информация, язык, интеллект № 3 (77) 2011

БИОНИКА ИНТЕЛЛЕКТА. 2011. № 3 (77). С. 54–59 хНурэ
N
system has only finite number of states (namely, K ),
there exists a positive integer T , for which the conditions
of periodicity hold
Aj( s) = Aj( s+ T ), ∀s ≥ s0,
for some integer s 0 > 0.
It is natural to call a number T the period of the
system. Let us extract the minor
⎛ A1( s) A1( s+ 1) … A1( s+
T −1)
⎞
⎜ ⎟
⎜
A2( s) A2( s+ 1) … A2( s+
T −1)
⎟ (2)
⎜ ⎟
⎜ ⎟
⎝AN( s) AN( s+ 1) … AN( s+
T −1)
⎠
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 Ω = { − ,0, + } , i. e. the set Ω consists
of three elements. We determine a relationship between
components i A and A j as an entry from the set Ω×Ω
and denote it as Λ ( Ai, Aj) = ( ω1, ω2)
, where ω1 ∈Ω,
ω2 ∈Ω . If Λ ( Ai, Aj) = ( ω1, ω2)
, this means of this relationship
following:
1. If ω1 ={ − } then large values of the component j A
implies decreasing the value of the component A i .
2. If 1 = {0} ω then the value of the component j A
doesn’t influence the value of the component A i .
3. If ω 1 ={ + } then the large values of the component
A j implies increasing the value of the component A i .
The relationship Λ is antisymmetric, i. e. if
Λ ( Ai, Aj) = ( ω1, ω2)
, then Λ ( Aj, Ai) = ( ω2, ω1)
. It is
obvious that all combinations ( ω1, ω 2)
correspond to
relationships (interspecific interactions) of neutralism,
competition, amensalism, predation, commensalism
and mutualism, widely used in ecology and biology. We
assume, that each component A j can have with itself
only following relationships — (0,0) , ( −, − ) and ( + , + ) ,
i. e. symmetric relationships.
Assume that all relationships Λ ( Aj, Ai)
between all
pairs ( Aj, A i)
of components A1, A 2 , … , A N are fixed.
Let us define for each A j the set of components, for
which A j has the relationship ( s, u ), s, u∈Ω, i. e. ( s, u )
is some fixed relationship from the set Ω×Ω
Lj( s, u) ={ Ai | Ω(
Aj, Ai) = ( s, u)}.
The sets L j ( + , + ) , L j ( −, − ) , L j (0,0) can have from
0 to N entries, other sets ( L j ( ω1, ω 2)
, ω1 = ω 2)
can
have from 0 to N − 1 entries. It is convenient to express
relationships by a relationships matrix. If we have N
components A1, A2, … , AN,
the relationships matrix is
called the following table
⎡ A1 A2 … AN⎤
⎢
A1
( ω1, ω1)
⎥
⎢ ⎥
⎢ A2
( ω2, ω1) ( ω2, ω2)
⎥.
⎢ ⎥
⎢ ⎥
⎢
⎣AN ( ωN, ω1) ( ωN, ω2) … ( ωN, ωN)
⎥
⎦
(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 = {1, 2, …,
K}
and Nj( 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 F j . Each function defines
the mapping
Nj( + , + ) + Nj( + ,0) + Nj( + , − ) + Nj( − , + ) + Nj( − ,0) + Nj(
−, −) → .
This mapping in symbolic form may be expressed by
the formula
Aj( t + 1) = Fj( Ak( t) ∈ Lj( + , + ), Ak( t) ∈ Lj(
+ ,0),
Aj( t) ∈ Lk( + , −), Ak( t) ∈Lj( − , + ),
(4)
Ak( t) ∈Lj( −,0), Ak( t) ∈Lj( −, −)), j = 1, 2, …,
N,
where Ak( t) ∈ Lj(
+ , + ) , Ak( t) ∈ Lj(
+ ,0) , … are the values
Ak( t ) of all A k , belonging to L j ( + , + ) , L j ( + ,0) , …
correspondingly.
The transition function, introduced by the above formula,
is quite natural in its structure. The given component
A j is influenced only by those components, which
indeed influence A j , i. e. the components from the sets
L j ( + , ω)
and L j ( − , ω)
for any ω ∈ 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
I nc( A) = min{ K, A + 1},
D ec( 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 A i 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 , 1 ()
s u
ϕ j
〈 〉 ,
⋅ , ,2 ()
s u
ϕ j
〈 〉 〈 s, u〉
⋅ , … , ϕ () ⋅ . These are the
j, Nj functions of interactions of those components, where
A j has relationships ( s, u ) , s ∈ { + , − } , u ∈ W . The
properties of these functions are the following:
1. The functions are defined on the discrete set .
,
2. ϕ j, k ()
〈+ +〉 ,0
⋅ , ϕ j, k ()
〈+ 〉 ,
⋅ , ϕ j, k ()
〈+ −〉 ⋅ are increasing functions.
,
3. ϕ j, k ()
〈− +〉 ,0
⋅ , ϕ j, k ()
〈− 〉 ,
⋅ , ϕ j, k ()
〈− −〉 ⋅ are decreasing functions.
55