## Introduction

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 *bio*economic. 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).

Parameter | Symbol |
---|---|

System-wide parameters | |

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 offtake | Q |

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 |

Discount rate | δ |

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:

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:

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.

#### Independent harvesting

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:

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:

#### Even effort

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

where *Q* is the total offtake and *c*(*x*)*q*(*x*) is the total cost of hunting at location *x*.

#### Optimal harvesting

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 *x*_{2} rather than *x*_{1} if the incremental cost at *x*_{2} is less than the incremental cost at *x*_{1}, i.e. if:

where *c*_{1} denotes *c*(*x*) at *x*_{1}, 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*,

Hence substituting equations 1 and 2:

In all three cases, total offtake *Q*, effort *E*, population size *N*, and carrying capacity *K* are given by: