Reserve with Central Part and Buffer Part

In this section, we consider reserves formed by two parts: a “central” part and a

“non-central”part. A zone of the reserve is said to be in the central part if it is“far enough”from the outside of the reserve. This central part is thus protected from the negative effects of activities outside the reserve. It can also simply, by being far enough from the outside of the reserve, ensures a certain climate in this part (heat, sunshine, humidity). The central part, which can be considered as the core of the DOI: 10.1051/978-2-7598-2521-9.c005

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reserve, therefore has no common border with the outside of the reserve. To be ecologically effective, taking into account the conservation objectives, the non-central part, called the buffer part, must be large enough. The distance that must be maintained between the part considered as central and the outside of the reserve depends on (1) the species that are to be protected in that central part, (2) the nature of the activities that take place outside the reserve, and (3) the ability of the buffer part to protect the central part from the effects of these activities. For example, the topography of the reserve can be taken into account. The buffer part completely surrounds the central part. The biodiversity protection within the buffer part itself may be relatively limited. On the other hand, this buffer part, which provides additional protection for the zones of the central part, may be fundamental for the protection of biodiversity in the central part. It should be noted that certain activities incompatible with the protection of biodiversity in the central part may be authorized in the buffer part.

Example 5.1. Let us consider a set of candidate zones represented by a grid of 1515 square and identical zones whose side lengths are equal to one unit.

Figure 5.1shows 2 reserves, defined on this grid, with a central part and a buffer part. In both cases, according to its definition, the central part never touches the outside of the reserve but, in case (b), the buffer part is larger than in case (a).

Indeed, in case (a) the smallest distance separating a point of the reserve from a point outside it is equal to one unit. In case (b), this distance is equal to two units.

The area of the reserve allocated to the buffer part depends, as we have said, on the desired level of protection for the central part but also on the size and shape of the reserve. Proportionally, the area of the buffer part is larger for a small and/or non-compact reserve than for a large and/or compact reserve. Figure5.2illustrates that for two relatively compact reserves, the buffer part is proportionally smaller in a large reserve than in a small one. In case (a), the area of the buffer part represents about 50% of the reserve while in case (b), it represents 60%.

(a) (b)

FIG. 5.1–Two examples of reserves including a central part and a buffer part. The zones in the central part are shown in black and those in the buffer part are shown in grey. In case (b), the buffer part is larger than in case (a).

From a species protection perspective, the zones in the central part of the reserve can be considered to be used to protect threatened species and the buffer part can be considered to only provide additional protection to the zones in the central part. It can also be considered that the characteristics of the buffer part are favourable to certain species and that these species are, therefore, protected if they live in the buffer part. These species can also generally be considered as protected if they live in the central part. In addition, as mentioned above, certain activities may be authorized in the buffer part, such as forest harvesting, environmentally friendly farming, and recreational activities.

5.2.1 Minimal Cost Protection of All the Considered Species

We examine here the selection of reserves with central parts and buffer parts. As in the previous chapters, we consider a set of candidate zones,Z= {z₁,z₂,…,z_n}, and we denote byZthe set of corresponding indices. To simplify the presentation, a zone of the reserve is considered to be in the central part if it is completely surrounded by other zones of the reserve. It should be noted that the model studied in the following could very easily be adapted to different and/or more elaborate definitions of the central part. We also consider a set of species to be protected, S= {s₁,s₂,…,s_m}.

This set is divided into two groups,S₁andS₂. To be protected, a species of the group S₁must occur in at least one protected zone belonging to the central part and a species of the groupS₂must occur in at least one protected zone belonging either to the central part or to the buffer part. We denote by S₁ and S₂ the set of indices corresponding to the sets of speciesS₁andS₂, respectively. As mentioned above, a selected zone is considered to be in the central part of the reserve if all the sur-rounding zones have also been selected, either in the central part or in the buffer

(a) (b)

FIG. 5.2 –Two examples of reserves of different sizes including a central part and a buffer part. The zones in the central part are shown in black and those in the buffer part are shown in grey. Proportionally, the buffer part in case (b) is larger than in case (a).

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part. Several problems may arise with regard to the protection of the species under consideration. The problem here is to determine a subset of zones, of minimal cost, that can protect all the species. For each zone, we know the list of the species present in that zone. If this zone is protected and is located in the central part, it is con-sidered to ensure the protection of all the species of this list; if this zone is protected and is located in the buffer part, it is considered to only ensure the protection of the species of the list belonging to the groupS₂. We denote byZ_kthe set of zones hosting speciess_k, andZ_k, the set of corresponding indices.

5.2.2 Mathematical Programming Formulation

Consider the Boolean variableti; i2Z, which is equal to 1 if and only if zonezi is selected and belongs to the central part of the reserve, and the Boolean variablexi, which is equal to 1 if and only if zonez_i is selected and belongs either to the buffer part or to the central part. Thus, variable xi is equal to 1 if and only if zonez_i is selected to form the reserve. It should be noted that the reserve may be made up of several separate “sub-reserves”. In this case, there will be several central parts and several buffer parts. Let ZCZ be the set of indices of the candidate zones for the central parts andI_iZ the set formed by the indexiand the set of indices of the neighbouring zones of zone z_i, and which must be selected, either in a central part or in a buffer part, to give zone z_ithe status of a zone belonging to a central part. The problem can be formulated as the linear program in Boolean variables P_5.1.

P5:1:

minP

i2Z

cixi

s.t: P

i2Z_kti1 k2S₁ ð5:1:1Þ j ti 2f0;1g i2ZC ð5:1:4Þ

P

i2Z_k

x_i1 k2S₂ ð5:1:2Þ j x_i 2f0;1g i2Z ð5:1:5Þ tixj i2ZC;j2I_i ð5:1:3Þ j

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The economic function expresses the total cost of the reserve. Constraints 5.1.1 express that, for each speciess_kof group S₁, at least one of the zones hosting that species must be, on the one hand, retained in the reserve and, on the other hand, located in the central part of that reserve. Constraints 5.1.2 express that, for each speciess_kof groupS₂, at least one of the zones hosting that species must be retained in the reserve,i.e., located either in the central part or in the buffer part. Constraints 5.1.3 express that if zonez_i is retained to constitute the central part of the reserve then all the surrounding zones,i.e., all zonesz_j,j2I_i, must also be retained in the reserve, either in the central part or in the buffer part. Constraints 5.1.4 and 5.1.5 specify the Boolean nature of variables t_iandx_i.

5.2.3 Example

Consider a set of 100 candidate zones for protection represented by a grid of 1010 square and identical zones (figure 5.3). It is considered here that a retained zone belongs to the central part of the reserve if and only if the 8 surrounding zones are also part of the reserve. This example concerns 10 species and figure5.3shows the names of the species that are hosted by each of the zones, and also the cost of protecting these zones.

The group of species S₁, i.e., those species which, in order to be protected, require to be present in at least one zone of the central part of the reserve, consists of the 3 speciess₁,s₂, ands₃; the groupS₂consists of the other 7 species,s₄,s₅,s₆,s₇, s₈,s₉, ands₁₀. Recall that the problem considered is to determine a subset of zones, of minimal cost, that allows all the species to be protected.

The optimal reserve is shown in figure5.4. Its cost is 113 units; it is made up of two parts, not connected, comprising a total of 24 zones, 3 of which are located in a central part. For example, zonez₉₉is in the central part because the 8 surrounding zones,z₈₈,z₈₉,z_8,10,z₉₈,z₉,₁₀,z_10,8,z_10,9, andz_10,10are part of the reserve.

FIG. 5.3–A set of 100 candidate zones for protection and, for each zone, the species that are present and the cost associated with the protection of the zone. For example, the zone at the intersection of row 8 and column 7 contains species s9 and s10, and the cost of its protection is 6.

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