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Поступила в редколлегию 11.09.2011

УДК 519.7

Про   метод   розшарування   кінцевого предикату/

Н. Є. Русакова // Біоніка інтелекту: наук.-техн. журнал. — 2011. — № 3 (77). — С. 50-53.

У статті ставиться задача розробки методу перетво­рення математичної моделі довільної кінцевої мозкопо-дібної структури у технічну эффективно діючу модель, що зветься реляційною мережею.

Табл. 2. Іл. 5. Бібліогр.: 4 найм.

UDC 519.7

About method of stratification of eventual predicate / N.E. Rusakova // Bionics of Intelligense: Sci. Mag. — 2011. — № 3

(77). —P. 50-53.

In the article the task of development of method of transformation of mathematical model of arbitrary eventual braine-like structure is put in its technical effectively operat­ing model, urgent a relation network.

Tabl. 2. Fig. 5. Ref.: 4 items.математическое моделирование. системный анализ. принятие решении

intelligence

УДК 004.942+271.47.15.14.21.21

Yu. Bespalov1, I. Gorodnyanskiy2, G. Zholtkevych3, I. Zaretskaya4, K. Nosov5, T. Bondarenko6, K. Kalinovskaya7, Y. Carrero8

3V. N. Karazin Kharkiv National University, Kharkiv, Ukraine, yuri.g.bespalov@univer.kharkov.ua; 2V. N. Karazin Kharkiv National University, Kharkiv, Ukraine, biolog@univer.kharkov.ua; 3V. N. Karazin Kharkiv National University, Kharkiv, Ukraine, zholtkevych@univer.kharkov.ua; 4V. N. Karazin Kharkiv National University, Kharkiv, Ukraine, zar@univer.kharkov.ua; 5V. N. Karazin Kharkiv National University, Kharkiv, Ukraine, k-n@nm.ru; 6Institute for Problems of Cryobiology and Cryomedicine of the National Academy of Sciences of Ukraine,

Kharkiv, Ukraine, cryo@online.kharkov.ua; 7Kharkiv Zoo, Kharkiv, Ukraine, kharkovzoo2010@gmail.com; 8College of Mount Saint Vincent, New York, USA, yafreisycarrero@ymail.com.

discrete dynamical modeling of system characteristics of a turtle's walk in ordinary situations and after slight stress

In the paper a class of discrete dynamical models, based on the intra- and between-specific relationships (interactions), which adopted in biology and ecology, is suggested. The relevance of the models in the form of a convergence in probability of sample correlation coefficients is grounded. One of the introduced models, ap­plied to the analysis of a turtle's walk under two states, allowed to reveal deep systemic factors of biomechanics of animal's walking.

DYNAMICAL SYSTEMS, BIOMECHANICS OF WALK, DISCRETE MODELS, SYSTEM IDENTI­FICATION, INTERSPECIFIC INTERACTIONS

Introduction

There are concepts that describe an animal's walk as a complex process of system character, which has been known at least since the first half of 20-th century, and are used to analyze the walking characteristics of dif­ferent animals from human beings to dinosaurs [1, 2, 6, 7, 8]. Beginning with reptiles, the relation between factors guaranteeing stability of an animal's body along with the motion rate in the process of motion is of very importance. Considerable difficulties arise when it is impossible to trace the entire sequence of an animal's motion during each phase. To get over these difficul­ties, in this paper we used a mathematical apparatus of discrete modeling and dynamic systems with feedback (DMDS), to develop with some of the authors partici­pation [3, 4, 5], a way to obtain the sequence of phases of a cycle of system changes (system trajectory) based on the partial information, when only separate phases are known but not their sequence. The objective of this paper is to investigate the relation between the rate of motion and body stability factors, in a rather simple case, of a turtle's walk using the DMDS method.

1. Theory

In this paper, the authors suggest an approach to determine the relationships between biological objects (say, species) in the framework of some discrete dy­namical model. In brief, this approach is presented in illustration [3], which we will discuss in more details.

Let a biological or ecological system be described by N components A1,A2 , ..., AN . These components can have different representations, for example, they can be express the numbers of animals or the amount of bio-mass of different species. We assume that components only take discrete values 1, 2, ..., K , i. e. K values. The value 1 means a minimum amount of a component, the value K means its maximum value, i. e. a component varies from 1 to K . Indeed, the real range of compo­nent's varying may differ from the range [1, K], but for our model the only important thing is that the compo­nent varies in some quantitative scale from minimum to maximum.

The value of each of the components is observed and measured at discrete instants of time t = 0,1,.... Thus, the values of the component Ai (i. e. i -th component) at the instants of time t = 0,1,... are numbers Ai(0),

Ai (1).

The trajectory of the system is described by an infin­itive-right matrix

(4(0)  4(1)  4(2) . >

A2(0)   A2(1)   A2(2) .

^An (0)

AN(1) AN(2)

(1)

This trajectory, as always, includes all states of the system at the instants of time t = 0,1,.... Hence the state of the system at the instant of time t is the vector (4 (0, A2 (t), ..., AN (t))T , where T means the matrix trans­position. We suppose that the system is strictly deter­mined, and its state at the instant of time t is fully de­termined by the state at the moment t -1. According to the theory of mathematical systems [9], such a system is called a free dynamical system with discrete time, but in our paper we shall use own terminology. Since the

БИОНИКА ИНТЕЛЛЕКТА. 2011. 3 (77). С. 54-59

ХНУРЭ

system has only finite number of states (namely, KN ), there exists a positive integer T , for which the condi­tions of periodicity hold

Aj (s) = Aj (s + T), Ms > s0,

for some integer s0 > 0.

It is natural to call a number T the period of the system. Let us extract the minor

ґ A1(s)    A1(s +1)   ...   A1(s + T -1) ^ A2(s)   A2(s +1)   ...   A2(s + T -1)

An (s)  An (s +1)  ...  An (s + T -1)

(2)

from the matrix (1) (s>s0), which gives full description of the behavior of the system.

Let us introduce a concept of relationships between components. Let Q = {-,0, +} , i. e. the set Q consists of three elements. We determine a relationship between components A and Aj as an entry from the set QxQ and denote it as A(Ai, Aj) = (ю1, ю2), where ю1 є Q, ю2 є Q . If A(Ai, Aj) = (ю1, ю2), this means of this rela­tionship following:

1. If ю1 = {-} then large values of the component Aj implies decreasing the value of the component Ai .

2. If ю1 = {0} then the value of the component Aj doesn't influence the value of the component Ai.

3. If ю1 = {+} then the large values of the component Aj implies increasing the value of the component Ai.

The relationship Л is antisymmetric, i. e. if Л(A, Aj) = (ю1, ro2), then , A) = (a>2, ю1). It is obvious that all combinations (ю1, ю2) correspond to relationships (interspecific interactions) of neutralism, competition, amensalism, predation, commensalism and mutualism, widely used in ecology and biology. We assume, that each component Aj can have with itself only following relationships — (0,0), (-, -) and (+, +), i. e. symmetric relationships.

Assume that all relationships Л(AJ■, Ai) between all pairs (Aj,A) of components A1,A2 , ..., AN are fixed. Let us define for each Aj the set of components, for which Aj has the relationship (s,u), s, u єQ, i. e. (s, u) is some fixed relationship from the set QxQ

Lj (s,u) = {A | Q(Aj, A) = (s, u)}.

The sets Lj (+, +), Lj (-, -), Lj (0,0) can have from 0 to N entries, other sets (Lj (ю1,ю2), ю1 = ю2) can have from 0 to N -1 entries. It is convenient to express relationships by a relationships matrix. If we have N

components

A1, Al,

AN , the relationships matrix is

called the following table

A1 A2

A1

(«>1, (01) ((02, (01)

A2

((2,(2)

A

N

An   ((»n,(01)  (o>n,ю2)       ((%,mn)

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