2. patch size and connectivity

2.1 POTENTIAL CONTRIBUTION OF INDIVIDUAL PLANNING UNITS

C-Plan already generates a number of indices (irreplaceability etc) that measure the potential contribution of unreserved planning units to achievement of areal feature targets. These indices use information only on the occurrence of features within the unit of interest, without any consideration of the distribution of features in the surrounding area or the proximity of existing reserves. C-Plan now allows spatial context to be automatically factored into the derivation of any of the existing indices. C-Plan can adjust an index to reflect spatial context by first estimating the index for individual planning units in the normal manner and then transforming the value for each unit according to the values of other units in the surrounding area. The extent of the ‘surrounding area’ considered in these calculations is determined by a user-specified radius. The transformed index assigned to each planning unit is calculated as a distance and area weighted average of the values (e.g. summed irreplaceabilities) of units falling within a specified radius of the unit of interest. The contribution of each unit within this radius decreases with increasing distance from the unit of interest and also decreases with decreasing unit area. If only a part of a unit lies within the radius the contribution of that unit to the weighted average is appropriately adjusted (this adjustment accounts for much of the complexity in the calculations given below). Reserved planning units are allocated a user-specified constant value reflecting the weight that existing reserves should carry in the calculations. This allows the user to control the extent to which the spatial index will reflect proximity to existing reserves in addition to proximity to unreserved areas of high potential conservation value.

By transforming an irreplaceability index to reflect spatial context the index is no longer just a measure of the potential contribution of each planning unit to achievement of areal feature targets but is instead a measure of the extent to which the unit is close to other units of high potential value (i.e. part of a large patch of high-value forest) and/or existing reserves. Spatially transformed indices can be derived for any specified subset of features, e.g. a single fauna species, all space-demanding fauna species or all features combined (forest ecosystems, old growth and species). When C-Plan is asked to derive an index (e.g. summed irreplaceability) adjusted for spatial context, both the original index and the transformed index are generated and made available for mapping and use in selection rules.

Spatial data used in deriving the index are extracted from a special-purpose spatial data structure (see Section 4) containing the area of each unit and the distances (edge to edge) between all possible pairs of units in the region. The shape of units is not considered in the approach – all units are assumed to be circular. The index, Spat_contrib_index, is calculated for a given planning unit x as follows (see Figure 1 for an illustration of some of these parameters):

where

d _max = the specified radius (m) around the unit of interest, x
n = the number of planning units within the specified radius
Vi = the value of planning unit (in terms of the specified irreplaceability index)
C = a user specified constant (between 0 and 1) that determines how quickly the weight applied to surrounding units diminishes with increasing distance from the unit of interest

r_i = the radius of planning unit i (assuming a circular shape)
d_ix = the distance (edge to edge) between planning unit i and the unit of interest, x

^{Figure 1. Parameters used in transformation of irreplaceability indices to reflect spatial context.}

2.2 CURRENT CONFIGURATION OF RESERVE SYSTEM

C-Plan can now calculate a patch size/connectivity configuration index for the reserve system at any point in negotiations. The index measures the extent to which a reserve system consists of large, well connected blocks of forest as opposed to small, isolated fragments. The analytical approach employed is largely new, but incorporates ideas from incidence function modelling of metapopulations (Hanski 1994, 1997) and gravity modelling of spatial interaction processes (Sen and Smith 1995). An overall index can be calculated for the reserve system as a whole as well as separate indices for any individual features nominated by the user. For example, if six fauna species are nominated then a separate index will be calculated for each of the six species, with each index measuring the configuration of the reserve system in terms of patch size/connectivity of reserved habitat for the species concerned.

Two basic sets of data are used to calculate the index for a given feature:

• The area of the feature of interest contained within each reserved planning unit. The total area of each unit (and therefore the proportion of the unit occupied by the feature) is also known. For the overall index, the area of the ‘feature of interest’ is set equal to the total area of the unit.
  
• The straight line distance (or separation) between each pair of reserved planning units. For example, if there are three reserved units A, B and C then the data consist of distances A to B, A to C and B to C. Each distance is initially measured as the minimum straight line distance between the edges of the two units involved. If two planning units are immediate neighbours (i.e. they abut) the distance between them is zero.

Calculating the index for a given feature involves the following steps:

1. The distance between each pair of planning units is adjusted upwards to account for the distribution of the feature of interest within the two units involved. It is assumed that the units are circular in shape and that the known area of the feature in each unit is distributed randomly across the total extent of that unit. The adjusted distance is an estimate of the mean distance across unreserved and/or unsuitable land traversed in travelling along a straight line between any two randomly selected occurrences of the feature (one in each unit). This adjusted distance therefore reflects both the separation of the two units and gaps in the distribution of the feature of interest within each unit. An adjusted distance is also calculated between each unit and itself, which is an estimate of the mean distance across gaps in distribution encountered along a straight line connecting any two randomly selected occurrences of the feature (both in the unit of interest).

2. The adjusted distances from Step 1 are used to derive a ‘minimum spanning tree’ connecting all reserved planning units containing the feature of interest. A minimum spanning tree is a special type of linked graph consisting of nodes (planning units in this case) and connections (or branches) such that 1) all nodes have at least one connection, 2) there are no loops in the tree and 3) the summed length of branches is minimal (see Gower and Ross 1969 for details).

A simple example of a minimum spanning tree connecting eight planning units is given in Figure 2. The circles depict planning units, with Ai denoting the area of the feature of interest within planning unit i. The values on the branches of the tree are the adjusted distances between each pair of connected units (i.e. the units at either end of a given branch).

^{Figure 2. Simple example of a minimum spanning tree connecting eight reserved planning units.}

3. The minimum spanning tree derived in Step 2 is used to estimate the ‘effective distance’ between all possible pairs of planning units in the reserve system. The effective distance d_ij between units i and j is calculated as the sum of the lengths of all branches connecting these two units. For example, in Figure 2 the effective distance between Units 2 and 7 is 3.1 (0.2+0.9+1.2+0.8).

4. An index of reserve configuration in terms of patch size/connectivity is then calculated as:

where n = the number of reserved planning units
d_ij = the effective distance between planning units i and j
A_i = the area of the feature of interest in planning unit i
c = a constant parameter specified by the user

The parameter c determines how quickly the spatial interaction of units declines with increasing separation. When dealing with individual species this parameter should be an estimate of the mean dispersal distance of the species of interest (see Hanski 1994). C-Plan allows the user to specify a c value to be used in calculating the overall configuration index, as well as separate c values for individual features.

The configuration index described above ranges between zero and one. A value of one indicates a reserve system configured as one contiguous area, with no internal gaps. The value of the index decreases with increasing fragmentation of reserves. As can be seen in the above equation, the index is expressed (or scaled) as a proportion of the total current area of the reserve system. The index is therefore a measure of reserve shape, not area. C-Plan also uses this index to derive an indicative measure of configuration that combines both the shape and the total area of a reserve system (or the reserved area of a specific feature). This is achieved by simply multiplying the initial index by the total reserved area of a feature, i.e. cancelling out the denominator in the above equation.

The indices described above can be used to compare different reservation scenarios/options in terms of spatial configuration of reservation. This can be done in real-time during negotiations, allowing the impact of reservation decisions on spatial configuration to be progressively tracked throughout the process. Unless some target is set for the index (in relation to a feature) such comparisons, while extremely useful, will be relative only. In other words we can say something about whether reserve system A is better than reserve system B in terms of spatial configuration for a feature, but not anything about how good scenario A or B is in relation to a desired or optimum configuration, i.e. a target or objective.

Setting realistic targets for the patch size/connectivity index is not easy. It is theoretically possible to derive a target value from parameters indicating optimum patch size, minimum and maximum spacing between patches and average proportional occupancy of the feature within patches. However, such a target may not be achievable if sufficient patches satisfying these parameters do not exist within the region of interest. An alternative approach involves using CPlan to manually configure a hypothetical reserve system for each feature (e.g. fauna species), containing the exact area of habitat targeted for the feature and exhibiting a spatial configuration which experts feel is ‘optimum’ for the feature in terms of patch size/connectivity. The configuration index calculated for this optimum reserve system can then serve as a target or objective for the feature concerned.

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