Doctor of technical sciences, full professor, M.K. Shuakayev¹

Ph.D, associate professor, S.T. Nazarbekova²

¹Kazakh British Technical University, Kazakhstan

²Al-Farabi Kazakh National University, Kazakhstan

Working - Out of Mathematical Nonlinear Models of Theory of Dynamics Population

In control theory there are two concepts for describing mathematical models of systems[1]. The first is the concept of "space of states", which describes a system in the form of systems of ordinary or stochastic differential equations. The second is Map "input-output", describes the system as a series of Volterra in nonlinear case. Study of the relationship between these concepts in various tasks is reflected in the works [2-12].

In the study of theories of population dynamics, all models were considered only in the concept of "space conditions". For example, examined management issues not limited environmental objects[13]. In [14], differential dynamics systems. .How to comment on works of theory of population dynamics, in our view, that have not been received analytical decision systems.

Therefore, this work is devoted to the development of mathematical models of population theories in the form of linear approximation deterministic model using a nonlinear transformation of variables R. Brockett and linearization by T. Karleman.

In [15] presented a specific biological example, model Mac Arthur, which is a special case of model Rosenzveig W.L., Mac Arthur R.H., which under certain assumptions the model is a special case of Kolmogorov, Kolmagorov A.N. from 1935, and published later in [16-18]. However, in General case, it has no place

Consider a model Rosenzveig W.L., Mac Arthur R.H. [17], which is the next system

(1)

Let the polynomials and p resented in the following form

, . (2)

According to [15], polynomial system (1) describes the rate of change of the number of victims in the absence of predators, and polynomial -intensity predation, coefficients - efficiency ratios, making victims of predator and mortality in predator.

Note that the system (1)-(2) is singular system of ordinary differential equations, which is, for example, by the extremely narrow class of matrices of Drazin [19], using numeric algorithms, but not the only way.. In the multidimensional case, when n is solvable system, since there is no multiplication of vectors and this is the main obstacle for her solution.

However, the problem of stochastic bilinear filtering, where was a multi-dimensional system with operation Kroneker product[20]. The authors intend to soon give the analytical solution of the system already in operation.

Statement of the problem

For (1)-(2) we must build a linearized model , retaining the original polynomial constant coefficients, and , .

Theorem 1. For nonlinear deterministic system (1) - (2) there is linearized, approximation model

, (3)

Где where

, (4)

,

The matrix contains all of the original coefficients of polynomials

, and parameters, .

Proof: On the base of the method of nonlinear conversion variables R. Brockett [21], we have - двух мерный вектор компоненты - 2-demensional vector of components ,

, . (5)

Through , we denote vector, consisting of all p-linear forms from

Then using the (2) and (5) we get to system (1) is equivalent to the following form

. ( 6)

Because

,

we get the model of type

. (7)

Matrixes of system (7), composed of the coefficients polynomials and , defined by (2) and parameters and .

Turning to the limit in (7) at û, we obtain infinite dimensional system, that after linearization method of T. Carleman, we have approximation model type (3) and (4) that in the end proves our theorem[22]. On the base of theorems 1 we can build the linearized an approximation model of system (3) - (4) for Lotka-Volterra models and Kolmogorov.

Example 1. For illustration purposes, the application of the theorem 1 demonstrate its use on Mac Arthur model.

Due to the fact that this model presented in [4] is 2 - dimensional, we have every reason to rewrite it as follows

(8)

.

According to [4], the system (8), means biomass of two species of insects. To update the values of the members of the right-wing parts of the equations, consider it relative.

Insect of larvae are eaten by larvae ( member +), but adults are eaten by larvae species of larvae at condition to a high number of species or , or of both types (members- ).

At low mortality rate of higher than its natural growth ( ).

( ).

In the second equation ember reflects the natural growth of the species, - - the self-restraint of this species, the larvae - of insect eating species by eating insect larvae look like adults. According to the theorem 1, the equation system (3) shall conform to the following ratios:

, (9)

, (10)

(11)

At , we have

- (12)

At , we have

- . (13)

Check the accuracy formulas (10), when

()+

() . (14)

Producing similar on the right side of (14), we obtain the formula (11) at .

Elements of the system (6), are calculated using formulas (8) -(13) in the following form

,

,

,

,

,

.

Still, you can get all the parameters of the model for approximation system (7).

As noted in [4], that the model of Mac Arthur [5] helps to answer the question: how should impact the bio enosis, manage to quickly destroy the harmful kind. In [4] assumes that the population of harmful species is compatible with surrounded of biocenoses. The introduction of management functions in the system of equations (1) may be in two forms. First, you may enter the control parameters that define the kind of features and. This is consistent with the use of a biological control, changing the character of interaction between populations.

Secondly, the authority may be confined to a brief, dramatic changes in variables and functions. This is a one-time destruction techniques for either or both populations of chemical means. From the formulated above shows that approval for the joint of the second method is ineffective, while changing the kinds of functions and can lead to the desired result is the destruction of populations of harmful type. It is interesting to note that sometimes effects to apply not to the pest, but to his partner.

Some of the ways to be more efficient in General can be said. It depends on the available means of control and of an explicit type functions and describing interaction between populations of researched conenoses. In connection with the aforementioned comments from [4], you can enter in future the classes of linear or bilinear deterministic and stochastic systems, based on the original system (6). In nonlinear deterministic and stochastic cases, they are described as an infinite uniformly convergent Volterra’s series. We note, that this work corresponds to the theoretical direction of research, it would be interesting to solve the following problems in the future for the dynamics of populations. The problem of Realization. Let the mean kernel type known deterministic or stochastic systems. We must to defined parameters of the original system.

Problems of the structural and parametric identification. Let the known values of the input and output of the system. We must to defined firstly structure and parameters of the original system. Solution of Realization Problem, structural and parametric identification are members assigned with the biological meaning, here we see a modern development of the theory of mathematical models of ecological systems. This paper investigates the problem of finding the analytical solutions to mathematical models of the dynamics of populations, due to the use of classical methods of management theory, which undoubtedly will make a definite contribution to and enrich the scope of this direction, and it is the nature of the theoretical result. The authors working - out algorithm representation of two-dimensional approximation models using a nonlinear transformation of variables and linearization of T. Karleman. The results are illustrated on the biological model of Mac Arthur [16].

Theory models for future studies, we can propose of another classes of bilinear, deterministic and stochastic systems, and also made mathematical problems of setting structural and parametric identification for the dynamics of populations.

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