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International Journal of Information
Systems Management Research and
Development (IJISMRD)
ISSN(P): 2250-236X; ISSN(E): 2319-4480
Vol. 4, Issue 2, Apr 2014, 27-34
© TJPRC Pvt. Ltd.
SPLITTING ALGORITHM IN KOLMOGOROV-FISHER TYPE
REACTION-DIFFUSION TASK
ARIPOV M. M. & MUHAMEDIYEVA D. K.
National University of Uzbekistan Named After Mirzo Ulugbek, Tashkent, Uzbekistan
ABSTRACT
Analyses show that the central place in the mathematical description of the spatial-temporal dynamics of one or
several interacting populations take the task of research of interaction of the migration population processes with
demographic. Population models that take into account only the demographic processes are well known and fairly
well-developed [3,4]. This so-called point model, the main assumption which is the assumption of "infinitely fast" stirring
of individuals in a given area. This assumption is true if modeled area is quite small compared with the mean free path of
individuals, or, equivalently, with a radius of individual activity [3]. If this provision violated, in population models must
account migration.
KEYWORDS: Biological Population, Nonlinear Model, Parabolic Type Equation, Self, Similar Solutions, Lower and
Upper Solutions
INTRODUCTION
The simplest and most widely used in the present situation is the hypothesis of randomness "walk" of individuals
in space. This assumption allows substantiating of using as a tool of modeling equations of reaction-diffusion type, where
as a reaction part was used the right part of the point models, and diffusion coefficients (mobility of individuals) are
assumed to be constant.
In the framework of these models it is possible to explain such effects as wave propagation of population at
settling the area and existence of complex spatio-temporal dynamics of population.
Consider in
N
T
R T Q × = ] , 0 [ generalized reaction – diffusion task of Kolmogorov – Fisher type in the
following form
) 1 ( ) (
2
β
u u t k
x
u
x
u
D
t t
u
m
p
m
- +
∂
∂
∂
∂
∂
∂
=
∂
∂
-
, (1)
+ ∞ < ∈ =
=
) ( sup , ), ( |
0 0 0
x u R x x u u
x
N
t
, (2)
which describes the process of biological populations with double nonlinearity, which diffusion coefficient is
equal to
2
1
-
-
∂
∂
p
m
m
x
u
Du . Here D, 1 , , ≥ β p m - are given constants,