Environmental Modeling & Assessment

, Volume 7, Issue 2, pp 107–114 | Cite as

Mathematical Methods for Spatially Cohesive Reserve Design

  • Mark D. McDonnell
  • Hugh P. Possingham
  • Ian R. Ball
  • Elizabeth A. Cousins


The problem of designing spatially cohesive nature reserve systems that meet biodiversity objectives is formulated as a nonlinear integer programming problem. The multiobjective function minimises a combination of boundary length, area and failed representation of the biological attributes we are trying to conserve. The task is to reserve a subset of sites that best meet this objective. We use data on the distribution of habitats in the Northern Territory, Australia, to show how simulated annealing and a greedy heuristic algorithm can be used to generate good solutions to such large reserve design problems, and to compare the effectiveness of these methods.

reserve design simulated annealing set covering problem spatial clustering fragmentation optimisation heuristics multiobjective optimisation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.A. McNeely, K.R. Miller, W.V. Reid, R.A. Mittermeier and T.B. Werner, Conserving the World's Biodiversity (IUCN Publication Services, Gland, 1990).Google Scholar
  2. [2]
    M.E. Soule, Conservation: Tactics for a constant crisis, Science 253 (1991) 744–750.Google Scholar
  3. [3]
    H. Possingham, I. Ball and S. Andelman, Mathematical methods for identifying representative reserve networks, in: Quantitative Methods for Conservation Biology, eds. S. Ferson and M. Burgman (Springer, New York, 2000) pp. 291–305.Google Scholar
  4. [4]
    H.P. Possingham, J.R. Day, M. Goldfinch and F. Salzborn, The mathematics of designing a network of protected areas for conservation, in: Decision Sciences: Tools for Today. Proceedings of 12th National ASOR Conference, eds. D.J. Sutton, C.E.M. Pearce and E.A. Cousins (ASOR, Adelaide, 1993) pp. 536–545.Google Scholar
  5. [5]
    H.A. Taha, Operations Research, An Introduction, 5th Ed. (MacMillan Publishing Company, 1992).Google Scholar
  6. [6]
    R.L. Pressey, H.P. Possingham and J.R. Day, Effectiveness of alternative heuristic algorithms for identifying indicative minimum requirements for conservation reserves, Biological Conservation 80 (1997) 207–219.Google Scholar
  7. [7]
    I. Ball, A. Smith, J.R. Day, R.L. Pressey and H.P. Possingham, Comparison of mathematical algorithms for the design of a reserve system for nature conservation: An application of genetic algorithms and simulated annealing, Journal of Environmental Management (1998), in press.Google Scholar
  8. [8]
    R.L. Pressey, H.P. Possingham and C.R. Margules, Optimality in reserve selection algorithms: When does it matter and how much?, Biological Conservation 76 (1996) 259–267.Google Scholar
  9. [9]
    W.F. Fagan, R.S. Cantrell and C. Cosner, How habitat edges change species interactions, American Naturalist (1999), in press.Google Scholar
  10. [10]
    A.O. Nicholls and C.R. Margules, An upgraded reserve selection algorithm, Biological Conservation 64 (1993) 165–169.Google Scholar
  11. [11]
    I.R. Ball, Mathematical applications for conservation ecology: The dynamics of tree hollows and the design of nature reserves, Ph.D. thesis, The University of Adelaide (2000).Google Scholar
  12. [12]
    J.L. Cohen, Multiobjective Programming and Planning (Academic Press, 1978).Google Scholar
  13. [13]
    J. Wright, C. ReVelle and J. Cohen, A multiobjective integer programming model for the land acquisition problem, Regional Science and Urban Economics 13 (1983) 31–53.Google Scholar
  14. [14]
    K.C. Gilbert, D.D. Holmes and R.E. Rosenthal, A mutliobjective discrete optimization model for land allocation, Management Science 31 (1985) 1509–1522.Google Scholar
  15. [15]
    M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Fransisco, CA, 1979).Google Scholar
  16. [16]
    S. Kirkpatrick, C.D. Gelatt, Jr. and M.P. Vecchi, Optimisation by simulated annealing, Science 220 (1983) 671–680.Google Scholar
  17. [17]
    R.H.J.M. Otten and L.P.P.P. Van Ginneken, The Annealing Algorithm (Kluwer Academic Publishers, 1989).Google Scholar
  18. [18]
    M. Lundy and A. Mees, Convergence of an annealing algorithm, Mathematical Programming 34 (1986) 111–124.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Mark D. McDonnell
    • 1
  • Hugh P. Possingham
    • 2
  • Ian R. Ball
    • 3
  • Elizabeth A. Cousins
    • 1
  1. 1.Department of Applied MathematicsThe University of AdelaideSouth AustraliaAustralia
  2. 2.Department of Mathematics and Zoology & EntomologyThe University of QueenslandSt Lucia QueenslandAustralia
  3. 3.Australian Antarctic DivisionTasmaniaAustralia

Personalised recommendations