ANALYSIS
Metapopulation dynamics and stochastic bioeconomic
modeling
Erwin H. Bulte
a,b,*, G. Cornelis van Kooten
c aDepartment of Economics,Tilburg Uni6ersity,Tilburg,The NetherlandsbDepartment of Economics and Management,Wageningen Agricultural Uni6ersity,6706KN Wageningen,The Netherlands cFaculty of Agricultural Sciences and Department of Forest Resources Management,Uni6ersity of British Columbia,Vancou6er,
BC,Canada
Received 14 July 1998; received in revised form 20 November 1998; accepted 25 November 1998
Abstract
We analyze the implications of metapopulation dynamics for optimal harvesting of stochastically fluctuating local subpopulations of a species. The effect of migrating individuals on harvest is twofold: a migration effect captures the possibility for ‘steering’ net migration flows toward the more valuable local population, while a risk term captures the possibility that stochastic fluctuations in different local subpopulations may not be independent. The sign of both effects is analytically ambiguous, implying that harvest intensity can both decrease and increase compared to the conventional, single-population benchmark. © 1999 Elsevier Science B.V. All rights reserved.
Keywords:Metapopulations; Migration; Optimal harvesting; Uncertainty
www.elsevier.com/locate/ecolecon
1. Introduction
Historical evidence of species loss indicates that hunting (harvest) and habitat destruction are prominent factors in loss of species (World Con-servation Monitoring Centre, 1992), although in-vasion of exotic species is now cited as a third
factor (Holmes, 1998). Local extinction of species is also common, but populations are sometimes replenished by immigration from other areas. Loss of a (local) population may not pose a problem if there exists a large mainland from which new members can be drawn. Species sur-vival becomes a particular problem, however, if all habitats are fragmented and there exists no mainland. In this case, the term metapopulation is used to describe a set of local populations of a single species that interact as members migrate among local populations (Hanski and Gilpin, 1991).
* Corresponding author. Department of Economics and Management, Wageningen Agricultural University, 6706 KN Wageningen, The Netherlands. Fax: +31-317-484037.
E-mail address:erwin.bulte@alg.oe.wau.nl (E.H. Bulte)
With metapopulations, survival of a local popu-lation, or the very survival of the species itself, is determined by the ability of migrants to re-colo-nize areas (known as ‘patches’) where a local population has gone extinct. One example is the re-colonization of Yellowstone National Park with wolves from (meta)populations outside the Park when those in the Park went extinct (Budi-ansky, 1995, p.179). Another is the plight of the Concho water snake (Nerodia harteri paucimacu
-lata) which is now threatened by a water develop-ment project. It survives only because of the migration and re-colonization of habitats where the snake went extinct as a result of the vagaries of population dynamics and external events (Quammen, 1996, pp.592 – 602).
Metapopulations can be useful to study both conservation of desirable species, and extirpation of ‘pests’. Concerning the latter, it is well-docu-mented that areas that have been cleared of pests have since been recolonised at greater densities by the same species. In many species, the metapopu-lation may be characterized by one or more core, or source populations with fairly stable popula-tions (in the absence of hunting), and several satellite, or sink populations. The manipulation of flow factors (e.g. barrier fences) could therefore be important in assisting pest eradication programs. Metapopulation models show how the elimination of a few core populations or reducing the poten-tial for migration could lead to the (local) extinc-tion of a species over a much wider area (Primack, 1998, p. 331).
While biologists have investigated metapopula-tions and their behavior, the harvest aspect has been ignored or excessively simplified. Until re-cently, spatial elements have received relatively little attention in resource economics (Sterner and van den Bergh, 1998; Deacon et al., 1998). Sanchirico and Wilen (1998) studied open access exploitation of ‘patchy environments’, concluding that findings are very different from conventional models. Brown and Roughgarden (1997) studied optimal management (including harvest) of a pop-ulation that has a two-stage life cycle. During the larval stage, the species forms a common property pool that provides recruits to several (private property) sites (patches) for the adult stage.
Set-tlement as adults, or colonization in terms of Levin’s model, takes place onto vacant space, and is thus density dependent. Due to biological in-creasing returns, Brown and Roughgarden (1997) then obtained the result that harvesting adults should concentrate on one site.
In this paper, we apply the metapopulation framework in a traditional fashion (i.e. for a species with a single stage lifecycle), and consider migration of a commercially harvestable species and its optimal management. The abundance of local populations is assumed to be a driving factor in migration of individuals from one subpopula-tion to another. To further enhance ecological realism, we assume that local populations are subject to stochastic fluctuations (Pindyck, 1984). The model is readily rewritten to study pest management.1
2. A model for two populations
Levins (1969, 1970) introduced the metapopula-tion concept in ecological science. For simplicity, Levins ignored local (or conventional) dynamics and assumed that any local population is either at zero or at its carrying capacity (K). Denote s(t) the fraction of habitat patches occupied by a species at time t, and assume that the spatial arrangement of patches is of no importance. Then, the rate of change in the fraction of occu-pied patches (ds/dt) is equal to the difference between the colonization rate and the extinction rate. (This is analogous to conventional popula-tion models where changes are described by the difference between the birth and death rates.) For a set of assumptions concerning colonization and extinction, a steady state for the fraction of occu-pied patches is readily computed.2
To analyze the consequences of metapopulation dynamics for harvesting regimes, we consider the simplest metapopulation conceivable; that is, a metapopulation consisting of just two local popu-lations, between which migration is possible (the model is readily generalized to a model of N
populations). Denote these populations as X and
Z, and the numbers of individuals in each by x
andz, respectively. Assume that influx of individ-uals in any population is a function of the vacant niche, defined as the difference between carrying capacity and actual abundance, and species abun-dance in the other population.
More specifically, assume that migration from population Z to population X is denoted as
a(KX−x)z, and that the process of individuals
moving fromXtoZis represented byb(KZ−z)x.
In these expressions,KXandKZdenote the
carry-ing capacities ofXandZ, respectively, andaand
bare population specific parameters. This specifi-cation subsumes most common biological inter-connections between subpopulations as described in the ecological literature, such as the sink-source, fully integrated and limited – distance cases (Sanchirico and Wilen, 1998). The sink-source case (i.e. one-way migration) is consistent with either a or b equal to zero. The fully integrated case is consistent witha andb greater than zero, and possibly of equal value.3
The limited-distance case, where migration is possible from some patches to others, but not to all, can only be treated within a generalized model of N
populations.4
The objective function of the resource manager can be written as:
MaxhE(
&
0
B(hX,hZ,x, z)e −rtd
t}, (1)
where Eis the expectations operator;Bare net benefits of exploitation (and possibly conserva-tion) of populations X and Z; hi (i=X,Z) is
harvesting of the respective populations; and ris the (constant) discount rate. Maximization takes place subject to the following stochastic processes:
dx=[G(x) –hX+a(KX–x)z–b(KZ–z)x] dt
+sX(x) dwX, (2)
and
dz=[F(z) –hZ+b(KZ–z)x–a(KX–x)z] dt
+sZ(z) dwZ. (3)
G(x) andF(z) describe net regeneration of the respective subpopulations. Since ecological cir-cumstances for the two local populations are po-tentially different, we allow for different regeneration functions. The terms si(·)dwi
repre-sent random disturbances in population abun-dance due to demographic or environmental stochasticity. The term dwiis an increment of the
stochastic Wiener process wi (with Brownian
mo-tion), such that dwi=o(t)dt, where o(t) is a
serially uncorrelated and normally distributed random variable with zero mean and unit vari-ance (Dixit and Pindyck, 1994). Assume dX/dx]
0 and dZ/dz]0.
We use Ito’s lemma and dynamic programming to maximize this model. As E(dwXdwZ)=rdt,
whereris the correlation coefficient between dwX
and dwZ, Bellman’s fundamental equation of
opti-mality can be written as:
rV(x,z)
=maxh{B(hX,hZ,x,z)
+Vx[G(x)−hX+a(KX−x)z–b(KZ–z)x]
+(1/2)sX2V xx
2Levins assumed that colonization is proportional tos, the fraction from which potential migrants are available, and to (1−s), or the empty patches available as targets. Hence, the colonization rate is described byas(1−s), whereais a scaling parameter. Since all local populations are at carrying capacity, Levins simply assumed a constant probability of extinction, and defined the extinction rate asbs. Hence ds/dt=as(1−s) –
bs. The equilibrium value of the occupied habitat patches is then simplys=1−b/a.
3The parametersaandb are potentially different because of geophysical conditions, say e.g., for certain fish species, downstream migration may occur more ‘easily’ than migration upstream.
+Vz[F(z) –hZ+b(KZ–z)x–a(KX–x)z]
+(1/2)sZ2V zz
+sXsZrVxz}, (4)
where V(·) is the optimal value function (see Dixit and Pindyck, 1994). For an optimal solu-tion, #B/#hX=Vx and #B/#hZ=Vz hold, or the marginal benefits of exploitation should equal the shadow price of the marginal individual in each of the respective local populations. Substituting opti-mal harvest levels in Eq. (4) and differentiating with respect to x gives:
rVx=(#B/#hX–Vx)#hX/#x+#B/#x
+Vxx[G(x) –hX+a(KX–x)z
–b(KZ–z)x] +Vx[G%(x) –az–b(KZ–z)]
+sX(#sX/#x)Vxx+(1/2)sX2V xxx
+Vxz[F(z) –hZ+b(KZ–z)x
–a(KX–x)z]+Vz[az+b(KZ–z)]
+(1/2)sZ2V
zzx +(#sX/#x)sZrVxz
+sXsZrVxxz, (5)
which is evaluated at hX*. Similarly
differentiat-ing with respect to zgives:
rVz=(#B/#hZ–Vz)#hZ/#z+#B/#z
+Vxz[G(x) –hX+a(KX–x)z
–b(KZ–z)x] +Vx[a(KX–x)+bx]
+(1/2)sX2Vxxz
+Vzz[F(z) –hZ+b(KZ–z)x
–a(KX–x)z] +Vz[F%(z) –bx–a(KX–x)]
+sZ(#sZ/#z)Vzz+(1/2)sZ2
Vzzz
+sX(#sZ/#z)rVxz+sZsXrVxzz, (6)
which is evaluated athZ*. Eqs. (5) and (6) can be
simplified. Taking a second-order Taylor series expansion ofV(x,z) and then differentiating with respect to x and z, respectively, gives:
dVx=Vxxdx+Vxzdz+(1/2)Vxxxdx 2
+(1/2)Vxzzdz2+Vxxzdx dz (7)
dVz=Vzzdz+Vzxdx+(1/2)Vzzzdz 2
+(1/2)Vzxxdx 2
+Vzzxdzdx. (8)
Next, substitute the right – hand sides of Eq. (2) and Eq. (3) for dx and dz, respectively, and use the knowledge that (dwi)
2=dt, dw
XdwZ=r, and
(dt)2=(dt)3/2=0 to obtain:
dVx=Vxx{[G(x) –hX+a(KX–x)z–b(KZ–z)x] dt
+sX(x) dwX}
+Vxz{[F(z)
–hZ+b(KZ–z)x
–a(KX–x)z] dt+sZ(z) dwZ}+(1/2)VxxxsX 2
dt
+(1/2)VxzzsZ2
dt+VxxzsZsXrdt, (9)
and
dVz=Vzz{[F(z) –hZ+b(KZ–z)x–a(KX–x)z] dt
+sZ(z) dwZ}
+Vzx{[G(x)
–hX+a(KX–x)z–b(KZ–z)x] dt+sX(x) dwX}
+(1/2)VzzzsZ2dt
+(1/2)VzxxsX2dt
+VzzxsXsZrdt. (10)
Taking the expectation of Eq. (9) and Eq. (10), noting thatE(dwi)=0, and dividing by dtgives us
the expected rate of change in the marginal value of the local populations, (1/dt)E(dVx) and (1/
dt)E(dVz). Substituting these expressions in Eq.
(5) and Eq. (6), we obtain optimal levels of popu-lation abundance for the two local popupopu-lations. First, consider subpopulation X. The optimal population level is implicit in the following equation:
r+[az*+b(KZ–z*)][1 – (Vz/Vx)]
– (Vxx/Vx)[sX(#sX/#x*)]
=G%(x*)
+(1/Vx)
[(#B/#x*)+(1/dt)E(dV
x)
Next, consider steady state abundance for population Z:
r+[a(KX–x*)+bx*][1 – (Vx/Vz)]
– (Vzz/Vz)[sZ(#sZ/#z*)]
=F%(z*)
+(1/Vz)
[(#B/#z*)+(1/dt)E(dV
z)
+sX(#sZ/#z*)rVzx]. (12)
To solve for the steady states, Eqs. (11) and (12) and the corresponding equations of motion should be solved simultaneously. Hence, typi-cally x* cannot be considered separately from, for example,F(z) and KZ. This finding is
consis-tent with Brown and Roughgarden (1997), who also conclude that the biology of one population affects harvesting of another.5 We discuss the
intertemporal non – arbitrage condition for popu-lation X, or Eq. (11), in detail. The intuition for local population Z, or Eq. (12), is perfectly analogous. Eqs. (11) and (12) state that, at the margin, the resource owner should be indifferent between current harvesting of an individual (unit) of the species (the LHS) and conserving that unit for future use (the RHS).
Now consider the LHS of Eq. (11). The op-portunity costs of conservation, or the benefits of harvesting, consist of (i) the opportunity cost of capital r; (ii) the ‘migration effect’; and (iii) Pindyck’s (Pindyck, 1984) well-known risk pre-mium (or compensation required for the in-crease in local sub population variance attributable to the marginal conserved unit, mul-tiplied by an implicit index of absolute risk aversion, −Vxx/Vx). Consider the migration
ef-fect. Harvesting individuals from population X
implies that migration to this local sub popula-tion will increase as the vacant niche is en-larged. Similarly, migration from X to Z will
decrease as there are fewer individuals to con-sider that option. The marginal value of individ-uals in the two subpopulations, Vx and Vz, may
differ due to stock effects in exploitation, acces-sibility, ‘image’, etc.6 Hence the sign of (
Vz/Vx)
in the second term on the LHS of Eq. (11) is ambiguous, and so is the entire term. This im-plies that the migration effect may contribute to both investment and disinvestment in population
X. For example, if, in the steady state and at the margin, individuals in local population Z
are more valuable than individuals in population
X, the term capturing the migration effect is negative, and hence contributes to investment in stock X.7 This may seem counterintuitive, but
building up population X implies enhancing net migration flows towards population Z, because of both an increased outflow of individuals and a reduced influx.
The RHS of Eq. (11) describes marginal benefits of conserving an individual of popula-tion X. Such benefits consist of (i) the marginal regeneration rate F%(z) (which can be positive or
negative, assuming a concave regeneration func-tion); (ii) perhaps marginal non-use values, or possibly the marginal stock effect on exploita-tion costs (#B/#z); (iii) the expected rate of
change in the marginal value of the species (or the expected ‘capital gain’); and (iv) an addi-tional adjustment factor, or risk premium. Analogous to Pindyck’s risk premium, the fourth term on the RHS introduces an adjust-ment to the required rate of return on conserva-tion, due to stochasticity. The term
sZ(#sX/#x)r(Vxz/Vx) captures the fact that
stochastic fluctuations in distinct local popula-tions may not be independent (r"0). Popula-tions can be affected similarly (e.g. fluctuaPopula-tions due to El Nin˜o affecting populations over vast areas) or in opposite fashion (e.g. abundance of migratory predator and prey species). Hence, the sign of this term is unknown. If rVxz\0, the
term has a positive sign and hence tends to
in-5We do not obtain the result of extreme specialization found by Brown and Roughgarden (1997). Our model is based on the conventional assumptionG%%(x)B0 andF%%(z)B0, such that, in general, marginal biological productivity across local populations can be equalized (G%(x)=F%(z)). The model thus lacks the feature of increasing biological returns.
6When#B/#h
X=#B/#hZin the steady state, the migration effect is effectively zero.
crease optimal abundance levels. The reverse holds when rVxzB0.8
The metapopulation concept, or the effect of migratory individuals, thus has two distinct effects on steady state harvesting. First, resource man-agers can partly manipulate migratory flows by determining the magnitude of the vacant niches. Increasing (decreasing) harvest effort depresses (increases) subpopulation levels and provokes a net influx (outflux) of migrants. When the mar-ginal value of both local populations is different, so (Vz/Vx)"1, managers are able to ‘favor’ the
preferred population by depressing its abundance. The population that is valued less at the margin is allowed to be relatively abundant to provide a sort of sanctuary or overflow area (see also Brown and Roughgarden, 1997). In contrast with the model by Brown and Roughgarden, who treat the biologicallymoreproductive population as a sanc-tuary, we find that the economically lessvaluable population should take that role. When potential migratory flows are symmetrical (i.e. a=b), the migration effect reduces to [bKZ][(Vz/Vx) – 1],
providing a simple benchmark. When inflows are more (less) likely than outflows, or in terms of Eq. (11) when a\b (aBb), the term becomes more (less) important.
One anonymous referee pointed out to us that some aboriginal communities have harvesting strategies similar to, or predicted by this result of the model. For example, some Canadian tribes adopt a strategy of clearing a hunting area of much of the fur-bearing animals, relying on mi-gration to make up for these losses. In part this is a product of the economics of spatially disparate wildlife populations; search costs are prohibitive across wide distances.
Second, migration is relevant when the stochas-tic processes that determine abundance over time are dependent. When upward fluctuations in one local population are matched by downward fluc-tuations in the other (rB0), resource owners’ overall exposure to shocks decreases. The risk term due to the metapopulation concept thus mitigates Pindyck’s risk premium (assuming
Vxz\0). As Pindyck’s risk term is interpreted as
an incentive to reduce stock size (local population abundance), this new risk term implies that the required premium declines, which is consistent with larger local populations, ceteris paribus. The reverse holds when subpopulations are affected in a similar fashion over time. In this case, investing in multiple local populations by refraining from harvesting can be considered extra risky, thus worthy of a higher premium. Due to concavity of the regeneration function, a higher risk premium typically implies reducing abundance.
We conclude that incorporating the metapopu-lation concept, or recognizing migration of indi-viduals between different local populations, is redundant only when the marginal value of local populations is equal and when stochastic pro-cesses are independent. If one of these conditions is violated, considering local populations in isola-tion will produce sub-optimal management regimes.
3. Conclusions
Many bioeconomic models suffer from lack of (biological) realism. For example, Deacon et al. (1998, p.391), write ‘‘(e)conomists have largely stuck with simple paradigms that most biologists regard as useful pedagogical metaphors, but of little practical value’’. Often cited shortcomings include the deterministic nature, and the single-species focus of many models.9 Sanchirico and
8The two risk terms capture the effect of stochasticity in abundance on the incentive to invest or disinvest in local populations. However, as clearly argued by Pindyck (1984), stochastic fluctuations imply additional effects on optimal management. SinceB(·),G(x) andF(z) are nonlinear, equally large upward and downward fluctuations in numbers of a subpopulation do not cancel out, which is known as Jensen’s inequality. This adds to the ambiguity of the results discussed in the text.
Wilen (1998) and Brown and Roughgarden (1997) made clear that the spatial aspect also deserves more attention, and that local popula-tions of one species should not be studied in isolation from other populations. Opportunistic individuals of a species may migrate between different local populations, thus giving rise to what ecologists consider metapopulation dynam-ics. Migration of individuals implies mixing of genes and possibly replenishment of locally ex-tinct populations, and it is partly for these rea-sons that metapopulation models are gaining importance in conservation biology.
Consistent with economists who study multi-species models, we found (perhaps rather unex-pectedly) that enlarging the scope of conventional bioeconomic models implies richer but more complex results. Migrating individuals have two effects on harvesting regimes: a migra-tion effect and a risk effect. The first is associ-ated with the resource managers’ ability to manipulate migration flows by choosing local harvest intensity, but our manager behaves dif-ferently from that of Brown and Roughgarden. The second effect captures the hedging possibil-ity that arises when stochastic fluctuations in lo-cal subpopulations are dependent. The net effect is analytically ambiguous and should be deter-mined empirically on a case-by-case basis.
The treatment of metapopulation dynamics in this paper is far from exhaustive. For exam-ple, one major implication of migration is that local extinction of a species does not constitute an irreversible event. After all, if habitat is un-affected, the vacant niche can potentially be replenished with individuals from elsewhere. This has repercussions for the concepts of option value and quasi option value, as con-ventionally measured by economists. Ex-ploring these complications is left for future re-search.
Acknowledgements
The authors are grateful to two anonymous referees for helpful comments. Remaining errors are our own.
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