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Sustainability and economic consequences of creating marine protected areas in multispecies multiactivity context
Journal of Theoretical Biology 318 (2013) 81–90
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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Sustainability and economic consequences of creating marine protected areas in multispecies multiactivity context T.K. Kar n, Bapan Ghosh Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, India
H I G H L I G H T S c c c c
Impacts of marine reserves have been investigated in multispecies fishery. Prey species possesses heterogeneous intrinsic growth rate and is harvested. Creation of reserve prevents species extinction under exploitation. Protection reduces the economic rent from fishery.
a r t i c l e i n f o
abstract
Article history: Received 21 August 2012 Received in revised form 30 October 2012 Accepted 1 November 2012 Available online 10 November 2012
The present study deals with harvesting of prey species in the presence of predator in a multispecies marine fishery. The total habitat is divided into two patches: one is reserve area where fishing is completely banned and other zone is called fishing area where only prey is exploited. We assume that the prey fish possesses heterogeneous intrinsic growth rate with uniform carrying capacity where as predator has constant intrinsic growth rate with prey dependent carrying capacity. The analytical conditions are derived to prevent the species extinction for larger employed effort in single (only prey) species fishery. Optimal equilibrium premium are presented for both monospecies and multispecies fishery for all degree of protection. Increasing standing stock (ISS) and protected standing stock (PSS) are measured in the presence of prey–predator interaction. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Multi-species interactions Bioeconomics Premium Increasing standing stock (ISS) Protected Standing stock (PSS)
1. Introduction Significant numbers of marine organisms, including mammals, birds and turtles, as well as some commercially harvested fish and shellfish are now threatened or endangered. Conventional management tools such as taxation, license fees, lease of property rights, seasonal harvesting etc. have failed to provide any significant protection of these resources. Clearly, new management approaches or options must be considered to stem the damage and ensure that marine ecosystems and their unique features are protected and restored. As management becomes more integrated and holistic, marine protected areas (MPAs) will take on greater importance as a tools for conserving marine resources. Although, the protected area concept, with its emphasis on management of spaces rather than species, is not new and has been used frequently on land, until recently there have been less support and few interagency
n
Corresponding author. Tel.: þ91 33 26684561. E-mail addresses: [email protected] (T.K. Kar), [email protected] (B. Ghosh). 0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.11.004
efforts to insight protected areas as a major marine management measure. MPA based approach will shift the focus from agency specific problem management to interagency cooperation for implementing marine politics that recognize the spatial heterogeneity of marine habitats and the need to preserve the structure of marine ecosystems. Various achievements are expected from the creation of marine protected areas. The objectives pursued can usually be classified under one of the following three categories: ecosystem conservation, fisheries management and development of nonextractive recreational activities. Promoting fishery management goals and objectives may require different criteria for designing and implementing MPAs, than for protecting unique habitats or biological diversity. Lauck et al. (1998) asserted that MPAs can be viewed as a kind of insurance against scientific uncertainty, stock assessments, or regulation errors. Conrad (1999) showed that, in the absence of ecological uncertainty and in the context of optimal harvesting, reserve generates no economic benefits to the fisherman. His results coincides with the perspective of many fisherman and also some economists. Hannesson (1998) developed two deterministic equilibrium models, one is continuous and the
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other is discrete. The conservation effect of a marine reserve is shown to be critically dependent on the size of the marine and the migration rate of the fish. Sumaila (2002a,b) has shown that MPA can protect the discounted economic rent from the fishery if the habitat is likely to face a shock and fishermen have a high discount rate. He concluded that the total standing stock biomass increases with the increasing size of the MPAs, but only up to a point. Lubchenco et al., (2003) has focused on the multiple benefits of MPAs which include protection of habitats, conservation of biodiversity, protection or enhancement of ecosystem services, recovery of depleted stock, insurance against environmental uncertainty and recreation etc. Making use of a single-species multi-cohort model incorporating a stock recruitment relationship, Holland and Brazee (1996) have shown that marine reserves could improve sustainable catches in overexploited fisheries, given a fixed level of fishing effort. Recently, Kar and Matsuda (2008) examined the impact of the creation of marine protected areas (MPAs), from both economic and biological perspectives. In particular, they examined the effects of protected patches and harvesting on resource populations and concluded that protected patches are an effective means of conserving resource populations, even though extinction cannot be prevented in all cases. Kvamsdal and Sandal (2008) also examined the consequences of marine protected areas on both economic and biological perspectives. They observed that the protected area does not produce any economic benefit, but the biological stock level increases as the size of the protection increases. Most management measures are directed at individual stocks of a single species and do not take into account species interactions, such as prey–predator relationships. A basic assumption of most models used to determine a catch level is that the catch rate a stock can sustain can be designed based upon the average productivity of the stock. Thus, maintaining the stock size that allows maximum sustainable yield (MSY) historically has been a major management goal. Fishing at the MSY level does not ensure constant catches in the future. Legovic et al. (2010) show that application of MSY policy will lead to extinction of a large number of fish species in most ecosystems. More precisely, they show: approaching MSY in ecosystems means that most likely fish species will be driven to extinction in every fishery that includes exploitation of at least one trophic level which is directly or indirectly used as food for a higher trophic level. Because such single and multispecies fisheries make up a great majority of existing fisheries, attempts to reach MSY should be discouraged instead of being legally prescribed as a goal. Recently, Kar and Misra (2006) consider a prey–predator system in a two patch environment: One accessible to both prey and predators and the other being a refuge for prey, and study the dynamics of the system. Baskett et al. (2007) highlight the importance of species interactions to reserve design and provide guidelines for how this complexity can begin to be incorporated into conservation planning. Kar and Chakraborty (2009) consider a prey–predator fishery model with prey dispersal in a two patch environment and their simulation outputs indicate that MPAs can substantially reduce the risk of fisheries collapse. Other possible benefits ranging from preservation of biodiversity and ecosystem integrity to increase tourism revenue in reserve zone are also considered. Boncoeur et al. (2002) considered a marine reserve in a multispecies, multi-activity context. This article investigates some economic consequences of creating a marine reserve on both fishing and ecotourism, when the range of controllability of fishing effort is limited and the impact of the reserve on ecosystem is taken into account. Globally, there has been a surge of interest in designating areas of the seas as marine protected areas to maintain and conserve marine species and habitats threatened by human activities. There is growing consensus that living marine resources require more stringent protections. Better approaches for utilizing and
protecting living marine resources are needed; however, choosing the best methods to maintain or restore the health of marine ecosystems is a difficult task for resource managers. Already, we have mentioned that conventional fishery management commonly focuses on single species. Whether or not these single species management strategies achieve their specific goals, their practice often neglects other important and pervasive problems. Furthermore, regulations designed for one fishery may negatively influence other species on the same fishing ground through prey– predator relationships (Legovic et al., 2010). The main purpose of this study is to develop further insights into biological as well as economics of the marine reserves, from a multispecies perspective and taking into account the heterogeneous intrinsic growth rate for prey species. Though our model system is not based on a case study, however, krill–whale (or seal) community could be a good example for such a system. Inspite of having some other resources of food such as zooplankton, copepods, and squids etc. krill, which are small shrimp, is the main (also favorite) source of food for whales and seals. In fact, seals have a relatively small fed area and without krill in their immediate area, their food supply becomes limited. Also in our prey–predator system, while the former is targeted by commercial fishing, the latter is not subject to harvest, as it has competing market values associated with the nonconsumptive use such as for ecotourism purposes (whale and seal watching) (see Boncoeur et al., 2002; NRC, 2001). Tourists are willing to pay significant sums for whale-watching tour, mainly to experience whales in their natural environment. It is likely, in fact, that the market value of a whale-watching trip exceeds the market value associated with whale meat. Whales also have value through their ecological role in maintaining the natural abundance of other marine species, including commercially valuable fisheries. For the above reasons we have not considered the presence of alternative prey and harvesting of predator species. However, our results give some significant differences from the single species results with MPA. Our multispecies modeling approach is also supported by the previous investigations of Boncoeur et al. (2002) and Reithe (2006). One of the controversial issues in designing MPAs is deciding where to put them. Kelleher (1999) identified several classes of related criteria that bear on choice of a site: biogeographic and ecological criteria; naturalness; economic, social and scientific importance; international or national significance; practically or feasibility; and duality or replication. However, these guidelines neither offer guidance on how to prioritize these criteria nor provide advice on how to rank candidate sites according to each criterion. If conservation of biodiversity is the goal, then ecological reserves must be located in places that will offer protection to the full spectrum of the species and habitats. Crowder et al. (2000) modeled a system of sources and sinks for reef fish and found that at high fishing effort, placement of reserves in sink areas not only reduced the capacity of the reserve to support the fished population, but also concentrated fishing on source populations. The model suggests that displacement of fishing effort to source population could actually further the decline of fish stock. Therefore, when reserves are established to benefit particular fish stocks, the relative productivity of different areas should be considered. Schnier (2005a) analyzes how the heterogeneity in the intrinsic growth rates and carrying capacities influence the optimal bioeconomic marine reserve formation within a fishery. The primary findings of this research is that first and foremost, in the presence of heterogeneity in growth functions within a fishery, a positively size optimal marine reserve exist. In the following Section 2, first we describe the main biological and technological assumptions of our model and then we construct the appropriate mathematical model. The condition of species
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2. The model To introduce the model equations we make the following biological and technical assumptions: (i) We suppose that the area under consideration is inhabited by two interacting stocks: a stock of prey (fish) and a stock of predators (marine mammals such as whales, seals etc.). (ii) Only one of the two interacting species is harvested. Here, we consider that the prey (fish) is targeted by commercial fishing and predators (whales, seals etc.) are not subject to harvest. It is likely, that market values of whale or seal watching trip is far exceed the market values associated with the whale or seal meat. (iii) The area under consideration is supposed to split into two subareas: a reserve area where fishing (for prey species) is prohibited and an open area for fishing. (iv) Fishing is done following the ‘catch per unit effort’ hypothesis (Clark, 1990). (v) We consider the heterogeneity in the intrinsic growth rates of prey species (Schnier, 2005a). (vi) Predators obey the logistic law of growth with carrying capacity of each area is supposed to be proportional to the prey biomass in that area. This assumption is unlike Boncoeur et al. (2002). Based on the above assumptions, our model takes the form: dx x x y ¼ r r x 1 s bxP x , dt sk sk ð1sÞk dy y x y ¼ r f y 1 þs byP y qey, ð1sÞk dt sk ð1sÞk dPx Px Px Py ¼ gP x 1 m , dt ax ax ay dPy Py Px Py ¼ gPy 1 þm , ð1Þ dt ay ax ay where, x (or Px) and y (or Py) are biomasses of the prey (or predator) fish in reserved and fishing zones respectively, at any time t. k is the carrying capacity of total prey species and s is the portion of marine under protection. rr and rf are the intrinsic growth rates in reserved and fishing area respectively. The predator intrinsic growth rate is measured by g, and a is proportionality constant regarding the predator carrying capacity. s (respectively, m) is the migration rate of the prey (respectively, predator) fish mostly depends on the mobile characteristic of the fish and the nature of the boundary between the two juxtaposed regions created artificially. Direction of the total flux s(x/(sk) y/((1 s)k)) or m(Px/ (ax) Py/(ay)) among prey (or predator) fish across the boundary completely determined by density difference of the respective species of the two areas. b measures the predation rate and h¼qey stands for harvested biomass followed by ‘catch per unit effort’ hypothesis with catchability coefficient q and effort level e. In our model, we have considered Type I functional response though the Type II functional response is more common. Type II functional response describes the average feeding rate of a predator where the predator spends some time for processing each captured prey item (i.e. handling time). Prey handling includes chasing, killing, eating and digesting. But as we have mentioned earlier, our model is based on a krill–whales (seals) community where whale is the largest marine mammal and krill is the small shrimp. Also whale
needs a very large amount of krill in a single day (see Reilly et al., 2004; NRC, 2001), so they nearly eat nonstop. For these reasons we think that Type I functional response is more appropriate in our model. Type I functional response is also taken into account by some other researches like Takashina et al. (2012), Boncoeur et al. (2002) and Reithe (2006). We, here, determine the intrinsic growth rates rr in protected zone and rf in the fishing area for a range of biological hot spots with the support of the base (minimum) intrinsic growth rate, rbase, for a majority of the fishery and the maximum rate, r max , for the location possessing the highest level of recruitment (for details see Schnier, 2005a and its erratum, Schnier, 2005b). In this aspect, we establish the degree of spatial heterogeneity as sn ¼ J/N, where N is the total number of distinct location within the fishery and J is the number of location incorporating higher intrinsic growth rate than the base intrinsic growth rate. To calculate the explicit form of intrinsic growth rate parameters, it is justified that resource manager should select reserve of those location possessing higher growth rate. The intrinsic growth rates are functions of rbase, r max , sn, s and measures the average of all specific intrinsic growth rates within the considered ground. Therefore, as Schnier (2005a,b) we define Z rðsÞ 1 rf ¼ rdGðrÞ ð2aÞ GðrðsÞÞ rbase and rr ¼
1 1GðrðsÞÞ
Z
rmax
rdGðrÞ,
ð2bÞ
rðsÞ
where G(r(s)) describes the distribution function of the intrinsic growth rate with the support of r(s) lies between rbase and r max : Fig. 1 represents the assumed distribution of the intrinsic growth rate which is continuous for all degree of protection and differentiable except at s¼sn. For simplicity and to make realistic model, we evaluate the intrinsic growth rates for sosn and s4sn. Case 1. sosn In this case the intrinsic growth rates take the following forms (Schnier, 2005b) r f ¼ r base þ
ðr max r base Þ ðsn sÞ2 2ð1sÞsn
ð3aÞ
ðr max r base Þs : 2sn
ð3bÞ
and r r ¼ r max
rmax
growth rate
persistence and premium are investigated in Section 3. Section 4 describes the increase standing stock (ISS), protected standing stock (PSS) and premium of prey species in presence of prey–predator interaction. Finally, the outcomes of our research work which differ from existing literatures are given in Section 5.
83
r
min
s=s
*
0
s
Fig. 1. Shows the graphical representation of the assumed distribution for heterogeneous growth rate.
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Case 2. s4 sn In a similar fashion we represent the intrinsic growth rates explicitly as r f ¼ r base
ð4aÞ
and
will occur and the extinction of the resource population is inevitable. The interior equilibria are the points of intersection of isoclines of the system (6a) in the positive quadrant. The two isoclines are parabolas passing through the origin presented as
g1 :
r r ¼ r base þ
ðr max r base Þsn : 2s
ð4bÞ
Over exploitation of system (1) leads to a decline in fish because it is fished faster than the fish have the opportunity to reproduce. Apart from this ecological damage, the net economic rent also starts to be downward due to higher level of applied effort with fixed demand and low price. These incidents motivate us to determine the optimal harvesting strategy of the system (1) to maximize the economic rent as Z 1 ch dt, ð5Þ Ps xs ð0Þ,ys ð0Þ,P sx ð0Þ,P sy ð0Þ ¼ max edt ðpdhÞh qy hZ0 0 where Ps is the economic rent with initial populations. Here suffix ‘s’ stands for the measure of the area under protection, ‘0’ within bracket refers as the initial time and d is the instantaneous annual rate of discount. We consider the total revenue as (p dh)h at any time t, where p is the maximum price with lowest availability of exploited resource and d represent the decline of the price in presence of sufficient harvested biomass. The total cost for exploitation is ce ¼ ch/qy, where c is the cost per unit effort. Such formulation of rent function is also observed in Kvamsdal and Sandal (2008) and Chakraborty et al. (2011).
y ¼ AxðxBÞ
and
g2 :
x ¼ CyðyDÞ,
where the parameters are defined as rf s r r ð1sÞ s A¼ 4 0, B ¼ sk 40 ,C¼ ss rr sð1sÞ and D ¼ ð1sÞk
s rf
qeð1sÞk : rf
If B40, then only one part of g1 enters into the positive quadrant cutting positive x-axis at (B, 0), otherwise it enters through the origin. Similar explanation also follows for the parabola g2. Hence parabolas cut each other at most once in the positive quadrant of the xy-plane or equivalently there exists at most one biological steady state (xn, yn) of the system. We examine all possible cases regarding the co-existence of the species for different signs of B and D. Case 1. B 40 and D4 0. The positive equilibrium exists as shown in Fig. 2a.
3. Reserve effects without prey–predator interaction
Case 2. B 40 and Do 0.
In the absence of prey–predator interaction, system (1) becomes dx x x y ¼ r r x 1 s , dt sk sk ð1sÞk dy y x y ¼ r f y 1 þs qey ð6aÞ dt ð1sÞk sk ð1sÞk
In this case also the interior equilibrium point exists as shown in Fig. 2b. We observe that the biological equilibrium point exists as shown in Fig. 2c. From the above three cases it can be concluded that the interior equilibrium exists if either B or D or both are non negative. In the following Case 4, we examine whether a positive equilibrium exists or not when both B and D are negative. We also derive the condition under which the equilibrium exists (if any).
with economic component Z 1 ch dt, ps xs ð0Þ,ys ð0Þ ¼ max edt ðpdhÞh qy hZ0 0
ð6bÞ
where ps is the net economic rent from the fishery. To study the effects of reserve, we also consider the following model having no protection as dY Y ¼ rY 1 qeY, ð7aÞ dt k
pðY ð0ÞÞ ¼ max HZ0
Z
1 0
cH dt, edt ðpdHÞH qY
ð7bÞ
where, Y is the biomass of the prey fish and H¼qeY is the harvested biomass, p refers as economic rent with reference to the initial stock Y(0) and r stands for intrinsic growth rate which is evaluated as r ¼ r base þ
ðr max r base Þ sn : 2
3.1. Persistence of equilibrium stock Now we shall study whether the model (6a) has an equilibrium point with positive subpopulations, as well as whether this equilibrium is stable. From system (7a), it is observed that if the effort e applied to the fishery exceeds the biotechnical productivity r/q of the fish, then a rapid collapse of the resource population
Case 4. B 40 and Do 0. In this situation, different subcases arise depending upon the inclinations of the parabolas at the origin. If the parabolas satisfy the following condition as:
dy dy ð0,0Þ
ð0,0Þ
, Z dx dx g1 g2 then no interior equilibrium exists (see Fig. 2d), while, for the converse of this inequality, there exists a positive equilibrium as shown in Fig. 2e. Thus for the existence of positive equilibrium eo ec ¼
rf ss : þ q qð1sÞ rsr ks
Now it is clear that, if the harvesting effort e4ec, the introduction of a protected patch is not sufficient to prevent extinction of harvested species. Therefore, we can conclude that, as the effort e increases in (0, ec),the equilibrium stock decreases and ultimately (xn, yn)-(0, 0). So, if the effort is large enough (e4ec), the population becomes extinct. On the otherhand, in the absence of any protected area, the population inevitably becomes extinct as effort exceeds r/q. Now we are interested to investigate whether the effort ec is greater than r/q.
T.K. Kar, B. Ghosh / Journal of Theoretical Biology 318 (2013) 81–90
85
(x , y ) *
*
(x , y )
γ2
*
y
y
*
γ
2
γ
1
(0, D)
γ1 (B, 0)
(B, 0) 0
x
0
x
(x , y ) *
*
(0, D)
γ1
y
y
γ2
γ
1
γ2 0
0
x
x
(x , y )
y
*
*
γ2 γ1
0
x
Fig. 2. (a) Existence of positive equilibrium point for B40 and D 40. (b) Existence of positive equilibrium point for B40 and D o 0. (c) Existence of positive equilibrium point for Bo 0 and D 40. (d) This figure shows the nonexistence of positive equilibrium point for B o0 and D o0. (e) Existence of positive equilibrium point instead Bo 0 and Do 0.
Let f¼
rf ss r : þ q q qð1sÞ rsr ks
Considering B o0, D o0, we cannot conclude analytically that f40 holds for all degree of protection as rf is also a nonlinear function of s. But, we can make some conclusion using simulation work. Suppose sn ¼0.5, rbase ¼1, r max ¼ 2:5, k ¼100 and q ¼0.3. Then harvester can employ an effort greater than r/q while migration rate is being kept fixed. We also calculate the optimal reserve size for species conservation even the employed effort is greater than r/q, but lies in (0, ec) associated with Bo0, D o0 (Table 1).
3.2. Optimal control problem In this section, our objective would be to maximize the discounted stream of profits realized within the fishery, subject to the state equations for the biomass located in the fishing ground and reserve. The discount rate would therefore determine the stock level maximizing the present value of the flow of resource rent over time. The problem (6b), subject to population Eq. (6a) and the control constraint 0 r hr hmax can be solved using Pontryagin’s maximum principle (Kamien and Schwartz, 1991). The current value Hamiltonian resulting from this optimization problem can be expressed as follows: ch H xðt Þ, yðt Þ, hðt Þ, f1 ðt Þ, f2 ðt Þ ¼ ðpdhÞh qy
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where Ps denotes the premium when the reserve size parameter is s,
Table 1 Exhibits the value of f for different migration rates as s ¼ 20, 30, 40 and 50.
p is the net economic rent from the fishery when there is no reserve and ps is the same when reserve size parameter is s.
sks-
20
30
40
50
0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25
0.211117 0.896269 4.4138 B, D 40
0.171863 0.529813 1.32702 3.65598 29.7577 B, D 40
0.155807 0.426285 0.894592 1.74888 3.52406 8.58198 74.392 B, D4 0
0.147075 0.377425 0.731641 1.2811 1.2811 3.6855 6.685 14.5903 77.1074 B, D 40
x x y s þ f1 r r x 1 sk sk ð1sÞk y x y þs h , þ f2 r f y 1 ð1sÞk sk ð1sÞk where f1 and f2 are the shadow values of the population within the fishery and the reserve area respectively. As we intend to derive an optimal equilibrium solution, we consider x and y constants in the subsequent steps. The adjoint equations are df1 @H 2x s ¼ df1 f1 r r 1 ¼ df1 f1 f2 ð8aÞ @x sk dt sk
Our simulations are made with respect to the specified ecological parameter sn ¼0.5, rbase ¼1, r max ¼ 2:5, k¼100, s ¼30; economic parameter as p¼8, c¼ 2, d ¼0.2 and technological parameter q¼0.3. Fig. 3 illustrates the impact of the size of the reserve on optimal harvesting of the fishery. It is observed that, as the reserve size increases, optimal harvesting decreases for any migration rate. However, as the migration rate increases, optimal harvesting also increases for any degree of protection as expected. The contribution of reserves to replenishing fish stocks depends on the export of fish to fishing areas. Potentially, due to higher migration rate, the stock level at the protected area is reduced through migration rate employing more recruitment in the fishing zone which permits higher rate of harvesting. The same result is also followed for the value function of the fishery, due to the fact that the optimal resource rent earned from the fishery is directly proportional to the harvesting (see Fig. 4). Also from Fig. 5, it can be concluded that there is a little bit
and df2 @H 2y s ch ¼ df2 f1 f2 2 , ¼ df2 f2 r f 1 þ dt ð1sÞk @y ð1sÞk qy
ð8bÞ and the transversility conditions are f1 tf ¼ 0 and f2 tf ¼ 0:
ð9Þ
Here tf is the final time for dynamic optimization. Now @H ¼0 @h leads to 1 c p f2 : h¼ 2d qy
ð10Þ Fig. 3. Optimal harvesting curves for different migration rates.
The optimal harvesting is obtained through the expression 1 c p f2 min hmax ,max 0, : 2d qy Whenever benefits of a management approach are discovered, measurements of success must take into account the time paths of benefits and cost as well as the possible changes in the distribution of the benefits and cost of different members of the community over time. For instances, reduced fishing access with institution of an ecological reserve may represent an immediate cost to fishers—a cost remaining in play until, and if, the fishery in question regenerates from the protective effects of the reserve, whereas members of the community supported by ecotourism may benefit immediately. If benefits are defined strictly in terms of market values, we intend to define premium of marine protected areas. It is obvious that if there is no reserve initially, the rent earned from the fishery should be greater than the rent earned from the fishery in the presence of certain reserve size. Thus following Kvamsdal and Sandal (2008), we define the premium as a function of the reserve area and presented as follows: Ps ¼
pps , p
s A ð0, 1Þ,
ð11Þ
Fig. 4. Net economic rent curves for different migration rates.
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Fig. 6 shows the variations of ISS for different predation rates. It is clearly observed that the ISS is always positive in the presence of predation and increases with the increasing size of the reserve. The positive value indicates that for any size of the reserve the sum of the standing stock in both the patches is more than the stock when there is no reserve. Hence increasing size of reserve in the presence of predation may be more effective for biological conservation. However, higher predation rate reduces the total stock, but it increases with the increasing reserve size. While the ISS measures how much the marine reserve strategy increases the standing stock, another biological measure known as protected standing stock (PSS) provides the share of the standing stock which is protected. We define PSS (Kvamsdal and Sandal, 2008) for any protection parameter, s, as PSS ¼
Fig. 5. Premium curves for different migration rates as 20, 30 and 50.
xs s: xs þys
Fig. 7 exhibits the measure of the PSS for different migration rates when b ¼ 0.1. It is clear from the figure that PSS is always positive and it increases with increasing size of the reserve but after about 35% of the reserve, the PSS gets decreased. Again PSS
change in economic rent when the protected area is less than half of the total area for different migration rate as s ¼20, 30 and 50. But, the economic rent gradually decreases when fishable area is less than the protected area. This phenomenon occurs as the intrinsic growth rate is heterogeneous. The migratory species with higher rate of migration upgrades the economic rent a little bit for small area protection, but for large area protection the increase of economic rent is significant.
4. Reserve effects with prey–predator interaction: biological and economic measures We now turn to the case of prey–predator interaction. In this section, we define two biological measures, namely increasing standing stock (ISS) and protected standing stock (PSS) from establishing marine protected areas. These biological measures incorporate the migration of the resource from protected area to fishing area and vice versa, as an important factor towards the standing stock assessment in both the areas which ultimately control the harvesting and enhance the fishing stock to reach its extinct limit. To analyze the economic perspective of the fishery, we shall again study the premium of the marine protected areas in a similar fashion as described for single species fishery. The analytical results for optimal harvesting of the considered system are given in the Appendix A. This section is mainly based on our simulation works due to the complexity of the analytical solutions. Using simulation works instead of real world data, which of course would be of great interest, has the advantage to isolate the effects of the multispecies interaction easily. It may also be considered that the simulations presented here are important indicators from a qualitative, rather than a quantitative point of view. We use the parameter values as sn ¼0.5, rbase ¼1, r max ¼ 2:5, k¼100, s ¼30, g ¼1, m ¼20, a ¼0.5, q¼ 0.3, p ¼8, c¼2, d ¼0.2 and the two different values of predation rate as b ¼0.01 and 0.1. Following Kvamsdal and Sandal (2008), we define the increasing standing stock (ISS) for any relative size of reserve area, s, as follows: ISS ¼
Fig. 6. ISS of prey for different predation rates.
xs þ ys Y , Y
where Y is the equilibrium biomass when no protection is established, and xsand ys are the optimal equilibrium level of prey biomass in reserve and fishing zone respectively.
Fig. 7. Variation of protected standing stock with respect to relative size of the reserve for different migration rates of prey species. For this simulation we take b ¼ 0.1.
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Fig. 8. Variation of protected standing stock with respect to relative size of the reserve for different predation rates.
Fig. 10. Premium curves for some different predation rates.
decreases with the migration rate. This is happened due to the fact that, a high rate of migration reduces the conservation effect. From Fig. 8, we observe that the protected standing stock also depends on multispecies interactions and it is increased with increasing rate of predation. Also it is observed that for b ¼0.1, around 30% reserve; for b ¼ 0.03, around 40% reserve; and for b ¼0.01, around 60% reserves are more beneficial for protected standing stock. Thus this simulation indicates that as the predation rate increases, lesser reserve size becomes more beneficial for the protected standing stock. Fig. 9a and b respectively represent the curve for optimal harvesting and economic rent for two different values of predation rate as b ¼ 0.01 and 0.1. For low predation rate (b ¼0.01), we observe that the harvesting curve (respectively, rent curve) is concave downward and decreases as protection rate increases which is consistent with monospecies fishery model. But for b ¼0.1 both the harvesting and economic rent curves increase for protection upto 20% and after that these measurements decrease. This phenomenon does not occur in single species marine fishery. Therefore, it may be concluded that the prey–predator interactions do matter when the implementation of a reserve is considered to achieve the optimal harvesting and economic rent of the fishery. Fig. 10 depicts the impact of the relative size of the reserve on premium for different values of predation rates. It is observed that in the presence of significant predation, upto 60% reserve is economically beneficial. When reserve is more than 60%, the return decreases due to the fact that we optimize our return in the fishing area only.
5. Conclusion
Fig. 9. (a) Optimal harvesting curves for two different predation rates as b ¼0.01 and 0.1. (b) Net economic rent curves for two different predation rates as b ¼ 0.01 and 0.1.
In the design of marine reserves, the complete spectrum of habitats supporting marine biodiversity should be included with emphasis on safeguarding ecosystem process. The ecosystem interactions are sometimes neglected because of its complexity and analysis of marine reserve is done keeping up with monospecies modeling. However, it is clear that these single-species models, no matter how complex they are in habitat structure and dispersal capacity, will not capture the community-wide effects of reduced fishing pressure and those effects to develop multispecies models are one of the next frontiers in modeling the biological properties of marine resources (Pikitch et al., 2004).
T.K. Kar, B. Ghosh / Journal of Theoretical Biology 318 (2013) 81–90
The aim of this study is to develop further insights into biological as well as economics of the marine reserves, from a multispecies perspective and taking into account the heterogeneous intrinsic growth rate for prey species. Predator which is protected by law from any extractive use, but which may provide benefits from non-extractive use. Though the model based on real world considerations, does not capture the entire complexity of the ecosystems interactions and simulations are based on arbitrary parameter values, however, numerous scenarios covering the breath of the biological and economic feasible parameter space were conducted and the results display the gamut of the dynamical results collected from all the scenarios tested. First we have considered the management measures, from both economic and biological perspective, based on single species fishery with heterogeneous intrinsic growth rate. We have discussed the analytical conditions under which a unique equilibrium exists as it requires for ecological managers to achieve a unique equilibrium for creating a harvesting guidelines and developing the ecosystem in a sustainable manner. In the absence of any reserve area, the population inevitably becomes extinct as long as the harvesting effort exceeds the biotechnical productivity of the species. Our analytical and simulation results suggest that creation of marine reserves is beneficial for the protection of species even though it may be collapsed while the harvesting effort crosses a certain critical value. This observation was also made by Kar and Matsuda (2008) for single species fishery with a marine reserve taking homogeneous intrinsic growth rate. Our analysis also shows that species with higher migration rate needs higher reserve area to protect the species from extinction. We have also calculated some economic measures of the single species fishery with heterogeneous intrinsic growth rate. The net economic profit decreases with increasing reserve size as shown in Fig. 4. These simulation contradicts Schnier (2005a), in the presence of heterogeneity in growth functions within a fishery, a positively size optimal marine reserve does exist. Our investigation is very much closure to the previous study of Kvamsdal and Sandal (2008) for single species fishery possessing homogeneous intrinsic growth rate. But, the increase of premium curve for small protection in our heterogeneous growth rate is less than the increase of premium in the case investigated by Kvamsdal and Sandal (see Fig. 5). Therefore, accounting the heterogeneous intrinsic growth rate, we can conclude that decrease in economic rent is slow process with respect to the relative size of the reserve while reserve size is small. We have also considered the consequences of prey–predator interactions in our study through various simulation works. The obtained results are not only important for resource conservation, but also very useful towards economic perspective of the fishery. We have discussed about two biological measures namely, increasing standing stock (ISS) and protected standing stock (PSS). Our analysis of the above biological measures (see Figs. 6 and 8) clearly indicate that prey–predator interaction do matter when the implementation of a reserve is considered. We have also studied some economic measures of the prey–predator fishery with marine reserve. Our simulation works (see Fig. 10) suggest that in the presence of significant predation, upto 60% reserve is also economically beneficial. In our analysis, both for single species fishery and the fishery with prey–predator interaction, we have considered only the economic benefits from harvested fish only, and it is established that the reserve reduces economic benefit. But reserve may provide some other benefits that may include preservation of biodiversity, research and education, noncomsumptive values, recreation, and tourism. Tourism and recreation could contribute significantly to the commercial value of marine reserve and give additional incomes through this channel. Marine reserves serve as
89
a catalyst for the development of a suit of nonconsumptive services that include much diverse elements as shore-based aquaria and seagoing activities such as whale, seal watching. Recreational activities also provide local communities with economic support, without threatening the protection of marine life. We did not consider rents earned from these resources. Total rent (fisheryþecotourism) may be higher than the rent obtained from the fishery having no reserve. We shall consider all these global cost-benefit analysis in our future research.
Acknowledgment The research work of T.K. Kar is supported by the University Grants Commission (UGC), India (File. no. 40-239/2011(SR), dated 29th June, 2011) and research of Bapan Ghosh is financed by the Council of Scientific and Industrial Research (CSIR), India (File no. 08/003(0077)/2011-EMR-I, dated 23rd March, 2011). The authors are very grateful to the anonymous referees and Janet Stein (Journal Manager, Journal of Theoretical Biology) for their careful reading, constructive comments, and helpful suggestions.
Appendix A The current value Hamiltonian of the systems (1) and (5) is formulated by ch H xðt Þ,yðt Þ,hðt Þ, f1 ðt Þ, f2 ðt Þ ¼ ðpdhÞh qy x x y s þ f1 r r x 1 sk sk ð1sÞk y x y þs h þ f2 r f y 1 ð1sÞk sk ð1sÞk Px Px Py m þ f3 gPx 1 ax ax ay Py Px Py þm : ðA:1Þ þ f4 gPy 1 ay ax ay The associated costate variables evolve with time as df1 @H 2x s ¼ df1 r r 1 bPx f1 ¼ df1 @x sk dt sk s Px mPx f2 2 m þ gP x f3 þ 2 f4 , sk ax ax df2 @H s ¼ df2 ¼ df2 f @y dt ð1sÞk 1 2y s bP y f2 r f 1 ð1sÞk ð1sÞk mPy Py ch þ 2 f3 2 m þ gP y f4 2 , qy ay ay
ðA:2Þ
ðA:3Þ
df3 @H 2P x m m ¼ df3 ¼ df3 þ bxf1 g 1 f f , @P x dt ax ax 3 ax 4 ðA:4Þ and 2P y df4 @H m m ¼ df4 ¼ df4 þ byf2 f3 g 1 f , @P y dt ay ay ay 4 ðA:5Þ with transversility conditions are given as fi tf ¼ 0 ði ¼ 1, 2, 3, 4Þ: Here tf is the final time for dynamic optimization.
ðA:6Þ
90
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Now @H ¼0 @h leads h¼
1 c p f2 : 2d qy
ðA:7Þ
The optimal harvesting is obtained through the expression 1 c p f2 : min hmax , max 0, 2d qy References Baskett, M.L., Micheli, F., Levin, S.A., 2007. Designing marine reserves for interacting species: insights from theory. Biol. Conserv. 137, 163–179. Boncoeur, J., Alban, F., Guyader, O., Thebaud, O., 2002. Fish, fishers, seals and tourists: Economic consequences of creating a marine reserve in a multispecies, multi-activity context. Nat. Resour. Model. 15 (4), 387–411. Chakraborty, K., Das, S., Kar, T.K., 2011. Optimal control of effort of a stage structured prey–predator fishery model with harvesting. Nonlinear Anal.: Real World Appl. 12, 3452–3467. Clark, C.W., 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd ed. John Wiley and Sons, New York. Conrad, J.M., 1999. The bioeconomics of marine sanctuaries. J. Bioecon. 1, 205–217. Crowder, L.B., Lyman, S.J., Figueira, W.F., Priddy, J., 2000. Source-sink population dynamics and the problem of siting marine reserves. Bull. Mar. Sci. 66 (3), 799–820. Hannesson, R., 1998. Marine reserves: What would they accomplish? Mar. Resour. Econ. 13, 159–170. Holland, D.S., Brazee, R.J., 1996. Marine reserves for fisheries management. Mar. Resour. Econ. 11, 157–171. Kamien, M.I., Schwartz, N.L., 1991. Dynamic optimization: the calculus of variations and optimal control in economics and management. In: Bliss, C.J., Intriligator, M.D. (Eds.), Advanced Textbook in Economics, 2nd ed Elsevier B.V, The Netherland.
Kar, T.K., Chakraborty, K., 2009. Marine reserves and its consequences as a fisheries management tool. World J. Modell. Simulation 5 (2), 83–95. Kar, T.K., Matsuda, H., 2008. A bioeconomic model of a single-species fishery with a marine reserve. J. Environ. Manage. 86, 171–180. Kar, T.K., Misra, S., 2006. Influence of prey reserve in a prey–predator fishery. Nonlinear Anal.: Theory Methods Appl. 65, 1725–1735. Kelleher, G., 1999. Guidelines for Marine Protected Areas. International Union for the Conservation of Natural and National Resources, Gland, Switzerland and Cambrize, UK. Kvamsdal, S.F., Sandal, L.K., 2008. The premium of marine protected areas: a simple valution model. Mar. Resour. Econ. 23, 171–197. Lauck, T., Clark, C.W., Mangel, M., Munro, G.R., 1998. Implementing the precautionary principle in fisheries management through marine reserves. Ecol. Appl. 8 (1), 72–78. Legovic, T., Klanjscek, J., Gecek, S., 2010. Maximum sustainable yield and species extinction in ecosystems. Ecol. Modell. 221, 1569–1574. Lubchenco, J., Palumbi, S.R., Gaines, S.D., Andelman, S., 2003. Plugging a hole in the ocean: the emerging science of marine reserves. Ecol. Appl. 13 (1), S3–S7. National Research Council (NRC), 2001. Marine Protected Areas: Tools for Sustaining Ocean Ecosystems. National Academy Press, Washington, DC. Pikitch, E.K., Santora, C., Babcock, E.A., Bakun, A., Bonfil, R., Conover, D.O., Dayton, P., Doukakis, P., Fluharty, D., Heneman, B., Houde, E.D., Link, J., Livingston, P.A., Mangel, M., McAllister, M.K., Pope, J., Sainsbury, K.J., 2004. Ecosystem-based fishery management. Science 305 (5682), 346–347. Reithe, S., 2006. Marine reserves as a measure to control bycatch problems: the importance of multispecies interactions. Nat. Resour. Model. 19 (2), 221–242. Reilly, S., Hedley, S., Borberg, J., Hewitt, R., Thiele, Deborah, Watkins, J., Naganobu, M., 2004. Biomass and energy transfer to baleen whales in the South Atlantic sector of the Southern Ocean. Deep-sea Res. Part II: Top. Stud. Oceanogr. 51, 1397–1409. Schnier, K.E., 2005a. Biological ‘‘hot spots’’ and their effect on optimal bioeconomic marine reserve formation. Ecol. Econ. 52, 453–468. Schnier, K.E., 2005b. Revisiting biological ‘‘hot spots’’ and marine reserve formation. Ecol. Econ. 54, 111–113. Sumaila, U.R., 2002a. Marine protected area performance in a model of the fishery. Nat. Resour. Model. 15 (4), 439–551. Sumaila, U.R., 2002b. Economic models of marine protected areas: an introduction. Nat. Resour. Model. 15 (3), 261–272. Takashina, N., Mougi, A., Iwasa, Y., 2012. Paradox of marine protected areas: suppression of fishing may cause species loss. Popul. Ecol. 54, 475–485.
Contents lists available at SciVerse ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Sustainability and economic consequences of creating marine protected areas in multispecies multiactivity context T.K. Kar n, Bapan Ghosh Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, India
H I G H L I G H T S c c c c
Impacts of marine reserves have been investigated in multispecies fishery. Prey species possesses heterogeneous intrinsic growth rate and is harvested. Creation of reserve prevents species extinction under exploitation. Protection reduces the economic rent from fishery.
a r t i c l e i n f o
abstract
Article history: Received 21 August 2012 Received in revised form 30 October 2012 Accepted 1 November 2012 Available online 10 November 2012
The present study deals with harvesting of prey species in the presence of predator in a multispecies marine fishery. The total habitat is divided into two patches: one is reserve area where fishing is completely banned and other zone is called fishing area where only prey is exploited. We assume that the prey fish possesses heterogeneous intrinsic growth rate with uniform carrying capacity where as predator has constant intrinsic growth rate with prey dependent carrying capacity. The analytical conditions are derived to prevent the species extinction for larger employed effort in single (only prey) species fishery. Optimal equilibrium premium are presented for both monospecies and multispecies fishery for all degree of protection. Increasing standing stock (ISS) and protected standing stock (PSS) are measured in the presence of prey–predator interaction. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Multi-species interactions Bioeconomics Premium Increasing standing stock (ISS) Protected Standing stock (PSS)
1. Introduction Significant numbers of marine organisms, including mammals, birds and turtles, as well as some commercially harvested fish and shellfish are now threatened or endangered. Conventional management tools such as taxation, license fees, lease of property rights, seasonal harvesting etc. have failed to provide any significant protection of these resources. Clearly, new management approaches or options must be considered to stem the damage and ensure that marine ecosystems and their unique features are protected and restored. As management becomes more integrated and holistic, marine protected areas (MPAs) will take on greater importance as a tools for conserving marine resources. Although, the protected area concept, with its emphasis on management of spaces rather than species, is not new and has been used frequently on land, until recently there have been less support and few interagency
n
Corresponding author. Tel.: þ91 33 26684561. E-mail addresses: [email protected] (T.K. Kar), [email protected] (B. Ghosh). 0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.11.004
efforts to insight protected areas as a major marine management measure. MPA based approach will shift the focus from agency specific problem management to interagency cooperation for implementing marine politics that recognize the spatial heterogeneity of marine habitats and the need to preserve the structure of marine ecosystems. Various achievements are expected from the creation of marine protected areas. The objectives pursued can usually be classified under one of the following three categories: ecosystem conservation, fisheries management and development of nonextractive recreational activities. Promoting fishery management goals and objectives may require different criteria for designing and implementing MPAs, than for protecting unique habitats or biological diversity. Lauck et al. (1998) asserted that MPAs can be viewed as a kind of insurance against scientific uncertainty, stock assessments, or regulation errors. Conrad (1999) showed that, in the absence of ecological uncertainty and in the context of optimal harvesting, reserve generates no economic benefits to the fisherman. His results coincides with the perspective of many fisherman and also some economists. Hannesson (1998) developed two deterministic equilibrium models, one is continuous and the
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other is discrete. The conservation effect of a marine reserve is shown to be critically dependent on the size of the marine and the migration rate of the fish. Sumaila (2002a,b) has shown that MPA can protect the discounted economic rent from the fishery if the habitat is likely to face a shock and fishermen have a high discount rate. He concluded that the total standing stock biomass increases with the increasing size of the MPAs, but only up to a point. Lubchenco et al., (2003) has focused on the multiple benefits of MPAs which include protection of habitats, conservation of biodiversity, protection or enhancement of ecosystem services, recovery of depleted stock, insurance against environmental uncertainty and recreation etc. Making use of a single-species multi-cohort model incorporating a stock recruitment relationship, Holland and Brazee (1996) have shown that marine reserves could improve sustainable catches in overexploited fisheries, given a fixed level of fishing effort. Recently, Kar and Matsuda (2008) examined the impact of the creation of marine protected areas (MPAs), from both economic and biological perspectives. In particular, they examined the effects of protected patches and harvesting on resource populations and concluded that protected patches are an effective means of conserving resource populations, even though extinction cannot be prevented in all cases. Kvamsdal and Sandal (2008) also examined the consequences of marine protected areas on both economic and biological perspectives. They observed that the protected area does not produce any economic benefit, but the biological stock level increases as the size of the protection increases. Most management measures are directed at individual stocks of a single species and do not take into account species interactions, such as prey–predator relationships. A basic assumption of most models used to determine a catch level is that the catch rate a stock can sustain can be designed based upon the average productivity of the stock. Thus, maintaining the stock size that allows maximum sustainable yield (MSY) historically has been a major management goal. Fishing at the MSY level does not ensure constant catches in the future. Legovic et al. (2010) show that application of MSY policy will lead to extinction of a large number of fish species in most ecosystems. More precisely, they show: approaching MSY in ecosystems means that most likely fish species will be driven to extinction in every fishery that includes exploitation of at least one trophic level which is directly or indirectly used as food for a higher trophic level. Because such single and multispecies fisheries make up a great majority of existing fisheries, attempts to reach MSY should be discouraged instead of being legally prescribed as a goal. Recently, Kar and Misra (2006) consider a prey–predator system in a two patch environment: One accessible to both prey and predators and the other being a refuge for prey, and study the dynamics of the system. Baskett et al. (2007) highlight the importance of species interactions to reserve design and provide guidelines for how this complexity can begin to be incorporated into conservation planning. Kar and Chakraborty (2009) consider a prey–predator fishery model with prey dispersal in a two patch environment and their simulation outputs indicate that MPAs can substantially reduce the risk of fisheries collapse. Other possible benefits ranging from preservation of biodiversity and ecosystem integrity to increase tourism revenue in reserve zone are also considered. Boncoeur et al. (2002) considered a marine reserve in a multispecies, multi-activity context. This article investigates some economic consequences of creating a marine reserve on both fishing and ecotourism, when the range of controllability of fishing effort is limited and the impact of the reserve on ecosystem is taken into account. Globally, there has been a surge of interest in designating areas of the seas as marine protected areas to maintain and conserve marine species and habitats threatened by human activities. There is growing consensus that living marine resources require more stringent protections. Better approaches for utilizing and
protecting living marine resources are needed; however, choosing the best methods to maintain or restore the health of marine ecosystems is a difficult task for resource managers. Already, we have mentioned that conventional fishery management commonly focuses on single species. Whether or not these single species management strategies achieve their specific goals, their practice often neglects other important and pervasive problems. Furthermore, regulations designed for one fishery may negatively influence other species on the same fishing ground through prey– predator relationships (Legovic et al., 2010). The main purpose of this study is to develop further insights into biological as well as economics of the marine reserves, from a multispecies perspective and taking into account the heterogeneous intrinsic growth rate for prey species. Though our model system is not based on a case study, however, krill–whale (or seal) community could be a good example for such a system. Inspite of having some other resources of food such as zooplankton, copepods, and squids etc. krill, which are small shrimp, is the main (also favorite) source of food for whales and seals. In fact, seals have a relatively small fed area and without krill in their immediate area, their food supply becomes limited. Also in our prey–predator system, while the former is targeted by commercial fishing, the latter is not subject to harvest, as it has competing market values associated with the nonconsumptive use such as for ecotourism purposes (whale and seal watching) (see Boncoeur et al., 2002; NRC, 2001). Tourists are willing to pay significant sums for whale-watching tour, mainly to experience whales in their natural environment. It is likely, in fact, that the market value of a whale-watching trip exceeds the market value associated with whale meat. Whales also have value through their ecological role in maintaining the natural abundance of other marine species, including commercially valuable fisheries. For the above reasons we have not considered the presence of alternative prey and harvesting of predator species. However, our results give some significant differences from the single species results with MPA. Our multispecies modeling approach is also supported by the previous investigations of Boncoeur et al. (2002) and Reithe (2006). One of the controversial issues in designing MPAs is deciding where to put them. Kelleher (1999) identified several classes of related criteria that bear on choice of a site: biogeographic and ecological criteria; naturalness; economic, social and scientific importance; international or national significance; practically or feasibility; and duality or replication. However, these guidelines neither offer guidance on how to prioritize these criteria nor provide advice on how to rank candidate sites according to each criterion. If conservation of biodiversity is the goal, then ecological reserves must be located in places that will offer protection to the full spectrum of the species and habitats. Crowder et al. (2000) modeled a system of sources and sinks for reef fish and found that at high fishing effort, placement of reserves in sink areas not only reduced the capacity of the reserve to support the fished population, but also concentrated fishing on source populations. The model suggests that displacement of fishing effort to source population could actually further the decline of fish stock. Therefore, when reserves are established to benefit particular fish stocks, the relative productivity of different areas should be considered. Schnier (2005a) analyzes how the heterogeneity in the intrinsic growth rates and carrying capacities influence the optimal bioeconomic marine reserve formation within a fishery. The primary findings of this research is that first and foremost, in the presence of heterogeneity in growth functions within a fishery, a positively size optimal marine reserve exist. In the following Section 2, first we describe the main biological and technological assumptions of our model and then we construct the appropriate mathematical model. The condition of species
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2. The model To introduce the model equations we make the following biological and technical assumptions: (i) We suppose that the area under consideration is inhabited by two interacting stocks: a stock of prey (fish) and a stock of predators (marine mammals such as whales, seals etc.). (ii) Only one of the two interacting species is harvested. Here, we consider that the prey (fish) is targeted by commercial fishing and predators (whales, seals etc.) are not subject to harvest. It is likely, that market values of whale or seal watching trip is far exceed the market values associated with the whale or seal meat. (iii) The area under consideration is supposed to split into two subareas: a reserve area where fishing (for prey species) is prohibited and an open area for fishing. (iv) Fishing is done following the ‘catch per unit effort’ hypothesis (Clark, 1990). (v) We consider the heterogeneity in the intrinsic growth rates of prey species (Schnier, 2005a). (vi) Predators obey the logistic law of growth with carrying capacity of each area is supposed to be proportional to the prey biomass in that area. This assumption is unlike Boncoeur et al. (2002). Based on the above assumptions, our model takes the form: dx x x y ¼ r r x 1 s bxP x , dt sk sk ð1sÞk dy y x y ¼ r f y 1 þs byP y qey, ð1sÞk dt sk ð1sÞk dPx Px Px Py ¼ gP x 1 m , dt ax ax ay dPy Py Px Py ¼ gPy 1 þm , ð1Þ dt ay ax ay where, x (or Px) and y (or Py) are biomasses of the prey (or predator) fish in reserved and fishing zones respectively, at any time t. k is the carrying capacity of total prey species and s is the portion of marine under protection. rr and rf are the intrinsic growth rates in reserved and fishing area respectively. The predator intrinsic growth rate is measured by g, and a is proportionality constant regarding the predator carrying capacity. s (respectively, m) is the migration rate of the prey (respectively, predator) fish mostly depends on the mobile characteristic of the fish and the nature of the boundary between the two juxtaposed regions created artificially. Direction of the total flux s(x/(sk) y/((1 s)k)) or m(Px/ (ax) Py/(ay)) among prey (or predator) fish across the boundary completely determined by density difference of the respective species of the two areas. b measures the predation rate and h¼qey stands for harvested biomass followed by ‘catch per unit effort’ hypothesis with catchability coefficient q and effort level e. In our model, we have considered Type I functional response though the Type II functional response is more common. Type II functional response describes the average feeding rate of a predator where the predator spends some time for processing each captured prey item (i.e. handling time). Prey handling includes chasing, killing, eating and digesting. But as we have mentioned earlier, our model is based on a krill–whales (seals) community where whale is the largest marine mammal and krill is the small shrimp. Also whale
needs a very large amount of krill in a single day (see Reilly et al., 2004; NRC, 2001), so they nearly eat nonstop. For these reasons we think that Type I functional response is more appropriate in our model. Type I functional response is also taken into account by some other researches like Takashina et al. (2012), Boncoeur et al. (2002) and Reithe (2006). We, here, determine the intrinsic growth rates rr in protected zone and rf in the fishing area for a range of biological hot spots with the support of the base (minimum) intrinsic growth rate, rbase, for a majority of the fishery and the maximum rate, r max , for the location possessing the highest level of recruitment (for details see Schnier, 2005a and its erratum, Schnier, 2005b). In this aspect, we establish the degree of spatial heterogeneity as sn ¼ J/N, where N is the total number of distinct location within the fishery and J is the number of location incorporating higher intrinsic growth rate than the base intrinsic growth rate. To calculate the explicit form of intrinsic growth rate parameters, it is justified that resource manager should select reserve of those location possessing higher growth rate. The intrinsic growth rates are functions of rbase, r max , sn, s and measures the average of all specific intrinsic growth rates within the considered ground. Therefore, as Schnier (2005a,b) we define Z rðsÞ 1 rf ¼ rdGðrÞ ð2aÞ GðrðsÞÞ rbase and rr ¼
1 1GðrðsÞÞ
Z
rmax
rdGðrÞ,
ð2bÞ
rðsÞ
where G(r(s)) describes the distribution function of the intrinsic growth rate with the support of r(s) lies between rbase and r max : Fig. 1 represents the assumed distribution of the intrinsic growth rate which is continuous for all degree of protection and differentiable except at s¼sn. For simplicity and to make realistic model, we evaluate the intrinsic growth rates for sosn and s4sn. Case 1. sosn In this case the intrinsic growth rates take the following forms (Schnier, 2005b) r f ¼ r base þ
ðr max r base Þ ðsn sÞ2 2ð1sÞsn
ð3aÞ
ðr max r base Þs : 2sn
ð3bÞ
and r r ¼ r max
rmax
growth rate
persistence and premium are investigated in Section 3. Section 4 describes the increase standing stock (ISS), protected standing stock (PSS) and premium of prey species in presence of prey–predator interaction. Finally, the outcomes of our research work which differ from existing literatures are given in Section 5.
83
r
min
s=s
*
0
s
Fig. 1. Shows the graphical representation of the assumed distribution for heterogeneous growth rate.
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Case 2. s4 sn In a similar fashion we represent the intrinsic growth rates explicitly as r f ¼ r base
ð4aÞ
and
will occur and the extinction of the resource population is inevitable. The interior equilibria are the points of intersection of isoclines of the system (6a) in the positive quadrant. The two isoclines are parabolas passing through the origin presented as
g1 :
r r ¼ r base þ
ðr max r base Þsn : 2s
ð4bÞ
Over exploitation of system (1) leads to a decline in fish because it is fished faster than the fish have the opportunity to reproduce. Apart from this ecological damage, the net economic rent also starts to be downward due to higher level of applied effort with fixed demand and low price. These incidents motivate us to determine the optimal harvesting strategy of the system (1) to maximize the economic rent as Z 1 ch dt, ð5Þ Ps xs ð0Þ,ys ð0Þ,P sx ð0Þ,P sy ð0Þ ¼ max edt ðpdhÞh qy hZ0 0 where Ps is the economic rent with initial populations. Here suffix ‘s’ stands for the measure of the area under protection, ‘0’ within bracket refers as the initial time and d is the instantaneous annual rate of discount. We consider the total revenue as (p dh)h at any time t, where p is the maximum price with lowest availability of exploited resource and d represent the decline of the price in presence of sufficient harvested biomass. The total cost for exploitation is ce ¼ ch/qy, where c is the cost per unit effort. Such formulation of rent function is also observed in Kvamsdal and Sandal (2008) and Chakraborty et al. (2011).
y ¼ AxðxBÞ
and
g2 :
x ¼ CyðyDÞ,
where the parameters are defined as rf s r r ð1sÞ s A¼ 4 0, B ¼ sk 40 ,C¼ ss rr sð1sÞ and D ¼ ð1sÞk
s rf
qeð1sÞk : rf
If B40, then only one part of g1 enters into the positive quadrant cutting positive x-axis at (B, 0), otherwise it enters through the origin. Similar explanation also follows for the parabola g2. Hence parabolas cut each other at most once in the positive quadrant of the xy-plane or equivalently there exists at most one biological steady state (xn, yn) of the system. We examine all possible cases regarding the co-existence of the species for different signs of B and D. Case 1. B 40 and D4 0. The positive equilibrium exists as shown in Fig. 2a.
3. Reserve effects without prey–predator interaction
Case 2. B 40 and Do 0.
In the absence of prey–predator interaction, system (1) becomes dx x x y ¼ r r x 1 s , dt sk sk ð1sÞk dy y x y ¼ r f y 1 þs qey ð6aÞ dt ð1sÞk sk ð1sÞk
In this case also the interior equilibrium point exists as shown in Fig. 2b. We observe that the biological equilibrium point exists as shown in Fig. 2c. From the above three cases it can be concluded that the interior equilibrium exists if either B or D or both are non negative. In the following Case 4, we examine whether a positive equilibrium exists or not when both B and D are negative. We also derive the condition under which the equilibrium exists (if any).
with economic component Z 1 ch dt, ps xs ð0Þ,ys ð0Þ ¼ max edt ðpdhÞh qy hZ0 0
ð6bÞ
where ps is the net economic rent from the fishery. To study the effects of reserve, we also consider the following model having no protection as dY Y ¼ rY 1 qeY, ð7aÞ dt k
pðY ð0ÞÞ ¼ max HZ0
Z
1 0
cH dt, edt ðpdHÞH qY
ð7bÞ
where, Y is the biomass of the prey fish and H¼qeY is the harvested biomass, p refers as economic rent with reference to the initial stock Y(0) and r stands for intrinsic growth rate which is evaluated as r ¼ r base þ
ðr max r base Þ sn : 2
3.1. Persistence of equilibrium stock Now we shall study whether the model (6a) has an equilibrium point with positive subpopulations, as well as whether this equilibrium is stable. From system (7a), it is observed that if the effort e applied to the fishery exceeds the biotechnical productivity r/q of the fish, then a rapid collapse of the resource population
Case 4. B 40 and Do 0. In this situation, different subcases arise depending upon the inclinations of the parabolas at the origin. If the parabolas satisfy the following condition as:
dy dy ð0,0Þ
ð0,0Þ
, Z dx dx g1 g2 then no interior equilibrium exists (see Fig. 2d), while, for the converse of this inequality, there exists a positive equilibrium as shown in Fig. 2e. Thus for the existence of positive equilibrium eo ec ¼
rf ss : þ q qð1sÞ rsr ks
Now it is clear that, if the harvesting effort e4ec, the introduction of a protected patch is not sufficient to prevent extinction of harvested species. Therefore, we can conclude that, as the effort e increases in (0, ec),the equilibrium stock decreases and ultimately (xn, yn)-(0, 0). So, if the effort is large enough (e4ec), the population becomes extinct. On the otherhand, in the absence of any protected area, the population inevitably becomes extinct as effort exceeds r/q. Now we are interested to investigate whether the effort ec is greater than r/q.
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85
(x , y ) *
*
(x , y )
γ2
*
y
y
*
γ
2
γ
1
(0, D)
γ1 (B, 0)
(B, 0) 0
x
0
x
(x , y ) *
*
(0, D)
γ1
y
y
γ2
γ
1
γ2 0
0
x
x
(x , y )
y
*
*
γ2 γ1
0
x
Fig. 2. (a) Existence of positive equilibrium point for B40 and D 40. (b) Existence of positive equilibrium point for B40 and D o 0. (c) Existence of positive equilibrium point for Bo 0 and D 40. (d) This figure shows the nonexistence of positive equilibrium point for B o0 and D o0. (e) Existence of positive equilibrium point instead Bo 0 and Do 0.
Let f¼
rf ss r : þ q q qð1sÞ rsr ks
Considering B o0, D o0, we cannot conclude analytically that f40 holds for all degree of protection as rf is also a nonlinear function of s. But, we can make some conclusion using simulation work. Suppose sn ¼0.5, rbase ¼1, r max ¼ 2:5, k ¼100 and q ¼0.3. Then harvester can employ an effort greater than r/q while migration rate is being kept fixed. We also calculate the optimal reserve size for species conservation even the employed effort is greater than r/q, but lies in (0, ec) associated with Bo0, D o0 (Table 1).
3.2. Optimal control problem In this section, our objective would be to maximize the discounted stream of profits realized within the fishery, subject to the state equations for the biomass located in the fishing ground and reserve. The discount rate would therefore determine the stock level maximizing the present value of the flow of resource rent over time. The problem (6b), subject to population Eq. (6a) and the control constraint 0 r hr hmax can be solved using Pontryagin’s maximum principle (Kamien and Schwartz, 1991). The current value Hamiltonian resulting from this optimization problem can be expressed as follows: ch H xðt Þ, yðt Þ, hðt Þ, f1 ðt Þ, f2 ðt Þ ¼ ðpdhÞh qy
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where Ps denotes the premium when the reserve size parameter is s,
Table 1 Exhibits the value of f for different migration rates as s ¼ 20, 30, 40 and 50.
p is the net economic rent from the fishery when there is no reserve and ps is the same when reserve size parameter is s.
sks-
20
30
40
50
0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25
0.211117 0.896269 4.4138 B, D 40
0.171863 0.529813 1.32702 3.65598 29.7577 B, D 40
0.155807 0.426285 0.894592 1.74888 3.52406 8.58198 74.392 B, D4 0
0.147075 0.377425 0.731641 1.2811 1.2811 3.6855 6.685 14.5903 77.1074 B, D 40
x x y s þ f1 r r x 1 sk sk ð1sÞk y x y þs h , þ f2 r f y 1 ð1sÞk sk ð1sÞk where f1 and f2 are the shadow values of the population within the fishery and the reserve area respectively. As we intend to derive an optimal equilibrium solution, we consider x and y constants in the subsequent steps. The adjoint equations are df1 @H 2x s ¼ df1 f1 r r 1 ¼ df1 f1 f2 ð8aÞ @x sk dt sk
Our simulations are made with respect to the specified ecological parameter sn ¼0.5, rbase ¼1, r max ¼ 2:5, k¼100, s ¼30; economic parameter as p¼8, c¼ 2, d ¼0.2 and technological parameter q¼0.3. Fig. 3 illustrates the impact of the size of the reserve on optimal harvesting of the fishery. It is observed that, as the reserve size increases, optimal harvesting decreases for any migration rate. However, as the migration rate increases, optimal harvesting also increases for any degree of protection as expected. The contribution of reserves to replenishing fish stocks depends on the export of fish to fishing areas. Potentially, due to higher migration rate, the stock level at the protected area is reduced through migration rate employing more recruitment in the fishing zone which permits higher rate of harvesting. The same result is also followed for the value function of the fishery, due to the fact that the optimal resource rent earned from the fishery is directly proportional to the harvesting (see Fig. 4). Also from Fig. 5, it can be concluded that there is a little bit
and df2 @H 2y s ch ¼ df2 f1 f2 2 , ¼ df2 f2 r f 1 þ dt ð1sÞk @y ð1sÞk qy
ð8bÞ and the transversility conditions are f1 tf ¼ 0 and f2 tf ¼ 0:
ð9Þ
Here tf is the final time for dynamic optimization. Now @H ¼0 @h leads to 1 c p f2 : h¼ 2d qy
ð10Þ Fig. 3. Optimal harvesting curves for different migration rates.
The optimal harvesting is obtained through the expression 1 c p f2 min hmax ,max 0, : 2d qy Whenever benefits of a management approach are discovered, measurements of success must take into account the time paths of benefits and cost as well as the possible changes in the distribution of the benefits and cost of different members of the community over time. For instances, reduced fishing access with institution of an ecological reserve may represent an immediate cost to fishers—a cost remaining in play until, and if, the fishery in question regenerates from the protective effects of the reserve, whereas members of the community supported by ecotourism may benefit immediately. If benefits are defined strictly in terms of market values, we intend to define premium of marine protected areas. It is obvious that if there is no reserve initially, the rent earned from the fishery should be greater than the rent earned from the fishery in the presence of certain reserve size. Thus following Kvamsdal and Sandal (2008), we define the premium as a function of the reserve area and presented as follows: Ps ¼
pps , p
s A ð0, 1Þ,
ð11Þ
Fig. 4. Net economic rent curves for different migration rates.
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Fig. 6 shows the variations of ISS for different predation rates. It is clearly observed that the ISS is always positive in the presence of predation and increases with the increasing size of the reserve. The positive value indicates that for any size of the reserve the sum of the standing stock in both the patches is more than the stock when there is no reserve. Hence increasing size of reserve in the presence of predation may be more effective for biological conservation. However, higher predation rate reduces the total stock, but it increases with the increasing reserve size. While the ISS measures how much the marine reserve strategy increases the standing stock, another biological measure known as protected standing stock (PSS) provides the share of the standing stock which is protected. We define PSS (Kvamsdal and Sandal, 2008) for any protection parameter, s, as PSS ¼
Fig. 5. Premium curves for different migration rates as 20, 30 and 50.
xs s: xs þys
Fig. 7 exhibits the measure of the PSS for different migration rates when b ¼ 0.1. It is clear from the figure that PSS is always positive and it increases with increasing size of the reserve but after about 35% of the reserve, the PSS gets decreased. Again PSS
change in economic rent when the protected area is less than half of the total area for different migration rate as s ¼20, 30 and 50. But, the economic rent gradually decreases when fishable area is less than the protected area. This phenomenon occurs as the intrinsic growth rate is heterogeneous. The migratory species with higher rate of migration upgrades the economic rent a little bit for small area protection, but for large area protection the increase of economic rent is significant.
4. Reserve effects with prey–predator interaction: biological and economic measures We now turn to the case of prey–predator interaction. In this section, we define two biological measures, namely increasing standing stock (ISS) and protected standing stock (PSS) from establishing marine protected areas. These biological measures incorporate the migration of the resource from protected area to fishing area and vice versa, as an important factor towards the standing stock assessment in both the areas which ultimately control the harvesting and enhance the fishing stock to reach its extinct limit. To analyze the economic perspective of the fishery, we shall again study the premium of the marine protected areas in a similar fashion as described for single species fishery. The analytical results for optimal harvesting of the considered system are given in the Appendix A. This section is mainly based on our simulation works due to the complexity of the analytical solutions. Using simulation works instead of real world data, which of course would be of great interest, has the advantage to isolate the effects of the multispecies interaction easily. It may also be considered that the simulations presented here are important indicators from a qualitative, rather than a quantitative point of view. We use the parameter values as sn ¼0.5, rbase ¼1, r max ¼ 2:5, k¼100, s ¼30, g ¼1, m ¼20, a ¼0.5, q¼ 0.3, p ¼8, c¼2, d ¼0.2 and the two different values of predation rate as b ¼0.01 and 0.1. Following Kvamsdal and Sandal (2008), we define the increasing standing stock (ISS) for any relative size of reserve area, s, as follows: ISS ¼
Fig. 6. ISS of prey for different predation rates.
xs þ ys Y , Y
where Y is the equilibrium biomass when no protection is established, and xsand ys are the optimal equilibrium level of prey biomass in reserve and fishing zone respectively.
Fig. 7. Variation of protected standing stock with respect to relative size of the reserve for different migration rates of prey species. For this simulation we take b ¼ 0.1.
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Fig. 8. Variation of protected standing stock with respect to relative size of the reserve for different predation rates.
Fig. 10. Premium curves for some different predation rates.
decreases with the migration rate. This is happened due to the fact that, a high rate of migration reduces the conservation effect. From Fig. 8, we observe that the protected standing stock also depends on multispecies interactions and it is increased with increasing rate of predation. Also it is observed that for b ¼0.1, around 30% reserve; for b ¼ 0.03, around 40% reserve; and for b ¼0.01, around 60% reserves are more beneficial for protected standing stock. Thus this simulation indicates that as the predation rate increases, lesser reserve size becomes more beneficial for the protected standing stock. Fig. 9a and b respectively represent the curve for optimal harvesting and economic rent for two different values of predation rate as b ¼ 0.01 and 0.1. For low predation rate (b ¼0.01), we observe that the harvesting curve (respectively, rent curve) is concave downward and decreases as protection rate increases which is consistent with monospecies fishery model. But for b ¼0.1 both the harvesting and economic rent curves increase for protection upto 20% and after that these measurements decrease. This phenomenon does not occur in single species marine fishery. Therefore, it may be concluded that the prey–predator interactions do matter when the implementation of a reserve is considered to achieve the optimal harvesting and economic rent of the fishery. Fig. 10 depicts the impact of the relative size of the reserve on premium for different values of predation rates. It is observed that in the presence of significant predation, upto 60% reserve is economically beneficial. When reserve is more than 60%, the return decreases due to the fact that we optimize our return in the fishing area only.
5. Conclusion
Fig. 9. (a) Optimal harvesting curves for two different predation rates as b ¼0.01 and 0.1. (b) Net economic rent curves for two different predation rates as b ¼ 0.01 and 0.1.
In the design of marine reserves, the complete spectrum of habitats supporting marine biodiversity should be included with emphasis on safeguarding ecosystem process. The ecosystem interactions are sometimes neglected because of its complexity and analysis of marine reserve is done keeping up with monospecies modeling. However, it is clear that these single-species models, no matter how complex they are in habitat structure and dispersal capacity, will not capture the community-wide effects of reduced fishing pressure and those effects to develop multispecies models are one of the next frontiers in modeling the biological properties of marine resources (Pikitch et al., 2004).
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The aim of this study is to develop further insights into biological as well as economics of the marine reserves, from a multispecies perspective and taking into account the heterogeneous intrinsic growth rate for prey species. Predator which is protected by law from any extractive use, but which may provide benefits from non-extractive use. Though the model based on real world considerations, does not capture the entire complexity of the ecosystems interactions and simulations are based on arbitrary parameter values, however, numerous scenarios covering the breath of the biological and economic feasible parameter space were conducted and the results display the gamut of the dynamical results collected from all the scenarios tested. First we have considered the management measures, from both economic and biological perspective, based on single species fishery with heterogeneous intrinsic growth rate. We have discussed the analytical conditions under which a unique equilibrium exists as it requires for ecological managers to achieve a unique equilibrium for creating a harvesting guidelines and developing the ecosystem in a sustainable manner. In the absence of any reserve area, the population inevitably becomes extinct as long as the harvesting effort exceeds the biotechnical productivity of the species. Our analytical and simulation results suggest that creation of marine reserves is beneficial for the protection of species even though it may be collapsed while the harvesting effort crosses a certain critical value. This observation was also made by Kar and Matsuda (2008) for single species fishery with a marine reserve taking homogeneous intrinsic growth rate. Our analysis also shows that species with higher migration rate needs higher reserve area to protect the species from extinction. We have also calculated some economic measures of the single species fishery with heterogeneous intrinsic growth rate. The net economic profit decreases with increasing reserve size as shown in Fig. 4. These simulation contradicts Schnier (2005a), in the presence of heterogeneity in growth functions within a fishery, a positively size optimal marine reserve does exist. Our investigation is very much closure to the previous study of Kvamsdal and Sandal (2008) for single species fishery possessing homogeneous intrinsic growth rate. But, the increase of premium curve for small protection in our heterogeneous growth rate is less than the increase of premium in the case investigated by Kvamsdal and Sandal (see Fig. 5). Therefore, accounting the heterogeneous intrinsic growth rate, we can conclude that decrease in economic rent is slow process with respect to the relative size of the reserve while reserve size is small. We have also considered the consequences of prey–predator interactions in our study through various simulation works. The obtained results are not only important for resource conservation, but also very useful towards economic perspective of the fishery. We have discussed about two biological measures namely, increasing standing stock (ISS) and protected standing stock (PSS). Our analysis of the above biological measures (see Figs. 6 and 8) clearly indicate that prey–predator interaction do matter when the implementation of a reserve is considered. We have also studied some economic measures of the prey–predator fishery with marine reserve. Our simulation works (see Fig. 10) suggest that in the presence of significant predation, upto 60% reserve is also economically beneficial. In our analysis, both for single species fishery and the fishery with prey–predator interaction, we have considered only the economic benefits from harvested fish only, and it is established that the reserve reduces economic benefit. But reserve may provide some other benefits that may include preservation of biodiversity, research and education, noncomsumptive values, recreation, and tourism. Tourism and recreation could contribute significantly to the commercial value of marine reserve and give additional incomes through this channel. Marine reserves serve as
89
a catalyst for the development of a suit of nonconsumptive services that include much diverse elements as shore-based aquaria and seagoing activities such as whale, seal watching. Recreational activities also provide local communities with economic support, without threatening the protection of marine life. We did not consider rents earned from these resources. Total rent (fisheryþecotourism) may be higher than the rent obtained from the fishery having no reserve. We shall consider all these global cost-benefit analysis in our future research.
Acknowledgment The research work of T.K. Kar is supported by the University Grants Commission (UGC), India (File. no. 40-239/2011(SR), dated 29th June, 2011) and research of Bapan Ghosh is financed by the Council of Scientific and Industrial Research (CSIR), India (File no. 08/003(0077)/2011-EMR-I, dated 23rd March, 2011). The authors are very grateful to the anonymous referees and Janet Stein (Journal Manager, Journal of Theoretical Biology) for their careful reading, constructive comments, and helpful suggestions.
Appendix A The current value Hamiltonian of the systems (1) and (5) is formulated by ch H xðt Þ,yðt Þ,hðt Þ, f1 ðt Þ, f2 ðt Þ ¼ ðpdhÞh qy x x y s þ f1 r r x 1 sk sk ð1sÞk y x y þs h þ f2 r f y 1 ð1sÞk sk ð1sÞk Px Px Py m þ f3 gPx 1 ax ax ay Py Px Py þm : ðA:1Þ þ f4 gPy 1 ay ax ay The associated costate variables evolve with time as df1 @H 2x s ¼ df1 r r 1 bPx f1 ¼ df1 @x sk dt sk s Px mPx f2 2 m þ gP x f3 þ 2 f4 , sk ax ax df2 @H s ¼ df2 ¼ df2 f @y dt ð1sÞk 1 2y s bP y f2 r f 1 ð1sÞk ð1sÞk mPy Py ch þ 2 f3 2 m þ gP y f4 2 , qy ay ay
ðA:2Þ
ðA:3Þ
df3 @H 2P x m m ¼ df3 ¼ df3 þ bxf1 g 1 f f , @P x dt ax ax 3 ax 4 ðA:4Þ and 2P y df4 @H m m ¼ df4 ¼ df4 þ byf2 f3 g 1 f , @P y dt ay ay ay 4 ðA:5Þ with transversility conditions are given as fi tf ¼ 0 ði ¼ 1, 2, 3, 4Þ: Here tf is the final time for dynamic optimization.
ðA:6Þ
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Now @H ¼0 @h leads h¼
1 c p f2 : 2d qy
ðA:7Þ
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