On The Exploitation Of A_two-patch Metapopulation With Delayed Juvenile Recruitment And Predation.

On the Exploitation of a Two-Pat
h Metapopulation
with Delayed Juvenile Re
ruitment and Predation
Majalah Ilmiah Himpunan Matematika Indonesia 8(2): 139-150

Asep K. Supriatna1, Georey N. Tu
k2 and Hugh P. Possingham3
1 Jurusan Matematika, Universitas Padjadjaran, Bandung 45363 - Indonesia
2 CSIRO Marine Resear
h GPO Box 1538 Hobart, Tasmania 7001 - Australia
3 Department of Zoology, The University of Queensland, St. Lu
ia Qld 4072 - Australia

Abstra
t
In this paper we dis
uss the appli
ation of the Lagrange multipliers method in natural resour
e modelling. We use the method to determine optimal harvesting strategies
for a two-pat

h metapopulation with delayed juvenile re
ruitment and predation. We
investigate the ee
ts of time-delay and predation on the optimal harvesting levels of
the metapopulation. We found that when the delays are the same for both subpopulations, the model in this paper suggests that we should harvest a relative sour
e
subpopulation more
onservatively than the other subpopulation. However, when the
delays are dierent, then there is a trade-o between the delays and the sour
e/sink
status of the subpopulations in determining the optimal harvesting strategies for the
metapopulation.

1 Introdu
tion
In this paper we develop a mathemati
al model for a predator-prey metapopulation
with delayed juvenile re
ruitment. We
onsider optimal harvesting strategies in exploiting the metapopulation using the method of Lagrange multipliers. The work in

this paper generalises the results of the previous authors who have developed optimal
harvesting strategies for various stru
tures of biologi
al populations, su
h as [2,4,12,14℄.
The results in this paper might be applied as a general guidan
e in the pra
ti
e of the
exploitation of marine living organisms, in whi
h delayed juvenile re
ruitment are often
to o

ur [9℄. Table 1 shows some known delay time for
ommer
ial marine populations.
In nature, a time delay for marine spe
ies may result from the need of the juveniles
of a spe

ies to travel from their origin/spawning habitat to the destination habitat and
also may re
e
t the time needed to mature before re
ruiting to the breeding sto
k [10℄.
1

2
Organism:

Red lip abalone
Sau
es s
allop
I
eland s
allop
Baleen whale
Sei whale

Fin whale
Orange roughy
Chinook salmon
Sturgeons
Pa
i
o
ean per
h
Atka ma
kerel
Squid

Age at maturity:

 3 years

1 year
6 years
 5 years

 9 years
8 years
 23 years
3 to 7 years
10 to 20 years
8 to 10 years
 3:6 years
 270 days

Table 1: Some known delay time for
ommer
ial marine populations
(Sour
e:[11,13℄).

Sometimes this time delay is longer than just ten or twenty years, as in the
ase of the
Australia's orange roughy. This spe
ies may take several years for juveniles to rea
h

sexual maturity. They be
ome sexually mature after about 23 years [6℄. In the next
se
tion we begin to develop a simple delayed juvenile re
ruitment model using
ouples
dieren
e equations.

2

A model for a predator-prey metapopulation with
juvenile re
ruitment delay

Assume that there is a predator and prey population in ea
h of two dierent pat
hes,
namely pat
h one and pat

h two (see Figure 1). As with the prey, let the movement of
predators between the lo
al populations be through the dispersal of juveniles. Adult
predators are assumed not to migrate from one pat
h to another pat
h. Let the
population size of the prey and predator on pat
h i at the beginning of period k
be denoted by Ni(k) and Pi(k) , respe
tively. The number of mature adults of the prey
and predator subpopulations i in the time period k + 1 is the sum of adult survival
from period k and re
ruitment from juveniles that were born
i periods ago for the
prey and i periods ago for the predator. In the absen
e of a predator-prey intera
tion,
the dynami
s of the prey is given by equation
Ni(k+1) = ai Ni(k) + pii Fi (Ni(k

i ) )

+ pjiFj (Nj (k

i ) );

(1)

where i = 1; 2 and Ni(k) is the sto
k abundan
e of prey subpopulation i in generation/year k, ai is the per generation/year adult survival of prey subpopulation i, and
the fun
tion Fi (.) is
alled the re
ruit produ
tion fun
tion of prey subpopulation i and

3

p

p

21

γ

1

12

N1

γ2

N2
α1

α2

p11

p

q11

q22
σ

τ1

1

22

σ

2

P1

P2

q21

q12

τ2

Figure 1: The gure illustrates a predator-prey metapopulation with juvenile re
ruitment delay. The delays for prey and predator populations are
i and i , respe
tively.
The boxes represent the immature sto
k before it joins the reprodu
tive adults (
ir
les).
The dots with the symbol i indi
ate that the predation rate is i . The dashes with
the symbol i indi
ate the resulting predator's osprings from predation i periods
ago.

denes the number of surviving juveniles produ
ed
generations/years ago that join
the mature sto
k in generation/year k + 1. The growth of the predator in the absen
e
of the prey is dened similarly, that is,

Pi(k+1) = bi Pi(k) + qii Gi (Pi(k

i )

) + qjiGj (Pj (k

i )

);

(2)

and hen
e we assume that the predator has another sour
e as a primary
onsumption.
This delay-dieren
e equation model is a simpli
ation of a more detailed Leslie matrix
model [3℄.
To in
lude predation into the system, we use the following fa
ts that generally
food supplies may ae
t predator reprodu
tion and adult survival of the predator [8℄.
To des
ribe prey mortality and predator reprodu
tion we use assumptions similar to
those in [15℄, i.e. adult prey mortality
aused by predation in period k is proportional
to the number of prey and predator in that period. Predator re
ruitment as a result of
biomass
onversion from the intera
tion is assumed to be proportional to the number
of
onta
ts between prey and predator, in whi
h the predator su

essfully kills the
prey some i periods ago. Mathemati
ally the prey mortality is given by i Nik Pik and

4
predator re
ruitment is iNik i Pik i , where jij  i > 0. With these additional
assumptions, a
omplete model of a predator-prey metapopulation
an be written as
Nik = pii Fi (Nik
i ) + pjiFj (Njk
i )
+aiNik + iNik Pik ;
(3)
+1

Pik+1 = qii Gi (Pik i ) + qji Gj (Pjk i )
+bi Pik + i Nik i Pik i ;

(4)
where all parameters retain the same meaning as in equations (1) and (2). Note that
for the remaining of the paper we simplify the notations Ni k and Pi k with Nik and
Pik . Equation (4) assumes that the delay i impa
ts on lo
al predator re
ruitment. In
this
ase, there is a delay of i time units between predation and benet to the lo
al
predator population. If predation only aids predator's adult survival then we would
expe
t i = 0.
( )

( )

3 An e
onomi
aspe
t on the exploitation of the
metapopulation
Suppose that to optimise the exploitation, the manager of the resour
es wants to maximise the resulting net present value, both from the predator and the prey populations.
To do this, we assume that at the end of period k subpopulation i is harvested with harvest HNik . The es
apements, SNik = Nik HNik , then grow a

ording to equations (3)
and (4) to Nik . Thus, in
luding harvesting, equations (1) and (2) be
ome
Nik = pii Fi (SNik
i ) + pjiFj (SNjk
i )
+aiSNik + iSNik SPik ;
(5)
+1

+1

Pik+1 = qii Gi (SPik i ) + qji Gj (SPjk i )
+biSPik + iSNik i SPik i ;

(6)

Next, we dene the net present value as
PV =

X1  X (
k

k=0

2

i=1

Ni

(Nik ; SNik ) + P i(Pik ; SPik )) :

(7)

We then maximise the net present value 7 over innite time subje
t to equations (5)
and (6), with non-negative es
apement less than, or equal to, the population size. We
also assume  = 1=(1 + Æ) where Æ denotes a periodi
dis
ount rate and
Xi (Xik ; HXik
X 2 fN; P g.

Z
)=

Xik
Xik HXik

(pX
Xi ( ))d;

(8)

5
Unlike [12℄, in whi
h the authors use dynami
programming approa
h, in this
paper we use the Lagrange multipliers to obtain optimal es
apements for the prey
subpopulation,

SN , and for the predator subpopulation, SP .
i

the optimal es
apements satisfy the following equations:

pN

N 1 (SN10 )


p

=

N 1 (N11 ))(a1 + p11 F10 (SN10 )
1 + 1 SP10 )

( N

p
+(pP

+( N

pN

N 2 (SN20 )


p

=

+( N

pP

P 1 (SP10 )


p

=

P 2 (SP20 )


p
+(pN
p

=

(9)

N 1 (N11 ))p21 F20 (SN20 )
1
P 2 (P21 ))2 SP20 2 ;

(10)

P 1 (P11 ))(b1 + q11 G01 (SP10 )1 + 1 SN10 1 )

( P
+( P

pP

N 2 (N21 ))p12 F10 (SN10 )
2
P 1 (P11 ))1 SP10 1 ;
N 2 (N21 ))(a2 + p22 F20 (SN20 )
2 + 2 SP20 )

( N

p
+(pP

Appendix 1 shows that

i

P 2 (P21 ))q12 G01 (SP10 )2
N 1 (N11 ))1 SN10 ;

(11)

P 2 (P21 ))(b2 + q22 G02 (SP20 )2 + 2 SN20 2 )

( P

p
+(pN
+( P

P 1 (P11 ))q21 G02 (SP20 )1
N 2 (N21 ))2 SN20 :

(12)

These equations are the general form of the optimal es
apement equations for a
two-pat
h predator-prey metapopulation with a time-delay. Note that in the absen
e of







the delay ( i = i = i = 0), we obtain optimal es
apement equations given in [12℄. If
i = i = 0, then [13℄ optimal es
apement equation for a single-spe
ies metapopulation





with time delay is obtained.
pat
hes,

pij

=

qij

= 0 for

i

On the other hand, if there is no migration between

6= j , and if Ni = Nj

=

N

Pi

and

=

Pj

=

P,

then the

impli
it optimal es
apements equation for pat
h one is

pN

N (SN 0 )


=

p

( N

p

+( P

pP

P (SP 0 )


=

p

( N

p

+( P

N (N1 ))(F1N + 
D1N )
i

P (P1 ))(G1N + 1 E1N );

(13)

N (N1 ))(F1P + 
D1P )
i

P (P1 ))(G1P + 1 E1P );

(14)

6
where F1N = a1 + 1SP ; G1N = 1SP  ; D1N = p11 F10 (SN ); E1N = 0; F1P = 1SN ;
G1P = b1 + 1 SN  ; D1P = 0 and E1P = q11 G01 (SP ): Optimal es
apements for
pat
h two
an be obtained similarly in this form. These equations are impli
it optimal
harvesting equations for two spe
ies derived by [2℄ in the presen
e of a time-delay in
the predator numeri
al response su
h as in [15℄. Finally, if both juvenile migration
and predator-prey intera
tion are ignored, equations (40) - (43)
ollapse to optimal
es
apement equation for a single-spe
ies with time-delay as in [4℄. The following se
tion
dis
usses further the optimal es
apements and gives some interpretations of the results
by
omparing them with other es
apements.
1

0

0

1

0

0

0

0

4 Optimal es
apement properties
To fa
ilitate interpretations of the optimal es
apements, we assume that the
osts of
harvesting are negligible or density and subpopulation independent. Furthermore, we
also assume that there are no dieren
es between the prey and predator pri
es and the
re
ruit produ
tion fun
tion for the prey is


Fi (Nik ; ri ; Ki ) = ri Nik

Nik 
:
Ki

1

(15)

Similarly, re
ruit produ
tion fun
tion for the predator is given by Gi (Pik ; si; Li). With
these assumptions, we
an obtain an expli
it form for the optimal es
apements for the
prey in ea
h pat
h
Ai (qi1  + qi2  ) 2Ls + Ci Bi

SN =
;
(16)

i
and the predator in ea
h pat
h
Bi (pi1 
+ pi2 
) 2Kr + Ci Ai

;
(17)
SP =

i
provided that
i 6= 0, where
(18)

= C 2 (p 
+ p 
) 2ri (q  + q  ) 2si ;
1

2

i

i

i

1

2

i

i

i

i

i1

i

1

2

i2

i1

Ki

1 (p 
+ p 
)r
i1
i2
i

1 (q  + q  2)s
Bi =
i1
i2
i


Ai =

1

2

1

and

1

i2

2

Li

ai ;

(19)

bi ;

(20)

Ci = i + i  :
(21)
If 1 = 2 or 1 = 2, we dene Ci as the dis
ounted predator eÆ
ien
y (see [12℄).
i

It
an be shown that if the following
onditions (22) and (23) are satised,
Ai < 0 and Bi < 0;

(22)

7
(23)
0  Ci > maxf 2KBi ; 2LAi g;
i
i
then
i ; SN i ; and SPi are positive. The same result
an be obtained if Ai, Bi, and Ci
una

eptable sin
e it means the predator
are positive. However, Ci > 0 is biologi
ally

i
eÆ
ien
y is more than 100% (i.e. jij > 1 or i + i > 0).
In the following part we will show that if prey subpopulation one is a sour
e
subpopulation with respe
t to its time delay, that is, satisfying inequality
r1 (p11 
+ p12 
) > r2 (p21 
+ p22 
):
(24)
then it should be harvested more
onservatively than the other subpopulation, provided
the
onditions (22) and (23) are both satised. To see this, let us assume that prey
subpopulation one is a relative sour
e, that is, (p11 + p12 )r1 > (p22 + p21)r2 , and also
r1m > r2m as in equation (24). All other parameters of the prey and the predator are
identi
al for both subpopulations ex
ept delay1 parameters1 for the prey.
Let us dene ai = a, bi = b, Ci = C , R =  a, S =  b, rim = (pi1
+ pi2 
)ri
and sim = (qi1  + qi2  )si. Using these assumptions and notations, the dieren
e
between the es
apement of prey in pat
h one to the es
apement of prey in pat
h two
is
SN = (SN SN ), whi
h
an be written as
2 
 1 
2
B  4s1m R 
(r2m r1m )

:
(25)

SN = s1m L C C K
KL
1 2
Sin
e C > 2KB (see
ondition (23)), then we have
SN > 0 if r2m < r1m , whi
h is satised
by inequality (24). Furthermore, sin
e
i is negative then
SN > 0 means SN > SN .
In other words, we should harvest prey subpopulation one more
onservatively than
prey subpopulation two if the per
apita larval produ
tion of prey subpopulation one,
whi
h is dis
ounted by its
ummulative death rate is larger than the dis
ounted per
apita larval produ
tion of prey subpopulation two. If both prey subpopulations have
the same delay,
1 =
2, then it simply restates the rule of thumb for single-spe
ies
metapopulation harvesting theory [14℄, that the relative sour
e prey subpopulation
should be harvested more
onservatively than the relative sink prey subpopulation.
Using the same method as above it
an be shown that the dieren
e between the
es
apement of predator in pat
h one to the es
apement of predator in pat
h two is

SP = (SP SP ), whi
h
an be written as
 
 1 

1m R

SP = C (r1m r2m ) C 2KB C + 4sKL
(26)

1
2
and has a non-positive value (sin
e the
onditions (22) and (23) are satised). If both
prey subpopulations have the same delay, then it simply states that the predator living
in the same pat
h with the relative sour
e prey subpopulation should be harvested
more heavily than the predator living in the other pat
h. This is
onsistent with the
rule of thumb for a non-delay predator-prey metapopulation harvesting theory [12℄.
Furthermore, if there is no predator-prey intera
tion (i = i = 0, and hen
e C = 0)
then both predator subpopulations should be harvested equally, whi
h is
onsistent
with the rule of thumb in single-spe
ies metapopulation harvesting theory [14℄.
1

2

1

2

1

1

1

2

2

2

1

1

2

2

8

5 Con
lusion
In this paper we have developed a mathemati
al model of delayed juvenile re
ruitment
predator-prey metapopulation. Optimal es
apements for the metapopulation were
derived using the method of Lagrange multipliers. Results depended not only on the
per
apita larval produ
tion as in the non-delay model [12℄, but also on the delays. This
means that dierent populations with dierent re
ruitment delays should be managed
dierently (see also [9℄).
The results showed that when the delays are the same for both subpopulations,
the model in this paper suggests that we should harvest a relative sour
e subpopulation more
onservatively than the other subpopulation. However, when the delays are
dierent, then there is a trade-o between the delays and the sour
e/sink status of the
subpopulations in determining the optimal harvesting strategies for the metapopulation. Furthermore, we also showed that even though all predator subpopulations have
the same delays, their optimal es
apements might be dierent if the delays of their
prey are dierent between subpopulations. This is not surprising sin
e the dynami
s
of the predator is in
uen
ed by the dynami
s of the prey, su
h as observed in many
populations [1,5℄.

A
knowledgement

We thank an anonymous referee who has made helpful suggestions in improving
the paper.

Referen
es
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osystem monitoring programme.
S
ien
e 9: 235-242.

Antar
ti

[2℄ T.T. Agnew (1982). Stability and exploitation in two-spe
ies dis
rete time population models with delay. E
ologi
al Modelling 15: 235-249.
[3℄ J.R. Beddington (1978). On the dynami
s of Sei whales under exploitation.
of the International Whaling Commision, S
/29/Do
7: 169-172.

Report

[4℄ C.W. Clark (1976). A delayed-re
ruitment model of population dynami
s, with
an appli
ation to baleen whale populations. Journal of Mathemati
al Biology 3:
381-391.
[5℄ R.J. Crawford and B.M. Dyer (1995). Responses by four seabird spe
ies to a
u
tuating availability of Cape an
hovy Engraulis
apensis o South Afri
a. Ibis
137: 329-339.
[6℄ R. Fran
is (1992). Use of risk analysis to assess shery management strategies:
A
ase study using orange roughy Hoplostethus atlanti
us on the Chatham Rise,
New Zealand. Canadian Journal of Fisheries and Aquati
S
ien
es 49: 922-930.
[7℄ P.H. Leslie (1945). On the use of matri
es in
ertain population mathemati
s.
Biometrika 35: 183-212.

9
[8℄ M. Mangel and P.V. Switzer (1998). A model at the level of the foraging trip
for the indire
t ee
ts of krill (Euphausia superba) sheries on krill predator.
E
ologi
al Modelling 105: 235-256.
[9℄ A. Mukhopadyay, J. Chattopadyay and P.K. Tapaswi (1997). Sele
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in a two spe
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ologi
al Modelling 94: 243-253.
[10℄ I.C. Potter and G.A. Hyndes. (1994). Composition of the sh fauna of a permanently open estuary on the Southern
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omparisons with
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[11℄ A.K. Supriatna (1998). Optimal Harvesting Theory for Predator-Prey Metapopulations. Thesis (Ph.D) University of Adelaide, Australia.
[12℄ A.K. Supriatna and H.P. Possingham (1998). Optimal harvesting for a predatorprey metapopulation. Bulletin of Mathemati
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[13℄ G.N. Tu
k (1994). Optimal Harvesting Models for Metapopulations. Thesis
(Ph.D) University of Adelaide, Australia.
[14℄ G.N. Tu
k and H.P. Possingham (1994). Optimal harvesting strategies for a
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al Biology 56: 107-127.
[15℄ P.J. Wangersky and W.J. Cunningham (1957). Time lag in prey-predator population model. E
ology 38: 136-139.
Appendix 1

The Lagrangian for the maximisation is

L =

X1 f [

(N k ; HN1k ) + N2 (N k ; HN2k )
+P1 (P k ; HP1k ) + P2 (P k ; HP2k )℄
 k [N k
a (N k HN1k ) p F (N k
1 HN1k
1 )
p F (N k
1 HN2k
1 )  (N k HN1k )(P k HP1k )℄
 k [N k
a (N k HN2k ) p F (N k
2 HN1k
2 )
p F (N k
2 HN2k
2 )  (N k HN2k )(P k HP2k )℄
 k [P k
b (P k HP1k ) q G (P k 1 HP1k 1 )
q G (P k 1 HP2k 1 )
 (N k 1 HN1k 1 )(P k 1 HP1k 1 )℄
 k [P k
b (P k HP2k ) q G (P k 2 HP1k 2 )
q G (P k 2 HP2k 2 )
 (N k 2 HN2k 2 )(P k 2 HP2k 2 )℄g:
k

N1

k=0

1

2

1

1

21

1( +1)

2

3

1

1

11

1

2

2

22

2

2( +1)

2

2

21

2

1

1

4

2( +1)

22

2

2

2

1

1

12

2

1

1

1

2

2

1( +1)

1

1

1

2

2

11

1

1

12

1

1

2

1

2

2

2

2

(27)

10
To nd the optimal es
apements we need to solve the ne
essary
onditions:
L
L
L
Pik = 0 for k  1 and HNik = HPik = 0. These
onditions are equivalent to

L
Nik

=

N1 
+  k a +  k
1 p F 0 (SN1k ) +  k  SP1k
0 = k N
k
k
+ k
2 p F 0 (SN1k ) +  k 1  SP1k ;

(28)

N2 
0 = k N
+  k a +  k
2 p F 0 (SN2k ) +  k  SP2k
k
k
+ k
1 p F 0 (SN2k ) +  k 2  SP2k ;

(29)

0 = k PP1  k +  k b +  k 1 q G0 (SP1k ) +  k 1  SN1k
k
+ k 2 q G0 (SP1k ) +  k  SN1k ;

(30)

0 = k PP2  k +  k b +  k 2 q G0 (SP2k ) +  k 2  SN2k
k
+ k 1 q G0 (SP2k ) +  k  SN2k ;

(31)

 N1
 k a  k
1 p F 0 (SN1k )  k  SP1k
0 = k H
N1k
 k
2 p F 0 (SN1k )  k 1  SP1k ;

(32)

 N2
 k a  k
2 p F 0 (SN2k )  k  SP2k
0 = k H
N2k
 k
1 p F 0 (SN2k )  k 2  SP2k ;

(33)

 P1
 k b  k 1 q G0 (SP1k )  k 1  SN1k
0 = k H
P1k
 k 2 q G0 (SP1k )  k  SN1k ;

(34)

 P2
0 = k H
 k b  k 2 q G0 (SP2k )  k 2  SN2k
P2k
 k 1 q G0 (SP2k )  k  SN2k ;

(35)

1(

1

2 +

12

4 +

21

1)

3

2

1

12

1

21

4 +

12

3 +

21

1

1 +

4 +

2

2

1)

11

2 +

1

2

=

k

2

2

2

3 +

1

1

4 +

2

2

2

3 +

Solving equations (28) to (35) produ
es
1(k

22

22

11

1

1

1

1

4 +

1

4

11

4 +

2

3

1

2

3 +

2

1

1

1

1

2

1 +

22

3 +

2

2

2 +

2 +

1

4

1)

11

1

2

1

4(

21

1 +

4 +

1

2

3 +

2

1)

2

3(

12

1

3 +

2(

1

1

1

2

1 +

1)

2

2

2

2

3 +

1

1

1

22

4 +

2

2

!

 N1  N1
+
;
N1k HN1k

2

(36)

2(k 1) = 

k

3(k 1) = k
4(k 1) = k

11

!

 N2  N2
+
;
N2k HN2k
!
  P1   P1
+
;
P1k HP1k
!
  P2   P2
+
:
P2k HP2k

(37)
(38)
(39)

Substituting 1k , 2k , 3k and 4k into equation (32) produ
es
!






N
N
N
1
1
1
a (k+1)
+
0 = k
HN1k

1

N1k HN1k
!
 N1
0
(
k+1+
1 )  N1
+
p11 F1 (SN1k )
N1k HN1k
!




N
N
1
1
1 SP1k (k+1)
+
N1k HN1k
!
 N2
0
(
k+1+
2 )  N2
+
p12 F1 (SN1k )
N2k HN2k
!




P
P
1
1
1 SP1k (k+1+1 )
:
+
P1k HP1k

 Ni
 Ni
 Ni
Divide by k+1, and re
all that H
Nik = p
N (SNi0 ) and Nik + HNik = p
N (Ni1 ),
then
pN
N 1 (SN10 )
= (pN
N 1 (N11))(a1 + p11 F10 (SN10 )
1 + 1SP10 )

+(pN
N 2 (N21))p12 F10 (SN10 )
2
+(pP
P 1(P11 ))1SP10 1 :
(40)
Similarly, substituting 1k , 2k , 3k and 4k into equations (33) to (35) produ
es
pN
N 2 (SN20 )
= (p
(N ))(a + p F 0 (S )
2 +  S )



pP

pP

N

N2

21

2

22 2

N20

2

P20

+(pN
N 1 (N11))p21 F20 (SN20 )
1
+(pP
P 2(P21 ))2SP20 2 ;

(41)

P 1 (SP10 )
= (pP
P 1(P11))(b1 + q11 G01(SP10 )1 + 1SN10 1 )

+(pP
P 2(P21))q12 G01(SP10 )2
+(pN
N 1 (N11))1 SN10 ;

(42)

P 2 (SP20 )
= (pP
P 2(P21))(b2 + q22 G02(SP20 )2 + 2SN20 2 )

+(pP
P 1(P11))q21 G02(SP20 )1
+(pN
N 2 (N21))2 SN20 :

(43)