Review of Decisions Taken at the Beginning of the Considered Horizon

Let us return to the constitution of a reserve discussed in section1.4. A disadvan-tage of the considered model is that the decisions are made, definitively, at the beginning of the time horizon even if some zones are effectively protected only from a certain period of this horizon. We discuss below the possibility of revising these decisions over time, taking into account possible changes in projected costs, budgets and capacities of zones to protect species. Suppose that in the solution to the problem in section 1.4, the list of zones to be protected at each period is defined by x_it¼~x_it,i2Z;t2T (~x_it is a constant which is 0 or 1). Let us also assume that we have arrived at the beginning of the periodT_jand that the forecasts for the coming periods are reviewed, including the available budget. Thus Z_kt becomes Z^kt; k2S;t2T;tj,Bt becomesB^t; t2T;tj, andcitbecomes^cit; i2ZR_j; t2T;tj, where R_j is the reserve already constituted and R_j the set of corre-sponding indices. The decisions that had been taken at the beginning of the horizon considered for the periods T_j,T_j+1,…,T_r can be abandoned and new optimal decisions can be sought, taking into account not only the new forecasts but also the reserve already constituted and the species that it allows to be protected. We are, therefore, in the case of establishing a reserve which must necessarily include certain

zones. One could also consider abandoning some zones, but this is not considered here (see chapter 12, section 12.3.3.2). One way to formulate the problem is to slightly modify program P_1.8: Constraints 1.8.3 are now to be taken into account only for tj and constraints 1.9.4 which stipulate that some zones have already been selected must be added. It is also necessary to set Z^kt¼Zkt; k2S;t2

1.6 Case Where Several Conservation Actions are Conceivable in Each Zone

1.6.1 The Problem

This section considers a slightly different case from the ones studied in the previous sections insofar as several different protection actions are possible for each zone.

Thus, for each candidate zone, a decision can be made to protect it or not to protect it, but if it is decided to protect it, several protection actions can be considered. We present below an example of reserve selection relevant to this issue and developed drawing on the references (Cattarinoet al.,2015; Salgado-Rojaset al.,2020). In this example it is considered that a species present in a given zone is exposed to different threats and that, if the protection of that zone is decided, different actions can be taken to remove all or part of these threats. An optimal reserve is a reserve that maximizes an “overall ecological benefit” for the species considered within an available budget. This benefit takes into account both protected zones and actions taken in these zones to eliminate certain threats.

Let S= {s₁,s₂,…,s_m} be the set of species, animal or plant, in which we are interested andZ= {z₁,z₂,…,z_n} be the set of zones that we can decide whether or not to protect.S_irefers to the set of species present in zonez_i. In addition, there are a number of threats, M= {µ1,µ2,…,µg}, affecting these species. We denote by M_ik ðMÞthe set of threats affecting speciess_kin zonez_iandM_ithe set of threats to be

Basic Problem and Variants 17

considered in zonez_i. We have thusMi¼ [k:s_k2SiMik. The protection of zonez_icosts c_iand the elimination of threatµjin zonez_icostsd_ij. As we have said, the protection strategy has two levels. It is defined by the set of zones that it has decided to protect and, for each of these zones, by the set of threats that it has decided to eliminate. For a speciess_kliving in zonez_iof the reserve, we consider that the degree of protection ofs_kinz_iis equal to the ratio between the number of eliminated threats weighing on s_kinz_iand the total number of threats weighing ons_kinz_i. The degree of protection of a speciess_kpresent in a protected zonez_iwhere it is not threatened is equal to 1 for that zone. The degree of protection of a species s_kin an unprotected zonez_i is equal to 0. We denote by w_ikthe square of the degree of protection of speciess_k in zonez_i. A we have seen, the value of this variable results from the strategy adopted for zone z_i: not protecting it or protecting it and eliminating a number of threats.

The problem is to determine the optimal strategy given the available budget. The value of a strategy is measured by the sum of the squares of the degrees of protec-tion,w_ik, for all pairs (z_i,s_k) wherez_iis a candidate zone ands_kis a species present in take the value 1 if and only if we decide to eliminate the threatµjfrom zonez_i. The problem considered can be formulated as program P_1.10.

P_1:10:

Since variablew_ikrepresents the square of the degree of protection of speciess_kin zonez_i, the economic function represents the sum, for all pairs (z_i,s_k) wheres_kis a species present in zonez_i, of the square of the degree of protection of speciess_k in zonez_i. If zonez_iis not selected–x_i = 0–then, due to constraints 1.10.3,w_ik= 0 for all the species living in this zone. If zonez_iis selected– x_i = 1– then two cases are possible: (1) species s_k is not threatened in this zone – jMikj ¼0 – and w_ik= 1 because of constraints 1.10.3 and the economic function to be maximized, (2) species s_kis threatened in this zone–jMikj[0–and because of constraints 1.10.2 and the

economic function to be maximized w_ik¼ P

j2M_iky_ij=jM_ikj

2

. The economic func-tion, therefore, expresses well the sum of the squares of the degrees of protecfunc-tion,w_ik, for all pairs (z_i,s_k) wherez_iis a protected zone ands_k, a species present in this zone.

Constraint 1.10.1 is the budget constraint and constraints 1.10.4 and 1.10.5 specify the Boolean nature of variablesx_iandy_ij.

1.6.3 Example

Consider the instance described in figure1.3(20 zones and 15 species). The optimal protection strategies are given in table1.2when the available budget is 25 units.

z

z

…,s15,living in these zones are concerned. For each zone, the species present and the threats associated, with their removal costs in brackets are indicated. The cost of protecting the white zones is equal to 1 unit, the cost of protecting the light grey zones is equal to 2 units and the cost of protecting the dark grey zones is equal to 4 units. For example, speciess7ands15are present in zonez₂₀, threatsµ2andµ9affect speciess₇in this zone and there are no threats to speciess₁₅. The cost of protecting this zone is equal to 4 units and the cost of removing threats µ2andµ9in this zone is equal to 7 and 6 units, respectively.

Basic Problem and Variants 19