Ecoinformatics Problems in the Future World - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 831

The matrix of this game has a saddle-shaped point in the case of following
conditions: M a (0) ≤ N Rb (0)exp[d b N Rb (0)], M b (0) ≤ N Ra (0)exp[d a N Ra (0)] and that it

takes place in the real system.
Thus for both systems it is advantageous during the first step to provide the

protective elements with a small effectiveness and to increase their effectiveness
during the second step. This is natural, since it is better to lose a greater number of
protective elements during the first step, thus securing a reliable defense of the

working elements during the second step of the game.
From Eq. ( 10.53 ) it follows that the loses of a-and b-elements by system H and
N, respectively will amount to:

Q 1 ¼ exp 1 þ d 1a d 2a f ð Þ N Ra ðÞþ d 2a M b ðÞ exp d 1a N Ra ðÞ ½ g= d 2a ;

ð 10 : 54 Þ

Q 2 ¼ exp 1 þ d 1b d 2b f ð Þ N Rb ðÞþ d 2b M a ðÞ exp d 1b N Rb ½ g= d 2b :

In order to determine the optimum value of the effectiveness of the R a -and R b -
elements for each step of the game, we must nd:

min
ð d 1a ; d 2a Þ

Q 1 ð d 1a ; d 2a Þ and min

ð d 1b ; d 2b Þ

Q 2 ð d 1b ; d 2b Þ

From Eq. ( 10.54 ) we obtain:

@ Q 1 =@ d 1a ¼ 1 þ 2d a d 1a ð Þ f 2N Ra ðÞ M b ðÞ

ð 10 : 55 Þ

½ 1 þ N Ra 2d a d 1a ð Þ exp d 1a N Ra ðÞ ½ g ¼ 0 ;

@ Q 2 =@ d 1b ¼ 1 þ 2d b d 1b f

1 þ N Rb ðÞ 2d b d 1b

ð Þ 2N Rb ðÞ M a ðÞ

ð 10 : 56 Þ

½ ð Þ exp d 1b N Rb ðÞ ½ g;

From Eq. ( 10.55 ) it is evident that if the equation:

f ð 2d a d 1a Þ= d a d 1a ð Þ g exp d 1a N Ra ðÞ ½ ¼ 0 : 5N Ra ðÞ= M b ðÞ ð 10 : 57 Þ

has a real root 0 d 1a 2d a then system H, by utilizing this root for its own
optimum strategy, can guarantee on the average losses of elements not exceeding
Q 1 ¼½e ð 2d a d 1a Þ 1 . If Eq. ( 10.57 ) does not have a root in the interval [0, 2d a ],
the optimum strategy is then determined either by the root of Eq. ( 10.55 )orby
d 1a ¼ 0. In particular, when M b (0) = N ra (0), then d 1a ¼ 0. Similar calculations are
realized for Eq. ( 10.56 ).