Designing connected nature reserve networks using a graph theory approach

Acta Ecologica Sinica 31 (2011) 235–240

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Acta Ecologica Sinica

journal homepage: www.elsevier .com/locate /chnaes

Designing connected nature reserve networks using a graph theory approach

Yi-cheng Wang a,⇑, Hayri Önal b

a College of Resources and Environment, Qingdao Agricultural University, Qingdao, Chinab Department of Agricultural and Consumer Economics, University of Illinois at Urbana-Champaign, Illinois, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 January 2011Revised 3 May 2011Accepted 9 June 2011

Keywords:Nature reserve designGraph theoryConnectedOptimizationIllinois

1872-2032/$ - see front matter � 2011 Ecological Socdoi:10.1016/j.chnaes.2011.06.001

⇑ Corresponding author. Tel.: +86 137 8065 7180.E-mail address: [email protected] (Y.-c. W

Habitat fragmentation has been cited as one of the critical reasons for biodiversity loss. Establishing con-nected nature reserve networks is an effective way to reduce habit fragmentation. However, theresources devoted to nature reserves have always been scarce. Therefore it is important to allocate ourscarce resources in an optimal way. The optimal design of a reserve network which is effective both eco-logically and economically has become an important research topic in the reserve design literature. Theproblem of optimal selection of a subset from a larger group of potential habitat sites is solved usingeither heuristic or formal optimization methods. The heuristic methods, although flexible and computa-tionally fast, can not guarantee the solution is optimal therefore may lead to scarce resources being usedin an ineffective way. The formal optimization methods, on the other hand, guarantees the solution isoptimal, but it has been argued that it would be difficult to model site selection process using optimiza-tion models, especially when spatial attributes of the reserve have to be taken into account. This paperpresents a linear integer programming model for the design of a minimal connected reserve networkusing a graph theory approach. A connected tree is determined corresponding to a connected reserve.Computational performance of the model is tested using datasets randomly generated by the softwareGAMS. Results show that the model can solve a connected reserve design problem which includes 100potential sites and 30 species in a reasonable period of time. As an empirical application, the model isapplied to the protection of endangered and threatened bird species in the Cache River basin area in Illi-nois, US. Two connected reserve networks are determined for 13 bird species.

� 2011 Ecological Society of China. Published by Elsevier B.V. All rights reserved.

1. Introduction

Establishing nature conservation reserves is considered aneffective and direct way to restore and maintain biodiversity, andthe issue of designing efficient nature reserves recently has at-tracted significant attention in the conservation biology literature[1–5]. The economic resources devoted to conservation effortshave always been scarce. This raises the question as how to allo-cate our scarce resources efficiently. Put it another way, the issueis how to select reserve sites from a larger set of potential sitesto establish a nature reserve which is efficient both economicallyand ecologically.

In the biological conservation literature, the issue of reserve siteselection was stated in one of the following two frameworks: (1) gi-ven a set of targeted species to protect, determine the least-cost sites(when land price is heterogeneous), or the minimum number of re-serve sites (when land price is homogeneous), that cover these spe-cies and (2) given a certain amount of conservation budget (ormaximum reserve area or number of reserve sites), determine the

iety of China. Published by Elsevie

ang).

maximum number of species that can be covered using this resource.These two problems are special cases of the prototypes ‘‘set coveringproblem (SCP)’’ and ‘‘maximal covering problem (MCP)’’ that wereintroduced in the operations research literature by Toregas et al.[6] and Church and ReVelle [7] and later were adapted to the reserveselection problem by other researchers [8–10].

Traditional SCP and MCP considered only species coverage con-straints while leaving out spatial configurations of the reserve.Such problems are easy to solve, but the selected reserve sitesare usually scattered across a large area and far from each other.The spatial configuration of a nature reserve, such as connected-ness, fragmentation, compactness, and reserve boundary, can becrucial for species dispersal and long term survival [11–14]. Takingspatial configuration into account in the site selection process hasattracted a significant number of studies in recent years [15]. How-ever, incorporating spatial configurations in the traditional SCP andMCP generally makes the problem much harder and sometimesleads to computational difficulties. Heuristic and formal optimiza-tion methods are usually employed to solve these problems. Heu-ristic methods, although flexible and computationally fast, cannotguarantee optimal solutions [16,17], meaning that scarce resourcesmay be allocated in an inefficient way. Formal optimizationmethods (specifically linear integer programming), on the other

r B.V. All rights reserved.

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hand, guarantees the solution is optimal but it cannot always findsuch a solution especially when the scale of the problem is large[18–20]. Therefore, improving our ability to formulate theseproblems and the model’s computational efficiency remainschallenging.

A graph theory approach was presented by Williams to deter-mine a minimal connected subgraph for land acquisition [21].His method can be adapted to address the nature reserve designproblem. To the author’s knowledge, that approach has not beenapplied in this area. Adapting that approach for the reserve selec-tion issue to design a minimal connected reserve is the major moti-vation of this paper. Exploring the computational performance ofthe modified model for larger reserve selection problems is an-other objective of this paper.

Fig. 1. Graphic representation of a connected reserve.

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6

Fig. 2. A disconnected reserve containing a cycle.

Fig. 3. Primal and dual graphs.

2. Method and model

2.1. Method

Suppose a potential conservation area is being considered fordevelopment as a reserve. It is assumed that the entire area ispartitioned into a set of geometric shapes, each of which is calleda site and considered as a decision making unit, i.e. a given site iseither selected and becomes part of the reserve or it is left out.Each site provides habit service to a known subset of targetspecies.

In order to use graph theory for reserve selection, a graph issuperimposed on the sites in the partition and an association isestablished between the partition and the graph [22,23]. In thisassociation, each site in the partition corresponds to a node in thegraph. Two sites are defined to be adjacent if they have a commonedge. An arc is defined for each pair of adjacent sites through whichnodes are linked. This allows us to view the partition as a graphformed by nodes and arcs, while the reserve can be viewed as a sub-graph formed by a set of selected nodes and corresponding arcs. Areserve is connected (or contiguous) if any two sites in it are con-nected by a chain of adjacent reserve sites that are also containedin the reserve. A connected graph can be employed to correspondto a connected reserve. From graph theory we know that the suffi-cient condition for a connected graph is that the number of arcsequals one less than the number of nodes and no cycle occurs (a cy-cle contains a number of nodes and the same number of arcs linkingthese nodes). A graph with these two properties, i.e. the number ofarcs equals one less than the number of nodes and no cycle occurs,is called a tree [24]. Therefore, the problem of selecting a minimalconnected reserve can be stated as determining a minimal networktree. Finally, although no orientation (flow direction) is actuallyneeded for reserve connectedness, for modeling reasons, we requirethat each arc in the graph is directed. Thus, for each pair of adjacentnodes i and j, we define an arc directed from node i (called origin) tonode j (called destination) and also an arc linking the same nodesbut in the reverse direction (namely it originates from node j andends at node i). Directed arcs are defined to satisfy the outflowrequirement for each node (except the sink node from which noarc directs) as will be explained in the model section.

An important feature of graphs is that associated with each pri-mal graph there is a dual graph [21]. Suppose a graph has n nodesand this graph is called primal graph. The nodes and edges of thisprimal graph partition the plane into a set r of zones, all but one ofwhich are enclosed (bounded) by primal edges joining the primalnodes in the graph. The dual graph is constructed by placing a dualnode in each of the zones, including the unbounded zone, and thenfor each edge in the primal graph, defining a dual edge that inter-sects with that primal edge and joins the two dual nodes separatedby that primal edge.

To illustrate the above concepts, consider Fig. 1–3. Fig. 1 depictsa 7 � 7 partition where 15 sites are selected to form a connectedreserve. The boundaries of the selected sites are shown with boldlines. Each solid dot represents a node in the graph while the ar-rows drawn from each site to an adjacent site represent arcs (noarc directs from node 31 because this node is arbitrarily specifiedas a sink node). This defines a graph where each site correspondsto a node and each arrow corresponds to an arc. The subgraphshown in the figure represents a fully connected reserve. Fig. 2illustrates a disconnected reserve where the number of arcs (whichis 12) equals one less the number of nodes (which is 13). This re-serve is disconnected because a cycle occurs. If no cycle occurs,the subgraph forms a connected reserve. Eliminating the cycles is

Y.-c. Wang, H. Önal / Acta Ecologica Sinica 31 (2011) 235–240 237

the main problem to satisfy the sufficient condition for a connectedreserve.

Fig. 3 depicts the primal and dual graphs. The primal graph has9 nodes (represented by black dots) and the dual graph has 5 nodes(represented by blank circles). The edges of primal and dual graphsare represented by solid lines and thin lines, respectively. Note thatthese edges form intersecting pairs, with one primal edge and onedual edge in each pair.

The special primal–dual structure of graphs described abovecan be used to eliminate cycles and enforce contiguity in trees (atree uses n � 1 edges to connect all n nodes of a graph). The modelwill generate both a tree in the primal graph and a second tree inthe dual graph. The two trees are complementary in the sense thatnone of the edges in the primal tree intersects any of the edges ofthe dual tree, and the two trees together form a partition of the setof intersecting edge pairs. Fig. 4 depicts this ‘‘interwoven’’ struc-ture where only the bold edges will be selected in the model whilethe value for the thin edges is zero.

2.2. The model

Given a set of target species and a set of potential reserve siteseach covering a known subset of those species, the following mod-el constructs a connected reserve network where the cost is mini-mized and the species coverage requirements are satisfied:

MinXn

i¼1

ciUi ð1Þ

such that:

Xn

i¼1

dsiUi � ks for all s 2 S ð2Þ

Xp

j¼1

Xij þXp

j¼1

Yij ¼ 1 for all primal nodes i ¼ 1; . . . ;n� 1 ð3Þ

Xd

l¼1

Zkl ¼ 1 for all dual nodes k ¼ 1; . . . ; r � 1 ð4Þ

Xij þ Yij þ Xji þ Yji þ Zkl þ Zlk ¼ 1for intersecting primal and dual arcs ð5Þ

Xij þ Xji 6 Ui

Xij þ Xji 6 Ujfor all primal arcsði; jÞ; i < j ð6Þ

Xp

j¼1

Xij 6 Ui for all primal nodes i ¼ 1; . . . ;n� 1 ð7Þ

Fig. 4. Interwoven structure of primal and dual graphs.

Xn

i¼1

Xp

j¼1

Xij ¼Xn

i¼1

Ui � 1 ð8Þ

In the above model: i, j are the indexes for primal cells (nodes); nthe number of primal cells; k, l the indexes for dual cells; r the num-ber of dual cells; s the index for individual species; S the set of spe-cies targeted to protect; dsi a parameter where dsi = 1 if species s ispresent at cell i, dsi = 0 otherwise; ks a user-specified parameterwhich represents the minimum number of reserve sites that con-tain species s. ci the land price of cell i. p and d are the number ofadjacent nodes of primal and dual node, respectively.

Ui is a binary variable where Ui = 1 if primal cell i is selected inthe reserve, and Ui = 0 otherwise; Xij is a binary variable whereXij = 1 if directed arc (i, j) in the primal graph is selected for the pri-mal tree and is also selected for the subtree (reserve), and Xij = 0otherwise; Yij is a binary variable where Yij = 1 if directed arc (i, j)in the primal graph is selected for the primal tree but is not se-lected for the subtree, and Yij = 0 otherwise; Zkl is a binary variablewhere Zkl = 1 if directed arc (k, l) in the dual graph is selected for thecomplementary dual tree, and Zkl = 0 otherwise.

The objective function (1) represents total cost of selected sites,which is to be minimized (when site prices are the same for allsites, ci can be dropped in which case the minimum number of se-lected sites will be determined). Constraint (2) is the species setcovering restraints, namely for each species s the reserve must in-clude at least ks sites which include that species (ks can be specifieddifferently for each species, 1 or 2 or more).

Constraint (3) says that for each cell i in the primal graph (ex-cept cell n which is designated as sink node), exactly one primalarc directed from it to an adjacent cell j must be selected for theprimal tree (outflow requirement). Furthermore, the selected pri-mal arc must either be part of the subtree, in which case Xij = 1,or not, in which case Ykl = 1. Constraint (4) is similar to (3) in thatit says that for each cell k in the dual graph (except the dual sinknode r), exactly one dual arc directed from it to an adjacent cell lmust be selected for the dual tree. Constraint (5) forces exactlyone arc to be selected from each set of intersecting primal and dualarcs. This constraint guarantees the complementarity of the primaland dual trees by ensuring that their respective edges do not inter-sect. This constraint, in conjunction with (3) and (4), prevent cyclesin the primal and dual trees. Together, constraints (3)–(5) createcomplementary spanning trees in the primal and dual graphs.

Constraints (6)–(8) create a subtree of the primal tree for the se-lected reserve. Constraint (6) stipulates that if one of the primalarcs Xij or Xji is selected for the subtree, then both of the incidentnodes must also be selected for the subtree (that is, cells i and jmust be selected for the reserve). Constraint (7) states that if celli is not selected (Ui = 0), then no arc is allowed to originate fromit (Xij = 0 for all j). Otherwise, i.e. if cell i is selected (Ui = 1), thenat most one arc can originate from it and direct to an adjacent cell.This constraint is not absolutely necessary in the above modelsince it does not enforce any extra condition not already enforcedby (3) and (6). However, as Williams argued, it is included becauseit was found to be effective in improving the model’s computa-tional efficiency. Constraint (8) equates the number of arcs in thesubtree to the number of nodes (selected sites) minus 1.

The model described above generates both a tree in the primalgraph and a second tree in the dual graph. Also, in the tree gener-ated in the primal graph, a minimum contiguous subtree whichcorresponds to the actual selected reserve is determined.

There are two important differences between the above modeland the original model developed by Williams (2002). The first isthe inclusion of the set covering constraint (2) which didn’t appearin Williams (2002) model. The second difference is the replace-ment of the right hand side of (8), which is fixed a priori as

20000

25000

econ

ds)

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(p � 1) in Williams (2002), by the endogenously determined num-ber of reserve sites that must be selected minus 1. Williams usedan exogenously given number, denoted by p, to represent the num-ber of sites that will be selected. In the above model, that numberis not known before the problem is solved.

0

5000

10000

15000

100 150 200 250 300 350 400Number of Sites

Proc

essi

ng T

ime

(s

Fig. 5. Changes of processing time as number of sites increases.

3. Experimental results

To test the computational performance of the model, experi-ments were carried out on a personal computer with Pentium 4processor and 1 GB RAM. CPLEX 9.0 incorporated in GAMS (GeneralAlgebraic Modeling System) 2.50 was used as the integer program-ming solver [25]. Several groups of hypothetical data sets including100 sites and 30 species were generated (See below).

In order to generate different reserve selection problems withdifferent habitat characteristics, the occurrence of 30 species overthe 100 sites is specified randomly using the built-in random num-ber generator of GAMS. More specifically, in each case, for an arbi-trarily specified maximum number (m) of species that can coexistin any given site, which is specified as a percentage of the totalnumber of species denoted by a (i.e. the number of species presentin each site can be at most m = 30 � a), a random integer is as-signed to each site as the number of species present in that site.Then the list of species coexisting in that site is determined by ran-dom selection from among the 30 species. In the first group of datasets, for instance, the capacity of each site is specified to be nomore than 80% of the total number of species (namely a = 0.8, orat most 24 different species can co-exist in any given site). In thesecond group of data sets, this capacity is reduced to70%, and soon, and eventually a is reduced down to 20% in the last group ofdata sets. It can be envisaged that as a goes down (i.e. species dis-tribution gets more sparse), more sites have to be selected in thereserve to get all species protected.

Also, in order to eliminate the effect of species distribution uponcomputational performance, for each group of data sets, speciesdistribution is generated randomly 20 times. For instance, in thefirst group of datasets where at most 24 species can coexist inone site, five species may occur in site 1 in the first data set; thenin the second data set, 10 species may occur in site 1, and so on. Itis specified that ci = 1 for all sites and each species must be pro-tected at least once (ks = 1). The model are solved for each of these20 data sets for each a value but with different species distribu-tions and average values for computational performance in 20 runsare reported (Table 1).

From Table 1 it can be seen that as a decreases, the objective va-lue increases, indicating that when species distribution goes moresparse, more sites have to be included in the reserve as expected.The average time needed to solve the model to exact optimal solu-tion also increases, from a little more than 1 s to nearly 93 s. This is

Table 1Computational results (number of sites = 100, number of species = 30).

a Objective valuea Time (s) Iterations Nodes

Mean s.d.b Mean s.d.b Mean s.d.b

0.8 2.60 1.1 0.9 3464 2821 47 530.7 3.05 1.7 1.6 5232 6869 70 1080.6 3.90 5.6 4.3 22743 19873 285 2640.5 4.65 9.5 6.6 42208 31302 532 4020.4 6.00 16.0 8.6 79933 51357 906 5030.3 7.90 40.5 29.8 217009 170693 2295 18160.2 11.20 92.7 73.0 513816 417168 5178 4297

a: species percentage, indicating maximum number of species that can coexist inone site.

a objective value is the average of the number of selected sites in 20 runs.b standard deviation.

because when more sites are included in the solution, the solverneeds to do more iterations and calculate more nodes in the branchand bound process. Standard deviations (s.d.) for solution time,iterations and nodes are also reported in Table 1. It can be seen thatthese standard deviations are generally large relative to the meanvalues. This is caused by data structures in these 20 runs. Datastructure can influence an IP model’s computational performancesignificantly but often in a very unpredictable way. For instance,when a = 0.2, the solver used the shortest time of 20 s to solvethe problem, while the longest time is nearly 300 s. Similar find-ings about data structure influence are also reported in other re-searches [19,22].

To further explore the model’s computational performance forlarger scale problems, more experiments are run for larger numberof potential sites, from 100 up to 400 with an increment of 50 sites.It is specified that each species has to be protected twice in the re-serve (ks = 2 for all s). Results are displayed in Fig. 5. The figure isnot intended to claim any functional form between computationalefficiency and the number of potential sites. Instead, they indicatethe significance of the number of variables in affecting the model’scomputational efficiency (as more sites are involved in the model,more variables are included). Three values of a are chosen: a = 0.4,0.5, and 0.6. These three a values are selected because when a = 0.7or 0.8 only a very short processing time is needed to solve theproblem, while for a = 0.2 or 0.3 the model generally does not re-trieve an optimal solution after running a long period of time (evenafter 20 h). Fig. 5 indicates that the processing time tends to in-crease as the number of potential sites increases and the increasebecomes more significant when a is reduced, i.e. when more sitesare needed to cover all species. For example, when a = 0.6 the in-crease shows a linear pattern in the processing time with regardto the number of sites while the solution time exhibits an exponen-tial pattern for a = 0.4,

4. An empirical application

4.1. The data

The US Government mandated all states to develop ‘‘Compre-hensive Wildlife Conservation Plans (CWCP)’’. Responding to thisrequirement, Illinois initiated a program in 1991 to compile andanalyze data on the State’s natural resources, ecosystems, andenvironment. A survey of the birds in Illinois was carried out overyears by the State’s Department of Natural Resources (IDNR) inwhich birds’ occurrence was recorded on a grid by grid basis. Thesize of each grid is about 6.4 km by 6.4 km (roughly about

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Fig. 7. Selected reserve (dark gray sites. ks = 1, i.e. each species protected at leastonce).

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

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29 30 31 32 33 34 35

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Fig. 8. Selected reserve (dark gray sites. ks = 2, i.e. each species protected at leasttwice).

Y.-c. Wang, H. Önal / Acta Ecologica Sinica 31 (2011) 235–240 239

41 km2).The state now has a recovering ecosystem, but some ani-mals particularly some birds are still listed as endangered or threa-tened (E/T). This application demonstrates the working of themodel using part of the data, sketching least-cost connectedconservation networks for these E/T bird species. 42(6 � 7) gridscorresponding to the boundary areas of the watersheds of Cache,Ohio, Saline, and Big Muddy in the south of the State were used(Fig. 6). These grids (sites) occupy an area of about 1722 km2. To-tally 13 E/T bird species are recorded in this area and they distrib-ute only in shaded sites. Among these 13 bird species, three areonly recorded in one site (three different sites), meaning that thesethree sites will have to be included in the reserve. The maximalnumber of sites in which a species occurs is 9, followed by 6. Eachsite contains only one to three species, mostly one, indicating thatthe distribution (dsi) of these bird species is sparse.

Land price (ci) of each site came from Illinois land transactionrecords, which included land area, total price, building asset,county, and town, etc. for each transaction. Most of these pricesare below $1.24 million per km2 ($5,000 per acre). It is assumedthat in practical reserve design natural and agricultural land siteswith relatively low price are potential sites, while industrial or re-sort sites with high price are usually out of considerations. There-fore to avoid the impact of especially high land transaction prices,the following entries were excluded from the records: transactionrecords for lands with an area less than 0.08 km2 (20 acres), trans-action records for lands with a building asset on them (prices forthese lands are hundreds of thousands or even millions of dollarsper acre), and transaction records for lands with a price higher than$1.24 million per km2. For each township, the average land pricewas calculated and used as land price of that township. Afterobtaining the land price for each township, a county map thatshows the township boundaries in that county was overlaid upona map of the same county which shows the locations of shadedsites in that county to determine the closest township to eachshaded site in Fig. 6. To determine the land price for each site, landprice of the nearest township was used. If a shaded site was in atownship for which a land price is not available, land price of thenearest township for which a land price is available was used.

4.2. The results

Two minimal connected reserve networks for these 13 bird spe-cies were determined (Figs. 7 and 8). In the first reserve network,each species was specified to be protected once (ks = 1), namelyeach species occurs in at least one site in the reserve. For the sec-ond network, each species occurs at least twice (where applicable)in the reserve (ks = 2). The cost for the first network was found tobe $223.8 million and for the second network the cost was$250.3 million.

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35

36 37 38 39 40 41 42

Fig. 6. Birds distribution (only occurred in gray sites).

In the first network, 15 sites were selected and they constituteda connected reserve. Note that three sites (site 9, 16 and 21) haveno bird species recorded but they were included in the network tomake the network connected. In the second network, the numberof selected sites increased to 18 and sites 9 and 16 were also in-cluded to make the network connected. It can also be seen thatby selecting three more sites with a cost increase of only about12% (from $223.8 million to $250.3 million), double coverage ofspecies was achieved.

A GAMS demon version was used to demonstrate the applicabil-ity of the model. The demon version has limits on both the numberof variables and the number of equations involved in the model(both are 300). That is why only 42 potential sites were includedin the application. Actually even with such a small scale, some siteshad to be deleted to make the problem size solvable. These sitesare unlikely to be selected because they are on the boundary areaand no birds are recorded there (such as sites 3–7). Due to thesmall scale of the problem, no computational difficulty wasencountered and in both scenarios (ks = 1 and ks = 2) the softwarereturned the optimal solution in less than one second. When morepotential sites are involved and therefore a GAMS full version hasto be used, computational difficulty is expected to be encounteredas tested in the experimental section.

5. Discussion

The model presented in this paper assumes the sites take regu-lar shapes such as squares and rectangles. This is because mostpractical data come in this way and this assumption also helpsmake the code writing process easier with the software GAMS.

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However, the method can also be used for sites of other shapessuch as triangles and even irregular shapes. As long as the adjacentrelationships of sites including primal and dual sites can be deter-mined, the model can be handled easily by the software.

The model can be guided to produce alternative solutions ifsuch solutions exist. For example, in the above empirical applica-tion, if site 1 in Fig. 8 cannot be selected due to some geographicor social or other reasons, U1 can be set as 0 then site 1 will notbe included in the selected reserve, and another site may be se-lected instead. This provides alternative solutions for decisionmakers to choose from.

The approach used here is deterministic in the sense that whena species is recorded in a site it is assumed that the species is pres-ent in that site with a probability of 1. Also, when a site is selectedit is assumed that all species in it are protected with certainty. Inreality, however, species exist in sites with a probability which isoften less than 1, and selecting a site can protect a species in itup to a certain degree of confidence which may not be 100%. Toincorporate uncertainty in an implicit way, several sites are re-quired to be selected to ensure multiple representation of eachspecies and improve the survival probability of targeted species.Incorporating survival probabilities in an explicit way, along witheconomic and spatial considerations, needs more studies andmay easily cause computational complexity. Dealing with theuncertainty in reserve design or conservation management is outof the scope of this article, but several recent studies explored thisissue and provide a good place to start with for further researches[26–29].

The connectivity model presented in this paper requires that allsites in the reserve must be connected to each other, which maynot always be necessary. In some situations, full connectivitymay be required for one group of species, but not for others. Siteselection for the latter group may be completely free of spatial con-siderations or a minimally fragmented reserve may be desirable.The models presented here can be modified to incorporate thistype of selection criteria. Briefly, the modified approach defines aseparate reserve network for these two groups of species and intro-duces new variables for each. Although this approach is essentiallythe same as the approach for designing one fully connected re-serve, the increased model size may severely limit the applicabilityof this approach in practical situations.

In most cases reserve networks are not established at one pointin time. Rather, they are developed gradually, as much as resourceconstraints and site availabilities allow, and sequentially over time.Thus, even if an optimal reserve network can be determined usinga mathematical model, this does not mean that the optimal solu-tion will be implemented immediately. Then the question becomesas which sites should be reserved at any point in time while stillincorporating spatial attributes such as connectivity which is ad-dressed here. Such dynamic considerations have been introducedrecently in the reserve site selection literature [30]. However, thefew existing studies did not incorporate both dynamic and spatialissues in a unified framework. This is an important and fertile re-search area that must be explored by future studies.

Finally, the successful application of the model in real reservedesign practices must rely on the complete and accurate datawhich are usually difficult to get. Available data often have theirown problems such as incomplete, out-of-date, and inaccurate.Some countries have collected a substantial amount of data withrespect to nature reserve design. The data situation in China maynot be as good in many aspects, therefore researchers, planners,conservationists, and other related institutions in China may needto work together to build a comprehensive, accurate, and up-to-date datasets for the optimal design of efficient nature reservesin China.

Acknowledgments

We thank the two anonymous reviewers for their helpful com-ments and suggestions. We also thank Dr. Douglas Austin, RobertGottfried, and Tara G. Kieninger of Illinois Department of NaturalResources, and Professor Bruce Sherrick of the University of Illinoisat Urbana-Champaign for providing the input data used in theempirical application.

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