Ecoinformatics Problems in the Future World - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 828

m a ð 0 Þ ¼m b ð 0 Þ ¼m Ra ð 1 Þ ¼m Rb ð 1 Þ ¼0 m a ð 1 Þ ¼d 1 ; exp½d a N Ra ð 0 Þ ;

a

m b ð 1 Þ ¼ d 1 exp½d b N Rb ð 0 Þ; m Ra ð 0 Þ ¼M b ð 0 Þ d 1

exp½d a N Ra ð 0 Þ ;

b a

m Rb ð 0 Þ ¼M a ð 0 Þ d b exp½b b N Rb ð 0 Þ

ð 10 :

42 Þ

The cost of the game is:

V 2 ¼ d a exp 1 þ d a M b ð 0 Þ exp½ d a N Ra ð 0 Þ f g

d b exp 1 þ d b M a ð 0 Þ exp½ d b N Rb ð 0 Þ

ð 10 :

43 Þ

f g

From Eq. ( 10.43 ) it is evident that for the system N to destroy the system H it is
necessary that the following inequality holds:

d a M b ðÞ exp d a N Ra ðÞ

½ l n ½ d a e ð 1 h a Þ N a ðÞ ð 10 :

44 Þ

from which we have:

M b ð 0 Þ d 1 exp½d a N Ra ð 0 Þ f 1 þ l n ½d a ð 1 h a Þ N a ð 0 Þ g ð 10 : 45 Þ

a

Similarly, we obtain the condition for system H:

M a ð 0 Þ d b exp½d b N Rb ð 0 Þ

f 1 þ l n ½d b ð 1 h b Þ N b ð 0 Þ g ð 10 : 46 Þ

Thus for the system N to destroy the system H it is necessary that the inequality
( 10.45 ) holds. In an analogous way, for the system H it is necessary that the
inequality ( 10.46 ) holds. The number of a-and b-elements, as a rule, is a xed

number, as follows from considerations related to the work of the system. The
number of R

elements, which perform protective functions, can be best selected
when certain physical parameters in the problem are

—

fixed and when a priori
information concerning the amount of C-elements of the opposite system is
available. For example, if the effectiveness of all R-elements is constant indepen-
dent of their number, then in order for the system H to survive, the necessary
number of protective elements must satisfy the inequality:

N ra ðÞ d a M b ðÞ l n =f d a ð 1 þ l n ½d a ð 1 h a Þ N a ð ÞÞg ð 10 : 47 Þ

It appears clear that when the number of C b -elements is constant and the sur-
vivability of the system H decreases, the necessary number of protective elements
increases rapidly. In this case, the greater their effectiveness, the smaller the number
of protective elements is required for carrying out one and the same task. With an
increase in survivability the necessary number of protective elements can be
decreased.