Present address: World Bank, East Asia and Pacific Region, Rural Development and Environment.
When does spatial structure matter in models of wildlife harvesting?
Article first published online: 18 OCT 2007
© 2007 The Authors
Journal of Applied Ecology
Volume 45, Issue 1, pages 63–71, February 2008
How to Cite
Ling, S. and Milner-Gulland, E. J. (2008), When does spatial structure matter in models of wildlife harvesting?. Journal of Applied Ecology, 45: 63–71. doi: 10.1111/j.1365-2664.2007.01391.x
- Issue published online: 18 OCT 2007
- Article first published online: 18 OCT 2007
- Received 13 December 2006; accepted 11 July 2007; Handling Editor: Philip Stephens
- bushmeat hunting;
- multispecies harvesting;
- open access;
- Schaefer model;
- sole owner
- 1The most broadly applied generalizations of bioeconomics derive from simple, non-spatial models. We develop a simple continuous spatial model in which harvesting costs are broken down into travel costs to a location and capture costs at that location within a harvested region.
- 2This is used to determine the conditions under which the spatial behaviour of harvesters is important and its implications, particularly for the assessment of maximum achievable sustainable yield (MASY) and the optimal management of exploited animal populations.
- 3The model suggests that, as a rule of thumb for harvested systems where density dependence is essentially a local process and harvesters act as independent agents, spatial structure will cause a significant impact on the dynamics and reduction in productivity of the system where the ratio of maximum travel cost to minimum capture cost is around 5 or more, and a very large impact where that ratio is around 10 or more.
- 4Also, in a spatial harvesting system, secure ownership rights will not result in socially efficient harvesting if the owner only exercises control over overall offtake levels. Efficient management requires control of the spatial pattern of harvesting as well.
- 5Synthesis and applications. These findings apply in cases where the spatial scale over which density dependence acts on the exploited population is limited in relation to the extent of the potential harvesting area, and where harvesters act as independent agents, choosing individually where to harvest. In most systems these conditions will not apply perfectly, and in many cases they may be seriously violated, but the rule of thumb above may be used quickly to establish whether spatial structure should warrant further investigation. In managed systems where harvesting decisions are regulated, the model may still be used to understand the gains produced by management.
Harvesting of wild animal populations often takes place in remote areas, where travel costs may make up a large proportion of overall hunting cost. Indeed, in many countries difficulty of access is typically a prime determinant of the persistence of populations of the large-bodied species preferred by hunters (Robinson & Bennett 2000). In addition to historical data on the geographical spread of the whaling and fur industries (Ponting 1991), a number of recent studies have noted strong spatial patterns of hunting and harvesting in developing countries, indicating that travel costs are often significant. Harvesting intensity is found to be much higher and/or exploited population densities much lower, closer to settlements and major transport routes in studies of hunting in tropical forests (Wilkie 1989; Wilkie & Curran 1991; Siren et al. 2004; Franzen 2006; Peres & Nasciemento 2006), artisanal fisheries (Petrere 1986; Cabrera & Defeo 2001), and non-timber forest product harvesting (Nantel et al. 1996; Ntamag 1997; Abbott & Mace 1999). Harvesters often cite travel costs as being significant components of their overall costs (e.g. Mendelson et al. 2003).
Harvesting models, by contrast, are rarely structured spatially, whether they are focused on the dynamics of exploited populations or on broader bioeconomic processes. Where space is included in models, it is introduced typically through biological rather than economic processes, most often through consideration of spatially structured metapopulations (e.g. Lundberg & Jonzen 1999). Two authors have previously provided spatial generalizations of the Schaefer model, but both focused on the impact of space on population dynamics, rather than on the costs of harvesting. Sanchirico & Wilen (1999) used a metapopulation structure, and Neubert (2003) looked at the effect of diffusive movement and an absorbing (fatal) boundary condition on the yield-maximizing harvesting strategy within a continuous linear model. Models examining the landscape economics of harvesting and explicitly including transport costs (e.g. Stone 1998a,b) do not typically include the dynamics of the exploited population and are not therefore bioeconomic. Clayton et al. (1997) and Keeling et al. (1999) provide an exception, but these papers make restrictive assumptions about the market structure and geography of the system and hence are not easily generalizable.
The divergence between empirical observations implying the importance of space and the brevity of its theoretical treatment is due perhaps to the fact that much of harvesting theory originates in commercial fisheries science (Gordon 1954; Schaefer 1954), where spatial structure has figured highly only recently with the explosion of interest in marine reserves as a management option (Gell & Roberts 2003). Even then, spatial structure tends to be relatively abstract. Typically, two-area models are used, and generally the focus is on effects on stock size within the reserve and not on economic components such as the likely effects of area closures on the behaviour of those fishing (Willis et al. 2003; Hilborn et al. 2004; Stefansson & Rosenberg 2005).
Most wild harvested systems are open access by default, and the most common use of simple harvesting models in terrestrial systems such as bushmeat harvesting is to assess whether actual offtake, or that expected in the absence of regulation, exceeds the sustainable yield. This typically involves the use of simple sustainability indices which provide some measure of the maximum sustainable yield (MSY) from an exploited population (Robinson & Redford 1991). The foremost policy insight from renewable resource harvesting theory is that over-harvesting is caused by open access, and economically efficient harvesting can be stimulated by sole-owner control of hunting levels. These common tenets, however, do not take the spatial structure of harvesting into account.
In this paper, we use a simple conceptual bioeconomic model to illustrate when space might be an important consideration in the harvesting of animal populations, and the impact that this has on expected yields and economic efficiency. Although the description of the model is couched in terms of terrestrial hunting, the conclusions are relevant to any harvested system where the conditions of high travel costs and locally acting density dependence apply.
the spatial harvesting model
This is an equilibrium model relying on simple algebraic functions (see Table 1 for parameter definitions). Hunters originate from a single point at one end of a linear territory of length λ and width w. The length and width of the area could be determined by any limiting constraint (such as maximum trip time, boundaries of permitted hunting zones, habitat type), and λ is the maximum potential distance travelled rather than the actual observed distance travelled. Hunting costs include the cost of travelling to and from the hunting location and the costs of capturing the prey once there. The capture cost may include both the time involved and the cost of any gear that is used, such as a cartridge or a snare. We make the standard assumption that the capture cost is inversely proportional to the local population density of prey (i.e. search costs rather than handling costs dominate capture cost). It is also assumed that on each trip a hunter travels from the origin to a single hunting location, where a fixed number of prey is captured (which for simplicity is set at 1 unit of prey), before returning to the origin. By assuming that per trip return is constant, we are able to optimize a single variable – trip cost or total revenues, depending on the situation – rather than maximizing rates of return on investment. Although this simplifies the model substantially it does not reduce significantly the generality of the approach (see discussion of diminishing returns in Table 2).
|Cost of travelling one unit of distance||T|
|Unit cost of capture effort||L|
|Average hunting cost||C|
|Cost of capturing a unit of prey at density n = 1||H|
|Total capture effort expended by hunters||E|
|Total population size||N|
|Total carrying capacity||K|
|Length of the hunted area||λ|
|Width of the hunted area||w|
|Intrinsic prey population growth rate||r|
|Carrying capacity density||k|
|Spatially varying parameters|
|Equilibrium rate of prey offtake per unit area||q(x)|
|Population density of prey at location x||n(x)|
|Capture effort expended by hunters at location x||e(x)|
|Cost of obtaining unit prey from location x||c(x)|
|Factor||Occurrence||Impact on spatial bias|
|Resource mobility||Random movement of prey may homogenize densities. Net directed movement may be into depleted regions or away from areas where harvesting is intense||Undermines the local impact of density dependence and therefore spatial structure, if passive or active net movement is counter to the stock density gradient. If prey respond to hunters such that movement is predominantly away from heavily hunted areas; however, spatial structure may be reinforced. In either case the effect could be considerable|
|Local extinction||Strong spatially structured harvesting patterns will depress prey densities close to settlements and markets to very low levels, at which they may become vulnerable to local extirpation||Local extinction of nearby populations will increase spatial bias, decrease overall supply, and may also increase the pressure on more remote populations|
|Non-linear geometry of harvesting region||Depending on the configuration of the points of origin of harvesters and the harvesting region, the width of the harvesting region might increase or decrease with distance||Predictable impacts on supply at increasing levels of cost, and therefore mean harvesting distance. Will not significantly change the difference between spatial and non-spatial MASYs; however, as these occur when the distribution of harvesting effort is already fairly even|
|Diminishing returns during a single hunt||Typically assumed in central place foraging models due to local prey depletion, but could also be caused by lower marginal value of additional prey items (i.e. satiation of demand) or accelerating time costs due to logistical complexity or opportunity costs||Would tend to lead to an increase in average catch per trip with distance and shorter maximum distance at any given average cost, producing a more concave prey density profile and hence an increased spatial bias, but effects would be subtle and case-specific|
|Different harvest techniques at different distances||Different techniques may be more efficient at different prey densities, e.g. setting traps close to a village, but hunting with guns at greater distance where densities are higher||Similar effects to diminishing returns|
|Harvesting during travel||May be possible to take prey opportunistically while traveling to preferred hunting locations||Would result in a steeper prey density profile, and hence increased spatial bias. If harvesting always occurs while moving in a straight line from the origin, then the resulting prey density profile would be sigmoidal (Ling 2004; Appendix 3–3)|
At any distance x from the hunters’ origin along this territory, and in the absence of dispersal, the equilibrium rate of prey offtake per unit area q(x) must be equal to the population growth rate per unit area at that point:
- (eqn 1)
where r is the intrinsic prey population growth rate, n(x) is the local population density, and k is the carrying capacity density, which for simplicity is assumed constant throughout the territory. Note, therefore, that density dependence is assumed to act locally, and for simplicity prey movement is not considered.
The cost of a harvesting trip to a location at distance x from the origin is split into two components – the spatial cost, which is incurred by travelling to the hunting location (xT) and the non-spatial cost, which is incurred by hunting at that location [H/n(x)]. Hence the unit cost of harvesting from a point at distance x is:
- (eqn 2)
where H is the cost of capturing one unit of prey at density n = 1 and T is the cost of travelling a unit distance.
To illustrate the importance of space, we examine outcomes under three different assumptions about hunter behaviour: (i) that they are individually free to choose where they hunt; (ii) that hunting effort is evenly distributed; and (iii) that they operate according to the dictates of a socially optimal harvesting strategy.
In open access systems, hunters would be expected to act as independent rational agents. Under the assumption that the revenue obtained from each hunt is the same, hunters would be expected to minimize the total cost of each trip. This implies that they choose locations so as to equalize the total cost of hunting across the whole territory, i.e. c(x) is equal to the average hunting cost C for 0 < x < λ. Hence equation 2 gives:
- (eqn 3)
At a particular location, the capture cost per unit of offtake is H/n(x) (eqn 2). Using the standard assumption from the Schaefer (1957) model that offtake is proportional to effort and population size, the capture cost is the effort expended in the location e(x) multiplied by the cost of one unit of effort L and divided by the amount of offtake at that location q(x). Rearranging, we obtain an expression for the amount of effort expended at a given location:
- (eqn 4)
In the simple case in which k is the same at all locations x, non-spatial harvesting models assume effectively that harvesting effort is also distributed evenly, i.e. that e(x) rather than c(x) is constant over the entire territory. In this case, average cost, C, is given by
- (eqn 5)
where Q is the total offtake and c(x)q(x) is the total cost of hunting at location x.
Assuming for simplicity that the discount rate δ is zero, a socially optimal harvesting strategy would seek to maximize offtake , and hence revenues, at any given level of total cost . It will therefore pay to harvest an additional unit of prey from x2 rather than x1 if the incremental cost at x2 is less than the incremental cost at x1, i.e. if:
- (eqn 6)
where c1 denotes c(x) at x1, etc.
The socially optimal harvester will therefore seek to equalize the marginal cost, (d(c(x)q(x)))/(dq(x)), over space rather than the average cost, c(x), which the independent harvester responds to.
Denoting marginal cost by F,
- (eqn 7)
Hence substituting equations 1 and 2:
- (eqn 8)
In all three cases, total offtake Q, effort E, population size N, and carrying capacity K are given by:
- (eqn 9)
We present the results in terms of the maximum achievable sustainable yield (MASY). In simple harvesting models, the MSY is purely a function of the biology of the prey species (rK/4 for the simple logistic model). However, the MASY is also a function of the institutional structure of the system, which determines the way in which individual harvesters distribute themselves in space. The MASY is not the same as the bioeconomic equilibrium of the system, which varies with the parameter values (harvesting costs and prices received), but is the maximum sustainable level of offtake that could be achieved for a given prey species within that institutional structure. In non-spatial models, MSY = MASY.
masy under independent and even-effort harvesting
Figure 1 illustrates profiles of prey population density, hunting effort and offtake along the length of the hunting territory for rational independent hunters, and compares it to the case when hunters are evenly distributed across space. The latter is the implicit assumption of non-spatial models, and the comparison is made to examine the influence of this assumption on the estimation of MASY in an open access system. The profiles represent harvesting at one particular cost point. As harvesting pressure, and so population depletion, increase, harvesters travel longer distances and encounter prey more slowly in any given location, leading to an increase in C. At higher values of C, however, the shapes of the curves for independent hunting remain the same; they are simply displaced further to the right along the distance axis. For instance, changes in C do not affect the shape of the prey density profile as the gradient of the curve (dn(x))/(dx) evaluated at any given value of n(x), v, is given by Tv2/H, and is therefore independent of C and x. The horizontal lines for the evenly distributed hunting case move up or down as C varies.
Figure 2 shows equilibrium average cost curves for both scenarios obtained by calculating Q over a range of C. Note the difference in magnitude of the maximum equilibrium (= sustainable) offtake for the two curves. The furthest point that the curve reaches to the right along the Q axis for the even-effort harvesting curve represents the standard (non-spatial) MSY for the whole population, and will always exceed the maximum sustainable offtake given by uneven harvesting (MASY). The difference between MSY and MASY is determined by the relative shape of the prey density profile for the independent hunting case at the point at which the MASY is achieved (the prey density profile for the even-effort hunting case at MASY is always the same, a horizontal line at n(x) = 0·5k, which is the non-spatial MSY level). The value of the growth rate parameter r is immaterial, as it affects both cases equally. The shape of the prey density profile is determined by the balance between capture and travel costs; the higher the cost of travel in comparison to capture costs, the steeper the curve towards the limit of the hunted region (Fig. 3a), and the greater the difference between the spatial (independent harvesting) and non-spatial (even-effort) cases. It can be shown that the shape of the curve at MASY depends only on the ratio of travel cost at the furthest point that could potentially be hunted, λ, i.e. the maximum travel cost, to the minimum capture cost, λT : H/k (see Supplementary material, Appendix 1).
The ratio of maximum travel cost to minimum capture cost, λT : H/k, is 5 in the example illustrated in Figs 1 and 2, which results in the non-spatial (even-effort) MSY overestimating the actual maximum yield (MASY) by nearly 29%. Figure 4a shows the percentage overestimate for different values of the ratio λT : H/k, when MSY is calculated without taking spatial structure into account, and Fig. 4b shows the effect of this ratio on the population size at which the MASY is obtained.
sole-owner and open-access supply in a spatial system
Here we compare the behaviour of independent and socially optimal harvesting patterns to examine the influence of ownership on supply in systems where space is important. Figure 3b illustrates prey density profiles under socially optimal spatial harvesting for various maximum travel cost to minimum capture cost ratios. As marginal cost is increased the curve shifts to the right along the distance axis, but retains the same shape, analogous to the independent hunting case under increasing average cost. As marginal cost becomes very large, the density profile tends to a horizontal line at n(x) = 0·5k, which maximizes production under the logistic growth function in equation 1. If δ = 0, no harvesting would be expected to occur if N fell below 0·5K, but if it did, then the optimal distribution would still be to harvest the population evenly across the hunting territory as no gains in productivity could be made from concentrating effort in one location at the expense of another. Figure 5a shows the average cost curves for independent and socially optimal harvesting when λT : H/k= 10, and 5b shows the equivalent marginal cost curves.
According to equations 1 and 2, removal of prey increases subsequent harvesting costs and affects the productivity of the remaining stock, i.e. it incurs social costs. Under open access harvesting, each individual maximizes their private gains, imposing externalities on all other harvesters of the same stock and leading to over-harvesting, a loss of profit from the resource and potentially even stock collapse (Gordon 1954). This is an example of the ‘tragedy of the commons’ (Hardin 1968). A sole owner of the stock, however, will internalize the costs of stock depletion and maximize the profits from the resource. Clark (1990) showed that open access harvesting will lead to a supply curve that coincides with the average cost curve for the stock in question (when supply price = average cost, no profits are earned), whereas the socially optimal supply would respond to stock marginal cost after adjustment for the appropriate discount rate (see also Ling 2004; Appendix 2–2). When actual or effective sole owners within a competitive market control the level of offtake from each individual stock or territory, they should also be expected to supply along the discounted marginal cost curve.
For the example illustrated in Fig. 5, an open access system would be expected to supply along curve 1. The introduction of a sole owner exercising control over overall offtake levels (but not over the spatial distribution of hunters) would be expected to shift supply from curve 1 to curve 3. This represents a social gain because the difference between the supply price and the unit cost of harvesting reflects profits earned from the resource, and because there is no incentive to harvest beyond the population size at which MASY is achieved. It does not, however, represent a Pareto optimum, because profits (and supply and consumer surplus) can be increased further by shifting supply to curve 4 when a sole owner controls the distribution of hunting effort as well as overall offtake levels. If the assumption that δ = 0 is relaxed, then the sole owner's supply curves lie somewhat closer to the open access average cost curve (Clark 1990). As δ tends to infinity, the sole owner would be expected to manage the stock as if it were subject to open access hunting regardless of whether or not the owner attempted to control the spatial distribution of harvesting. At more modest values of δ, however, the fundamental result, that sole owner control of overall offtake levels alone does not produce a social optimum, still holds.
The importance of space for the economics of renewable resource harvesting depends on the relative magnitudes of capture and travel costs within the harvesting area. Non-spatial estimates of MSY represent a theoretical maximum, and could overestimate significantly the MASY of open access systems. Especially in terrestrial settings, where the travel costs across harvesting territories may be significant, the MASY could be significantly lower than the MSY estimated using non-spatial models. The model used here suggests as a rule of thumb that if the ratio of maximum travel cost to minimum capture cost is less than 5, the overestimate bias will be relatively modest (< 30%) in comparison to the other sources of uncertainty typically inherent to these calculations. If the ratio is greater than 10, the effect is likely to be large, resulting in non-spatial MSY estimates being two or more times the actual value. Note that the maximum travel cost and minimum capture costs discussed are the imputed limits for the system, not maximum and minimum observed values.
The lower MASYs of spatially structured harvesting also correspond to lower equilibrium stock sizes. Where travel to capture cost ratios are very high, independent harvesting will greatly depress stock sizes if the stock is being maintained at its most productive level, which has important implications for cases of conservation interest. For example, Fig. 4b shows that when the maximum travel cost to minimum capture cost ratio is 10, the stock size at MASY is about 23% of K, rather than the 50% that would be expected if spatial structure were not considered. Stock density is particularly depressed close to settlements.
Introducing sole-owner control of overall offtake levels may be expected to address the familiar economic inefficiency associated with open access – over-harvesting. However, it will not deal with the potentially much larger inefficiency caused by spatial patterns of harvesting, unless the owner also regulates the distribution of harvesting effort. Control of overall offtake levels increases the probability of maintaining a yield close to the MASY, but it does nothing to counter the reduction in MASY caused by the spatial structure.
In a competitive industry, sole owners should automatically be incentivized to optimize spatial harvesting patterns as well as supply, but in reality few renewable resource markets are fully competitive. If a village manages its hunting grounds inefficiently, it is unlikely to sell or lease those grounds to a neighbouring village with improved harvesting techniques, even if both do supply the same market. Ensuring effective, consolidated ownership is an enabling measure for efficient natural resource management, but does not guarantee it. The development of community resource management should pay attention to the management of spatial patterns well as overall harvesting levels.
Crude regulation of the spatial pattern of harvesting already occurs in fisheries management, for example the International Council for the Exploration of the Sea (ICES), divides the sea into boxes, which are used by governments as the basis for fisheries management (ICES 2004). Zonation of harvest areas in wildlife management is also aimed at providing some spatial control of harvesting, but again in a relatively crude way (McCullough 1996). However, thus far there has been no formal consideration of the optimal distribution of harvesting that has taken travel cost into account in optimizing economic yields.
influence of assumptions and model structure
The model described above is purposefully very simple. A range of alternative and more complex functions could have been used to describe stock growth and harvesting costs, and all would have some influence on the shapes of the average and marginal cost curves obtained. Inclusion of fixed costs would increase values of C, shifting cost curves upwards, but would have no qualitative effect on the results. The assumption that capture cost is inversely proportional to prey density is based notionally on a random search and is common to many simple analytical hunting and fishing models, but is a very strong assumption at low prey density, where it produces accelerating hunting costs that are extremely sensitive to further prey reductions (Ling 2004). In central place foraging models, diminishing returns, due typically to local prey depletion within a single visit, limit the foraging time spent in any one location and result in the number of prey items collected per trip increasing with distance.
In general, however, the most critical assumption is that density dependence primarily acts locally in comparison to the maximum travel distances involved. If the resource is highly mobile, or if density dependence acts at a broad spatial scale rather than locally for other reasons, spatial effects of hunting will be undermined (Gerber et al. 2005). Observations of pronounced density gradients in wildlife with distance from villages (e.g. Marshall et al. et al. 2006) and of hunters travelling long distances (e.g. Clayton et al. 1997) suggest that there are many cases where the assumption has validity. It is most defensible where the harvested territory stretches over a large area, e.g. commercial markets being supplied at the landscape level, or where prey species are relatively sedentary, e.g. hunting of territorial duikers in Central/West Africa (Bousquet et al. 2001). There will be many cases where density dependence is not localized, however. If the distinction is unclear, then the rule of thumb can be applied anyway to assess whether the potential for strong spatial influence warrants further investigation. Table 2 summarizes the probable impacts of a number of processes that could alter spatial structure.
multispecies harvesting and space
A further potential complexity, common in the harvesting of wild animal populations, especially in regard to the exploitation of bushmeat from tropical forests, is the concurrent harvest of multiple species (Fa et al. 2006). Multispecies systems are under-represented in bioeconomic models and have never received a comprehensive treatment because of the complexity of potential biological and economic interactions. The economic essentials of multispecies harvests can be readily understood, however. Separate species within the same harvesting system have an indirect interaction through a subsidy effect (harvesting one species allows opportunistic harvest of another beyond the point of rarity at which this would otherwise become uneconomic), and a crowding-out effect (consumption of one species saturates and so reduces demand for the other). The bushmeat and conservation literature has focused traditionally on the former (negative) effect, i.e. bycatch and piggyback extinctions, rather than the latter (positive) effect. This is largely for two reasons:
- 1Authors pay little attention to demand, typically assuming perfect own-price elasticity (e.g. Clark 1990, chapter 10) either through convenience or oversight. Clayton et al. (1997) and Winterhalder & Lu (1997) present models with conflicting implications as to the effect of resilient prey species on the vulnerable because they make very different implicit assumptions regarding the elasticity of demand.
- 2The spatial aspect of harvesting is not considered explicitly, which may hide the crowding-out effect. For instance, a vulnerable species may be extirpated locally within an actively hunted region where more resilient species are exploited, only remaining within more distant areas. The vulnerable species appears to be a victim of the subsidy effect, but if resilient prey species were not meeting the demand of hunters, then the hunted region might extend further into the remaining refuges of the vulnerable species, extirpating it over a wider area. Taking the spatial distribution of harvesting as a given, rather than including it within the analysis, undervalues the importance of the crowding-out effect.
An explicitly bioeconomic and spatial approach to multispecies systems reveals the limitations of using simple models to generalize about the probable outcome of harvesting (see also Ling & Milner-Gulland 2006). In a similar manner, it has been shown that simple harvesting models capture inadequately the effects of harvesting on social species (Stephens et al. 2002). When spatial structure is likely to be important, simple models should be replaced by models which take space explicitly into consideration. Spatially explicit simulation models are being applied more frequently as computing limitations are relaxed. For example, Clayton et al. (1997) use a coupled map lattice model to represent landscape and prey dynamics, while Bousquet et al. (2001) use a multi-agent model to simulate hunter behaviour in a spatial system. In many harvested systems, particularly in marine settings, the assumption of local density dependence is violated and spatial structure is not of major significance, just as in many systems demand can be safely assumed to be perfectly elastic. It is judicious, however, to consider explicitly these simplifying assumptions before making them, and the rule of thumb which we present here aims to aid that judgement with regard to space.
S. L. was supported by a Beit Scientific Research Fellowship at Imperial College and a Leverhulme Study Abroad Studentship.
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Appendix S1. Relative shape of the prey density profile for independent hunting at MASY depends only on the ratio &lgr;T : H/k.
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