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(3)

The relationships above the main diagonal are omit­ted, since they are recovered by the relationships below the diagonal (according to the antisymmetric proper­ty).

Let ж = {1,2,..., K} and N j (s, u) is the number of components in the set Lj (s, u), j = 1,2,... ,N .A tran­sition from the state (A1 (t), A2 (t), ..., An (t))T to the state (A1(t +1), A2(t +1), ..., An(t + 1))T is described by N transition functions Fj . Each function defines the mapping

N j (+,+)+N j (+,0)+N j (+,-)+N j (-,+)+N j (-,0)+N f (-,-) ж J J J J J J —> ж.

This mapping in symbolic form may be expressed by the formula

Aj (t +1) = Fj (Ak (t) є Lj (+, +), Ak (t) є Lj (+, 0),

Aj (t) є Lk (+, -), Ak (t) є Lj (-, +), (4)

Ak (t) є Lj (-, 0), Ak (t) є Lj (-, -)), j = 1,2,..., N,

where Ak (t) є Lj (+, +), Ak (t) є Ly (+,0), ... are the val­ues Ak(t) of all Ak , belonging to Lj(+, +), Lj(+,0), ... correspondingly.

The transition function, introduced by the above for­mula, is quite natural in its structure. The given compo­nent Aj is influenced only by those components, which indeed influence Aj , i. e. the components from the sets Lj (+, го) and Lj (-, го) for any ю є W .

Now, let us describe types of relationships, inherent to real biological and ecological systems.

The formula (4) expresses a general form of transi­tion of the system from the state at the moment t to the moment t +1. For a more detailed description of the behavior of biological or ecological system, we have to specify an explicit form of transitional functions, which express dynamical properties of the system.

We suggest two approaches to such a dynamics, which are based on concepts of biological interactions.

Let us introduce the following functions defined on the set ж

Inc( A) = min{ K, A +1}, Dec(A) = max{1, A -1}.

First we define a type of relationships, which takes into account the weighted sum of all Aj (t) (inclusive Ai (t)) for calculating the value of component Ai at the instant of time t +1. We call this type of relationships a weight functions' approach. Now, this is the exact defi­nition.

For each j (j = 1,2 , ..., N) we introduce a set of

functions (pyf' (), ф j2u> (), ..., Ф J'N*. (). These are the functions of interactions of those components, where Aj has relationships (s, u), s є {+, -} , u є W . The properties of these functions are the following:

1. The functions are defined on the discrete set ж .

2. ф <jtk+> (), Ф(), Ф j+t } () are increasing func

,<+,0)

<+,->,

tions.

3. ф < k+> (), Ф <k°> (), Ф <k ; () are decreasing func-

tions.

<- 0)(

<- -)

4 ,k

4. ф a+> о

Ф (1)

ф

<- -)

(1)=0.

Now define a set of numbers 8j >0 (j = 1,2,...,N) and call them thresholds of sensivity. For the system's state at the instant of time t the following value is cal­culated

d

S    Ф <++> (Ak (t)) +     S    ф jf' (Ak (t)) +

Ak єLj (+,+)

7,k

<+,0)(

Ak єLJ (+,0)

S    Ф <+"> (Ak (t)) +

ф

<-,+) <-,-)

(Ak(t))+ (5)

(Ak(t)),

Ak єLJ (-,0) J' Ak єLJ (-,-)

(it is clear that dj depends on t, but t is omitted for short in the left side).

The value of the component Aj changes according to the value dj by the following rules

1. if dj > 8 j , then Aj (t +1) = Inc(Aj (t));

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

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