SOLUTION OF BIOLOGICAL POPULATION TASK BY TAKING INTO
ACCOUNT THE REACTION-DIFFUSION
Rustam Sh. Melikuziev
1
, M.Sh. Melikoziev
2
1
University of Tashkent for Applied Sciences, Tashkent, Uzbekistan
2
Tashkent University of Information Technologies named after Muhammad al-Khorazmi
(rustam.timur4323@gmail.com, melikuziyevmurodbek7@gmail.com)
https://doi.org/10.5281/zenodo.10471370
Keywords:
Cross-diffusion, biological population, double nonlinearity, variable density, model, algorithm, Cauchy problem.
Abstract:
The work examines the solution of problems of a biological population taking into account reaction-diffusion. This research
work uses a method for analyzing the qualitative characteristics of solutions to a system of differential equations in order to
solve the problem of choosing the optimal initial approximation for effective iterative convergence to the solution of the
Cauchy problem. The effectiveness of this method depends on the numerical parameters and initial data of the system.
Researchers use the asymptotic representation of the solution as an initial approximation, which allows them to conduct a
numerical experiment and visualize the evolution of the system depending on the parameter values. The results of the study
show that the proposed nonlinear mathematical model successfully reflects the process of competing biological populations
with double nonlinearity. Analysis of the obtained solution estimates confirms the preservation of localization in a finite
range and the size of the source in the system. Thus, this work provides a complete, comprehensive picture of the process
and can be a valuable contribution to the study of two-component competing systems in biological populations.
1 INTRODUCTION
Research on the development and application of
nonlinear mathematical models is carried out in many
developed countries, including the USA, Japan, Spain,
Germany, Great Britain, France, the Russian Federation,
Uzbekistan and many others. Nonlinear mathematical models
play an important role in various fields of science and
industry, such as physics, biology, economics, engineering,
artificial intelligence, climatology and others. These models
are used to analyze complex systems that cannot always be
described by linear equations. For example, in medical
research, nonlinear models can be used to study disease
dynamics, in economics to analyze market trends and forecast
economic development, and in climatology to model changes
in climate. Research in this area may include the development
of new methods and algorithms for analyzing and solving
nonlinear mathematical models, as well as their application to
specific applied problems. It is important to note that with
increasing computing power and data availability, interest in
nonlinear models continues to grow, and they play a key role
in modern research and development [1-5].
Mathematical modeling of nonlinear processes is of
fundamental importance in modern science, and this is also
true in biology and ecology, where it is important to
understand and predict the complex interactions between
biological populations and their environment. Modeling
biological populations using nonlinear models can help in the
study of population dynamics and in the development of
natural resource management and conservation strategies. For
example, nonlinear models can be useful for analyzing the
effects of climate change on the distribution and behavior of
species, studying predator-prey interactions or competition
between species in ecosystems. Cross-diffusion, or
interactions between different species or subpopulations in
biological systems, is one of the key characteristics of
nonlinear biological models. Research into this phenomenon
can be important for understanding ecological processes,
including the spread of disease, species migration, and
population dynamics. The development of methods and
algorithms for solving nonlinear problems in biological
populations is an important area of research. These methods
may include numerical algorithms for solving systems of
differential equations, optimizing model parameters, and
analyzing the sensitivity of models to changes in parameters.
Given the complexity and diversity of biological systems,
further development of nonlinear models and methods for
their solution will be important for science and practical
applications in biology and ecology [6,7].
As a result of research on numerical modeling of cross-
diffusion systems carried out around the world, a number of
scientific results have been obtained and represent important
contributions to the field of mathematical modeling and
understanding of complex processes in nature. The proof of
the Turing bifurcation in a one-dimensional stationary model
is an important result in the theory of nonlinear diffusion
systems. This type of bifurcation can lead to the formation of
structural patterns in a system, such as the formation of
patterns and structures, and has wide applications in biology
and chemistry. The study of the conditions for the existence
and uniqueness of solutions in diffusion systems is of great
importance for substantiating the mathematical correctness of
models and their applicability in real problems. Studying the
stability of stationary solutions and diffusion instability helps
determine which structures and solutions can be stable in the
long term. The Maxwell-Stephan system of equations
describes diffusion processes in multicomponent mixtures. Its
study is important for understanding the transport of
molecules in gases and liquids and has applications in
chemical engineering and physics. Studying the correctness
and existence of both weak and strong solutions helps to
understand the qualitative properties of models and their
ability to predict the behavior of a system. The Shigezada-
Kawasaki-Teramoto model is an example of a nonlinear
diffusion system that can have interesting dynamic properties.
Her research could lead to a better understanding of the
dynamics in complex physical systems. All these results show
how important it is to study mathematical models and their
application in various fields of science. They can be useful for
developing new control strategies and solving complex
problems associated with diffusion and interaction in various
systems [4–8].
The purpose of the study is to analyze the qualitative
properties of nonlinear mathematical models with double
nonlinearity and develop numerical schemes for describing
multicomponent cross-diffusion systems of a biological
population, which is an important direction in mathematical
modeling and has practical significance in biology and
ecology. Studying the qualitative properties of mathematical
models allows us to understand what types of dynamics can
arise in a system. This is important for determining the
stability of populations, the possibility of the existence of
stable stationary solutions and other key characteristics of the
system. Analysis of dual nonlinearity models can be complex,
but can reflect more realistic interactions between different
species or components in a biological system. This may
include competition, predatory relationships, and other types
of interactions. The study of multicomponent systems is
important for understanding complex ecosystems and
biological communities. This may involve the interaction of
several types of populations and analysis of the influence of
some species on others. The development of numerical
schemes and methods for solving complex mathematical
models makes it possible to conduct numerical experiments
and analyze various scenarios of population dynamics under
different conditions [9-14].
The diffusion equation can be used to describe the
distribution of various substances, both in biological systems
(for example, the diffusion of chemicals in tissues or the
migration of populations) and in physical systems (for
example, the diffusion of heat in materials or the diffusion of
matter in liquids). This property of the diffusion equation
allows it to be used in various fields of science and
engineering. Thus, diffusion equations represent a powerful
tool for describing various processes and allow scientists to
transfer knowledge and methods from one field of science to
another, which contributes to a deeper understanding of
nature and the development of scientific and engineering
solutions [15-17].
The numerical parameters that appear in these equations
can have different physical meanings and depend on the
specific conditions of the problem. For example, in the
thermal conductivity equation, the parameters may represent
the thermal conductivity of the material and the initial
temperature, while in the equation for species distribution, the
parameters may correspond to the diffusion rate and the initial
concentration of the species [18-20].
2.Methods
In this paper we investigate the properties of solutions of the
problem of a biological population of the Fisher-Kolmogorov
type in the case of variable density. The main method of
investigation is the self-similar approach. Consider in the
field
Q={(t,x): 0< t <
, x
𝑅
𝑁
}
parabolic system of two
quasilinear reaction-diffusion equations which describes the
process of a biological population of the Kolmogorov-Fisher
type in a nonlinear two-component medium,
{
𝜕𝑢
1
𝜕𝑡
= 𝛻(𝐷
1
𝑢
2
𝑚
1
−1
|𝛻𝑢
1
𝑘
|
𝑝−2
𝛻𝑢
1
) + 𝑘
1
𝑢
1
(1 − 𝑢
1
𝛽
1
),
𝜕𝑢
2
𝜕𝑡
= 𝛻(𝐷
2
𝑢
1
𝑚
2
−1
|𝛻𝑢
1
𝑘
|
𝑝−2
𝛻𝑢
2
) + 𝑘
2
𝑢
2
(1 − 𝑢
2
𝛽
2
),
(1)
𝑢
1
|𝑡 = 0 = 𝑢
10
(𝑥)
,
𝑢
2
|𝑡 = 0 = 𝑢
20
(𝑥)
(2)
the coefficients of mutual diffusion are respectively
equal
𝐷
1
𝑢
2
𝑚
1
−1
|𝛻𝑢
1
𝑘
|
𝑝−2
𝛻𝑢
1
,
𝐷
2
𝑢
1
𝑚
2
−1
|𝛻𝑢
2
𝑘
|
𝑝−2
𝛻𝑢
2
.
Numeric parameters
𝑚
1
, 𝑚
2
, 𝑛, 𝑝, 𝛽
1
, 𝛽
2
,
D
1
,
𝐷
2
- positive real
numbers,
𝛻(. ) − 𝑔𝑟𝑎𝑑(. )
𝑥
,
𝛽
1
, 𝛽
2
≥ 1
,
𝑥 ∈ 𝑅
𝑁
𝑙 > 0
;
𝑢
1
= 𝑢
1
(𝑡, 𝑥) ≥ 0
,
𝑢
2
= 𝑢
2
(𝑡, 𝑥) ≥ 0
- sought solutions.
We will study the properties of solutions of problem (1),
(2) on the basis of a self-similar analysis of solutions of the
system of equations constructed by the method of nonlinear
splitting and standard equations.
We note that the substitution in (1)
𝑢
1
(𝑡, 𝑥) = 𝑒
−𝑘
1
𝑡
𝑣
1
(𝜏(𝑡), 𝑥)
,
𝑢
2
(𝑡, 𝑥) = 𝑒
−𝑘
2
𝑡
𝑣
2
(𝜏(𝑡), 𝑥)
will bring it to mind:
{
𝜕𝑣
1
𝜕𝜏
= 𝛻(𝐷
1
𝑣
2
𝑚
1
−1
|𝛻𝑣
1
𝑘
|
𝑝−2
𝛻𝑣
1
) + 𝑎
1
(𝑡)𝑣
1
𝛽
1
+1
,
𝜕𝑣
2
𝜕𝜏
= 𝛻(𝐷
2
𝑣
1
𝑚
2
−1
|𝛻𝑣
2
𝑘
|
𝑝−2
𝛻𝑣
2
) + 𝑎
2
(𝑡)𝑣
2
𝛽
2
+1
,
(3)
where
𝑎
1
= 𝑘
1
((𝑝 − 2)𝑘𝑘
1
+ (𝑚
1
− 1)𝑘
2
)
𝑏
1
,
𝑏
1
=
(𝛽
1
−(𝑝−2)𝑘)𝑘
1
−(𝑚
1
−1)𝑘
2
(𝑝−2)𝑘𝑘
1
+(𝑚
1
−1)𝑘
2
,
𝑎
2
= 𝑘
2
((𝑚
2
− 1)𝑘
1
+ (𝑝 − 2)𝑘𝑘
2
)
𝑏
2
,
𝑏
2
=
(𝛽
2
−(𝑝−2)𝑘)𝑘
2
−(𝑚
2
−1)𝑘
1
(𝑚
2
−1)𝑘
1
+(𝑝−2)𝑘𝑘
2
.
If
𝑏
𝑖
= 0
,
and
𝑎
𝑖
(𝑡) = 𝑐𝑜𝑛𝑠𝑡
,
𝑖 = 1,2
, then the system has the
form:
{
𝜕𝑣
1
𝜕𝜏
= 𝛻(𝐷
1
𝑣
2
𝑚
1
−1
|𝛻𝑣
1
𝑘
|
𝑝−2
𝛻𝑣
1
) + 𝑎
1
𝑣
1
𝛽
1
+1
,
𝜕𝑣
2
𝜕𝜏
= 𝛻(𝐷
2
𝑣
1
𝑚
2
−1
|𝛻𝑣
2
𝑘
|
𝑝−2
𝛻𝑣
2
) + 𝑎
2
𝑣
2
𝛽
2
+1
,
(4)
𝑣̄
2
(𝜏) = (𝜏(𝑡))
−𝛾
1
, 𝛾
1
=
𝑏
1
+1
𝛽
1
,
𝑣̄
2
(𝜏) = (𝜏(𝑡))
−𝛾
2
, 𝛾
2
=
𝑏
2
+1
𝛽
2
,
Then the solution of system (3) is sought in the form
𝑣
1
(𝑡, 𝑥) = 𝑣̄
1
(𝜏)𝑤
1
(𝜏(𝑡), 𝑥),
𝑣
2
(𝑡, 𝑥) = 𝑣̄
2
(𝜏)𝑤
2
(𝜏(𝑡), 𝑥),
(5)
Then we obtain a system of equations:
{
𝜕𝑤
1
𝜕𝜏
= 𝛻(𝐷
1
𝑤
2
𝑚
1
−1
|𝛻𝑤
1
𝑘
|
𝑝−2
𝛻𝑤
1
) + 𝜓
1
(𝑤
1
− 𝑤
1
𝛽
1
+1
),
𝜕𝑤
2
𝜕𝜏
= 𝛻(𝐷
2
𝑤
1
𝑚
2
−1
|𝛻𝑤
2
𝑘
|
𝑝−2
𝛻𝑤
2
) + 𝜓
2
(𝑤
2
− 𝑤
2
𝛽
2
+1
),
(6)
If
1 − [𝛾
1
(𝑝 − 2)𝑘 + 𝛾
2
(𝑚
1
− 1) = 0
,
self-similar solution of
system (9) has the form
𝑤
𝑖
(𝜏(𝑡), 𝑥) = 𝑓
𝑖
(𝜉), 𝑖 = 1,2, 𝜉 = 𝑥/[𝜏(𝑡)]
1/𝑝
. (7)
Then substituting (7) in (6) with respect to
𝑓
𝑖
(𝜉)
we
obtain a system of self-similar equations
{
𝜉
1−𝑁 𝑑
𝑑𝜉
(𝜉
𝑁−1
𝑓
2
𝑚
1
−1
|
𝑑𝑓
1
𝑘
𝑑𝜉
|
𝑝−2
𝑑𝑓
1
𝑑𝜉
) +
𝜉
𝑝
𝑑𝑓
1
𝑑𝜉
+ 𝜇
1
𝑓
1
(1 − 𝑓
1
𝛽
1
) = 0,
𝜉
1−𝑁 𝑑
𝑑𝜉
(𝜉
𝑁−1
𝑓
1
𝑚
2
−1
|
𝑑𝑓
2
𝑘
𝑑𝜉
|
𝑝−2
𝑑𝑓
2
𝑑𝜉
) +
𝜉
𝑝
𝑑𝑓
2
𝑑𝜉
+ 𝜇
2
𝑓
2
(1 − 𝑓
2
𝛽
2
) = 0.
(8)
where
𝜇
1
=
1
(1−[𝛾
1
𝑘(𝑝−2)+𝛾
2
(𝑚
1
−1)])
and
𝜇
2
=
1
(1−[𝛾
2
𝑘(𝑝−2)+𝛾
1
(𝑚
2
−1)])
.
The system (8) has an approximate solution of the form
𝑓̄
1
= 𝐴(𝑎 − 𝜉
𝛾
)
𝑛
1
, 𝛾 = 𝑝/(𝑝 − 1)
,
𝑓̄
2
= 𝐵(𝑎 − 𝜉
𝛾
)
𝑛
2
,
where А and В constant and
𝑛
1
=
(𝑘(𝑝−2)+1)(𝑘(𝑝−2)−(𝑚
1
+1))
𝑘
2
(𝑝−2)
2
−(𝑚
1
−1)(𝑚
2
−1)
,
𝑛
2
=
(𝑘(𝑝−2)+1)(𝑘(𝑝−2)−(𝑚
2
+1))
𝑘
2
(𝑝−2)
2
−(𝑚
1
−1)(𝑚
2
−1)
.
Let us construct an upper solution for system (1).
Note that the functions
𝑓̄
1
(𝜉), 𝑓̄
2
(𝜉)
have properties
𝑓̄
2
𝑚
1
−1
|
𝑑𝑓̄
1
𝑘
𝑑𝜉
|
𝑝−2
𝑑𝑓̄
1
𝑑𝜉
− 𝐴
𝑘(𝑝−2)+1
𝐵
𝑚
1
−1
(𝛾𝑛
1
𝑘)
(𝑝−2)
𝛾𝑛
1
𝜉𝑓̄
1
∈ 𝐶(0,
∞
)
𝑓̄
1
𝑚
2
−1
|
𝑑𝑓̄
2
𝑘
𝑑𝜉
|
𝑝−2
𝑑𝑓̄
2
𝑑𝜉
− 𝐴
𝑚
2
−1
𝐵
𝑘(𝑝−2)+1
(𝛾𝑛
2
𝑘)
(𝑝−2)
𝛾𝑛
2
𝜉𝑓̄
2
∈ 𝐶(0,
∞
)
due to the fact that
(𝛾 − 1)(𝑝 − 1) = 1,
𝛾 =
𝑝
𝑝−1
,
and
Let us choose A and B from the system of nonlinear algebraic
equations
Then functions
𝑓̄
1
, 𝑓̄
2
are Zeldovich-Kompaneyets type
solutions for system (3.1) and in the region
|𝜉| < (𝑎)
(𝑝−1)/𝑝
,
which satisfy the system of equations
{
𝜉
1−𝑁 𝑑
𝑑𝜉
(𝜉
𝑁−1
𝑓̄
2
𝑚
1
−1
|
𝑑𝑓̄
1
𝑘
𝑑𝜉
|
𝑝−2
𝑑𝑓̄
1
𝑑𝜉
) +
𝜉
𝑝
𝑑𝑓̄
1
𝑑𝜉
+
𝑁
𝑝
𝑓̄
1
= 0
𝜉
1−𝑁 𝑑
𝑑𝜉
(𝜉
𝑁−1
𝑓̄
1
𝑚
2
−1
|
𝑑𝑓̄
2
𝑘
𝑑𝜉
|
𝑝−2
𝑑𝑓̄
2
𝑑𝜉
) +
𝜉
𝑝
𝑑𝑓̄
2
𝑑𝜉
+
𝑁
𝑝
𝑓̄
2
= 0
(9)
in classic sence.
Thus,
let
𝑢
𝑖
(0, 𝑥) ≤ 𝑢
𝑖±
(0, 𝑥), 𝑥 ∈ 𝑅.
Then in the domain Q the
following estimate holds for solving system (9):
𝑢
1
(𝑡, 𝑥) ≤ 𝑢
1+
(𝑡, 𝑥) = 𝑒
𝑘
1
𝑡
𝑓̄
1
(𝜉),
𝑢
2
(𝑡, 𝑥) ≤ 𝑢
2+
(𝑡, 𝑥) = 𝑒
𝑘
2
𝑡
𝑓̄
2
(𝜉),
𝜉 = 𝑥/[𝜏(𝑡)]
1/𝑝
.
Here functions
𝑓̄
1
(𝜉), 𝑓̄
2
(𝜉)
и
𝜏(𝑡)
- defined above.
3.Result.
Table 1 looked at fast diffusion. As an initial
approximation we took:
Table 1
Fast diffusion process analysis results.
𝑢
1
(𝑥, 𝑡) = (𝑇 + 𝜏(𝑡))
−𝛾
1
(𝑎 + 𝜉
𝛾
)
𝑛
1
,
𝑢
2
(𝑥, 𝑡) = (𝑇 + 𝜏(𝑡))
−𝛾
2
(𝑎 +
𝜉
𝛾
)
𝑛
2
,
𝛾
1
=
1
𝛽
1
,
𝛾
2
=
1
𝛽
2
,
𝛾 =
𝑝
𝑝−1
,
𝑛
𝑖
=
(𝑝−1)[𝑘(𝑝−2)−(𝑚
𝑖
−1)]
𝑞
,
𝑖 = 1,2
,
𝑞 = 𝑘
2
(𝑝 − 2)
2
− (𝑚
1
− 1)(𝑚
2
− 1)
.
Parameter values must be
𝑛
1
> 0, 𝑛
2
> 0, 𝑞 < 0
,
1 − [𝛾
1
(𝑝 − 2)𝑘 + 𝛾
2
(𝑚
1
− 1)] = 0
:
𝜏(𝑡) = 𝑙𝑛( 𝑡)
.
Table 2.
Speed diffusion process analysis results.
Table 3 looked at slow diffusion. The initial approximation should be:
𝑢
1
(𝑥, 𝑡) = (𝑇 + 𝜏(𝑡))
−𝛾
1
(𝑎 − 𝜉
𝛾
)
+
𝑛
1
,
𝑢
2
(𝑥, 𝑡) = (𝑇 + 𝜏(𝑡))
−𝛾
2
(𝑎 −
𝜉
𝛾
)
+
𝑛
2
,
𝛾
1
=
1
𝛽
1
,
𝛾
2
=
1
𝛽
2
,
𝛾 =
𝑝
𝑝−1
,
𝑛
𝑖
=
(𝑝−1)[𝑘(𝑝−2)−(𝑚
𝑖
−1)]
𝑞
,
𝑖 = 1,2
,
𝑞 =
𝑘
2
(𝑝 − 2)
2
− (𝑚
1
− 1)(𝑚
2
− 1)
.
The parameter values satisfy the inequalities
𝑛
1
> 0, 𝑛
2
> 0, 𝑞 > 0
,
1 − [𝛾
1
(𝑝 − 2)𝑘 + 𝛾
2
(𝑚
1
− 1)] ≠ 0
:
𝜏(𝑡) =
(𝑇+𝜏)
1−[𝛾1(𝑝−2)𝑘+𝛾2(𝑚1−1)]
1−[𝛾
1
(𝑝−2)𝑘+𝛾
2
(𝑚
1
−1)]
.
Table 3.
Slow diffusion
process analysis results.
Значения
параметров
max
1max
2 max
0.5,
1.229,
1.229
t
x
x
=
=
=
max
1max
1max
10,
2.972,
2.972
t
x
x
=
=
=
max
1max
2 max
15,
3.488,
3.488
t
x
x
=
=
=
1
2
4.1,
4.0,
4.4
m
m
p
=
=
=
3
10
eps
−
=
1
0.822
0
n
=
2
0.779
0
n
=
7.86
0
q
= −
1
2
1,
1
=
=
0.5
k
=
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
1
2
5.7,
5.4,
3
m
m
p
=
=
=
3
10
eps
−
=
1
0.291 0
n
=
2
0.24
0
n
=
11.68
0
q
= −
1
2
2,
2
=
=
3
k
=
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
1
2
3.7,
3.3,
4
m
m
p
=
=
=
3
10
eps
−
=
1
1.216
0
n
=
2
1.021 0
n
=
6.17
0
q
= −
1
2
2,
0.5
=
=
0.1
k
=
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
Table 4 shows the slow diffusion results. Initial
approximation receiving
:
Table 4.
Slow diffusion process analysis results.
𝑢
1
(𝑥, 𝑡) = (𝑇 + 𝜏(𝑡))
−𝛾
1
(𝑎 − 𝜉
𝛾
)
+
𝑛
1
,
𝑢
2
(𝑥, 𝑡) = (𝑇 +
𝜏(𝑡))
−𝛾
2
(𝑎 − 𝜉
𝛾
)
+
𝑛
2
,
𝛾
1
=
1
𝛽
1
,
𝛾
2
=
1
𝛽
2
,
𝛾 =
𝑝
𝑝−1
,
𝑛
𝑖
=
(𝑝−1)[𝑘(𝑝−2)−(𝑚
𝑖
−1)]
𝑞
,
𝑖 = 1,2
,
𝑞 = 𝑘
2
(𝑝 − 2)
2
− (𝑚
1
−
1)(𝑚
2
− 1)
.
The parameter values satisfy the inequalities
𝑛
1
>
0, 𝑛
2
> 0, 𝑞 > 0
.
Here
1 − [𝛾
1
(𝑝 − 2)𝑘 + 𝛾
2
(𝑚
1
− 1)] = 0
:
𝜏(𝑡) = 𝑙𝑛( 𝑡)
.
CONCLUSIONS
4.
Thus, on the basis of the above described
methodology, the qualitative properties of solutions to system
(1) were studied, on its basis the problem of choosing an
initial approximation for the iterative one was solved, leading
to rapid convergence to the solution of the Cauchy problem,
depending on the values of the numerical parameters and
initial data. For this purpose, the asymptotic representations
of the solution we found were used as an initial
approximation. This made it possible to perform a numerical
experiment and visualize the process described by system (1),
depending on the values included in the system of numerical
parameters.
In this paper, based on the method described above, the
qualitative properties of the solutions of the system (1) are
investigated, and on this basis the problem of choosing the
initial approximation for the iterative solution is solved,
leading to rapid convergence to the solution of the Cauchy
problem (1), (2), depending on the value of the numerical
parameters and initial data. For this purpose, the asymptotic
representation of the solution found by us was used as the
initial approximation. This allowed us to perform a numerical
experiment and visualization of the process, described by the
system (1), depending on the values entering into the system
of numerical parameters.
Thus, the proposed nonlinear mathematical model of a
biological population with a double nonlinearity correctly
reflects the process under study. Carrying out the analysis of
the results on the basis of the obtained estimates of the
solutions gives an exhaustive picture of the process in two-
component competing systems of the biological population
with preservation of the localization properties in the final
range and the size of the outbreak. It makes it possible to
estimate the propagation velocity of diffusion waves.
REFERENCES
1.
Aripov M. Method of Reference Equations for Solving
Nonlinear Boundary-Value Problems Tashkent, Fan,
1988, 137 p.
2.
Belotelov N.V, Lobanov A.I Population models with
nonlinear diffusion. // Math modeling. -M .; 1997, No.
12, pp. 43-56.
3.
Lin X. and Wang M. The critical exponent of doubly
singular parabolic equations. // J. Math. Anal. Appl.,
257:1, 2001, pp.170-188.
4.
Гаузе Г.Ф. О процессах уничтожения одного вида
другим в популяциях инфузорий // Зоологический
журнал,1934, т.13, №1.
5.
Shigesada N., Kawasaki K., Biological Invasions:
Theory and Practice (Oxford University Press, Oxford,
1997)
6.
С. П. Курдюмов, Е. С. Куркина, “Спектр
собственных функций автомодельной задачи для
нелинейного уравнения теплопроводимости с
источником”, Ж. вычисл. матем. и матем. физ., 44:9
Значения
параметров
max
1max
2 max
0.5,
1.229,
1.229
t
x
x
=
=
=
max
1max
1max
10,
2.972,
2.972
t
x
x
=
=
=
max
1max
2 max
15,
3.488,
3.488
t
x
x
=
=
=
1
2
3,
3.5,
5
m
m
p
=
=
=
3
10
eps
−
=
1
1 0
n
=
,
2
0.5
0
n
=
4
0
q
=
,
1
2
5,
5
=
=
,
1
k
=
time1 FRAME
0
+
(
) time2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
1
2
3,
3.5,
7
m
m
p
=
=
=
3
10
eps
−
=
,
1
0.505
0
n
=
,
2
0.474
0
n
=
95
0
q
=
1
2
14,
7
=
=
,
2
k
=
time1 FRAME
0
+
(
) time2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
t ime1 FRAME
0
+
(
) t ime2 FRAME
0
+
(
)
(2004), 1619–1637; Comput. Math. Math. Phys., 44:9
(2004), 1539–1556.
7.
Самарский А.А., Галактионов В.А., Курдюмов С.П.,
Михайлов А.П. Режимы с обострением в задачах для
квазилинейных параболических уравнений. – М.:
Наука, 1987, 480с.
8.
Galaktionov V. A., Vazquez J. L. The problem of blow-
up in nonlinear parabolic equations. //J. Discrete and
continuous dynamical systems. 2002. V.8. № 2. P. 399-
433.
9.
Mittal R. C. and Arora G. Quintic B-spline collocation
method for numerical solution of the extended Fisher-
Kolmogorov equation. // Int. J. of Appl. Math and Mech.
6 (1): 74-85, 2010.
10. Трифонов А.Ю., А.В. Шаповалов Одномерное
уравнение
Фишера
–
Колмогорова
с
нелокальнойнелинейностью в квазиклассическом
приближении. // Известия высших учебных
заведений. 2009. № 9. C. 14-23
11. Kombe Ismail. Doubly nonlinear parabolic equations
with singular lower order term. // Nonlinear Anal., 56,
2004, №2, pp.185-199.
12. Jong-Sheng Guo, Bei Hu. Quenching profile for a
quasilinear parabolic equation // Quarterly of applied
mathematics. v. LVIII, № 4, 2000, pp.613-626.
13. Afanas’eva N.V., Tedeev A.F. Fujita type theorems for
quasilinear parabolic equations with initial data slowly
decaying to zero. //Sbornik Mathematics. 195:4 (2004),
pp.459-478.
14. Мухидинов
Н.
Газогидродинамическое
исследование нелинейной фильтрации жидкости и
газа. Ташкент: Фан, 1977, 152с.
15. Usmanov R.N., Khamidov V.S. To the Question of
Fuzzy Evalution of Quality of Trainees Knowledge in
the System of Distance Learning // Computer Science
and
Information
Technology1(2),2013.
DOI:10.13189/csit.2013.010209. – Р 42-45.
16. Рахмонов З.Р. О повеление решений одной задачи
нелинейной фильтрации с переменной плотностью и
с нелокальным граничным условием. Узб. Матем.
Журнал, №1, 2015, 75-85.
17. Садуллаева Ш. О неограниченных решениях одного
параболического
уравнения
с
двумя
нелинейностями. Ўзбекистон математика журнали,
№3, 2015, 121-128.
18. Aripov M. 1988 Methods of reference equations for
solving nonlinear boundary value problems. 137
19. Muhamediyeva D. K. 2019 IOP Conf.Journal of
Physics: Conf. Series 1210
20. Muhamediyeva D K 2019 International Journal of
Mechanical and Production Engineering Research and
Development 9 1095
