Harvesting A Twopatch Predator Prey Metapopulation.
NATURAL RESOURCE MODELING
Volume 12, Number 4, Winter 1999
HARVESTING A TWO-PATCH
PREDATOR-PREY METAPOPULATION
ASEP K. SUPRIATNA
Jurusan Matematika
FMIPA Univesitas Padjadjaran
Km 21 Jatinangor-Bandung, Indonesia
Fax: 62-22-4218676
Current address: Dept. of Applied Mathemati
s
University of Adelaide
SA 5005, Australia
E-mail:
asupriatmaths.adelaide.edu.au
HUGH P. POSSINGHAM
Dept. of Environmental S
ien
e and Management
University of Adelaide
Roseworthy SA 5371, Australia
ABSTRACT. A mathemati
al model for a two-pat
h
predator-prey metapoplation is developed as a generalization
of single-spe
ies metapopulation harvesting theory. We nd
optimal harvesting strategies using dynami
programming and
Lagrange multipliers. If predator e
onomi
eÆ
ien
y is relatively high, then we should prote
t a relative sour
e prey
subpopulation in two di erent ways: dire
tly, with a higher
es
apement of the relative sour
e prey subpopulation, and indire
tly, with a lower es
apement of the predator living in the
same pat
h as the relative sour
e prey subpopulation. Numeri
al examples show that if the growth of the predator is relatively low and there is no di eren
e between prey and predator
pri
es, then it may be optimal to harvest the predator to extin
tion. While, if the predator is more valuable
ompared to
the prey, then it may be optimal to leave the relative exporter
prey subpopulation unharvested. We also dis
uss how a `negative' harvest might be optimal. A negative harvest might be
onsidered a seeding strategy.
KEY WORDS: Fisheries, harvesting strategies, predatorprey metapopulation, seeding strategy.
1. Introdu
tion. This paper studies optimal harvesting strategies for a two-pat
h predator-prey metapopulation. The dynami
s of
the predator-prey metapopulation is de ned by four
oupled di eren
e
This paper was presented at the World Conferen
e on Natural Resour
e Modelling, CSIRO Division of Marine Resear
h, Hobart, Australia, De
ember 15 18,
1997.
Copyright
1999 Ro
ky Mountain Mathemati
s Consortium
1
2
A.K. SUPRIATNA AND H.P. POSSINGHAM
equations. Optimal harvesting strategies for the metapopulation are
derived using dynami
programming and Lagrange multipliers. The
theory presented here is a generalization of single-spe
ies metapopulation harvesting theory developed by Tu
k and Possingham [1994℄.
The optimal strategy for managing a dynami
renewable resour
e
was not established until Clark [1971, 1973 and 1976℄, explored optimal harvesting strategies for a single-spe
ies population using dynami
programming. Clark showed that if the growth rate of the resour
e is
less than the dis
ount rate, then a rational sole owner maximizing the
net e
onomi
gain of a resour
e should exploit the resour
e to extin
tion. Clark's work [1971, 1973 and 1976℄ has been very in
uential in
the development of an e
onomi
theory of renewable resour
e exploitation and has been extended to in
lude various e
onomi
and biologi
al
omplexities (Reed [1982℄, Agnew [1982℄, Gatto et al. [1982℄, Clark and
Tait [1982℄, Ludwig and Walters [1982℄, Chaudhuri [1986 and 1988℄,
Mesterton-Gibbons [1996℄, Tu
k and Possingham [1994℄, Ganguly and
Chaudhuri [1995℄, Supriatna and Possingham [1998℄).
Spatial heterogeneity is re
ognized as a fa
tor that needs to be
taken into a
ount in population modelling in general (Dubois [1975℄,
Goh [1975℄, Hilborn [1979℄, Lefkovit
h and Fahrig [1985℄, Matsumoto
and Seno [1995℄), and in sheries modelling in parti
ular (Beverton
and Holt [1957℄, Brown and Murray [1992℄, Frank [1992℄, Frank and
Leggett [1994℄, Parma et al. [1998℄). In the o
ean, population pat
hes
may exist from s
ales of meters to thousands of kilometers and often
o
ur in response to physi
al and biologi
al pro
esses, like adve
tion,
temperature and food quality (Let
her and Ri
e [1997℄). The in
lusion
of spatial heterogeneity may
hange the de
isions that should be made
to manage a shery (Tu
k and Possingham [1994℄, Pelletier and Magal
[1996℄, Brown and Roughgarden [1997℄). With the in
lusion of spatial
heterogeneity into Clark's [1976℄ model, Tu
k and Possingham [1994℄
found some rules of thumb for optimal e
onomi
harvesting of a twopat
h single-spe
ies metapopulation system. One of their rules is that
a relative sour
e subpopulation, that is a subpopulation with a greater
per-
apita larval produ
tion, should be harvested more
onservatively
than a relative sink subpopulation. We de ne the terms `relative sour
e'
and `relative sink' subpopulation more pre
isely in the next se
tion.
Supriatna and Possingham [1998℄ showed that, in some
ir
umstan
es, Tu
k and Possingham's [1994℄ single-spe
ies harvesting rules
PREDATOR-PREY METAPOPULATION
3
of thumb are preserved in the presen
e of a predator. Our previous
model (Supriatna and Possingham [1998℄) assumed that predation affe
ts the predator survival. In this paper, we modify our previous
model to assume that predation a e
ts predator re
ruitment, that is,
prey
onsumption primarily in
reases the predator population through
produ
tion and survival of young rather than adult survival. The results in this paper show that the most signi
ant rule, that we should
harvest a relative sour
e subpopulation more
onservatively than a relative sink subpopulation, is robust, regardless of the biologi
al stru
ture
of the population. We also explore the situation in whi
h the optimal
harvest for one of the populations is negative. While at rst glan
e
this appears unlikely, a negative harvest
ould be implemented in some
ases by seeding a population.
2.
The model.
Consider a predator-prey metapopulation that
oexists in two di erent pat
hes, pat
h 1 and pat
h 2. The movement
of individuals between the lo
al populations is a result of dispersal by
juveniles. Adults are assumed to be sedentary, and they do not migrate
from one pat
h to another pat
h. Let the number of the prey (predator)
in pat
h i at the beginning of period k be denoted by Nik (Pik ) and the
survival rate of adult prey (predator) in pat
h i be denoted by ai (bi ),
respe
tively. The proportion of juvenile prey and predator from pat
h
i that su
essfully migrate to pat
h j are pij and qij , respe
tively, see
Figure 1. Let SNik = Nik HNik (SPik = Pik HPik ) be the es
apement
of the prey (predator) in pat
h i at the end of that period, with HNik
(HPik ) the harvest taken from the prey (predator). Furthermore, let
the dynami
s of exploited metapopulation of these two spe
ies be given
by the equations:
(1)
(2)
Ni(k+1) = ai SNik + i SNik SPik + pii Fi (SNik ) + pji Fj (SNjk );
Pi(k+1) = bi SPik + qii (Gi (SPik ) + i SNik SPik )
+ qji (Gj (SPjk ) + j SNjk SPjk );
where the fun
tions Fi (Nik ) and Gi (Pik ) are the re
ruit produ
tion
fun
tions of the prey and predator in pat
h i at time period k. We
will assume that the re
ruit produ
tion fun
tions are logisti
for the
remainder of this paper, that is, Fi (Nik ) = ri Nik (1 Nik =Ki ) and
4
A.K. SUPRIATNA AND H.P. POSSINGHAM
Patch 1
Patch 2
p21
N1
p11
N2
p22
p12
β1 N 1 P1
β2 N 2 P2
q21
q 12
q 22
q 11
q 12
q11
P1
P2
q
22
q 21
FIGURE 1. The relationships between the dynami
s of the populations in a
two-pat
h predator-prey metapopulation. The number of prey and predator
are Ni and Pi , respe
tively. The prey and predator juvenile migration rates are
pij and qij , respe
tively. The number of predator's o springs in pat
h i from
the
onversion of eaten prey is i Ni Pi , whi
h is distributed into pat
h i and
j with proportion qii and qij , respe
tively, while some of them (1 qii qij ),
either die or are lost from the system.
Gi (Pik ) = si Pik (1 Pik =Li ), where ri (si ) denotes the intrinsi
growth
of the prey (predator) and Ki (Li ) denotes the lo
al
arrying
apa
ity
of the prey (predator), with i < 0 and i > 0.
R ik
If Xi (Xik ; Sik ) = SXXik
(pX
X i( )) d represents the present value
of net revenue from harvesting subpopulation Xi in period k, where
X = NorP , pX is the pri
e per unit harvested population X ,
Xi is
the
ost to harvest subpopulation Xi (it may depend on lo
ation), and
is a dis
ount fa
tor, then to obtain an optimal harvest from the shery
we should maximize net present value
(3)
PV =
T
X
k=0
k
2
X
i=1
Ni (Nik ; SNik ) +
T
X
k=0
k
2
X
i=1
Pi (Pik ; SPik );
subje
t to equations (1) and (2), with nonnegative es
apement less than
PREDATOR-PREY METAPOPULATION
5
or equal to the population size. We will assume = 1=(1 + Æ ) for the
remainder of this paper, where Æ denotes a periodi
dis
ount rate, e.g.,
Æ = 10%.
Supriatna and Possingham [1998℄ used dynami
programming to obtain optimal harvesting strategies for a similar predator-prey metapopulation. They generalized the method in Clark [1976℄ and Tu
k and
Possingham [1994℄, and they found that in some
ir
umstan
es harvesting strategies for a single-spe
ies metapopulation
an be generalized in
the presen
e of predators. Following Supriatna and Possingham [1998℄
we use dynami
programming to obtain optimal harvesting strategies
by maximizing net present value in equation (3) and we nd impli
it
and S in the form:
expressions for optimal es
apements SN
Pi0
i0
pN
)
Ni (SNi
0
0
= (ai + i SPi
0 + pii Fi (SNi0 ))(pN
+ pij Fi0 (SNi
Nj (Nj 1 ))
0 )(pN
+ qii i SPi0 (pP
Pi (Pi1 ))
+ qij i SPi
Pj (Pj 1 ));
0 (pP
pP
)
Pi (SPi
0 = (b + q S
i ii i Ni0 + qii G0i (SPi
0 ))(pP
+ qij i SNi
Pj (Pj 1 ))
0 (pP
0
+ qij Gi (SPi0 )(pP
Pj (Pj 1 ))
+ i SNi
Ni (Ni1 )):
0 (p N
(4)
(5)
Ni (Ni1 ))
Pi (Pi1 ))
These equations are the general form of the optimal harvesting equation
for a two-pat
h predator-prey metapopulation. If there is no predator
mortality asso
iated with migration qii +qij = 1, and
osts of harvesting
are spatially independent, then these equations are the same as in
Supriatna and Possingham [1998℄. If i = i = 0, then the optimal
harvesting equation for a single-spe
ies metapopulation (Tu
k and
Possingham [1994℄) is obtained. Furthermore, if there is no migration
between pat
hes, pij = qij = 0 for i 6= j , and F 0 (S ) = ai +
pii Fi0 (SNi0 ) together with i = i = 0, then the equation redu
es to
the optimal harvesting equation for a single-spe
ies population (Clark
[1976℄). Although we
an show that the es
apements SXi
0 found by
solving these impli
it equations are independent of the time horizon
6
A.K. SUPRIATNA AND H.P. POSSINGHAM
onsidered, we do not show it in this paper. Consequently, due to the
time independen
e, there is a notational
hange for the remainder of
the paper, that is, we simply use SX to denote optimal es
apement
for subpopulation X in pat
h i. We dis
uss some properties of these
es
apements in the following se
tion.
i
In this se
tion we dis
uss some
properties of the optimal es
apements de ned by equations (4) and (5).
We
ompare the optimal es
apements between the two subpopulations.
For the remainder of the paper we assume that market pri
e for the
predator is higher than or equal to the pri
e for the prey, that is,
pP = mpN with m 1, and prey vulnerability is the same in both
pat
hes, that is, 1 = 2 = .
To fa
ilitate some
omparisons of the properties of our optimal es
apements, we adopt the following de nitions from Tu
k and Possingham [1994℄ and Supriatna and Possingham [1998℄:
1. Prey subpopulation i is a relative sour
e subpopulation if its per
apita larval produ
tion is greater than the per
apita larval produ
tion
of prey subpopulation j , that is, ri(pii + pij ) > rj (pjj + pji). If this is
the
ase, then prey subpopulation j is a relative sink subpopulation.
2. Prey subpopulation i is a relative exporter subpopulation if it exports more larvae to prey subpopulation j than it imports (per
apita),
that is, r1p12 > r2p21 . If this is the
ase, then prey subpopulation j is
alled a relative importer subpopulation.
3. The fra
tion i=j i j is
alled the biologi
al eÆ
ien
y and the
fra
tion m(qii + qij ) i=j i j is
alled the e
onomi
eÆ
ien
y of predator
subpopulation i.
3.
Results and dis
ussion.
To simplify the analysis, the
osts
of harvesting are assumed to be negligible. Using these assumptions,
and substituting all derivatives of the logisti
re
ruitment fun
tions, Fi
and Gi, into equations (4) and (5), we
an nd expli
it expressions for
and S :
the optimal es
apements SNi
Pi
3.1. Negligible
osts analysis.
(6)
si =Li ) + Ci Bi
= Aim(qi1 + qi2 )(2
SNi
;
i
PREDATOR-PREY METAPOPULATION
(7)
7
SPi
= Bi(pi1 + pi2)(2ri=Ki ) + CiAi ;
i
provided i = Ci2 m(pi1 + pi2)(2ri=Ki)(qi1 + qi2)(2si=Li) 6= 0, with
Ai = (1=) (pi1 + pi2 )ri ai , Bi = (m=) m(qi1 + qi2 )si mbi , and
Ci = i + m(qi1 + qi2 ) i .
It is well known, in harvesting a single-spe
ies population, that if the
growth rate of the exploited population is less than the dis
ount rate,
then it is e
onomi
ally optimal to harvest the population to extin
tion
(Clark [1976℄, S
hmitt and Wissel [1985℄, Dawid and Kopel [1997℄).
This is also true in harvesting a one- or two-spe
ies metapopulation
(Tu
k and Possingham [1994℄, Supriatna and Possingham [1998℄). We
an show that if Ai and Bi are negative and Ci is nonpositive with
Ci > maxf(2Bi =Ki ); (2mAi =Li )g, then i < 0 and all resulting
es
apements, SN and SP , are positive. If this is the
ase, we
an
also establish the following result.
i
i
Result 1. Assume prey subpopulation
(p11 + p12)r1 > (p22 + p21 )r2
1
is a relative sour
e, that is,
, while all other parameters of the prey
Ai and Bi are
Ci is nonpositive with Ci > maxf(2B=K ); (2mA=L)g,
then (i) SN > SN and (ii) SP SP . Furthermore if, in addition,
1
2
1
2
pi1 pi2 , qi1 = qi2 , SNi Ki , SPi Li with SN 1 SP 1 > SN 2 SP 2 , then
(iii) HN 1 < HN 2 and (iv) HP1 HP2 .
and the predator are identi
al for both subpopulations. If
negative, and
Result 1 suggests that if the growth rate of the populations is
higher than the dis
ounting rate 1= (indi
ated by Ai < 0 and
Bi < 0) and Ci > maxf(2B=K ); (2mA=L)g, then we should prote
t the relative sour
e prey subpopulation in two di erent ways:
dire
tly, with a higher es
apement of the relative sour
e prey subpopulation, and indire
tly, with a lower es
apement of the predator living in the same pat
h with the relative sour
e prey subpopulation. Sin
e Ci > maxf(2B=K ); (2mA=L)g
an be written as
m(qii + qij )( =j j) > 1 + maxf(2B=K ); (2mA=L)g=j j, then we
an
interpret Ci > maxf(2B=K ); (2mA=L)g as a relatively high predator
e
onomi
eÆ
ien
y. Furthermore, if every es
apement is less than ea
h
subpopulation's
arrying
apa
ity, lower es
apement means higher harvest.
8
A.K. SUPRIATNA AND H.P. POSSINGHAM
3.2.
Numeri
al example.
Let us assume that there is a twopat
h predator-prey metapopulation and the prey in both pat
hes
have
arrying
apa
ities K1 = K2 = 50000000, intrinsi
growth
r1 = r2 = 10, and adult survivals per period are a1 = a2 = 0:001.
Prey juveniles migrate with migration fra
tions p11 = p12 = 0:3 and
p21 = p22 = 0:1, hen
e prey subpopulation 1 is a relative sour
e
and exporter subpopulation. Let the dis
ount rate Æ be 10%. Now
suppose predators are present in both pat
hes with intrinsi
growth
s1 = s2 = 4,
arrying
apa
ities L1 = L2 = 50000, and adult
survival per period b1 = b2 = 0:001. Suppose the predator juvenile
migration is symmetri
al and high with q11 = q12 = q21 = q22 = 0:5.
Let j i j = 0:000001 and i = 0:0000001, that is, we assume the
biologi
al predator eÆ
ien
y is 10%. Using equations (1) and (2),
we
an show that one of the positive equilibrium population sizes
for this two-pat
h predator-prey metapopulation is (N1 ; N2 ; P1 ; P2 ) =
(36473692; 36473692; 83105; 83105). We assume harvesting begins
with this equilibrium as the initial population size.
Using equations (6) and (7) and assuming the predator is 10 times
more valuable than the prey, i.e., m = 10, we nd the optimal
es
apement for the system SN 1 = 20420833, SN 2 = 11262500, and
SP 1 = SP 2 = 18131 with the rst period optimal harvests HN 1 =
16052859, HN 2 = 25211192, HP1 = HP2 = 64973, and the equilibrium
optimal harvests HN 1 = 24196828, HN 2 = 33512055, HP1 = HP2 =
56835. As suggested by Result 1, we should harvest the relative
exporter and sour
e prey subpopulation more
onservatively than the
relative importer and sink prey subpopulation (in term of es
apement
SN 1 > SN 2 and in term of harvest HN 1 < HN 2 ). There is no di eren
e
in es
apement and harvest between the predator subpopulations. This
is be
ause the predator biologi
al eÆ
ien
y is exa
tly the same as the
inverse of m, (m = 10 and =j j = 0:1). Figure 2 shows that if
0 < m < 10, then all rules in Result 1 are satis ed. However, if m
is suÆ
iently large, in our example if m > 10, these rules may be
violated. This is be
ause a large m means that a predator's e
onomi
eÆ
ien
y is more than 100% or C > 0, see Result 1.
Figure 2 shows es
apements and harvests whi
h are plotted as fun
tions of the ratio of predator market pri
e to prey market pri
e, m. The
gure suggests that, in this example where the growth of the predator
is relatively low (si = 4 while ri = 10), if there is no di eren
e between
PREDATOR-PREY METAPOPULATION
9
Prey Escapements
22000
SN1
20000
SN1=SN2
18000
16000
14000
SN2
12000
10000
m
2
4
6
8
10
12
14
16
18
20
(2a)
Predator Escapements
SP1=SP2
25000
SP1
SP2
20000
15000
10000
5000
m
0
2
4
6
8
10
12
14
16
18
20
(2b) Prey numbers are in thousands.
FIGURE 2. Es
apements (2a, 2b) and harvests (2
, 2d) are plotted as fun
tions of the ratio of predator market pri
e to prey market pri
e
. Lines indi
ate results if prey subpopulation 1 is a relative sour
e subpopulation and dots
indi
ate results if prey subpopulation 1 is a relative exporter subpopulation.
m
10
A.K. SUPRIATNA AND H.P. POSSINGHAM
Prey Harvests
50000
HN2
40000
30000
HN1
20000
10000
HN2
HN1
m
0 400
450
500
550
600
(2
)
Predator Harvests
40000
30000
HP2
HP1
20000
10000
HP1=HP2
m
00000400
450
500
(2d)
550
600
PREDATOR-PREY METAPOPULATION
11
the market pri
es (m = 1) then it is optimal to harvest all the predator down to extin
tion (upper right gure). While, if m is suÆ
iently
large (m is approximately more than 550) then it is optimal to leave
the relative sink and importer prey subpopulation unharvested, and
eventually it is optimal to leave both prey subpopulations unharvested
if m is even larger (lines in lower left gure). This rule is also observed
for a single pat
h predator-prey system (Ragozin and Brown [1985℄).
This situation is di erent if there is no relative sour
e/sink prey subpopulation. For example, if p11 = p21 = 0:12 and p12 = p22 = 0:28,
then we should not harvest the relative exporter prey subpopulation
(dots in lower left gure).
Furthermore, even when m is very small, if the growth of the predator
is suÆ
iently large, then predator extin
tion is not optimal. For
example, if si = 20 with m = 1, then predator optimal es
apements
are SN 1 = 656 and SN 2 = 11096. Predator es
apement in pat
h 1
is less than in pat
h 2; this is be
ause prey subpopulation 1 is a
relative exporter and sour
e subpopulation whi
h should be prote
ted
by leaving less predators in that pat
h, see Result 1.
3.3. Dealing with negative harvest. As seen in the example
above, the optimal strategy may be a negative harvest for one or more
populations. A negative harvest
ould be interpreted as a seeding or
resto
king strategy. However, in many situations, su
h a strategy is
not pra
ti
al. In this
ase, as in single population exploitation, we
an
use the harvest fun
tion
X S if X S ,
i Xi
i
Xi
(8)
HX =
.
i
0
if Xi < SX
i
Another alternative is suggested by Tu
k and Possingham [1994℄. To
avoid a negative harvest, they used the following pro
edure. Assume
that, using the metapopulation harvesting theory, optimal equilibrium
harvest for subpopulation i is negative. They set HXi = 0 and found a
new optimal es
apement from the maximization of the value fun
tion
under this zero harvest
onstraint. We apply the same pro
edure if the
method presented in the previous se
tion produ
es a negative harvest.
A negative harvest may be optimal for the subpopulation that exports
a high proportion of larvae but only
ontributes a low proportion of the
larvae to its own subpopulation. For example, if the juvenile migration
12
A.K. SUPRIATNA AND H.P. POSSINGHAM
parameters for the prey in the previous example are
p
p
:
11 = 21 = 0 2
= 10, then the optimal equilibrium
=
harvest for the prey subpopulation 2 is
1427582, while all
2
other subpopulations have a positive harvest. This strategy suggests
and
p12
=
p22
:
= 0 065 with
m
HN
that we should seed prey into subpopulation 2 and harvest the results
from prey subpopulation 1 and both predator subpopulations. To avoid
a negative harvest for prey subpopulation 2, we set
HN2 = 0 and, using
the method of Lagrange multipliers, we maximize the present value
in equation (3) with this additional
onstraint. The new equilibrium
, and satisfy equations:
1
2
1
2
optimal es
apements
pN
(
N1 (SN10 ))
SN ; SN ; SP
pN
=(
SP
N1 (N11 ))[a1 + p11 F10 (SN10 ) + 1 SP10 ℄
pP
+ (pP
P1 (P11 ))[q11 1 SP10 ℄
P2 (P21 ))[q12 1 SP10 ℄
p F 0 (S )(1 1=)
+ (pN
N2 (SN20 )) 12 1 N10
Z
p12 p21 F10 (SN10 )F20 (SN20 )
+ (p N
N1 (N11 ))
Z
p12 F10 (SN10 )q22 2 SP20
+ (p P
P2 (P21 ))
Z
p12 F10 (SN10 )q21 2 SP20
+ (pP
P1 (P11 ))
;
Z
+(
(9)
pP
(
P2 (SP20 ))
pP
=(
P1 (P11 ))[q21 G02 (SP20 ) + q21 2 SN20 ℄
S (1 1=)
+ (pN
N2 (SN20 )) 2 N20
Z
0
p F (S ) S
+ (pN
N1 (N11 )) 21 2 N20 2 N20
Z
2 SN20 q22 2 (SP20 )
+ (pP
P2 (P21 ))
Z
2 SN20 q21 2 (SP20 )
+ (pP
P1 (P11 ))
;
Z
+(
pP
(10)
P2 (P21 ))[b2 + q22 G02 (SP20 ) + q22 2 SN20 ℄
PREDATOR-PREY METAPOPULATION
13
30000000
25000000
20000000
15000000
10000000
5000000
0
-5000000
Seeding
SN1
Zero Harvest
FIGURE 3a.
(pP
(11)
SN2
HN1
HN2
Modified zero harvest
Prey es
apements and harvests.
P1 (SP10 ))
= (pP
P1 (P11 ))[b1 + q11 G01 (SP10 ) + q11 1 SN10 ℄
+ (pP
P2 (P21 ))[q12 G01 (SP10 ) + q12 1 SN10 ℄
+ (pN
N1 (N11 ))[ 1 SN10 ℄;
with Z = 1 (a2 + p22 F20 (SN20 ) + 2 SP20 ). Solving the last three
equations together with SN20 = N21 , and assuming Z 6= 0, produ
es
a nonnegative harvest for prey subpopulation 2. Figure 3 shows total
pro ts derived from the three methods, i.e., negative harvest from (6)
and (7), zero harvest from (8), and modi ed zero harvest from (9), (10)
and (11).
If it is possible to implement a negative harvest, we nd equilib = 26517750, H =
rium optimal harvests are HN
1427582 and
N2
1
HP1 = HP2 = 54642. However, if we
annot use a negative harvest in management, then using the rst method (equation (8)) we
= 24720979, H = 0 and
nd equilibrium optimal harvests are HN
N2
1
HP1 = HP2 = 52849. Using the se
ond method (equation (9) to (11))
= 15249769, S = 12923857,
we nd new optimal es
apements SN
N2
1
SP1 = 18131, and SP2 = 16045 with equilibrium optimal harvests
HN 1 = 24852974, HN 2 = 0, HP1 = 50985 and HN 2 = 53069, see Figure 3. If we assume the
ost to put a unit of sh into the shery is equal
14
A.K. SUPRIATNA AND H.P. POSSINGHAM
60000
50000
40000
30000
20000
10000
0
SP1
Seeding
SP2
Zero Harvest
HP1
HP2
Modified zero harvest
FIGURE 3b. Predator es
apements and harvests.
26300000
26200000
26100000
26000000
25900000
25800000
25700000
25600000
25500000
Seeding
Zero Harvest
Modified
zero harvest
FIGURE 3
. Total pro t from three di erent strategies. Es
apements,
harvests, and pro ts
omparison for the three di erent methods of dealing
with a negative harvest.
PREDATOR-PREY METAPOPULATION
15
to the pro t per unit sh from harvesting, then, negle
ting all asso
iated
osts, the total revenue from the harvest is, if a negative harvest
is allowable, HN1 + HN2 + 10(HP1 + HP2 ) = 26183008
urren
y units.
This revenue is above the revenue if we use zero harvest from either
the rst or se
ond method, i.e., 25777959 from the rst method and
25893514 from the se
ond method. This suggests a negative harvest is
optimal.
4. Con
lusion. Harvesting strategies for a predator-prey metapopulation are established as a generalization of harvesting strategies for
a single-spe
ies metapoplation. Some properties of the es
apements
for a single-spe
ies metapopulation (Tu
k and Possingham [1994℄) are
preserved in the presen
e of the predator, su
h as the strategies on how
to harvest relative sour
e and sink subpopulations. We found that if
there are no di eren
es between the biologi
al parameters of the lo
al
populations, ex
ept migration parameters, we should harvest a relative
sour
e prey subpopulation more
onservatively than a relative sink subpopulation. This is the same as the rule of thumb derived from work
on the single-spe
ies metapopulation (Tu
k and Possingham [1994℄).
In addition, we should harvest the predator subpopulation living in the
same pat
h with the relative sour
e prey subpopulation more heavily
than the other predator subpopulation.
In this paper we have dis
ussed how a `negative' harvest might
o
ur. Under some
ir
umstan
es our equations show that a negative
harvest is optimal. A negative harvest might be
onsidered a seeding
strategy. In many situations a seeding strategy is impra
ti
al, so in
this
ase an alternative strategy of imposing zero harvest, for the
population whi
h has a negative harvest, is the best that
an be
done. If it is possible to implement a negative harvest, numeri
al
examples show that if the market pri
e of the predator is mu
h greater
than the market pri
e of the prey, then it may be optimal to feed
the predator by seeding the prey populations, espe
ially the exporter
prey subpopulation. Numeri
al examples show that this strategy
ould
in
rease the total net revenue
ompared to zero prey harvest strategies.
A
knowledgments. We thank two anonymous referees who made
helpful suggestions. We also thank Drew Tyre, Brigitte Tenhumberg,
Shane Ri
hards, Ian Ball and Kristin Munday who read an earlier
16
A.K. SUPRIATNA AND H.P. POSSINGHAM
version of the manus
ript and provided many useful suggestions. The
rst author gratefully a
knowledges nan
ial support from AusAID in
undertaking this work.
Appendix
Proof of Result 1. Let R = (1=) ai , S = m((1=) bi ), rim =
(pii + pij )ri, sim = m(qii + qij )si.
1. If SN = (SN 1 SN 2 )12 , then using Equations 6 and 7 we
obtain
4
s2m
2
R
S =
(r2m r1m)
C
N
KL
2C C
L
L
2S
sm (r1m r2m )
2 K 2B 4s R
m
= sm L C C K
(r2m r1m):
KL
Clearly, SN 1 > SN 2 , sin
e (2B=K ) C 0 and i < 0.
2. We
an prove sP1 SP2 similarly, sin
e we have
Sp = C
2B 4smR (r
C C
K
2m
KL
r1m ):
3. Re
all that equilibrium harvests are given by
HNi = aSNi + pii rSNi
HPi = bSPi + qii sSPi
+ qji
sSPj
1
1
SP i
L
SN j
SN i
+
SN i SP i
K
+ pjirSN j 1
and
1
SN i
K
+ SNiSPi
SP j
SP i :
+
SN j SP j
L
PREDATOR-PREY METAPOPULATION
17
The di eren
e between these two harvests, HN 1 and HN 2 , is
HN 1
HN 2 = (a 1)(SN 1
+ r (p11
SN 2 )
p12 )SN 1 1
+ (p21
+ (SN 1 SP 1
< 0:
4. Similarly, we
an prove HP1
SN 1
K
p22 )SN2 1
SN 2 SP 2 )
SN 2
K
HP2 0 if qi1 = qi2 .
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Optimal Harvesting for a Predator65.
Optimal Harvesting Strategies for a
127.
Volume 12, Number 4, Winter 1999
HARVESTING A TWO-PATCH
PREDATOR-PREY METAPOPULATION
ASEP K. SUPRIATNA
Jurusan Matematika
FMIPA Univesitas Padjadjaran
Km 21 Jatinangor-Bandung, Indonesia
Fax: 62-22-4218676
Current address: Dept. of Applied Mathemati
s
University of Adelaide
SA 5005, Australia
E-mail:
asupriatmaths.adelaide.edu.au
HUGH P. POSSINGHAM
Dept. of Environmental S
ien
e and Management
University of Adelaide
Roseworthy SA 5371, Australia
ABSTRACT. A mathemati
al model for a two-pat
h
predator-prey metapoplation is developed as a generalization
of single-spe
ies metapopulation harvesting theory. We nd
optimal harvesting strategies using dynami
programming and
Lagrange multipliers. If predator e
onomi
eÆ
ien
y is relatively high, then we should prote
t a relative sour
e prey
subpopulation in two di erent ways: dire
tly, with a higher
es
apement of the relative sour
e prey subpopulation, and indire
tly, with a lower es
apement of the predator living in the
same pat
h as the relative sour
e prey subpopulation. Numeri
al examples show that if the growth of the predator is relatively low and there is no di eren
e between prey and predator
pri
es, then it may be optimal to harvest the predator to extin
tion. While, if the predator is more valuable
ompared to
the prey, then it may be optimal to leave the relative exporter
prey subpopulation unharvested. We also dis
uss how a `negative' harvest might be optimal. A negative harvest might be
onsidered a seeding strategy.
KEY WORDS: Fisheries, harvesting strategies, predatorprey metapopulation, seeding strategy.
1. Introdu
tion. This paper studies optimal harvesting strategies for a two-pat
h predator-prey metapopulation. The dynami
s of
the predator-prey metapopulation is de ned by four
oupled di eren
e
This paper was presented at the World Conferen
e on Natural Resour
e Modelling, CSIRO Division of Marine Resear
h, Hobart, Australia, De
ember 15 18,
1997.
Copyright
1999 Ro
ky Mountain Mathemati
s Consortium
1
2
A.K. SUPRIATNA AND H.P. POSSINGHAM
equations. Optimal harvesting strategies for the metapopulation are
derived using dynami
programming and Lagrange multipliers. The
theory presented here is a generalization of single-spe
ies metapopulation harvesting theory developed by Tu
k and Possingham [1994℄.
The optimal strategy for managing a dynami
renewable resour
e
was not established until Clark [1971, 1973 and 1976℄, explored optimal harvesting strategies for a single-spe
ies population using dynami
programming. Clark showed that if the growth rate of the resour
e is
less than the dis
ount rate, then a rational sole owner maximizing the
net e
onomi
gain of a resour
e should exploit the resour
e to extin
tion. Clark's work [1971, 1973 and 1976℄ has been very in
uential in
the development of an e
onomi
theory of renewable resour
e exploitation and has been extended to in
lude various e
onomi
and biologi
al
omplexities (Reed [1982℄, Agnew [1982℄, Gatto et al. [1982℄, Clark and
Tait [1982℄, Ludwig and Walters [1982℄, Chaudhuri [1986 and 1988℄,
Mesterton-Gibbons [1996℄, Tu
k and Possingham [1994℄, Ganguly and
Chaudhuri [1995℄, Supriatna and Possingham [1998℄).
Spatial heterogeneity is re
ognized as a fa
tor that needs to be
taken into a
ount in population modelling in general (Dubois [1975℄,
Goh [1975℄, Hilborn [1979℄, Lefkovit
h and Fahrig [1985℄, Matsumoto
and Seno [1995℄), and in sheries modelling in parti
ular (Beverton
and Holt [1957℄, Brown and Murray [1992℄, Frank [1992℄, Frank and
Leggett [1994℄, Parma et al. [1998℄). In the o
ean, population pat
hes
may exist from s
ales of meters to thousands of kilometers and often
o
ur in response to physi
al and biologi
al pro
esses, like adve
tion,
temperature and food quality (Let
her and Ri
e [1997℄). The in
lusion
of spatial heterogeneity may
hange the de
isions that should be made
to manage a shery (Tu
k and Possingham [1994℄, Pelletier and Magal
[1996℄, Brown and Roughgarden [1997℄). With the in
lusion of spatial
heterogeneity into Clark's [1976℄ model, Tu
k and Possingham [1994℄
found some rules of thumb for optimal e
onomi
harvesting of a twopat
h single-spe
ies metapopulation system. One of their rules is that
a relative sour
e subpopulation, that is a subpopulation with a greater
per-
apita larval produ
tion, should be harvested more
onservatively
than a relative sink subpopulation. We de ne the terms `relative sour
e'
and `relative sink' subpopulation more pre
isely in the next se
tion.
Supriatna and Possingham [1998℄ showed that, in some
ir
umstan
es, Tu
k and Possingham's [1994℄ single-spe
ies harvesting rules
PREDATOR-PREY METAPOPULATION
3
of thumb are preserved in the presen
e of a predator. Our previous
model (Supriatna and Possingham [1998℄) assumed that predation affe
ts the predator survival. In this paper, we modify our previous
model to assume that predation a e
ts predator re
ruitment, that is,
prey
onsumption primarily in
reases the predator population through
produ
tion and survival of young rather than adult survival. The results in this paper show that the most signi
ant rule, that we should
harvest a relative sour
e subpopulation more
onservatively than a relative sink subpopulation, is robust, regardless of the biologi
al stru
ture
of the population. We also explore the situation in whi
h the optimal
harvest for one of the populations is negative. While at rst glan
e
this appears unlikely, a negative harvest
ould be implemented in some
ases by seeding a population.
2.
The model.
Consider a predator-prey metapopulation that
oexists in two di erent pat
hes, pat
h 1 and pat
h 2. The movement
of individuals between the lo
al populations is a result of dispersal by
juveniles. Adults are assumed to be sedentary, and they do not migrate
from one pat
h to another pat
h. Let the number of the prey (predator)
in pat
h i at the beginning of period k be denoted by Nik (Pik ) and the
survival rate of adult prey (predator) in pat
h i be denoted by ai (bi ),
respe
tively. The proportion of juvenile prey and predator from pat
h
i that su
essfully migrate to pat
h j are pij and qij , respe
tively, see
Figure 1. Let SNik = Nik HNik (SPik = Pik HPik ) be the es
apement
of the prey (predator) in pat
h i at the end of that period, with HNik
(HPik ) the harvest taken from the prey (predator). Furthermore, let
the dynami
s of exploited metapopulation of these two spe
ies be given
by the equations:
(1)
(2)
Ni(k+1) = ai SNik + i SNik SPik + pii Fi (SNik ) + pji Fj (SNjk );
Pi(k+1) = bi SPik + qii (Gi (SPik ) + i SNik SPik )
+ qji (Gj (SPjk ) + j SNjk SPjk );
where the fun
tions Fi (Nik ) and Gi (Pik ) are the re
ruit produ
tion
fun
tions of the prey and predator in pat
h i at time period k. We
will assume that the re
ruit produ
tion fun
tions are logisti
for the
remainder of this paper, that is, Fi (Nik ) = ri Nik (1 Nik =Ki ) and
4
A.K. SUPRIATNA AND H.P. POSSINGHAM
Patch 1
Patch 2
p21
N1
p11
N2
p22
p12
β1 N 1 P1
β2 N 2 P2
q21
q 12
q 22
q 11
q 12
q11
P1
P2
q
22
q 21
FIGURE 1. The relationships between the dynami
s of the populations in a
two-pat
h predator-prey metapopulation. The number of prey and predator
are Ni and Pi , respe
tively. The prey and predator juvenile migration rates are
pij and qij , respe
tively. The number of predator's o springs in pat
h i from
the
onversion of eaten prey is i Ni Pi , whi
h is distributed into pat
h i and
j with proportion qii and qij , respe
tively, while some of them (1 qii qij ),
either die or are lost from the system.
Gi (Pik ) = si Pik (1 Pik =Li ), where ri (si ) denotes the intrinsi
growth
of the prey (predator) and Ki (Li ) denotes the lo
al
arrying
apa
ity
of the prey (predator), with i < 0 and i > 0.
R ik
If Xi (Xik ; Sik ) = SXXik
(pX
X i( )) d represents the present value
of net revenue from harvesting subpopulation Xi in period k, where
X = NorP , pX is the pri
e per unit harvested population X ,
Xi is
the
ost to harvest subpopulation Xi (it may depend on lo
ation), and
is a dis
ount fa
tor, then to obtain an optimal harvest from the shery
we should maximize net present value
(3)
PV =
T
X
k=0
k
2
X
i=1
Ni (Nik ; SNik ) +
T
X
k=0
k
2
X
i=1
Pi (Pik ; SPik );
subje
t to equations (1) and (2), with nonnegative es
apement less than
PREDATOR-PREY METAPOPULATION
5
or equal to the population size. We will assume = 1=(1 + Æ ) for the
remainder of this paper, where Æ denotes a periodi
dis
ount rate, e.g.,
Æ = 10%.
Supriatna and Possingham [1998℄ used dynami
programming to obtain optimal harvesting strategies for a similar predator-prey metapopulation. They generalized the method in Clark [1976℄ and Tu
k and
Possingham [1994℄, and they found that in some
ir
umstan
es harvesting strategies for a single-spe
ies metapopulation
an be generalized in
the presen
e of predators. Following Supriatna and Possingham [1998℄
we use dynami
programming to obtain optimal harvesting strategies
by maximizing net present value in equation (3) and we nd impli
it
and S in the form:
expressions for optimal es
apements SN
Pi0
i0
pN
)
Ni (SNi
0
0
= (ai + i SPi
0 + pii Fi (SNi0 ))(pN
+ pij Fi0 (SNi
Nj (Nj 1 ))
0 )(pN
+ qii i SPi0 (pP
Pi (Pi1 ))
+ qij i SPi
Pj (Pj 1 ));
0 (pP
pP
)
Pi (SPi
0 = (b + q S
i ii i Ni0 + qii G0i (SPi
0 ))(pP
+ qij i SNi
Pj (Pj 1 ))
0 (pP
0
+ qij Gi (SPi0 )(pP
Pj (Pj 1 ))
+ i SNi
Ni (Ni1 )):
0 (p N
(4)
(5)
Ni (Ni1 ))
Pi (Pi1 ))
These equations are the general form of the optimal harvesting equation
for a two-pat
h predator-prey metapopulation. If there is no predator
mortality asso
iated with migration qii +qij = 1, and
osts of harvesting
are spatially independent, then these equations are the same as in
Supriatna and Possingham [1998℄. If i = i = 0, then the optimal
harvesting equation for a single-spe
ies metapopulation (Tu
k and
Possingham [1994℄) is obtained. Furthermore, if there is no migration
between pat
hes, pij = qij = 0 for i 6= j , and F 0 (S ) = ai +
pii Fi0 (SNi0 ) together with i = i = 0, then the equation redu
es to
the optimal harvesting equation for a single-spe
ies population (Clark
[1976℄). Although we
an show that the es
apements SXi
0 found by
solving these impli
it equations are independent of the time horizon
6
A.K. SUPRIATNA AND H.P. POSSINGHAM
onsidered, we do not show it in this paper. Consequently, due to the
time independen
e, there is a notational
hange for the remainder of
the paper, that is, we simply use SX to denote optimal es
apement
for subpopulation X in pat
h i. We dis
uss some properties of these
es
apements in the following se
tion.
i
In this se
tion we dis
uss some
properties of the optimal es
apements de ned by equations (4) and (5).
We
ompare the optimal es
apements between the two subpopulations.
For the remainder of the paper we assume that market pri
e for the
predator is higher than or equal to the pri
e for the prey, that is,
pP = mpN with m 1, and prey vulnerability is the same in both
pat
hes, that is, 1 = 2 = .
To fa
ilitate some
omparisons of the properties of our optimal es
apements, we adopt the following de nitions from Tu
k and Possingham [1994℄ and Supriatna and Possingham [1998℄:
1. Prey subpopulation i is a relative sour
e subpopulation if its per
apita larval produ
tion is greater than the per
apita larval produ
tion
of prey subpopulation j , that is, ri(pii + pij ) > rj (pjj + pji). If this is
the
ase, then prey subpopulation j is a relative sink subpopulation.
2. Prey subpopulation i is a relative exporter subpopulation if it exports more larvae to prey subpopulation j than it imports (per
apita),
that is, r1p12 > r2p21 . If this is the
ase, then prey subpopulation j is
alled a relative importer subpopulation.
3. The fra
tion i=j i j is
alled the biologi
al eÆ
ien
y and the
fra
tion m(qii + qij ) i=j i j is
alled the e
onomi
eÆ
ien
y of predator
subpopulation i.
3.
Results and dis
ussion.
To simplify the analysis, the
osts
of harvesting are assumed to be negligible. Using these assumptions,
and substituting all derivatives of the logisti
re
ruitment fun
tions, Fi
and Gi, into equations (4) and (5), we
an nd expli
it expressions for
and S :
the optimal es
apements SNi
Pi
3.1. Negligible
osts analysis.
(6)
si =Li ) + Ci Bi
= Aim(qi1 + qi2 )(2
SNi
;
i
PREDATOR-PREY METAPOPULATION
(7)
7
SPi
= Bi(pi1 + pi2)(2ri=Ki ) + CiAi ;
i
provided i = Ci2 m(pi1 + pi2)(2ri=Ki)(qi1 + qi2)(2si=Li) 6= 0, with
Ai = (1=) (pi1 + pi2 )ri ai , Bi = (m=) m(qi1 + qi2 )si mbi , and
Ci = i + m(qi1 + qi2 ) i .
It is well known, in harvesting a single-spe
ies population, that if the
growth rate of the exploited population is less than the dis
ount rate,
then it is e
onomi
ally optimal to harvest the population to extin
tion
(Clark [1976℄, S
hmitt and Wissel [1985℄, Dawid and Kopel [1997℄).
This is also true in harvesting a one- or two-spe
ies metapopulation
(Tu
k and Possingham [1994℄, Supriatna and Possingham [1998℄). We
an show that if Ai and Bi are negative and Ci is nonpositive with
Ci > maxf(2Bi =Ki ); (2mAi =Li )g, then i < 0 and all resulting
es
apements, SN and SP , are positive. If this is the
ase, we
an
also establish the following result.
i
i
Result 1. Assume prey subpopulation
(p11 + p12)r1 > (p22 + p21 )r2
1
is a relative sour
e, that is,
, while all other parameters of the prey
Ai and Bi are
Ci is nonpositive with Ci > maxf(2B=K ); (2mA=L)g,
then (i) SN > SN and (ii) SP SP . Furthermore if, in addition,
1
2
1
2
pi1 pi2 , qi1 = qi2 , SNi Ki , SPi Li with SN 1 SP 1 > SN 2 SP 2 , then
(iii) HN 1 < HN 2 and (iv) HP1 HP2 .
and the predator are identi
al for both subpopulations. If
negative, and
Result 1 suggests that if the growth rate of the populations is
higher than the dis
ounting rate 1= (indi
ated by Ai < 0 and
Bi < 0) and Ci > maxf(2B=K ); (2mA=L)g, then we should prote
t the relative sour
e prey subpopulation in two di erent ways:
dire
tly, with a higher es
apement of the relative sour
e prey subpopulation, and indire
tly, with a lower es
apement of the predator living in the same pat
h with the relative sour
e prey subpopulation. Sin
e Ci > maxf(2B=K ); (2mA=L)g
an be written as
m(qii + qij )( =j j) > 1 + maxf(2B=K ); (2mA=L)g=j j, then we
an
interpret Ci > maxf(2B=K ); (2mA=L)g as a relatively high predator
e
onomi
eÆ
ien
y. Furthermore, if every es
apement is less than ea
h
subpopulation's
arrying
apa
ity, lower es
apement means higher harvest.
8
A.K. SUPRIATNA AND H.P. POSSINGHAM
3.2.
Numeri
al example.
Let us assume that there is a twopat
h predator-prey metapopulation and the prey in both pat
hes
have
arrying
apa
ities K1 = K2 = 50000000, intrinsi
growth
r1 = r2 = 10, and adult survivals per period are a1 = a2 = 0:001.
Prey juveniles migrate with migration fra
tions p11 = p12 = 0:3 and
p21 = p22 = 0:1, hen
e prey subpopulation 1 is a relative sour
e
and exporter subpopulation. Let the dis
ount rate Æ be 10%. Now
suppose predators are present in both pat
hes with intrinsi
growth
s1 = s2 = 4,
arrying
apa
ities L1 = L2 = 50000, and adult
survival per period b1 = b2 = 0:001. Suppose the predator juvenile
migration is symmetri
al and high with q11 = q12 = q21 = q22 = 0:5.
Let j i j = 0:000001 and i = 0:0000001, that is, we assume the
biologi
al predator eÆ
ien
y is 10%. Using equations (1) and (2),
we
an show that one of the positive equilibrium population sizes
for this two-pat
h predator-prey metapopulation is (N1 ; N2 ; P1 ; P2 ) =
(36473692; 36473692; 83105; 83105). We assume harvesting begins
with this equilibrium as the initial population size.
Using equations (6) and (7) and assuming the predator is 10 times
more valuable than the prey, i.e., m = 10, we nd the optimal
es
apement for the system SN 1 = 20420833, SN 2 = 11262500, and
SP 1 = SP 2 = 18131 with the rst period optimal harvests HN 1 =
16052859, HN 2 = 25211192, HP1 = HP2 = 64973, and the equilibrium
optimal harvests HN 1 = 24196828, HN 2 = 33512055, HP1 = HP2 =
56835. As suggested by Result 1, we should harvest the relative
exporter and sour
e prey subpopulation more
onservatively than the
relative importer and sink prey subpopulation (in term of es
apement
SN 1 > SN 2 and in term of harvest HN 1 < HN 2 ). There is no di eren
e
in es
apement and harvest between the predator subpopulations. This
is be
ause the predator biologi
al eÆ
ien
y is exa
tly the same as the
inverse of m, (m = 10 and =j j = 0:1). Figure 2 shows that if
0 < m < 10, then all rules in Result 1 are satis ed. However, if m
is suÆ
iently large, in our example if m > 10, these rules may be
violated. This is be
ause a large m means that a predator's e
onomi
eÆ
ien
y is more than 100% or C > 0, see Result 1.
Figure 2 shows es
apements and harvests whi
h are plotted as fun
tions of the ratio of predator market pri
e to prey market pri
e, m. The
gure suggests that, in this example where the growth of the predator
is relatively low (si = 4 while ri = 10), if there is no di eren
e between
PREDATOR-PREY METAPOPULATION
9
Prey Escapements
22000
SN1
20000
SN1=SN2
18000
16000
14000
SN2
12000
10000
m
2
4
6
8
10
12
14
16
18
20
(2a)
Predator Escapements
SP1=SP2
25000
SP1
SP2
20000
15000
10000
5000
m
0
2
4
6
8
10
12
14
16
18
20
(2b) Prey numbers are in thousands.
FIGURE 2. Es
apements (2a, 2b) and harvests (2
, 2d) are plotted as fun
tions of the ratio of predator market pri
e to prey market pri
e
. Lines indi
ate results if prey subpopulation 1 is a relative sour
e subpopulation and dots
indi
ate results if prey subpopulation 1 is a relative exporter subpopulation.
m
10
A.K. SUPRIATNA AND H.P. POSSINGHAM
Prey Harvests
50000
HN2
40000
30000
HN1
20000
10000
HN2
HN1
m
0 400
450
500
550
600
(2
)
Predator Harvests
40000
30000
HP2
HP1
20000
10000
HP1=HP2
m
00000400
450
500
(2d)
550
600
PREDATOR-PREY METAPOPULATION
11
the market pri
es (m = 1) then it is optimal to harvest all the predator down to extin
tion (upper right gure). While, if m is suÆ
iently
large (m is approximately more than 550) then it is optimal to leave
the relative sink and importer prey subpopulation unharvested, and
eventually it is optimal to leave both prey subpopulations unharvested
if m is even larger (lines in lower left gure). This rule is also observed
for a single pat
h predator-prey system (Ragozin and Brown [1985℄).
This situation is di erent if there is no relative sour
e/sink prey subpopulation. For example, if p11 = p21 = 0:12 and p12 = p22 = 0:28,
then we should not harvest the relative exporter prey subpopulation
(dots in lower left gure).
Furthermore, even when m is very small, if the growth of the predator
is suÆ
iently large, then predator extin
tion is not optimal. For
example, if si = 20 with m = 1, then predator optimal es
apements
are SN 1 = 656 and SN 2 = 11096. Predator es
apement in pat
h 1
is less than in pat
h 2; this is be
ause prey subpopulation 1 is a
relative exporter and sour
e subpopulation whi
h should be prote
ted
by leaving less predators in that pat
h, see Result 1.
3.3. Dealing with negative harvest. As seen in the example
above, the optimal strategy may be a negative harvest for one or more
populations. A negative harvest
ould be interpreted as a seeding or
resto
king strategy. However, in many situations, su
h a strategy is
not pra
ti
al. In this
ase, as in single population exploitation, we
an
use the harvest fun
tion
X S if X S ,
i Xi
i
Xi
(8)
HX =
.
i
0
if Xi < SX
i
Another alternative is suggested by Tu
k and Possingham [1994℄. To
avoid a negative harvest, they used the following pro
edure. Assume
that, using the metapopulation harvesting theory, optimal equilibrium
harvest for subpopulation i is negative. They set HXi = 0 and found a
new optimal es
apement from the maximization of the value fun
tion
under this zero harvest
onstraint. We apply the same pro
edure if the
method presented in the previous se
tion produ
es a negative harvest.
A negative harvest may be optimal for the subpopulation that exports
a high proportion of larvae but only
ontributes a low proportion of the
larvae to its own subpopulation. For example, if the juvenile migration
12
A.K. SUPRIATNA AND H.P. POSSINGHAM
parameters for the prey in the previous example are
p
p
:
11 = 21 = 0 2
= 10, then the optimal equilibrium
=
harvest for the prey subpopulation 2 is
1427582, while all
2
other subpopulations have a positive harvest. This strategy suggests
and
p12
=
p22
:
= 0 065 with
m
HN
that we should seed prey into subpopulation 2 and harvest the results
from prey subpopulation 1 and both predator subpopulations. To avoid
a negative harvest for prey subpopulation 2, we set
HN2 = 0 and, using
the method of Lagrange multipliers, we maximize the present value
in equation (3) with this additional
onstraint. The new equilibrium
, and satisfy equations:
1
2
1
2
optimal es
apements
pN
(
N1 (SN10 ))
SN ; SN ; SP
pN
=(
SP
N1 (N11 ))[a1 + p11 F10 (SN10 ) + 1 SP10 ℄
pP
+ (pP
P1 (P11 ))[q11 1 SP10 ℄
P2 (P21 ))[q12 1 SP10 ℄
p F 0 (S )(1 1=)
+ (pN
N2 (SN20 )) 12 1 N10
Z
p12 p21 F10 (SN10 )F20 (SN20 )
+ (p N
N1 (N11 ))
Z
p12 F10 (SN10 )q22 2 SP20
+ (p P
P2 (P21 ))
Z
p12 F10 (SN10 )q21 2 SP20
+ (pP
P1 (P11 ))
;
Z
+(
(9)
pP
(
P2 (SP20 ))
pP
=(
P1 (P11 ))[q21 G02 (SP20 ) + q21 2 SN20 ℄
S (1 1=)
+ (pN
N2 (SN20 )) 2 N20
Z
0
p F (S ) S
+ (pN
N1 (N11 )) 21 2 N20 2 N20
Z
2 SN20 q22 2 (SP20 )
+ (pP
P2 (P21 ))
Z
2 SN20 q21 2 (SP20 )
+ (pP
P1 (P11 ))
;
Z
+(
pP
(10)
P2 (P21 ))[b2 + q22 G02 (SP20 ) + q22 2 SN20 ℄
PREDATOR-PREY METAPOPULATION
13
30000000
25000000
20000000
15000000
10000000
5000000
0
-5000000
Seeding
SN1
Zero Harvest
FIGURE 3a.
(pP
(11)
SN2
HN1
HN2
Modified zero harvest
Prey es
apements and harvests.
P1 (SP10 ))
= (pP
P1 (P11 ))[b1 + q11 G01 (SP10 ) + q11 1 SN10 ℄
+ (pP
P2 (P21 ))[q12 G01 (SP10 ) + q12 1 SN10 ℄
+ (pN
N1 (N11 ))[ 1 SN10 ℄;
with Z = 1 (a2 + p22 F20 (SN20 ) + 2 SP20 ). Solving the last three
equations together with SN20 = N21 , and assuming Z 6= 0, produ
es
a nonnegative harvest for prey subpopulation 2. Figure 3 shows total
pro ts derived from the three methods, i.e., negative harvest from (6)
and (7), zero harvest from (8), and modi ed zero harvest from (9), (10)
and (11).
If it is possible to implement a negative harvest, we nd equilib = 26517750, H =
rium optimal harvests are HN
1427582 and
N2
1
HP1 = HP2 = 54642. However, if we
annot use a negative harvest in management, then using the rst method (equation (8)) we
= 24720979, H = 0 and
nd equilibrium optimal harvests are HN
N2
1
HP1 = HP2 = 52849. Using the se
ond method (equation (9) to (11))
= 15249769, S = 12923857,
we nd new optimal es
apements SN
N2
1
SP1 = 18131, and SP2 = 16045 with equilibrium optimal harvests
HN 1 = 24852974, HN 2 = 0, HP1 = 50985 and HN 2 = 53069, see Figure 3. If we assume the
ost to put a unit of sh into the shery is equal
14
A.K. SUPRIATNA AND H.P. POSSINGHAM
60000
50000
40000
30000
20000
10000
0
SP1
Seeding
SP2
Zero Harvest
HP1
HP2
Modified zero harvest
FIGURE 3b. Predator es
apements and harvests.
26300000
26200000
26100000
26000000
25900000
25800000
25700000
25600000
25500000
Seeding
Zero Harvest
Modified
zero harvest
FIGURE 3
. Total pro t from three di erent strategies. Es
apements,
harvests, and pro ts
omparison for the three di erent methods of dealing
with a negative harvest.
PREDATOR-PREY METAPOPULATION
15
to the pro t per unit sh from harvesting, then, negle
ting all asso
iated
osts, the total revenue from the harvest is, if a negative harvest
is allowable, HN1 + HN2 + 10(HP1 + HP2 ) = 26183008
urren
y units.
This revenue is above the revenue if we use zero harvest from either
the rst or se
ond method, i.e., 25777959 from the rst method and
25893514 from the se
ond method. This suggests a negative harvest is
optimal.
4. Con
lusion. Harvesting strategies for a predator-prey metapopulation are established as a generalization of harvesting strategies for
a single-spe
ies metapoplation. Some properties of the es
apements
for a single-spe
ies metapopulation (Tu
k and Possingham [1994℄) are
preserved in the presen
e of the predator, su
h as the strategies on how
to harvest relative sour
e and sink subpopulations. We found that if
there are no di eren
es between the biologi
al parameters of the lo
al
populations, ex
ept migration parameters, we should harvest a relative
sour
e prey subpopulation more
onservatively than a relative sink subpopulation. This is the same as the rule of thumb derived from work
on the single-spe
ies metapopulation (Tu
k and Possingham [1994℄).
In addition, we should harvest the predator subpopulation living in the
same pat
h with the relative sour
e prey subpopulation more heavily
than the other predator subpopulation.
In this paper we have dis
ussed how a `negative' harvest might
o
ur. Under some
ir
umstan
es our equations show that a negative
harvest is optimal. A negative harvest might be
onsidered a seeding
strategy. In many situations a seeding strategy is impra
ti
al, so in
this
ase an alternative strategy of imposing zero harvest, for the
population whi
h has a negative harvest, is the best that
an be
done. If it is possible to implement a negative harvest, numeri
al
examples show that if the market pri
e of the predator is mu
h greater
than the market pri
e of the prey, then it may be optimal to feed
the predator by seeding the prey populations, espe
ially the exporter
prey subpopulation. Numeri
al examples show that this strategy
ould
in
rease the total net revenue
ompared to zero prey harvest strategies.
A
knowledgments. We thank two anonymous referees who made
helpful suggestions. We also thank Drew Tyre, Brigitte Tenhumberg,
Shane Ri
hards, Ian Ball and Kristin Munday who read an earlier
16
A.K. SUPRIATNA AND H.P. POSSINGHAM
version of the manus
ript and provided many useful suggestions. The
rst author gratefully a
knowledges nan
ial support from AusAID in
undertaking this work.
Appendix
Proof of Result 1. Let R = (1=) ai , S = m((1=) bi ), rim =
(pii + pij )ri, sim = m(qii + qij )si.
1. If SN = (SN 1 SN 2 )12 , then using Equations 6 and 7 we
obtain
4
s2m
2
R
S =
(r2m r1m)
C
N
KL
2C C
L
L
2S
sm (r1m r2m )
2 K 2B 4s R
m
= sm L C C K
(r2m r1m):
KL
Clearly, SN 1 > SN 2 , sin
e (2B=K ) C 0 and i < 0.
2. We
an prove sP1 SP2 similarly, sin
e we have
Sp = C
2B 4smR (r
C C
K
2m
KL
r1m ):
3. Re
all that equilibrium harvests are given by
HNi = aSNi + pii rSNi
HPi = bSPi + qii sSPi
+ qji
sSPj
1
1
SP i
L
SN j
SN i
+
SN i SP i
K
+ pjirSN j 1
and
1
SN i
K
+ SNiSPi
SP j
SP i :
+
SN j SP j
L
PREDATOR-PREY METAPOPULATION
17
The di eren
e between these two harvests, HN 1 and HN 2 , is
HN 1
HN 2 = (a 1)(SN 1
+ r (p11
SN 2 )
p12 )SN 1 1
+ (p21
+ (SN 1 SP 1
< 0:
4. Similarly, we
an prove HP1
SN 1
K
p22 )SN2 1
SN 2 SP 2 )
SN 2
K
HP2 0 if qi1 = qi2 .
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