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In this paper, we discussed population model of two competing populations with non-linear double diffusion and variable density which described by nonlinear system of competing individuals. We identify new properties, such as finite speed of propagation, and localization of the outbreaks in a specific area.

Population models are studied for a long time. The first such work was done by Gause G.F. and Fisher R.D., and mathematical studies were performed by Kolmogorov, Petrovskii (KPP) and Piskunov (1937) in the famous paper [

Then there were other models of the population [

In this paper, we investigate the properties of solutions of biological population task of Fisher-Kolmogorov type in the case of variable density. The main research method is a self-similar approach. Considering in the field

which describes the process of biological population of Kolmogorov-Fisher in a nonlinear two-component environment, and mutual diffusion coefficients which are respectively equal to

We study properties of solutions to problem (1), (2) based on the self-similar analysis of solutions of a system of equations constructed by the method of nonlinear splitting and a reference equations and bringing the system (1) for radially symmetric mind. Note that replacing in (1)

leads to the form

If

we get the following system of equations:

where

If

A significant role in the study of the Cauchy problem and boundary problems for Equations (1) has self- similar solutions. Under self-similar solution we will understand as particular solutions of Equation (1), depending on the combination of t and x. Knowledge of them plays a sometimes crucial role in the study of various properties of solutions of the original equations.

Below we describe one way of obtaining self-similar system for the system of Equations (5). It consists in the following. We find first the solution of a system of ordinary differential equations

in the form

for the case of

in the form

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

and

if

Then for

where

If

Then substituting (10) into (8) with respect to

where

System (11) has an approximate solution of the form

where А and В are constants and

In this paper, on the basis of the aforesaid methods, we studied qualitative properties of solutions of the system (1), solved the problem of choosing the initial approximation for iterative, leading to fast convergence to the solution of the Cauchy problem (1), (2), depending on the values of numerical parameters and initial data. For this purpose, as the initial approximation was used, we found the asymptotic representation of the solution. This has allowed to perform numerical experiments and visualization of the process described by system (1), depending on the values included in the system of numeric parameters.

Let us build an upper solution for system (11).

Note that the functions

and

We choose A and B from the system of nonlinear algebraic equations

Then functions

in the classical sense.

Due to the fact that

function

We choose A and B such that the inequality of inequality

Since then

It is due to the fact that

from (12) we have

Then in the field Q according to the comparison principle of solutions have

Theorem 1. Let

where

Note that the solution of system (1) when

where B(a, b)-Euler Beta function.

It is proved that this view is the self-similar asymptotics of solutions of systems (1).

Here

Case

where

and

Let’s take the function

Theorem 2. The finite solution of system (11) when

Proof. We seek a solution of Equation (8) in the following form

where

where

Note that the study of the solution of the last equation is equivalent to examining the solution of Equation (11), each of which in a certain period

Let us show first of all that decision

Then for the Equation (14) has the form

To analyze the last expression we introduce a new helper function

where

And so for the function

Hence, given that

get the following algebraic equation

The latter system gives

Theorem 2 is proved.

Case

where

Theorem 3. At

Proof. In the proof of theorem used the transform

where

Substituting (15) into (11) for

where

Note that the study of the solution of the last equation is equivalent to examining the solution of Equation (11), each of which in a certain period

Let us show first of all that solution

troduce the notation

To analyze the last expression we introduce a new helper function

where

Therefore for the function

Hence, when

The calculation of the last equation gives

Theorem 3 is proved.

Investigation of qualitative properties of system (1) has allowed to perform numerical experiment depending on the values included in the system of numeric parameters. For this purpose, the initial approximation was used to construct asymptotic solutions. The numerical solution of the problem for the linearization of system (2) was used linearization methods of Newton and Picard. To build self-similar system of equations of biological population used the method of nonlinear splitting [

For the numerical solution of the problem (1) we will construct a uniform grid

and temporal grid

Replace the problem (1) implicit difference scheme and receive differential task with the error

It is known that the main problem for the numerical solution of nonlinear problems is the appropriate choice of the initial approximation and the method of linearization of system (1).

Consider the function:

where

Record

Created on input language Matlab the program allows you to visually trace the evolution process for different values of the parameters and data.

Numerical calculations show that in the case of arbitrary values

Thus, the proposed nonlinear mathematical model of biological populations with double nonlinearity and variable density properly describes the studied process. Numerical study of nonlinear processes described by equations with a double nonlinearity and analysis results on the basis of evaluation solutions provides a comprehensive picture of the process in two-component systems competing biological population with the preservation of localization properties in the target area and the size of the flash.

Results in future will provide an opportunity to evaluate the speed of propagation of diffusive waves.

Muhamediyeva Dildora, (2016) Properties of Solutions of Kolmogorov-Fisher Type Biological Population Task with Variable Density. Journal of Applied Mathematics and Physics,04,903-913. doi: 10.4236/jamp.2016.45099