Ecoinformatics Problems in the Future World - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 830

being independent of the distribution of C a -elements during the last two steps. In
this case Q 1 ð 3 Þ ¼Q 1 ð 2 Þ . An analogous result is also obtained for system N.
Thus for an identical game, for instance, system N can destroy at the most
Q 1 ð 2 Þ ¼M b ð 0 Þ exp½ d a N Ra ð 0 Þ of a-elements; when the number of steps is greater
than 2, system N can destroy not more than Q 1 ð 2 Þ ¼ ð ed a Þ 1 exp½d a Q 1 ð 1 Þ of a-
elements. Therefore by comparing Q 1 ð 1 Þ with Q 1 ð 2 Þ we obtain that in the analyzed
antagonistic situation the best strategy for both systems is the performance of all
allowable operations in two steps. And that is, during the first step the action of a

portion of each system is C-elements is to be set against the protective elements of

the opposite system, and during the second step the action of the remaining force is
to be set against the working elements of the opposite system. This conclusion
completely agrees with the conclusions reached by Krapivin (1978), which were
obtained by different methods.
In the above-examined models of the interaction of the two systems, it was
assumed that the effectiveness of the protective elements does not change with
respect to time. However, this assumption in many real situations must be with-
drawn. Let us consider a case where both systems can vary the effectiveness of the
protective elements from step to step, so that at each step the effectiveness is
independent of the number of protective elements. Let the effectiveness of R a -and
R b -elements be equal to d 1a and d 1b respectively during the

'

first step in the two-step
case. Similarly, during the second step the effectiveness acquires values d 2a and d 2b
so that d 1a +d 2a = 2d a ,d 1b +d 2b = 2d b ; that is, the summed value of the
effectiveness does not exceed a constant value. As a result, we obtain the following
matrix game:

d 1a ¼ d 2a d 1a \ d 2a d 1a [ d 2a

d 1b ¼ d 2b
d 1b \

Q 11 Q 12 Q 13
Q 21 Q 22 Q 23
Q 31 Q 32 Q 33

d 2b
d 1b [ d 2b

where,

Q 11 ¼ ð ed a Þ 1 exp f M b ðÞ d a exp½ d a N Ra ð Þg ð ed b Þ 1 exp f M a ðÞ d b exp½ d b N Rb ð Þg;
Q 22 ¼ Q 33 ¼ Q 23 ¼ Q 32 ¼ ed 2a Þ 1 exp d 1a d 2a ð f ð Þ N Ra ðÞþ d 2a M b ðÞ exp d 1a N Ra ðÞ ½ g

Þ 1 exp ed 2b ð f ð d 1b d 2b Þ N Rb ðÞþ d 2b M a ðÞ exp d 1b N Rb ðÞ ½ g;

Þ 1 exp d 1a d 2a Q 12 ¼ Q 13 ¼ ed 2a ð f ð Þ N Ra ðÞþ d 2a M b ðÞ exp d 1a N Ra ðÞ ½ g

ed ðÞ

1 exp M b ðÞ d b exp d b N Rb ðÞ

f ½ g;

Q 21 ¼ Q 31 ¼ ed ðÞ

1 exp M a ðÞ d a exp d a N Ra ðÞ

f ½ g

Þ 1 exp d 1b d 2b ed 2b ð f ð Þ N Rb ðÞþ d 2b M b ðÞ exp d 1b N Rb ðÞ ½ g

ð 10 : 53 Þ