Application Of Dynamic Programming In Harvesting A Predator-prey Metapopulation.

Application of Dynamic Programming

in Harvesting a Predator-Prey Metapopulation

Asep K. Supriatna

Department of Mathematics, University of Padjadjaran, Indonesia

Abstract

In this paper I use dynamic programming theory to obtain optimal harvesting strategies for a two-patch predator-prey metapopulation. I found that if predator economic efficiency is relatively high then we should protect a relative source prey sub-population in two different ways. Directly, with a higher escapement of the relative source prey sub-population, and indirectly, with a lower escapement of the predator living in the same patch with the relative source prey sub-population. Other rules are also found as generalisations of rules to harvest a single-species metapopulation.

Abstrak

Di dalam paper ini, penulis menggunakan pemograman dinamik untuk mendapatkan strategi yang optimal dalam eksploitasi sumber alam yang mempunyai struktur metapopulasi dan mempunyai relasi biologi ‘predator-prey’. Teori di dalam paper ini menyebutkan bahwa dalam keadaan tertentu, proteksi populasi prey yang relatif ‘source’ bisa dilakukan dengan dua cara. Pertama dengan sisa tangkapan yang lebih banyak dibandingkan sisa tangkapan untuk prey yang relatif ‘sink’ dan dengan eksploitasi predator yang lebih besar dibandingkan dengan eksploitasi predator di tempat lainnya. beberapa aturan umum juga diperoleh sebagai perluasan dari teori eksploitasi untuk species tunggal.

Keywords: Dynamic Programming, Natural Resource Modelling, Harvesting Theory, Predator-Prey Metapopulation

1. Introduction

Many marine organisms which have commercial value are known to have a

metapopulation structure, for example abalone, Haliotis rubra. The adults are sedentary

occupying suitable space in reefs that are separated by some distance from other suitable reefs.

These sub-populations are connected by the dispersal of their larvae (Prince et al., 1987). In

addition, many of these marine metapopulations are part of predator-prey interactions. For

example, the abalone are eaten by crabs, lobsters, octopi and many species of fish (Kojima,

1990). In this paper I develop a model to describe a predator-prey metapopulation in which the

predator-prey interaction takes place in the adult life stage of the prey.

Predation on adult life stage is not uncommon in nature. Zaret (1980) divides predators

in freshwater communities into two types: ‘gape-limited predators’ and ‘size-dependent

predators’. The first type of predator eats prey by swallowing it whole. Hence the prey needs to

be smaller than the predator's mouth diameter. The probability of prey with body size larger

than the predator's gape being eaten by the predator is zero. The second type of predator eats

prey by piercing, crushing or sucking it, and hence can eat prey with a relatively larger body

size than the predator's mouth diameter. In general, the second type of predators also can be

found both in freshwater and marine communities. For example, sea lumprey (Petromyzon

marinus) that prey on many species of fish, such as lake trout, salmon, rainbow trout, whitefish,

burbot, walleye and catfish, is an example in freshwater communities, and octopus that prey on

rock lobster is an example in marine communities.

Some predators of both types have preferential feeding habits. For example, several

species of Coregonus and many planktivorous fish feed on the largest individuals of their prey

(De Bernardi and Giussani, 1975; Vanni, 1987). The maximum body size of the prey that is

captured by the gape-limited predators is limited by the diameter of the predator's mouth, while

the maximum body size of the prey that is captured by the size-dependent predators is only

limited by the predator capability in capturing and handling the prey (Zaret, 1980). Although it

is not regarded as feeding habits, large crabs can prey on large abalone up to 200 mm (Shepherd

and Breen, 1992), and large octopi eat large mussels by drilling the shells (McQuaid, 1994).

Body size in many aquatic organisms is often related to age of maturity; a larger individual

often means an older individual. Hence the predator-prey interaction can be regarded as

The model in this paper has a similar structure and assumptions to the model in

Supriatna and Possingham (1997a, 1998). These papers assume that the juveniles of the

predator as a result of food conversion from the captured prey are sedentary. In this paper, I

modify the model in Supriatna and Possingham (1997a, 1998) to allow some proportion of

these juveniles to migrate between patches. Using dynamic programming theory I found

optimal harvesting strategies to harvest the predator-prey metapopulation. The results show that

some properties of the optimal escapements for a single-species metapopulation are preserved

in the presence of predators, such as the strategies on how to harvest a relative source/sink and

exporter/importer sub-population.

2. The Model

Consider a predator-prey metapopulation that coexists in two different patches; patch

one and patch two. The movement of individuals between the local populations is a result of

dispersal by juveniles. Adults are assumed to be sedentary, and they do not migrate from one

patch to another patch. Assume that the dynamics of the prey metapopulation is given by

Prey in patch i now = the number of surviving adult prey from last period

+ prey recruitment from patch i

+ prey recruitment from patch j

- the number of prey killed by the predator in patch i,

where i =1,2 and j =1,2. If we assume that predation affects the predator recruitment then

Predator in patch i now = the number of surviving adult predator from last period

+ predator recruitment from patch i

+ predator recruitment from patch j.

Let the number of prey (predator) in patch i (where i =1,2) at the beginning of period k

be denoted by Nik and Pik and the survival rate of adult prey (predator) in patch i be denoted by ai and bi, respectively. If the proportion of juvenile prey and predator from patch i that

successfully migrate to patch j are pijand qij , respectively, then the above equation can be

written as

Ni(k+1) = aiNik + piiFi(Nik) + pjiFj(Njk) + αiNikPik, (1)

Pi(k+1) = biPik + qii [Gi(Pik)+βiNikPik] + qji [Gj(Pjk)+βjNjkPjk], (2)

where the functions Fi(Nik) and Gi(Pik) are the recruit production functions of the prey and

predator in patch i at time period k, αinegative and βi positive.

If SNik=Nik-HNik (SPik=Pik-HPik) is the escapement of the prey (predator) in patch i at the

end of that period, and HNik (HPik) is the harvest taken from the prey (predator), then we obtain

the dynamic of exploited population

Ni(k+1) = aiSNik + piiFi(SNik) + pjiFj(SNjk) + αiSNikSPik, (3) Pi(k+1) = biSPik + qii [Gi(SPik)+βiSNikSPik] + qji [Gj(SPjk)+βjSNjkSPjk]. (4)

Furthermore, we assume that in the absence of predator-prey interaction, there is an

environmental carrying capacity in the recruitment of each species, and that recruitment is

linearly dependent on population size, whenever the population size is far below the carrying

capacity. With these assumptions, we can choose a logistic function as a recruit production

function, that is Fi(Nik)=riNik(1-Nik/Ki) and Gi(Pik)=siPik(1-Pik/Li), where ri (si) denotes the

intrinsic growth of the prey (predator) and Ki (Li) denotes the carrying capacity of the prey

(predator), respectively.

To find optimal escapements for the metapopulation, I use dynamic programming as in

2 _{( , )}

1 0

Xik ik i

P

N X

Xi T

k k

S X PV

∑

∑ ∑

= = =

Π

=

ρ

(5)

subject to the equations (3) and (4), and ignoring all costs of harvesting, to obtain optimal

escapements:

,

2 )

( 1 2

*

i

i i i

i i i i

Ni

B C L

s q q m A S

∆ + +

= (6)

,

2 )

( 1 2

*

i

i i i i i i i

Pi

A C K

r p p B S

∆ + +

= (7)

where: ρ=1/(1+δ) with δ≥0 denoting a periodic discount rate, ∆i=Ci2-m(pi1+pi2)[2ri/Ki]

(qi1+qi2)[2si/Li] is not zero, X S p cXiξ dξ

Xik

SXik X Xik ik

Xi( , )= ( − ( ))

Π

∫

is the net revenue from the

harvest HXik of the local population Xi in period k, Ai=1/ρ-[pi1+pi2]ri-ai, Bi=m(1/ρ-[qi1+qi2]si-bi), Ci=αi+m(qi1+qi2)βi,sim=m(qi1+qi2)si, rim=(pi1+pi2)ri, and m is the relative price of the predator

to the prey.

Appendix A shows the derivation of the optimal escapements (6) and (7) and Appendix B

shows that these escapements are independent of the time horizon considered. It can also be

shown that if Aiand Bi are negative and Ci is non-positive with Ci>max{2B_i/K_i,2mA_i/L_i} then

∆i<0 and all resulting escapements, S*Ni and S*Pi,are positive. For the remaining part of this

paper I will assume that Aiand Biare negative, so that extinction is not optimal.

3. Results and Conclusion

If we use escapements in (6) and (7) as a policy to harvest the metapopulation, then we

have the following rules (see Appendix C):

S1. If prey sub-population one is a relative source sub-population, i.e. r1(p11+p12) > r2(p21+p22), and both predator sub-populations are identical with predator efficiency Ci

satisfying max{2B_i/K_i,2mA_i/L_i}<C_i≤0, (i=1,2), then we should protect the relative source

prey sub-population in two different ways. Directly, with a higher escapement of the

relative source prey sub-population, and indirectly, with a lower escapement of the

predator living in the same patch with the relative source prey sub-population.

S2. If prey population one is relatively more vulnerable to predation than the other sub-population, i.e. |α1|>|α2|, and both predator sub-populations are identical with predator

efficiency Ci satisfying max{(-r_mB_i)/(A_iK_i),(-s_mA_i)/(B_iL_i)}<C_i≤0, (i=1,2), then we should

harvest the relatively less vulnerable prey sub-population more conservatively than the

other prey sub-population.

S3. If predator sub-population one is relatively more efficient than the other sub-population, i.e. β1/|α1|>β2/|α2|, and both prey sub-populations are identical with predator efficiency Ci

satisfying max{(-r_mB_i)/(A_iK_i),(-s_mA_i)/(B_iL_i)}<C_i≤0, (i=1,2), then we should harvest the

relatively more efficient predator sub-population more conservatively than the other

predator sub-population.

Furthermore, Supriatna and Possingham (1997b) show that the economic return from

the exploitation using these strategies is higher than if we use other methods such as the

unconnected two-patch predator-prey harvesting theory or well-mixed predator-prey harvesting

theory. The unconnected two-patch predator-prey harvesting theory assumes that there is no

migration between sub-populations, and the well-mixed predator-prey harvesting theory

assumes that there is only one ‘big’ homogeneous patch rather than two connected patches.

To summarise, in this paper I developed a model for a metapopulation with adult

for the population and established the rules described in S1 to S3 to harvest the population optimally.

Acknowledgement. I thank (an) anonymous referee(s) who made helpful suggestions. I

gratefully acknowledge support from AusAID in undertaking this study at the University of

Adelaide, South Australia, and thank my supervisor, Professor Hugh P. Possingham from the

Department of Environmental Science and Management - University of Adelaide, for his

guidance and helpful discussion during the preparation of the paper.

Appendix A: Optimal escapements derivation

Recall that the net present value from harvesting the metapopulation is

( , ).

2

1

0

∑ ∑

∑

_{= = =} Π = i P N X Xik ik Xi T k k S X

PV

ρ

(8)

We use the net revenue for a two-patch metapopulation in the form

X S p cXi ξ dξ

Xik

SXik X Xik ik

Xi( , )= ( − ( ))

Π

∫

(9)

where pX is the price of the harvested stock X which is assumed to be constant, and cXi is the

unit cost of harvesting. Furthermore, we define the value function

(

∑

∑ ∑

)

= = = ≤ ≤ Π = 2 1 0 0 20 10 20

10, , , ) max ( , )

( 0 0 _i P N X Xik ik Xi T k k X S

T N N P P X S

J

i

Xi

ρ

(10)

which is the sum of the discounted net revenue resulting from harvesting both populations in

both patches up to period t=T. We need to maximise this function by choosing appropriate

optimal escapements SXik for each patch and each time period. Equation (10) can be used

recursively to obtain the value function at time T+1, that is

_{( , , , ) max}

(

_{( , , , )} 2 _{( , )}

)

_.

1 0 0 21 11 21 11 0 20 10 20 10 1 0

0≤

∑ ∑

_{= =}

≤ + = + Π i P N X Xi i Xi T X S

T N N P P J N N P P X S

J

i Xi

ρ (11)

To obtain the optimal harvesting strategy, first we will look at the optimal escapement

for the final period, that is T=0. In this case, we would maximise immediate net revenue taken from immediate harvests without considering the future value of the immediate harvested stock

which means no discount factor is applied, and hence the best strategy is the strategy that

maximises PV in (10) for T=0, that is

( , , , ) max

(

( , )

)

.

2 1 0 0 0 20 10 20 10 0 0

0≤

∑ ∑

_{= =}

≤ Π = i P N X Xi i Xi X

S X S

P P N N J i Xi (12)

Let us assume that the optimal escapements are SXi∞, then the maximum revenue is given by

_{( , , , ) ( , ).} 0 2 1 20 10 20 10 0 ∞ = = Π

=

∑ ∑

Xi i Xi

i P N X S X P P N N

J (13)

Next let us consider the next time horizon, T=1. Rewrite equation (11) for T=0 and obtain

J0(N11,N21,P11,P21) in a similar way to J0 in equation (13), to give

( , , , ) max

(

( , ) ( , )

)

.

2 1 0 0 2 1 1 0 20 10 20 10 1 0

0≤

∑ ∑

_{= =} ∞

∑ ∑

_{= =}

≤ Π + Π = i P N X Xi i Xi i P N X Xi i Xi X

S X S X S

P P N N J i Xi

ρ (14)

The optimum value will be given by the condition

( , , , ) 0. 0 20 10 20 10 1 ₌ ∂ ∂ Xi S P P N N

J ₍₁₅₎

Solving the last equations will generate the following implicit expression of the optimal

escapements S*Ni0 and S*Pi0:

_{( ( )) ( ( )),} )) ( )( ( )) ( ))( ( ( ) ( 1 * 0 1 * 0 1 * 0 ' 1 * 0 ' * 0 * 0 j Pj p Pi i ij i Pi p Pi i ii j Nj N Ni i ij i Ni N Ni i ii Pi i i Ni Ni N P c p S q P c p S q N c p S F p N c p S F p S a S c p − + − + − + − + + = − β β α ρ (16) )). ( ( )) ( ( )) ( )( ( )) ( ))( ( ( ) ( 1 * 0 1 * 0 1 * 0 ' 1 * 0 ' * 0 * 0 i Ni N Ni i j Pj p Ni i ij j Pj P Pi i ij i Pi P Pi i ii Ni i ii i Pi Pi P N c p S P c p S q P c p S G q P c p S G q S q b S c p − + − + − + − + + = − α β β ρ (17)

The escapements S*Xi0 found by solving these equations are the optimum escapements

of the prey and the predator on each patch that maximise revenue provided the Hessian matrix

Sx=(SN10,SN20,SP10,SP20) and S*x=(S*N10,S*N20,S*P10,S*P20). These optimal escapements are

independent of time horizon considered, for T≥1, as shown in the Appendix B. Furthermore if we ignore all costs of harvesting then we find explicit form of the optimal escapements as in (6)

and (7).

Appendix B: Time-horizon independence

To prove the claim, first consider the time horizon T=2. Let us define

∑ ∑

= = Π ∞ = 2 1 2 1 2

1, , , ) ( , )

( i P N X Xi ik Xi k k k

k N P P X S

N

V (18)

and rewrite the net revenue with time horizon T=1, J1(N10,N20,P10,P20) in the equation (14)

above using the equations (18) for k=1 into the first term of the equation (14). Using equation (18) for k=0 together with the equation (9) into the second term, we have an expression of the net revenue

(

).

) , , , ( ) , , , ( ) , , , ( max ) , , , ( 20 10 20 10 20 10 20 10 21 11 21 11 0 20 10 20 10 1 0 0 P P N N X S S S S S V P P N N V P P N N V P P N N J i Xi − + = ≤ ≤ ρ (19)

If the optimal escapements S*Xi0 can be obtained from equations (16) and (17) then the above

equation becomes _{( , , , )} ) , , , ( ) , , , ( ) , , , ( * 20 * 10 * 20 * 10 20 10 20 10 * 21 * 11 * 21 * 11 20 10 20 10 1 P P N

N S S S

S V P P N N V P P N N V P P N N J − + = ρ (20)

where each stock abundance X*i1 is a function of the escapement S*Xi0 of the previous period.

To produce the net revenue for the next time horizon, T=2, use the following procedure.

Rewrite J2(N10,N20,P10,P20) in equation (11) in the following form

_{( , , , ) max}

(

_{( , , , )} 2 _{( , )}

).

1 0 0 21 11 21 11 1 0 20 10 20 10 2 0

0≤

∑ ∑

_{= =}

≤ + Π = i P N X Xi i Xi X

S J N N P P X S

P P N N J i Xi

ρ (21)

Rewrite J1(N11,N21,P11,P21) in the equation (11) as follow

(

)

(

)

(

)

). , , , ( ) , , , ( ) , , , ( ) , , , ( ) , , , ( ) , , , ( max ) , ( ) , ( max ) , ( ) , , , ( max ) , , , ( * 21 * 11 * 21 * 11 21 11 21 11 * 22 * 12 * 22 * 12 21 11 21 11 21 11 21 11 22 12 22 12 0 2 1 1 1 2 1 2 0 2 1 1 1 22 12 22 12 0 0 21 11 21 11 1 0 0 0 0 0 0 P P N N P P N N X S i P N X Xi i Xi i P N X Xi i Xi X S i P N X Xi i Xi X S S S S S V P P N N V P P N N V S S S S V P P N N V P P N N V S X S X S X P P N N J P P N N J i Xi i Xi i Xi + = − + = Π + Π = Π + = ≤ ≤ = = = = ∞ ≤ ≤ = = ≤ ≤

∑ ∑

∑ ∑

∑ ∑

ρ ρ ρ ρ (22)

Substitute this result into the equation (21) to produce

[ (

)

]

. ) , , , ( ) , , , ( ) , , , ( ) , , , ( ) , , , ( max ) , , , ( 20 10 20 10 20 10 20 10 * 21 * 11 * 21 * 11 21 11 21 11 * 22 * 12 * 22 * 12 0 20 10 20 10 2 0 0 P P N N P P N N X S S S S S V P P N N V S S S S V P P N N V P P N N V P P N N J i Xi − + − + = ≤ ≤ ρ ρ (23)

All terms with stars are constant with respect to SXi0, hence

[

]

. ) , , , ( ) , , , ( ) , , , ( max ) , , , ( 20 10 20 10 20 10 20 10 21 11 21 11 0 20 10 20 10 2 0 0 C S S S S V P P N N V P P N N V P P N N J P P N N X SXi i

ρ ρ + − + = ≤

≤ (24)

Then equation (19) is used to obtain

J2(N10,N20,P10,P20)=J1(N10,N20,P10,P20)+ρC. (25)

Hence the maximisation of the discounted net revenue resulting from harvesting two periods

from the end is given by the same first period escapements of the maximisation with only one

period, that is given by S*Xi0 which are resulted from solving the equations (16) and (17). We

can use mathematical induction to show that expected net revenue T periods from the end is

( , , , ) ( , , , ) 1 ,

20 10 20 10 1 20 10 20

10 N P P J N N P P C

N J T T T − − +

= ρ (26)

which proves that the optimal first-period escapements independent on the choice of time

Appendix C: Results S1 - S3

To prove the rule S1, we only need to show S*N1>S*N2 and S*P1≤S*P2. To prove the

rules S2 and S3, we only need to show S*N1>S*N2 and S*P1>S*P2. It can be shown that if Ai

and Bi are negative and Ci is non-positive with Ci>max{2B_i/K_i,2mA_i/L_i} then ∆i<0 and all

resulting escapements, S*Ni and S*Pi , are positive. Let Ri=(1/ρ)-ai,Si=m[(1/ρ)-bi],rim=(pii+pij)ri

and sim=m(qii+qij)si. Since both sub-populations are identical, except p11+p12>p21+p22, then let R1=R2=R, S1=S2=S, and s1m=s2m=sm. Let us define ∆SN=(S*N1-S*N2)∆1∆2. Using equations (6)

and (7) we obtain

(

)

(

)

.

4 2 2

2 2 2

) ( 4

1 2

2 1 1

2 2

m m m m

m m m m

m m SN

r r KL

R s K

B C C L s

r r s K

S C L

C C L

R r r KL

s

−    

 

  

 ₋

      − =

−       − −     

 ₋

  



− −

= ∆

Since 2B/K<C≤0 and ∆i<0 then we have S*N1>S*N2. Similarly, we can obtain

(

)

0

4 2

1 2 − ≤ 

 

 ₋

      − =

∆ m m

m

SP r r

KL R s K

B C C C

or S*P1≤S*P2. This proves S1. The rules S2 and S3 can be proved analogously.

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