Reserve Design to Maximize Species Persistence

Abstract

We develop a reserve design strategy to maximize the probability of species persistence predicted by a stochastic, individual-based, metapopulation model. Because the population model does not fit exact optimization procedures, our strategy involves deriving promising solutions from theory, obtaining promising solutions from a simulation optimization heuristic, and determining the best of the promising solutions using a multiple-comparison statistical test. We use the strategy to address a problem of allocating limited resources to new and existing reserves. The best reserve design depends on emigration, dispersal mortality, and probabilities of movement between reserves. When movement probabilities are symmetric, the best design is to expand a subset of reserves to equal size to exhaust the habitat budget. When movement probabilities are not symmetric, the best design does not expand reserves to equal size and is strongly affected by movement probabilities and emigration rates. We use commercial simulation software to obtain our results.

Appendix

This Appendix gives Kuhn–Tucker necessary conditions for optimality for a special case of the reserve design problem (Eqs. 1–4) in which there is no dispersal between patches. In this version of the problem, the objective is to minimize extinction risk, which depends on the sizes of habitat patches. We do not specify a function form for the extinction risk function and only require that the risk function be strictly positive with strictly negative slope. Given these assumptions, it is possible to derive a rule to obtain solutions that satisfy all but one of the Kuhn–Tucker conditions. Because we can obtain many solutions that satisfy this rule, we describe a heuristic to find solutions among them that are likely to satisfy the final Kuhn–Tucker condition. We call the solutions found in this way “solutions suggested by theory,” and we use them to initialize the simulation optimization of more general problems with dispersal.

Suppose there exists an extinction risk function p such that 1 − P(y ₁, y ₂, ... y _n )=$\prod\limits_{i = 1}^n {p\left( {y_i } \right)}$. Furthermore, p(·)>0 and p′(·) exists and is strictly negative. These assumptions will be satisfied when the patches are independent and identical and p(y _i ) is the probability of extinction in patch i over the time horizon. Given independence of the fate of the individual patches, the product form of the objective function gives the probability of metapopulation extinction. Note that the absence of dispersal is required. Under these assumptions and when d _i  = ∞ and c _i  = 1, the reserve design problem (Eqs. 1–4) is equivalent to:

$$ \min \sum\limits_{i = 1}^n {{\text{ln}}\left( {p\left( {y_i } \right)} \right)} $$

(A1)

subject to:

$${\sum\limits_{i = 1}^n {y_{i} - B \leqslant 0} }$$

(A2)

$$a_{i} - y_{i} \leqslant 0\quad i = 1, \ldots ,n$$

(A3)

Eq. A1 comes from applying a monotonic transformation ln(·) to the objective $\min {\prod\limits_{i = 1}^n {p{\left( {y_{i} } \right)}} }$. Eq. A2 comes from substituting Eq. 2 into Eq. 3 and defining $B = b + {\sum\nolimits_{i = 1}^n {a_{i} } }$. Eq. A3 comes from substituting Eq. 2 into Eq. 4 with d _i  = ∞.

Defining y = (y ₁,...,y _n ) and γ = (γ ₁,...,γ _n ), the Lagrangian function for Eqs. A1–A3 is $L\left( {{\mathbf{y}},\lambda ,{\mathbf{\gamma }}} \right) = \sum\limits_{i = 1}^n {\ln \left( {p\left( {y_i } \right)} \right)} + \lambda \left( {\sum\limits_{i = 1}^n {y_i - B} } \right) + \sum\limits_{i = 1}^n {\gamma _i \left( {a_i - y_i } \right)}$, where λ and γ are the dual variables corresponding to the budget constraints and the existing reserve size constraints, respectively. Because Eqs. A2 and A3 are linear, the Kuhn–Tucker constraint qualification holds, and the Kuhn-Tucker conditions for Eqs. A1–A3 are necessary conditions for optimality. These conditions are:

$$\frac{{p\prime {\left( {y_{i} } \right)}}}{{p{\left( {y_{i} } \right)}}} + \lambda - \gamma _{i} = 0\quad \quad i = 1, \ldots ,n$$

(A4)

$$ {\left( {{\sum\limits_{i = 1}^n {y_{i} - B} }} \right)}\lambda = 0 $$

(A5)

$${\left( {a_{i} - y_{i} } \right)}y_{i} = 0\quad \quad i = 1, \ldots ,n$$

(A6)

$$ {\sum\limits_{i = 1}^n {y_{i} - B \leqslant 0} } $$

(A7)

$$a_{i} - y_{i} \leqslant 0\quad \quad i = 1, \ldots ,n$$

(A8)

$$ \lambda \geqslant 0 $$

(A9)

$$\gamma _{i} \geqslant 0\quad \quad i = 1, \ldots ,n$$

(A10)

Eq. A4 comes from setting the partial derivatives of the Lagrangian function to 0. Eqs. A5 and A6 comprise the complementary slackness condition; each constraint must either be binding at optimum or its corresponding dual variable must be zero. Eqs. A7 and A8 give primal feasibility, and Eqs. A9 and A10 state that the duals must be positive at optimum.

We cannot derive sets of (y,λ,γ) that satisfy Eqs. A4–A10 without specifying the functional form of p(·). We do not specify the form of the extinction risk function because we use a stochastic simulation model to determine extinction risk. Nevertheless, we can derive (y,λ,γ) that satisfy Eqs. A4–A9 and examine them to see if they are likely to satisfy Eq. A10 and be Kuhn–Tucker points.

To find a solution (y,λ,γ) to Eqs. A4–A9, select an arbitrary set of indices A∈{1, 2, ... n}. These are indices corresponding to the patches whose size will be augmented. Let I be the set of indices corresponding to patches that are left at their initial value. Now consider the point y where:

$$\begin{array}{*{20}c} {y_{i} = a_{i} }{i \in I} \\ {y_{j} = \frac{{B - {\sum\nolimits_{i \in l} {a_{i} } }}}{{{\left| A \right|}}}}{j \in A} \\ \end{array} $$

(A11)

where ${\left| A \right|}$ is the number of elements in A. Notice that the above solution corresponds to leaving the patches with subscripts in I equal to their initial values, and dividing the remaining resources among the patches with subscripts in A. The numerator in Eq. A11 represents the resources remaining after allocation to the patches in set I. The set A must be selected so that:

$$\frac{{B - {\sum\nolimits_{i \in I} {a_{i} } }}}{{{\left| A \right|}}} > a_{j} \quad \forall j \in A$$

(A12)

If this is not the case, the set of augmented patches A is simply too large relative to the budget B, and dividing the resources among the set does not suffice to bring the common resulting size up to the largest initial size of the patches corresponding to the set A. In this case, removing one or more indices from A will suffice to generate a set A such that Eq. A12 is satisfied. Note that there will always be some sets A that satisfy Eq. A12 because any set of size 1 will work.

Theorem A1

Given the problem described by Eq. A1 – A3 , a set of indices A satisfying Eq. A12 , and a solution y defined by Eq. A11 , there exist λ and γ such that Eq. A4 – A9 are satisfied.

Proof

$$ {\sum\nolimits_{k \in I \cup A} {y_{k} } } = {\sum\nolimits_{i \in I} {a_{i} } } + {\left| A \right|}{\left( {\frac{{B - {\sum\nolimits_{i \in I} {a_{i} } }}} {{{\left| A \right|}}}} \right)} = B $$

so Eqs. A5 and A7 are satisfied. Equation A8 follows from Eq. A12. Note that p′(y _k ) / p′(y _k ) < 0 for all k ∈ I ∪ A because we assumed that p′(•) > 0, and p′(•) < 0. Furthermore, p′(y _j ) / p(y _j ) is equal for all j ∈ A because y _j is equal for all j ∈ A by Eq. A11. Define

$$\begin{array}{*{20}c} {\lambda = \frac{{ - p\prime {\left( {y_{j} } \right)}}}{{p{\left( {y_{j} } \right)}}}}{j \in A} \\ {\gamma _{j} = 0}{j \in A} \\ {\gamma _{i} = \frac{{p\prime {\left( {y_{i} } \right)}}}{{p{\left( {y_{i} } \right)}}} + \lambda }{i \in I} \\ \end{array} $$

With these definitions, Eqs. A4, A6, and A9 hold. □

Using Eq. A11, we can find solutions (y,λ,γ) that satisfy necessary conditions (Eqs. A4–A9) by selecting a subset of reserves and expanding them to equal size to exhaust the habitat area budget. There are many ways to select the subset of reserves to be expanded and hence a large number of solutions satisfying Eqs. A4–A9. To reduce the number of solutions evaluated, we order the reserves according to initial patch size and augment a subset of consecutive reserves from this ranking. Specifically, we set upper (u) and lower (l) bounds on initial patch sizes, and augment those patches between l and u until they are size u (see body of paper). By varying l and setting u to exhaust budget, we construct a set of “solutions suggested by theory,” all of which satisfy Eq. A11 and hence Eqs. A4–A9. Setting u so as not to overextend the budget guarantees that Eq. A12 will be satisfied.

The focus on a subset of consecutive reserves is heuristic and motivated by the likelihood (and experimental observation) that ln(p(y)) will have only one steep area, where marginal returns to reserve expansion are greatest. The location of this steep area will be a function of problem parameter values, so we use all possible lower bounds l to generate our set of promising points. One or more of these points is likely to have a good match between the augmented patch sizes and the steep area of ln(p(y)) and hence satisfy:

$$\frac{{p\prime {\left( {y_{i} } \right)}}}{{p{\left( {y_{i} } \right)}}} > \frac{{p\prime {\left( {y_{j} } \right)}}}{{p{\left( {y_{j} } \right)}}}\quad \quad \forall i \in I,j \in A$$

(A13)

Intuitively, when p′(y _j ) / p(y _j ) is large in absolute value (highly negative), the marginal impact on the objective function of augmenting patch j is large. If the point is selected so that the marginal impact of augmenting the chosen patches is larger than those left at their initial values, then the final Kuhn–Tucker condition, Eq. A10, will be satisfied.

It is important to note that, without additional assumptions on the function form of p(·), we cannot prove that the solutions suggested by theory contain all possible solutions to the Kuhn–Tucker Eqs. A4–A10. Furthermore, our application includes dispersal of individuals between habitat patches, which violates one of the assumptions on which Theorem A1 is based. Therefore, although the set of solutions suggested by theory is a convenient and promising place to start the simulation optimization, there is no guarantee that the true optimal solution lies within this set.