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