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

with Delayed Juvenile Reruitment and Predation

MajalahIlmiah HimpunanMatematika Indonesia8(2): 139-150

Asep K. Supriatna 1

, Georey N. Tuk 2

and Hugh P. Possingham 3

1

JurusanMatematika,UniversitasPadjadjaran,Bandung 45363 -Indonesia

2

CSIROMarineResearh GPOBox 1538 Hobart,Tasmania7001 - Australia

3

Department of Zoology,The Universityof Queensland,St. Luia Qld4072 - Australia

Abstrat

InthispaperwedisusstheappliationoftheLagrangemultipliersmethodin

nat-uralresouremodelling. Weusethemethodtodetermineoptimalharvestingstrategies

for atwo-path metapopulationwith delayed juvenile reruitment and predation. We

investigatethe eets of time-delay and predation on the optimal harvesting levelsof

the metapopulation. We found that when the delays are the same for both

subpop-ulations, the model in this paper suggests that we should harvest a relative soure

subpopulation more onservatively than the other subpopulation. However, when the

delays are dierent, then there is a trade-o between the delays and the soure/sink

status of the subpopulations in determining the optimal harvesting strategies for the

metapopulation.

1 Introdution

In this paper we develop a mathematial model for a predator-prey metapopulation

with delayed juvenile reruitment. We onsider optimal harvesting strategies in

ex-ploiting the metapopulation using the method of Lagrange multipliers. The work in

this papergeneralises the resultsof the previousauthors who havedeveloped optimal

harvestingstrategiesforvariousstruturesofbiologialpopulations,suhas[2,4,12,14℄.

The resultsinthis papermightbeapplied asageneralguidane inthe pratie of the

exploitationofmarinelivingorganisms,inwhihdelayed juvenilereruitmentareoften

toour[9℄. Table 1showssomeknown delaytimeforommerialmarinepopulations.

Innature,atimedelayformarinespeiesmayresultfromtheneedofthejuveniles

ofaspeiestotravelfromtheirorigin/spawninghabitattothe destinationhabitatand

Organism: Age at maturity:

Red lipabalone 3 years

Saues sallop 1 year

Ieland sallop 6 years

Baleenwhale 5 years

Seiwhale 9 years

Fin whale 8years

Orangeroughy 23 years

Chinooksalmon 3to 7 years

Sturgeons 10 to 20 years

Pai oeanperh 8 to 10 years

Atka makerel 3:6years

Squid 270 days

Table 1: Some known delay time for ommerial marine populations

(Soure:[11,13℄).

Sometimesthis timedelay islongerthanjust tenortwenty years, asinthe aseof the

Australia's orange roughy. This speies may take several years for juveniles to reah

sexual maturity. They beome sexually mature after about 23 years [6℄. In the next

setionwe begintodevelop a simpledelayed juvenile reruitmentmodelusing ouples

diereneequations.

2 A model for a predator-preymetapopulation with

juvenile reruitment delay

Assume that there isa predator and prey populationineah of two dierentpathes,

namelypathoneandpathtwo (seeFigure1). Aswiththe prey,letthemovementof

predators between the loal populations be through the dispersal of juveniles. Adult

predators are assumed not to migrate from one path to another path. Let the

population size of the prey and predator on path i at the beginning of period k

be denoted by N

i(k)

and P

i(k)

, respetively. 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 reruitment from juveniles that were born

i

periods ago for the

preyand

i

periodsagoforthepredator. Intheabsene ofapredator-preyinteration,

the dynamis of the prey is given by equation

N

i(k+1) =a

i N

i(k) +p

ii F

i (N

i(k

i )

)+p

ji F

j (N

j(k

i )

); (1)

where i = 1;2 and N

i(k)

is the stok abundane of prey subpopulation i in

genera-tion/year k, a

i

is the per generation/year adult survival of prey subpopulationi, and

thefuntion F( .

p

p

₂₂

γ

₁

γ

2

p

p

21

12

N2

N1

P1

P2

1

σ

σ

₂

11

q

₁₁

q

₂₂

q

12

q

21

τ1

τ2

α

1

α

2

Figure1: The gure illustrates apredator-prey metapopulationwith juvenile

reruit-ment delay. The delays for prey and predator populations are

i and

i

, respetively.

Theboxesrepresenttheimmaturestokbeforeitjoinsthereprodutiveadults(irles).

The dots with the symbol

i

indiatethat the predation rate is

i

. The dashes with

the symbol

i

indiate the resulting predator's osprings from predation

i

periods

ago.

denes the number of surviving juveniles produed generations/years ago that join

the maturestok ingeneration/year k+1. The growth of the predator inthe absene

of the prey isdened similarly,that is,

P

i(k+1) =b

i P

i(k) +q

ii G

i (P

i(k

i )

)+q

ji G

j (P

j(k

i )

); (2)

andhene weassumethat the predatorhas anothersoureasa primaryonsumption.

Thisdelay-diereneequationmodelisasimpliationofamoredetailedLesliematrix

model[3℄.

To inlude predation into the system, we use the following fats that generally

foodsupplies may aet predator reprodution and adult survivalof the predator [8℄.

To desribe prey mortality and predator reprodution we use assumptions similar to

those in[15℄, i.e. adultprey mortality ausedby predation inperiod k isproportional

tothenumberofprey and predatorinthatperiod. Predatorreruitmentasaresultof

biomassonversion from the interation is assumed tobeproportionalto the number

of ontats between prey and predator, in whih the predator suessfully kills the

predator reruitment is

i N

ik

i P

ik

i

, where j

i

j

i

> 0. With these additional

assumptions, aomplete model ofa predator-prey metapopulationan bewritten as

N

ik+1

= p

ii F

i (N

ik

i )+p

ji F

j (N

jk

i )

+a

i N

ik +

i N

ik P

ik

; (3)

P

ik+1

= q

ii G

i (P

ik

i )+q

ji G

j (P

jk

i )

+b

i P

ik +

i N

ik

i P

ik

i

; (4)

where allparameters retain the same meaning asin equations (1)and (2). Note that

for the remaining of the paper we simplify the notations N

i(k)

and P

i(k)

with N

ik and

P

ik

. Equation(4)assumesthat thedelay

i

impatsonloalpredatorreruitment. In

this ase, there is a delay of

i

time units between predation and benet to the loal

predator population. If predation only aids predator's adult survival then we would

expet

i =0.

3 An eonomi aspet on the exploitation of the

metapopulation

Suppose thattooptimisethe exploitation,the managerofthe resoureswantsto

max-imisetheresultingnetpresentvalue, bothfromthepredator andthepreypopulations.

Todothis,weassumethatattheendofperiodksubpopulationiisharvestedwith

har-vest H

N

ik

. The esapements, S

N

ik =N

ik H

N

ik

,then growaording toequations(3)

and (4)to N

ik+1

. Thus, inludingharvesting, equations (1)and (2)beome

N

ik+1

= p

ii F

i (S

N

ik

i )+p

ji F

j (S

N

jk

i )

+a

i S

N

ik +

i S

N

ik S

P

ik

; (5)

P

ik+1

= q

ii G

i (S

P

ik

i )+q

ji G

j (S

P

jk

i )

+b

i S

P

ik +

i S

N

ik

i S

P

ik

i

; (6)

Next, wedene the net present value as

PV = 1

X

k=0

k 2

X

i=1 (

Ni (N

ik ;S

N

ik )+

Pi (P

ik ;S

P

ik

)): (7)

We then maximise the net present value 7 over innite time subjet to equations (5)

and (6), with non-negative esapementless than,orequal to,the populationsize. We

alsoassume =1=(1+Æ)where Æ denotes aperiodidisount rate and

Xi (X

ik ;H

X

ik )=

Z

X

ik

X

ik H

X

ik (p

X

Xi

Unlike [12℄, in whih the authors use dynami programming approah, in this

paper we use the Lagrange multipliers to obtain optimal esapements for the prey

subpopulation, S

N

i

, and for the predator subpopulation, S

P

i

. Appendix 1 shows that

the optimal esapements satisfy the followingequations:

p

These equations are the general form of the optimal esapement equations for a

two-pathpredator-preymetapopulationwithatime-delay. Notethatintheabseneof

the delay (

=0),we obtain optimalesapement equationsgiven in [12℄. If

i =

i

=0,then[13℄optimalesapementequationforasingle-speiesmetapopulation

with time delay is obtained. On the other hand, if there is no migration between

pathes, p

ij

impliitoptimal esapements equation for path one is

where F

1N =a

1 +

1 S

P

0 ; G

1N =

1 S

P

0

1

; D

1N =p

11 F

0

1 (S

N

0 ); E

1N

=0;F

1P =

1 S

N

0 ;

G

1P = b

1 +

1 S

N0

1

; D

1P

= 0 and E

1P = q

11 G

0

1 (S

P0

): Optimal esapements for

pathtwoanbeobtainedsimilarlyinthisform. Theseequationsareimpliitoptimal

harvesting equations for two speies derived by [2℄ in the presene of a time-delay in

the predator numerial response suh as in [15℄. Finally, if both juvenile migration

and predator-prey interation are ignored, equations (40) - (43) ollapse to optimal

esapementequationforasingle-speieswithtime-delayasin[4℄. Thefollowingsetion

disussesfurtherthe optimalesapementsandgivessomeinterpretationsoftheresults

by omparing them with other esapements.

4 Optimal esapement properties

To failitate interpretations of the optimal esapements, we assume that the osts of

harvesting are negligibleor density and subpopulationindependent. Furthermore,we

alsoassumethatthereare nodierenes between theprey andpredator priesandthe

reruitprodution funtionfor the prey is

F

i (N

ik ;r

i ;K

i )=r

i N

ik

1 N

ik

K

i

: (15)

Similarly,reruit produtionfuntion forthe predatorisgiven byG

i (P

ik ;s

i ;L

i

). With

theseassumptions, wean obtainanexpliitformfor theoptimalesapements forthe

prey in eahpath

S

Ni =

A

i (q

i1

1

+q

i2

2

) 2si

L

i +C

i B

i

i

; (16)

and the predator in eahpath

S

P

i =

B

i (p

i1

1

+p

i2

2

) 2r

i

K

i +C

i A

i

i

; (17)

provided that

i

6=0, where

i =C

2

i (p

i1

1

+p

i2

2

) 2r

i

K

i (q

i1

1

+q

i2

2

) 2s

i

L

i

; (18)

A

i =

1

(p

i1

1

+p

i2

2

)r

i a

i

; (19)

B

i =

1

(q

i1

1

+q

i2

2

)s

i b

i

; (20)

and

C

i =

i +

i

i

: (21)

If

1 =

2 or

1 =

2

,we deneC

i

as the disounted predator eÆieny(see [12℄).

It an be shown that if the following onditions (22) and (23) are satised,

0C

i

>maxf 2B

i

K

i ;

2A

i

L

i

g; (23)

then

i ; S

N

i

; andS

P

i

are positive.Thesame resultanbeobtained ifA

i ,B

i

,and C

i

are positive. However, C

i

>0is biologiallyunaeptable sine it means the predator

eÆieny is more than 100% (i.e.

i

j

i j

>1 or

i +

i >0).

In the following part we will show that if prey subpopulation one is a soure

subpopulationwith respet toits time delay, that is,satisfying inequality

r

1 (p

11

1

+p

12

2

)>r

2 (p

21

1

+p

22

2

): (24)

thenitshouldbeharvestedmoreonservativelythantheothersubpopulation,provided

the onditions (22) and (23) are both satised. To see this, let us assume that prey

subpopulation one is a relative soure, that is, (p

11 +p

12 )r

1 >(p

22 +p

21 )r

2

, and also

r

1m >r

2m

asin equation (24). All other parameters of the prey and the predator are

idential forboth subpopulations exept delay parameters for the prey.

Letusdene a

i

=a, b

i

=b, C

i

=C,R = 1

a,S = 1

b, r

im =(p

i1

1

+p

i2

2

)r

i

and s

im = (q

i1

1

+q

i2

2

)s

i

. Using these assumptions and notations, the dierene

between the esapement of prey inpath one to the esapement of prey inpath two

is

S

N =(S

N1 S

N2

), whih an be writtenas

S

N =s

1m

2

L C

C 2B

K

4s

1m R

KL

(r

2m r

1m )

1

1

2

: (25)

SineC > 2B

K

(seeondition(23)),thenwehave

S

N

>0ifr

2m <r

1m

,whihissatised

by inequality (24). Furthermore,sine

i

isnegativethen

S

N

>0meansS

N1 >S

N2 .

In other words, we should harvest prey subpopulation one more onservatively than

prey subpopulationtwo if the perapita larvalprodutionofprey subpopulationone,

whih is disounted by its ummulative death rate is larger than the disounted per

apitalarvalprodution ofprey subpopulationtwo. If both prey subpopulations have

the same delay,

1 =

2

, then it simply restates the rule of thumb for single-speies

metapopulation harvesting theory [14℄, that the relative soure prey subpopulation

should beharvested moreonservatively than the relativesink prey subpopulation.

Usingthe same method as above it an be shown that the dierene between the

esapement of predator in path one to the esapement of predator in path two is

S

P =(S

P

1 S

P

2

), whih an be writtenas

S

P

=C(r

1m r

2m )

C

2B

K C

+ 4s

1m R

KL

1

1

2

(26)

and has anon-positivevalue (sine the onditions (22) and(23) are satised). If both

preysubpopulationshavethe samedelay,thenitsimplystatesthatthepredatorliving

in the same path with the relative soure prey subpopulation should be harvested

more heavily than the predator living in the other path. 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 interation (

i =

i

=0, and hene C = 0)

5 Conlusion

Inthispaperwehave developedamathematialmodelofdelayed juvenilereruitment

predator-prey metapopulation. Optimal esapements for the metapopulation were

derived using the method of Lagrangemultipliers. Results depended not only on the

perapitalarvalprodutionasinthenon-delaymodel[12℄,butalsoonthedelays. This

meansthat dierent populations withdierentreruitment delaysshould bemanaged

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 soure

subpopula-tion moreonservativelythan the other subpopulation. However, whenthe delays are

dierent,then thereisatrade-obetween the delays andthe soure/sink statusofthe

subpopulations in determining the optimal harvesting strategies for the

metapopula-tion. Furthermore,we alsoshowed thateven thoughall predatorsubpopulations have

the same delays, their optimal esapements might be dierent if the delays of their

prey are dierent between subpopulations. This is not surprising sine the dynamis

of the predator is inuened by the dynamis of the prey, suh as observed in many

populations [1,5℄.

Aknowledgement

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

the paper.

Referenes

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Appendix 1

The Lagrangian for the maximisationis

L = 1

X

k=0 f

k

[

N

1 (N

1k ;H

N

1k )+

N

2 (N

2k ;H

N

2k )

+

P

1 (P

1k ;H

P

1k )+

P

2 (P

2k ;H

P

2k )℄

1k [N

1(k+1) a

1 (N

1k H

N

1k ) p

11 F

1 (N

1k

1 H

N

1k

1 )

p

21 F

2 (N

2k

1 H

N

2k

1

)

1 (N

1k H

N

1k )(P

1k H

P

1k )℄

2k [N

2(k+1) a

2 (N

2k H

N

2k ) p

12 F

1 (N

1k

2 H

N

1k

2 )

p

22 F

2 (N

2k 2 H

N

2k

2

)

2 (N

2k H

N

2k )(P

2k H

P

2k )℄

3k [P

1(k+1) b

1 (P

1k H

P

1k ) q

11 G

1 (P

1k 1 H

P

1k

1 )

q

21 G

2 (P

2k

1 H

P

2k

1 )

1 (N

1k

1 H

N

1k

1 )(P

1k

1 H

P

1k

1 )℄

4k [P

2(k+1) b

2 (P

2k H

P

2k ) q

12 G

1 (P

1k

2 H

P

1k

2 )

q

22 G

2 (P

2k

2 H

P

2k

2 )

2 (N

2k

2 H

N

2k

2 )(P

2k

2 H

P

2k

2

To nd the optimal esapements we need to solve the neessary onditions:

=0. These onditionsare equivalentto

0 =

Solving equations(28) to (35) produes

Substituting

1k

intoequation (32) produes

0 =

,and reall that

Similarly,substituting

1k

into equations(33) to (35) produes