Computational Management Science

, Volume 10, Issue 4, pp 331–351 | Cite as

Spatial control of invasive species in conservation landscapes

  • Christopher M. Baker
  • Michael BodeEmail author
Original Paper


Areas of high conservation value are often baited with poison to reduce the density of invasive predators, and consequentially reduce the impact on native animals. Poison baits are typically distributed uniformly across a landscape, but there has been no work to determine if this is a cost-effective way to protect endangered species. We consider a small, high-value conservation asset within a broader landscape that is baited to control invasive predators. The predator density in the baited region is modelled with a reaction-diffusion equation, and we use optimal control theory to solve for the long term baiting strategy which minimises the predator density at the conservation asset.


Optimal control Pontryagin Bait distribution  Invasive predators Conformal transformation Exclusion fencing 

Mathematics Subject Classification (2000)

92D25 92D40 



The authors would like to thank Steve Carnie and Tamsin Lee for their assistance.


  1. Algar D, Burrows ND (2004) Feral cat conrol research: Western Shield review—February 2003. Conserv Sci West Aust 5(2):131Google Scholar
  2. Balogh S, Gentle M, Brown S (2001) In: Proceedings 12th Australasian vertebrate pest control conference, Department of Natural Resources and Environment, Melbourne, VictoriaGoogle Scholar
  3. Bode M, Burrage K, Possingham HP (2008) Using complex network metrics to predict the persistence of metapopulations with asymmetric connectivity patterns. Ecol Model 214(2–4):201. doi: 10.1016/j.ecolmodel.2008.02.040
  4. Bode M, Wintle B (2010) How to build an efficient conservation fence. Conserv Biol 24(1):182–188. doi: 10.1111/j.1523-1739.2009.01291.x Google Scholar
  5. Canada A, Gamez JL, Montero JA (1998) Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type. SIAM J Control Optim 36(4):1171Google Scholar
  6. Cantrell S, Cosner C, Ruan S (2010) Spatial ecology. Chapman and Hall/CRCGoogle Scholar
  7. Carrasco LR, Baker R, MacLeod A, Knight JD, Mumford JD (2010) Optimal and robust control of invasive alien species spreading in homogeneous landscapes. J Royal Soc Interface 7(44):529. doi: 10.1098/rsif.2009.0266 Google Scholar
  8. Carritt R (1999) Fox control in wildlife habitat. In: Technical Report, NSW National Parks and Wildlife Service, SydneyGoogle Scholar
  9. Carter A, Luck GW, McDonald SP (2011) Fox-baiting in agricultural landscapes in south-eastern Australia: a case-study appraisal and suggestions for improvement. Ecol Manag Restor 12(3):214–223. doi: 10.1111/j.1442-8903.2011.00604.x
  10. Chisholm RA, Taylor R (2010) Body size and extinction risk in Australian mammals: an information-theoretic approach. Aust Ecol 35(6):616–623. doi: 10.1111/j.1442-9993.2009.02065.x Google Scholar
  11. Clark CW, Munro GR (1975) The economics of fishing and modern capital theory: a simplified approach. J Environ Econ Manag 2(2):92. doi: 10.1016/0095-0696(75)90002-9 Google Scholar
  12. Clobert J, Le Galliard JF, Cote J, Meylan S, Massot M (2009) Informed dispersal, heterogeneity in animal dispersal syndromes and the dynamics of spatially structured populations. Ecol Lett 12(3):197–209. doi: 10.1111/j.1461-0248.2008.01267.x Google Scholar
  13. Cohen H (2007) In: complex analysis with applications in science and engineering. Springer, US, pp 249–366Google Scholar
  14. Coleman J (1995) Department of conservation, p 11Google Scholar
  15. Coman B, Staples LS, McPhee SR, Walker A (1995) In: 10th Australian vertebrate pest conference proceedings: Department of Primary Industry and Fisheries, Tasmania, Hobart, Tasmania, pp 234–237Google Scholar
  16. Connolly T, Day T, King C (2009) Estimating the potential for reinvasion by mammalian pests through pest-exclusion fencing. Wildl Res 36(5):410Google Scholar
  17. Courchamp F, Chapuis JL, Pascal M (2003) Mammal invaders on islands: impact, control and control impact. Biol Rev 78(3):347–383. doi: 10.1017/S1464793102006061 Google Scholar
  18. de Preu N (2000) Operation bounceback interim report. In: Technical report, Department for Environment and Heritage, South AustraliaGoogle Scholar
  19. Denny E, Dickman CR (2010) Review of cat ecology and management strategies in Australia. In: Technical report, Invasive Animals Cooperative Research CentreGoogle Scholar
  20. Driscoll TA, Trefethen LN (2002) Schwarz–Christoffel mapping, 1st edn. Cambridge University PressGoogle Scholar
  21. Ecosystems and human well-being (2005) Biodiversity synthesis. In: Technical report, World Resources Institute, Washington, DCGoogle Scholar
  22. Hauser CE, McCarthy MA (2009) Streamlining ‘search and destroy’: cost-effective surveillance for invasive species management. Ecol Lett 12(7):683–692. doi: 10.1111/j.1461-0248.2009.01323.x Google Scholar
  23. Hayward MW, Kerley GI (2009) Fencing for conservation: restriction of evolutionary potential or a riposte to threatening processes?. Biol Conserv 142(1):1. doi: 10.1016/j.biocon.2008.09.022 Google Scholar
  24. Holmes EE (1993) Are diffusion models too simple? A comparison with telegraph models of invasion. Am Nat 142(5):779. doi: 10.2307/2462716 Google Scholar
  25. Hone J (1990) Predator-prey theory and feral pig control, with emphasis on evaluation of shooting from a helicopter. Aust Wildl Res 17(2):123–130Google Scholar
  26. Jordan BW, Polak E (1964) Optimal control of aperiodic discrete-time systems. J Soc Ind Appl Math Ser A Control 2(3):332–346Google Scholar
  27. Kot M (2001) Elements of mathematical ecology. Cambridge University PressGoogle Scholar
  28. Lenhart S, Protopopescu V, Stojanovic S (1993) A minimax problem for semilinear nonlocal competitive systems. Appl Math Optim 28(2):113. doi: 10.1007/BF01182976 Google Scholar
  29. Lenhart S, Workman JT (2007) Optimal control applied to biological models. In: Mathematical and computational biology series, Chapman & Hall/CRC, LondonGoogle Scholar
  30. Leung A, Stojanovic S (1993) Optimal control for elliptic Volterra-Lotka type equations. J Math Anal Appl 173(2):603. doi: 10.1006/jmaa.1993.1091 Google Scholar
  31. Matthysen E (2005) Density-dependent dispersal in birds and mammals. Ecography 28(3):403–416. doi: 10.1111/j.0906-7590.2005.04073.x Google Scholar
  32. McLennan JA, Potter MA, Robertson HA, Wake GC, Colbourne R, Dew L, Joyce L, McCann AJ, Miles J, Miller PJ, Reid J (1996) Role of predatation in the decline of kiwi, Apteryx spp., in New Zealand. N Z J Ecol 20(1):27Google Scholar
  33. Miller Neilan R, Lenhart S (2011) Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons. J Math Anal Appl 378(2):603. doi: 10.1016/j.jmaa.2010.12.035 Google Scholar
  34. Moore JL, Rout TM, Hauser CE, Moro D, Jones M, Wilcox C, Possingham HP (2010) Protecting islands from pest invasion: optimal allocation of biosecurity resources between quarantine and surveillance. Biol Conserv 143(5):1068. doi: 10.1016/j.biocon.2010.01.019 Google Scholar
  35. Murdoch W, Polasky S, Wilson KA, Possingham HP, Kareiva P, Shaw R (2007) Maximizing return on investment in conservation. Biol Conserv 139(3–4):375. doi: 10.1016/j.biocon.2007.07.011 Google Scholar
  36. Murray AJ, Poore RN, Dexter N (2006) Project deliverance the response of critical-weight-range mammals to effective fox control in mesic forest habitats in far east gippsland. In: Technical report, Department of Sustainability and Environment, Victoria, MelbourneGoogle Scholar
  37. Murray JD, Stanley EA, Brown DL (1986) On the spatial spread of Rabies among foxes. Proc Royal Soc Lond Ser B Biol Sci 229(1255):111. doi: 10.1098/rspb1986.0078
  38. Murray R, Poore R (2001) In: Proceedings: Australian Wildlife Management Society, 14th annual conference. NSW National Parks and Wildlife Service, DubboGoogle Scholar
  39. Neubert MG (2003) Marine reserves and optimal harvesting. Ecol Lett 6(9):843–849. doi: 10.1046/j.1461-0248.2003.00493.x Google Scholar
  40. Neubert MG, Herrera GE (2008) Triple benefits from spatial resource management. Theor Ecol 1(1):5. doi: 10.1007/s12080-007-0009-6 Google Scholar
  41. Papamichael N, Stylianopoulos N (2010) Numerical conformal mapping. World Scientific, SingaporeGoogle Scholar
  42. Pollard A (2000) In: Proceedings of the NSW pest animal control conference. NSW Agriculture, Orange, New South WalesGoogle Scholar
  43. Pontryagin LS (1987) In: Pontryagin LS(ed) Mathematical theory of optimal processes, english edn. CRC PressGoogle Scholar
  44. Possingham H, Jarman P, Kearns A (2004) Independent review of Western Shield-February 2003. Conserv Sci West Aust 5(2):2Google Scholar
  45. Priddel D, Wheeler R (1997) Efficacy of fox control in reducing the mortality of released captive-reared Malleefowl, Leipoa ocellata. Wildl Res 24(4):469Google Scholar
  46. Pulliam HR (1988) Sources, sinks, and population regulation. Am Nat 132(5):652Google Scholar
  47. Richards J, Short J (2003) Reintroduction and establishment of the western barred bandicoot Perameles bougainville (Marsupialia: Peramelidae) at Shark Bay, Western Australia. Biol Conserv 109(2):181. doi: 10.1016/S0006-3207(02)00140-4 Google Scholar
  48. Roa AV (2009) A survey of numerical methods for optimal control. Adv Astronaut Sci 135(1):497Google Scholar
  49. Roache PJ (1972) On artificial viscosity. J Comput Phys 10(2):169. doi: 10.1016/0021-9991(72)90058-7 Google Scholar
  50. Robley A (2011) Review of fox control measures in box ironbark conservation management networks. In: Technical report 221, Arthur Rylah Institute for, Environmental Research, WedderburnGoogle Scholar
  51. Robley A, Wright J, Gormley A, Evans I (2008) Adaptive experimental management of foxes. Final report, Technical Report 59, Parks Victoria, MelbourneGoogle Scholar
  52. Rout TM, Kirkwood R, Sutherland D, Murphy S, McCarthy M (2013) When to declare successful eradication of an invasive predator? Anim Conserv. doi: 10.1111/acv.12065
  53. Rout TM, Hauser CE, Possingham HP (2009) Optimal adaptive management for the translocation of a threatened species. Ecol Appl 19(2):515. doi: 10.1890/07-1989.1 Google Scholar
  54. Salinas RA, Lenhart S, Gross LJ (2005) Control of a metapopulation harvesting model for black bears. Nat Resour Model 18(3):307–321. doi: 10.1111/j.1939-7445.2005.tb00160.x Google Scholar
  55. Saunders G, McLeod L (2007) Improving fox management strategies in Australia. Bureau of Rural Sciences, Canberra, AustraliaGoogle Scholar
  56. Symm DGT (1969) Numerische Mathematik. Numerische Mathematik 13(5):448. doi: 10.1007/BF02163272
  57. Thomson PC, Algar D (2000) The uptake of dried meat baits by foxes and investigations of baiting rates in Western Australia. Wildl Res 27(5):451Google Scholar
  58. Thompson JF, Thames FC, Mastin C (1974) Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies. J Comput Phys 15(3):299. doi: 10.1016/0021-9991(74)90114-4 Google Scholar
  59. Tores PJd, Marlow N (2012) In: Somers MJ, Hayward M (eds) Fencing for conservation. Springer, New York, pp 21–42Google Scholar
  60. Victorian Department of Sustainability, Environment (2003) Southern Ark project: benefiting the biodiversity of East Gippsland through fox control. In: Department of Sustainability and Environment, MelbourneGoogle Scholar
  61. Wayne A (2008) Diagnosis of recent woylie (bettongia penicillata ogilbyi) declines in southwestern Australia. In: Progress report, Woylie conservation research projectGoogle Scholar
  62. Wiens JA (2001) In: Clobert J, Danchin E, Dhondt AA, Nicholas JD (eds) Dispersal. Oxford University Press, USAGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of BotanyThe University of MelbourneMelbourneAustralia

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