Harvesting A Twopatch Predator Prey Metapopulation.

HARVESTING A TWO-PATCH

PREDATOR-PREY METAPOPULATION

ASEPK.SUPRIATNA

JurusanMatematika

FMIPAUnivesitasPadjadjaran

Km21Jatinangor-Bandung,Indonesia

Fax: 62-22-4218676

Currentaddress: Dept.ofAppliedMathematis

UniversityofAdelaide

SA5005,Australia

E-mail:asupriatmaths.adelaide.edu.au

HUGHP.POSSINGHAM

Dept.ofEnvironmentalSieneandManagement

UniversityofAdelaide

RoseworthySA5371,Australia

ABSTRACT. A mathematial model for a two-path

predator-preymetapoplationisdevelopedasageneralization

of single-speiesmetapopulation harvestingtheory. Wend

optimalharvestingstrategiesusingdynamiprogrammingand

Lagrange multipliers. Ifpredator eonomieÆienyis

rel-ativelyhigh, then weshould protet a relative soureprey

subpopulationintwo dierentways: diretly,withahigher

esapementoftherelativesourepreysubpopulation,and

in-diretly,withaloweresapementofthepredatorlivinginthe

samepathastherelativesourepreysubpopulation.

Numer-ialexamplesshowthatifthegrowthofthepredatoris

rela-tivelylowandthereisnodierenebetweenpreyandpredator

pries,thenitmaybeoptimaltoharvestthepredatorto

ex-tintion.While,ifthepredatorismorevaluableomparedto

theprey,thenitmaybeoptimaltoleavetherelativeexporter

preysubpopulationunharvested.Wealsodisusshowa

`neg-ative'harvestmightbeoptimal.Anegativeharvestmightbe

onsideredaseedingstrategy.

KEY WORDS:Fisheries, harvesting strategies,

predator-preymetapopulation,seedingstrategy.

1. Introdution. This paper studies optimal harvesting

strate-giesfor a two-path predator-prey metapopulation. Thedynamis of

thepredator-preymetapopulationisdenedbyfouroupleddierene

Thispaperwas presentedattheWorldConfereneon NaturalResoure

Mod-elling,CSIRO DivisionofMarine Researh, Hobart,Australia, Deember15 18,

1997.

equations. Optimal harvesting strategies for the metapopulation are

derived using dynami programming and Lagrange multipliers. The

theorypresentedhere isageneralizationofsingle-speies

metapopula-tionharvesting theorydevelopedbyTukandPossingham [1994℄.

The optimal strategy for managing a dynami renewable resoure

was not established until Clark [1971, 1973and 1976℄, explored

opti-malharvestingstrategiesforasingle-speiespopulationusingdynami

programming. Clarkshowed that ifthegrowthrateof theresoureis

lessthanthedisountrate,thena rationalsoleownermaximizingthe

neteonomigain ofa resoure should exploitthe resoureto

extin-tion. Clark's work [1971, 1973 and 1976℄ hasbeenvery inuentialin

thedevelopmentofaneonomitheoryofrenewableresoure

exploita-tionandhasbeenextendedtoinludevariouseonomiandbiologial

omplexities(Reed[1982℄,Agnew[1982℄,Gattoetal.[1982℄,Clarkand

Tait [1982℄, Ludwig and Walters [1982℄, Chaudhuri [1986 and 1988℄,

Mesterton-Gibbons[1996℄, TukandPossingham[1994℄, Gangulyand

Chaudhuri [1995℄,SupriatnaandPossingham[1998℄).

Spatial heterogeneity is reognized as a fator that needs to be

taken intoaountin population modellingin general(Dubois [1975℄,

Goh [1975℄, Hilborn [1979℄, Lefkovith and Fahrig [1985℄, Matsumoto

and Seno [1995℄), and in sheries modelling in partiular (Beverton

and Holt [1957℄, Brown and Murray [1992℄, Frank [1992℄, Frankand

Leggett[1994℄,Parma etal.[1998℄). Intheoean, population pathes

may exist from sales ofmeters to thousandsof kilometers and often

our in response to physial and biologial proesses, likeadvetion,

temperatureandfoodquality(LetherandRie[1997℄). Theinlusion

ofspatialheterogeneitymayhangethedeisionsthat shouldbemade

tomanageashery(TukandPossingham[1994℄,PelletierandMagal

[1996℄,BrownandRoughgarden [1997℄). Withtheinlusionofspatial

heterogeneity into Clark's [1976℄ model, Tukand Possingham [1994℄

foundsome rules ofthumb for optimaleonomi harvesting ofa

two-path single-speiesmetapopulation system. Oneoftheirrulesisthat

arelativesouresubpopulation,thatisasubpopulationwithagreater

per-apita larvalprodution,should beharvestedmore onservatively

thanarelativesinksubpopulation. Wedenetheterms`relativesoure'

and`relativesink'subpopulation morepreiselyin thenextsetion.

irum-of thumb are preserved in the presene of a predator. Our previous

model(Supriatnaand Possingham[1998℄) assumedthat predation

af-fets the predator survival. In this paper, we modify our previous

model to assumethat predation aets predatorreruitment, that is,

preyonsumptionprimarilyinreasesthepredatorpopulationthrough

produtionand survivalof youngrather thanadult survival. The

re-sultsinthispapershowthatthemostsigniantrule,that weshould

harvestarelativesouresubpopulationmoreonservativelythana

rela-tivesinksubpopulation,isrobust,regardlessofthebiologialstruture

ofthepopulation. Wealsoexplorethesituation in whih theoptimal

harvest for one of the populations is negative. While at rst glane

thisappearsunlikely,anegativeharvestouldbeimplementedinsome

asesbyseedinga population.

2. The model. Consider a predator-prey metapopulation that

oexistsin twodierentpathes,path 1andpath 2. Themovement

ofindividualsbetweentheloalpopulations isaresultofdispersal by

juveniles. Adultsareassumedtobesedentary,andtheydonotmigrate

fromonepathtoanotherpath. Letthenumberoftheprey(predator)

inpathiatthebeginningofperiodkbedenotedbyN

ik (P

ik

)andthe

survivalrateofadultprey(predator) inpathibedenoted bya

i (b

i ),

respetively. Theproportionofjuvenilepreyandpredatorfrompath

ithat suessfullymigratetopath j arep

ij andq

ij

, respetively,see

Figure1. LetS

Nik =N

ik H

Nik (S

Pik =P

ik H

Pik

)betheesapement

oftheprey(predator)in pathiat theend ofthatperiod, withH

Nik

(H

Pik

)the harvest taken from the prey(predator). Furthermore, let

thedynamisofexploitedmetapopulationofthesetwospeiesbegiven

bytheequations:

(1) N

i(k+1) =a

i S

Nik +

i S

Nik S

Pik +p

ii F

i (S

Nik )+p

ji F

j (S

Njk );

(2)

P

i(k+1) =b

i S

Pik +q

ii (G

i (S

Pik )+

i S

Nik S

Pik )

+q

ji (G

j (S

Pjk )+

j S

Njk S

Pjk );

where the funtions F

i (N

ik

) and G

i (P

ik

) are the reruit prodution

funtions of the prey and predator in path i at time period k. We

will assume that the reruit prodution funtions are logisti for the

p

p

p

21

22

11

N

1

N

2

p

12

P

2

q

21

P

1

q

11

q

12

q

12

q

22

β

1

N

1

P

1

β

2

N P

2

2

Patch 1

Patch 2

q

22

q

11

q

21

FIGURE1. Therelationshipsbetweenthedynamisofthepopulationsina

two-pathpredator-preymetapopulation. Thenumberof preyand predator

areNiandPi,respetively.Thepreyandpredatorjuvenilemigrationratesare

p

ij andq

ij

,respetively. Thenumberofpredator'sospringsinpathifrom

theonversionofeatenpreyis

i N

i P

i

,whihisdistributedintopathi and

jwithproportionq

ii andq

ij

,respetively,whilesomeofthem(1 q

ii q

ij ),

eitherdieorarelostfromthesystem.

G

i (P

ik )=s

i P

ik (1 P

ik =L

i

),wherer

i (s

i

)denotestheintrinsigrowth

oftheprey(predator)andK

i (L

i

)denotestheloalarryingapaity

of theprey(predator),with

i

<0and

i >0.

If

Xi (X

ik ;S

ik )=

R

X

ik

S

Xik (p

X

X

i())drepresentsthepresentvalue

of net revenue from harvesting subpopulation X

i

in period k, where

X =NorP, p

X

is theprie perunit harvested population X,

Xi is

theosttoharvestsubpopulationX

i

(itmaydependonloation),and

isadisountfator,thentoobtainanoptimalharvestfromtheshery

weshould maximizenetpresentvalue

(3) PV = T

X

k=0

k 2

X

i=1

Ni (N

ik ;S

Nik )+

T

X

k=0

k 2

X

i=1

Pi (P

ik ;S

orequal to thepopulation size. Wewillassume=1=(1+Æ)for the

remainderofthispaper,whereÆdenotesaperiodidisountrate,e.g.,

Æ=10%.

SupriatnaandPossingham[1998℄useddynamiprogrammingto

ob-tainoptimalharvestingstrategiesforasimilarpredator-prey

metapop-ulation. Theygeneralized the method in Clark [1976℄ and Tuk and

Possingham[1994℄,andtheyfoundthatinsomeirumstanes

harvest-ingstrategiesforasingle-speiesmetapopulationanbegeneralizedin

thepreseneofpredators. FollowingSupriatnaandPossingham[1998℄

weuse dynamiprogramming to obtain optimal harvesting strategies

by maximizing netpresentvalue in equation (3)and wend impliit

expressionsforoptimalesapementsS

Ni0 andS

Pi0

in theform:

(4) p N Ni (S Ni0 ) =(a i + i S Pi0 +p ii F 0 i (S Ni0 ))(p N Ni (N i1 )) +p ij F 0 i (S Ni0 )(p N Nj (N j1 )) +q ii i S Pi0 (p P Pi (P i1 )) +q ij i S Pi0 (p P Pj (P j1 )); (5) p P Pi (S Pi0 ) =(b i +q ii i S Ni0 +q ii G 0 i (S Pi0 ))(p P Pi (P i1 )) +q ij i S Ni0 (p P Pj (P j1 )) +q ij G 0 i (S Pi0 )(p P Pj (P j1 )) + i S Ni0 (p N Ni (N i1 )):

Theseequationsarethegeneralformoftheoptimalharvestingequation

foratwo-path predator-preymetapopulation. Ifthere isnopredator

mortalityassoiatedwithmigrationq

ii +q

ij

=1,andostsofharvesting

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-speies metapopulation (Tuk and

Possingham [1994℄)is obtained. Furthermore, ifthere isnomigration

between pathes, p

ij = q

ij

= 0 for i 6= j, and F 0

(S) = a

i + p ii F 0 i (S Ni0

)together with

i =

i

= 0,then the equation redues to

theoptimalharvesting equationfora single-speiespopulation (Clark

[1976℄). Although we an show that the esapements S

Xi0

onsidered, wedonotshowitin this paper. Consequently, dueto the

time independene, there is a notational hange for the remainderof

the paper, that is, we simply use S

Xi

to denote optimal esapement

for subpopulation X in path i. We disuss some properties of these

esapementsin thefollowingsetion.

3. Results and disussion. In this setion we disuss some

propertiesoftheoptimalesapementsdenedbyequations(4)and(5).

Weomparetheoptimalesapementsbetweenthetwosubpopulations.

For the remainder of the paper we assumethat market prie for the

predator is higher than or equal to the prie for the prey, that is,

p

P = mp

N

with m 1, and prey vulnerability is the same in both

pathes,that is,

1 =

2 = .

To failitate some omparisons of the properties of our optimal

es-apements,weadopt thefollowingdenitionsfromTukand

Possing-ham [1994℄andSupriatnaandPossingham[1998℄:

1. Preysubpopulation i isa relative soure subpopulation ifitsper

apitalarvalprodutionisgreaterthantheperapitalarvalprodution

ofpreysubpopulation j,that is,r

i (p

ii +p

ij )>r

j (p

jj +p

ji

). Ifthisis

thease,thenpreysubpopulationj isa relative sink subpopulation.

2. Preysubpopulationiisa relative exporter subpopulationifit

ex-portsmorelarvaetopreysubpopulationjthanitimports(perapita),

thatis,r

1 p

12 >r

2 p

21

. Ifthisisthease,thenpreysubpopulationj is

alledarelative importer subpopulation.

3. The fration

i =j

i

j is alled the biologial eÆieny and the

frationm(q

ii +q

ij )

i =j

i

jisalledtheeonomieÆieny ofpredator

subpopulationi.

3.1. Negligibleosts analysis. Tosimplifytheanalysis,theosts

ofharvesting areassumed to be negligible. Using these assumptions,

andsubstitutingallderivativesofthelogistireruitmentfuntions,F

i

andG

i

,intoequations(4)and(5),wean ndexpliitexpressionsfor

theoptimalesapementsS

Ni and S

Pi :

S

Ni =

A

i m(q

i1 +q

i2 )(2s

i =L

i )+C

i B

i

S Pi = B i (p i1 +p i2 )(2r i =K i )+C

i A i i ; (7) provided i =C 2 i m(p i1 +p i2 )(2r i =K i )(q i1 +q i2 )(2s i =L i

)6=0,with

A

i

=(1=) (p

i1 +p i2 )r i a i ,B i

=(m=) m(q

i1 +q i2 )s i mb i ,and C i = i +m(q i1 +q i2 ) i .

Itiswellknown,inharvestingasingle-speiespopulation,thatifthe

growthrateoftheexploited populationis lessthanthedisount rate,

thenitiseonomiallyoptimaltoharvestthepopulationtoextintion

(Clark [1976℄, Shmitt and Wissel [1985℄, Dawid and Kopel [1997℄).

This is also true in harvesting a one- or two-speies metapopulation

(Tukand Possingham [1994℄, SupriatnaandPossingham[1998℄). We

an show that if A

i and B

i

are negative and C

i

is nonpositive with

C i > maxf(2B i =K i );(2mA i =L i )g, then i

< 0 and all resulting

esapements, S Ni and S Pi

, are positive. If this is the ase, we an

alsoestablishthefollowingresult.

Result 1. Assumepreysubpopulation1 isarelative soure, thatis,

(p 11 +p 12 )r 1 > (p 22 +p 21 )r 2

, while all other parameters of the prey

andthepredatorareidentialforbothsubpopulations. IfA

i andB

i are

negative, and C

i

is nonpositive with C

i

> maxf(2B=K);(2mA=L)g,

then (i)S

N1 > S

N2

and (ii) S

P1 S

P2

. Furthermore if, in addition,

p i1 p i2 ,q i1 =q i2 , S Ni K i ,S Pi L i with S N1 S P1 >S N2 S P2 , then (iii)H N1 <H N2

and(iv)H P1 H P2 .

Result 1 suggests that if the growth rate of the populations is

higher than the disounting rate 1= (indiated by A

i

< 0 and

B

i

< 0) and C

i

> maxf(2B=K);(2mA=L)g, then we should

pro-tet the relative soure prey subpopulation in two dierent ways:

diretly, with a higher esapement of the relative soure prey

sub-population, and indiretly, with a lower esapement of the

preda-tor living in the same path with the relative soure prey

subpop-ulation. Sine C

i

> maxf(2B=K);(2mA=L)g an be written as

m(q

ii +q

ij

)(=j j) > 1+maxf(2B=K);(2mA=L)g=j j, then we an

interpret C

i

>maxf(2B=K);(2mA=L)g as a relatively high predator

eonomieÆieny. Furthermore,ifeveryesapementislessthaneah

har-3.2. Numerial example. Let us assume that there is a

two-path predator-prey metapopulation and the prey in both pathes

have arrying apaities K

1 = K

2

= 50000000, intrinsi growth

r

1 = r

2

= 10, and adult survivals per period are a

1 = a

2

= 0:001.

Prey juvenilesmigrate with migration frations p

11 =p

12

= 0:3 and

p

21 = p

22

= 0:1, hene prey subpopulation 1 is a relative soure

and exporter subpopulation. Let the disount rate Æ be 10%. Now

suppose predators are present in both pathes with intrinsi growth

s

1 = s

2

= 4, arrying apaities L

1 = L

2

= 50000, and adult

survival per period b

1 = b

2

= 0:001. Suppose the predator juvenile

migration is symmetrial and high with q

11 =q

12 =q

21 =q

22 =0:5.

Let j

i

j = 0:000001 and

i

= 0:0000001, that is, we assume the

biologial predator eÆieny is 10%. Using equations (1) and (2),

we an show that one of the positive equilibrium population sizes

forthistwo-path predator-prey metapopulation is(

N

1 ;

N

2 ;

P

1 ;

P

2 )=

(36473692; 36473692; 83105; 83105). We assume harvesting begins

withthisequilibriumastheinitialpopulationsize.

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

esapement for the system S

N

1

= 20420833, S

N

2

= 11262500, and

S

P

1 = S

P

2

= 18131 with the rst period optimal harvests H

N

1 =

16052859, H

N

2

=25211192,H

P

1 =H

P

2

=64973, andtheequilibrium

optimal harvests H

N

1

= 24196828, H

N

2

= 33512055, H

P

1 = H

P

2 =

56835. As suggested by Result 1, we should harvest the relative

exporterandsoureprey subpopulation more onservativelythan the

relativeimporterandsink preysubpopulation(in termofesapement

S

N1 >S

N2

andin termofharvestH

N1 <H

N2

). Thereisnodierene

in esapementandharvestbetweenthepredatorsubpopulations. This

is beausethepredatorbiologial eÆienyis exatlythesame 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 satised. However, if m

is suÆiently large, in our example if m > 10, these rules may be

violated. Thisis beausea largemmeansthat a predator'seonomi

eÆieny ismorethan100%orC>0,seeResult1.

Figure 2showsesapementsandharvests whih areplottedas

fun-tionsoftheratioofpredatormarketprietopreymarketprie,m. The

guresuggeststhat,inthisexamplewherethegrowthofthepredator

isrelativelylow(s

i

=4whiler

i

20

18

16

14

12

10

8

6

4

2

22000

20000

18000

16000

14000

12000

10000

Prey Escapements

S

N1

=S

N2

S

N1

S

N2

m

(2a)

20

18

16

14

12

10

8

6

4

2

25000

20000

15000

10000

5000

0

Predator Escapements

S

P1

S

P2

S

P1

=S

P2

m

(2b) Preynumbersareinthousands.

FIGURE2. Esapements(2a,2b)andharvests(2,2d)areplottedas

fun-tionsoftheratioofpredatormarketprietopreymarketpriem.Lines

600

550

500

450

400

50000

40000

30000

20000

10000

0

Prey Harvests

H

N2

H

N2

H

N1

H

N1

m

(2)

600

550

500

450

400

40000

30000

20000

10000

00000

Predator Harvests

H

P1

H

P2

H

P1

=H

P2

the marketpries(m=1)then itisoptimalto harvest allthe

preda-tor down toextintion (upperrightgure). While,ifmis suÆiently

large (m is approximatelymore than 550) then it is optimalto leave

the relative sink and importer prey subpopulation unharvested, and

eventuallyitisoptimaltoleavebothpreysubpopulationsunharvested

ifmisevenlarger(linesinlowerleftgure). Thisruleisalsoobserved

fora singlepath predator-prey system(Ragozin and Brown [1985℄).

Thissituation is dierent ifthere isno relative soure/sink prey

sub-population. For example, ifp

11 = p

21

= 0:12and p

12 =p

22

= 0:28,

then we should not harvest the relative exporter prey subpopulation

(dotsinlowerleftgure).

Furthermore,evenwhenmisverysmall,ifthegrowthofthepredator

is suÆiently large, then predator extintion is not optimal. For

example, ifs

i

= 20 with m = 1, then predator optimal esapements

are S

N1

= 656 and S

N2

= 11096. Predator esapement in path 1

is less than in path 2; this is beause prey subpopulation 1 is a

relativeexporterandsouresubpopulation whih should beproteted

byleavinglesspredatorsinthat path,seeResult1.

3.3. Dealing with negative harvest. As seen in the example

above,theoptimalstrategymaybea negativeharvestforoneormore

populations. A negativeharvest ould beinterpreted as a seeding or

restoking strategy. However, in many situations, suh a strategy is

notpratial. Inthisase,asinsinglepopulation exploitation,wean

usetheharvestfuntion

(8) H

X

i =

X

i S

Xi ifX

i S

Xi ,

0 ifX

i <S

Xi .

Another alternativeis suggested by Tuk andPossingham [1994℄. To

avoid a negative harvest,they used thefollowingproedure. Assume

that,usingthemetapopulationharvestingtheory,optimalequilibrium

harvestforsubpopulationiisnegative. TheysetH

X

i

=0andfounda

new optimalesapementfrom themaximization of thevaluefuntion

underthiszeroharvestonstraint. Weapplythesameproedureifthe

methodpresentedintheprevioussetionproduesanegativeharvest.

Anegativeharvestmaybeoptimalforthesubpopulationthatexports

parameters for theprey in the previous exampleare p 11 =p 21 =0:2 and p 12 = p 22

= 0:065 with m = 10, then the optimal equilibrium

harvest for the prey subpopulation 2 is H

N2

= 1427582, while all

other subpopulations have a positive harvest. This strategy suggests

that weshould seedpreyintosubpopulation 2and harvesttheresults

frompreysubpopulation1andbothpredatorsubpopulations. Toavoid

anegativeharvestforpreysubpopulation2,wesetH

N2

=0and,using

the method of Lagrange multipliers, we maximize the present value

in equation (3) with this additional onstraint. Thenew equilibrium

optimalesapementsS N 1 ;S N 2 ;S P 1 ,andS

P 2 satisfyequations: (p N N 1 (S N 10 )) =(p N N 1 (N 11 ))[a 1 +p 11 F 0 1 (S N 10 )+ 1 S P 10 ℄ +(p P P 1 (P 11 ))[q 11 1 S P 10 ℄ +(p P P2 (P 21 ))[q 12 1 S P10 ℄ +(p N N 2 (S N 20 )) p 12 F 0 1 (S N10 )(1 1=) Z (9) +(p N N1 (N 11 )) p 12 p 21 F 0 1 (S N10 )F 0 2 (S N20 ) Z +(p P P2 (P 21 )) p 12 F 0 1 (S N10 )q 22 2 S P20 Z +(p P P1 (P 11 )) p 12 F 0 1 (S N 10 )q 21 2 S P 20 Z ; (p P P 2 (S P 20 )) =(p P P 2 (P 21 ))[b 2 +q 22 G 0 2 (S P 20 )+q

22 2 S N 20 ℄ +(p P P 1 (P 11 ))[q 21 G 0 2 (S P 20 )+q

-5000000

0

5000000

10000000

15000000

20000000

25000000

30000000

SN1

SN2

HN1

HN2

Seeding

Zero Harvest

Modified zero harvest

FIGURE3a. Preyesapementsandharvests.

(p

P

P

1 (S

P

10 ))

=(p

P

P

1 (P

11 ))[b

1 +q

11 G

0

1 (S

P

10 )+q

11

1 S

N

10 ℄

+(p

P

P

2 (P

21 ))[q

12 G

0

1 (S

P

10 )+q

12

1 S

N

10 ℄

+(p

N

N1 (N

11 ))[

1 S

N10 ℄; (11)

with Z = 1 (a

2 +p

22 F

0

2 (S

N20 )+

2 S

P20

). Solving the last three

equations together with S

N20 = N

21

, and assuming Z 6= 0, produes

a nonnegativeharvest forpreysubpopulation 2. Figure3 showstotal

protsderivedfromthethree methods, i.e.,negativeharvest from(6)

and(7),zeroharvestfrom(8),andmodiedzeroharvestfrom(9),(10)

and (11).

If it is possible to implement a negative harvest, we nd

equilib-rium optimal harvests are H

N1

= 26517750, H

N2

= 1427582 and

H

P1 = H

P2

= 54642. However, if we annot use a negative

har-vest in management, then using the rst method (equation (8)) we

nd equilibrium optimal harvests areH

N

1

=24720979, H

N

2

=0 and

H

P

1 =H

P

2

=52849. Using theseondmethod (equation(9)to (11))

wend new optimal esapements S

N

1

=15249769, S

N

2

= 12923857,

S

P

1

= 18131, and S

P

2

= 16045 with equilibrium optimal harvests

H

N

1

=24852974, H

N

2

=0, H

P

1

=50985 andH

N

2

Fig-0

10000

20000

30000

40000

50000

60000

SP1

SP2

HP1

HP2

Seeding

Zero Harvest

Modified zero harvest

FIGURE3b. Predatoresapementsandharvests.

25500000

25600000

25700000

25800000

25900000

26000000

26100000

26200000

26300000

Seeding

Zero Harvest

Modified

zero harvest

FIGURE 3. Total prot from three dierent strategies. Esapements,

to theprotperunitsh from harvesting, then,negletingall

assoi-atedosts,thetotalrevenuefrom theharvestis, ifa negativeharvest

isallowable,H

N

1 +H

N

2

+10(H

P

1 +H

P

2

)=26183008urrenyunits.

This revenue is above the revenue if we use zero harvest from either

therst or seond method, i.e., 25777959 from the rst method and

25893514fromtheseondmethod. Thissuggestsanegativeharvestis

optimal.

4. Conlusion. Harvestingstrategiesforapredator-prey

metapop-ulationare established as a generalizationof harvesting strategies for

a single-speies metapoplation. Some properties of the esapements

forasingle-speies metapopulation (Tuk andPossingham[1994℄) are

preservedinthepreseneofthepredator,suhasthestrategiesonhow

to harvest relativesoure and sink subpopulations. We foundthat if

therearenodierenesbetweenthebiologialparameters oftheloal

populations,exeptmigrationparameters,weshouldharvestarelative

sourepreysubpopulationmoreonservativelythanarelativesink

sub-population. This is thesame as therule ofthumb derived from work

on the single-speies metapopulation (Tuk and Possingham [1994℄).

Inaddition,weshouldharvestthepredatorsubpopulationlivinginthe

same path withthe relative soureprey subpopulation more heavily

thantheotherpredatorsubpopulation.

In this paper we have disussed how a `negative' harvest might

our. Under some irumstanesourequationsshow that a negative

harvest isoptimal. Anegativeharvest mightbeonsidered a seeding

strategy. In many situations a seeding strategy is impratial, so in

this ase an alternative strategy of imposing zero harvest, for the

population whih has a negative harvest, is the best that an be

done. If it is possible to implement a negative harvest, numerial

examplesshowthatifthemarketprieofthepredatorismuhgreater

than the market prie of the prey, then it may be optimal to feed

thepredatorby seeding the preypopulations, espeially the exporter

preysubpopulation. Numerialexamplesshowthatthisstrategyould

inreasethetotalnetrevenueomparedtozeropreyharveststrategies.

Aknowledgments. Wethanktwo anonymousreferees whomade

version ofthemanusriptand providedmanyuseful suggestions. The

rstauthorgratefully aknowledgesnanialsupportfrom AusAIDin

undertaking thiswork.

Appendix

Proof of Result 1. Let R = (1=) a

i

, S = m((1=) b

i ), r im = (p ii +p ij )r i ,s im =m(q ii +q ij )s i . 1. If SN = (S N 1 S N 2 ) 1 2

, then using Equations 6 and 7 we

obtain SN = 4s 2 m KL (r 2m r 1m ) 2R L C 2C L C 2S K s m (r 1m r 2m ) =s m 2 L C C 2B K 4s m R KL (r 2m r 1m ): Clearly,S N1 >S N2

,sine(2B=K)C0 and

i <0.

2. Wean proves P 1 S P 2

similarly, sinewehave

Sp =C C C 2B K 4s m R KL (r 2m r 1m ):

3. Reallthatequilibriumharvests aregivenby

H N i = aS N i +p ii rS N i 1 S N i K +p ji rS Nj 1 S N j K

+ S Ni S Pi S Ni and H P i =bS P i +q ii sS P i 1 S P i L

+S Ni S Pi +q ji sS Pj 1 S P j

Thedierenebetweenthesetwoharvests, H

N1 and H

N2 ,is

H

N1 H

N2

=(a 1)(S

N1 S

N2 )

+r

(p

11 p

12 )S

N1

1 S

N1

K

+(p

21 p

22 )S

N2

1 S

N2

K

+ (S

N1 S

P1 S

N2 S

P2 )

<0:

4. Similarly,wean proveH

P1 H

P2

0 ifq

i1 =q

i2 .

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