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3. if -8 j < dj < 8 j , then Aj (t +1) = Aj (t).

Now we can explain the mean of introduced tran-

sition functions. For example, the functions ф

<-,+)

(•)

(k = 1,2,..., Nj (-, +)) reflects the influence upon com­ponent Aj of those components, which related with Aj by relationship (-,+), i. e. the components from the set Lj (-, +). The greater this influence (i. e. the greater val­ues of Ai (t) from the set Lj (-, +)), the less values of dj. An influence of other components, where Aj has other relations, are "weighted" in similar way. If cumulative influence of the components, interacting with Aj and expressed by (5), exceeds the threshold value 8j , then the value of Aj changes by unit.

From the rules for definition of transitional function one can observe, that an increment of the value of Aj is less or equal to 1 (| Aj (t +1) - Aj (t) |< 1). This means, that the rate of changes of Aj is invariable. It is possible to avoid such an unnatural restriction, for example, by introducing a dependence of the increment on | dj | /8 j . However, in this paper we do not consider such extensions.

It is clear that the threshold 8 j influences the dynam­ics of the system in following way: the greater 8 j , the greater absolute value of the weighted sum dj required for overcoming 8 j in changing the value of Aj . So, if 8 j is very large, the system becomes very inert. When

8 j >max{     S    ф <■++■> (K) +     S    ф <+°> (K) +

AkєLJ (+,+) J' AkєLJ (+,0) J'

+ s ф j+~> ( k ), -( s ф <~k+> ( k ) + +   s   ф <~k°> (к) +   s   ф <~k~> (k ))},

AkєLJ (-,0) J' AkєLJ (-,-) J'

the value of Aj never changes (Aj (t) = const, t = 0,1,...). If we wish to avoid this trivial case, the value of 8 j should be not very large.

A second approach, proposed here, is based on the famous Justus von Liebich's law of limiting factors. This concept was originally applied to plant or crop growth. This approach is described in brief [3] and now we give detailed description of this approach. Our following re­sults are based on it.

Assume, that the system of relationships between A1, A2 , . , AN is given. Let us introduce two constant ma­trices C and C of size N x N . The transition functions are based on the following algorithm.

Let the system in the instant of time t has the state (A1 (t), A2 (t), ..., AN (t))T and Aj is an arbitrary fixed component. Let i runs from 1 to N , by u we denote any entry from the set W .

1. If for the current i the equality Л(Aj, Ai) = (-, u) holds, we assume

-1,   ifAi (t) > с*,

0, ifcji +1 < Aj (t) < с* -1,

1, if Aj (t) < .

Note, that no matter the specific value of u, only the influence on Aj from the side of Ai plays the role.

2. If for the current i the equality Л(Aj, Ai) = (+, u) holds, we assume

1,   if Aj (t) < ,

0, ifcji +1 < Aj (t) < c* -1,

1, if Aj (t) > c*i.

3. If for the current i the equality , Ai) = (0,u) holds, we assume fi =1.

After the cycle termination, we obtain the sequence

f1 , f2

fN . Then we can calculate the value

Af (t +1) according to the following rule:

Aj (t +1)

Dec(Aj (t)), Inc(Aj (t)),

if min {fi} :

1<i< N

if min {fi} :

1<i<N

if min {fi} :

1<i<N

1.

(6)

N

Applying this algorithm for each j = 1, 2 , we shall obtain the system's state at the instant of time

t+1.

Now we can explain the mean of an introduced tran­sition from t to t +1. E. g., stating that the given compo­nent Aj has the relationship (+, -), which is the current component A (see algorithm). According to the mean of relation (+,-), large values of Ai lead to decreasing Aj . Indeed, according to the item 1 of the algorithm, if Ai (t) > c*ji (i. e. Ai (t) is "large enough"), f = -1 and, according to (6), Aj will decrease if Aj (t) > 1. Other cases of this transition works analogically.

When we investigate real data, we do not observe the dynamics, described by relationships (3), by the matrix of the trajectory (1) or by its minor (2).

j

>

DISCRETE DYNAMICAL MODELING OF SYSTEM CHARACTERISTICS OF A TURTLE'S WALK IN ORDINARYSITUATIONS AND AFTER SLIGHT STRESS

The result of this observation is the following table of cases

M

12

C Л

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