Ecoinformatics Problems in the Future World - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 825

where V 1 ¼ M b ðÞ M a ðÞ , and
M a ðÞ ¼max 0

f ; M a ðÞ max 0 ½ ; b C ðÞ N Ra ðÞ p 1 N Ra ðÞ ½ g p 1 N ra ðÞ ½ ;

M b ðÞ ¼max 0 f ; M b ðÞ max 0 ½ ; a C ðÞ N Rb ðÞ p 2 N Rb ðÞ ½ g p 2 N Rb ðÞ ½

ð 10

: 31 Þ

It is obvious that at this step of the game the participants have no pure strategies.
Therefore the solution of this game is impossible in an analytic form, and it can be
obtained in a concrete case only by the numerical method. The modelling of the
game provides some understanding of the nature of its solution. Indeed with the aid
of a computer it is possible either to construct a model of the game and to gather
statistics or to solve the functional Eq. ( 10.23 ) numerically and with the aid of
heuristic concepts to investigate the dependence of strategies on the initial condi-
tions. Of course, such an approach for a short interval of modeling cannot give any
signi

cant information concerning the solution. Nevertheless, this is the only
possible approach at the present time. The feeling of hopelessness in a speci c

situation should not deter us from seeking a solution by analytical methods. The
importance of obtaining analytical solutions is obvious, since they have the
advantage over the numerical solutions in that they make possible the detection of
the general regularities of the optimal behavior of complex systems in antagonistic
situations. The importance of the analytical solutions has been pointed out by
Krapivin (1978), who has shown that a single numerical solution cannot replace an
analytical solution in which the quantitative description of the phenomenon is most
concentrated. In the case examined here, when the participants have no information
concerning the action of the opponent in the process of the entire game, the solution
of particular cases with the aid of a computer enables us to obtain the following
quantitative description of optimal strategies.
Let M b ðÞ= M a ðÞ ¼ d n . Then if

d n 1, system N in the initial stage of the
game has a pure strategy, but for system H it is more advantageous to adopt a mixed
strategy, using the tactics of deception. During the first steps system N destroys only

C a - elements and only during the last steps does it destroy the a-elements. By using
the corresponding probability mechanism, system H must direct all its C a -elements,
with probability p 1 towards the destruction of the C b -elements, with probability p 2
towards the defense of its own elements, and with probability 1

− p 1 − p 2 towards

the attack of the b-elements. In the case when 1, then at this stage of the game

p 1 +p 2 =1,m b (n) = 0, and the behavior of the system becomes symmetrical.
During the last steps of the game systems H and N, independent of the magnitude of
ʴ n , change over to the strategies: m a (t)

ʴ n ≅

≠ 0, m b (t) ≠ 0, and b C =a C =N Ra =N Rb = 0,

that is, to the destruction of working elements.
Let us consider a particular case where the systems have no protective elements,
that is, N Ra =N Rb = 0, and therefore, p 1 =p 2 = 1. Then we obtain: