT Q={(t,x):T 0<T tT <,T x
N
R
T }T parabolicTsystemT ofT twoT quasilinearT reaction-diffusionT equations
1 1 2 2 2 1 11 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 01
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 parameters2 1 2
1
,
m
,
n
,
p
,
,
m
,D
1,D
2- positive real numbers,x
grad
(.)
(.)
,
1, 21,x
R
Nl
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 tu t x
e
v
t x
,2
2
( , )
2( ( ), )
k tu 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 01
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 11 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
11 1 ( 2) 1 ( 1 1) 2
b
a k p kk m k ,
1 1 1 2
1
1 1 2
(
(
2) )
(
1)
,
(
2)
(
1)
p
k k
m
k
b
p
kk
m
k
22 2 ( 2 1) 1 ( 2) 2
b
a k m k p kk ,
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
, anda
i(
t
)
const
,i
1
,
2
, then thesystem has the form:
1 1 2 2 2 1 1 11 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
, anda
i(
t
)
const
,i
1
,
2
. And in1 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 so1 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 [ (
1p
2)
k
2(
m
1
1)
0
, self-similar solution of system (9) has the form1/
( ( ), )
( ),
1, 2,
/ [ ( )]
pi i
w
t x
f
i
x
t
, . (10)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 22 1 2
1
(1 [
k p
(
2)
(
m
1)])
.The system (11) has an approximate solution of the form
1
1
(
) ,
/ (
1)
nf
A a
p
p
, 22
(
)
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( )
haveproperties
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
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
Nf
f
f
C
d
d
2
2 1
1 2 2
2 1 2
0
( ),
0,
p k m
N
df
df
Nf
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)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,
20
(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 ofcomparing 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/
/ [ ( )]
px
t
where
f
1( ),
f
2( )
и
( )
t
- the functions definedabove.
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
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