Ecoinformatics Problems of the World Ocean - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 289

Substituting ( 4.39 ) into ( 4.38 ) and integrating both parts with respect to z,we
have upon simpli cations:

<
:

=
; ¼

<
:

2
4

3
5

=
;

Z Z Z

y

z z

p ð z Þ exp ½ F ðnÞm N ðnÞ w ð y Þ exp ½ F ð x Þm N ð x Þ dx dy þ C

0 0 0

ð 4 : 40 Þ

Substituting z = 0 into ( 4.40 ), we obtain: C = p(0). Finally, from ( 4.40 ) we obtain
the formula of the vertical distribution of the phytoplankton biomass:

8
<

8
<

9
=

9
=

Z Z

z y

p ð z Þ ¼ p ð 0 Þ w ð y Þ exp ½ F ð x Þm N ð x Þ dx dy

: : ; ;

8
<
:

0 0

9
=
;

exp Z

z

½ F ðnÞm N ðnÞ d n ð 4 : 41 Þ

0

The character of the p(z) distribution from the viewpoint of the number of
maxima and their location is important for the speci

cation of the role of various
depths in the functioning of the ocean carbonic system. So, it will be investigated
based on ( 4.41 ). For:

Z Z

h h

g ð h Þ ¼ N ð z Þ dz ; wð h Þ ¼ F ðÞ dz ;

0 0

then the expression ( 4.41 ) is re-written as:

Z

h

p ð h Þ ¼ f p ð 0 Þ w ð y Þ exp ½ m g ð y Þwð y Þ dy g exp ½ wð h Þm g ð h Þ ð 4 : 42 Þ

0

Thus, p(h) is the functional of g(h) and ˈ (h). The g function has a simple

interpretation if h 2 > h 1 , then the difference g(h 2 ) − g(h 1 ) represents the zooplankton

biomass located in the layer h 1 ≤ z ≤ h 2 . The functions g(h) and ˈ (h) are mono-

tonically non-decreasing, since T(h) and F(h) are not negative.
Before proceeding to a qualitative analysis of the function p(h), let us obtain
some auxiliary results. The following sets are introduced:

H þ ¼ f h 0 : F ðÞ= N ðÞ ¼v = F 0 ðÞ= N 0 ðÞ [ v g;

H ¼ f h 0 F 0 ðÞ= N 0 ðÞ \

: F ðÞ= N ðÞ ¼v = v g