Bulletin of Mathematical Biology

, Volume 79, Issue 1, pp 88–116 | Cite as

Optimal Culling and Biocontrol in a Predator–Prey Model

Original Article

Abstract

Invasive species cause enormous problems in ecosystems around the world. Motivated by introduced feral cats that prey on bird populations and threaten to drive them extinct on remote oceanic islands, we formulate and analyze optimal control problems. Their novelty is that they involve both scalar and time-dependent controls. They represent different forms of control, namely the initial release of infected predators on the one hand and culling as well as trapping, infecting, and returning predators on the other hand. Combinations of different control methods have been proposed to complement their respective strengths in reducing predator numbers and thus protecting endangered prey. Here, we formulate and analyze an eco-epidemiological model, provide analytical results on the optimal control problem, and use a forward–backward sweep method for numerical simulations. By taking into account different ecological scenarios, initial conditions, and control durations, our model allows to gain insight how the different methods interact and in which cases they could be effective.

Keywords

Eco-epidemiology Biocontrol Optimal control Invasive pest Predator–prey 

Mathematics Subject Classification

34D20 49K15 92D30 92D40 

References

  1. Allen LJS (2007) An introduction to mathematical biology. Pearson Prentice Hall, New JerseyGoogle Scholar
  2. Anderson R, May RM (1986) The invasion, persistence and spread of infectious diseases within animal and plant communities. Phil Trans R Soc Lond 314:533–570CrossRefGoogle Scholar
  3. Bate AM, Hilker FM (2013) Complex dynamics in an eco-epidemiological model. Bull Math Biol 75:2059–2078MathSciNetCrossRefMATHGoogle Scholar
  4. Bester MN, Bloomer JP, van Aarde RJ, Erasmus BH, van Rensburg PJJ, Skinner JD, Howell PG, Naude TW (2002) A review of the successful eradication of feral cats from sub-Antarctic Marion Island, Southern Indian Ocean. S Afr J Wildl Res 32:65–73Google Scholar
  5. Chaphuis J, Bousses P, Barnaud G (1994) Alien mammals, impact and management in the French sub-antarctic islands. Biol Conserv 67:97–104CrossRefGoogle Scholar
  6. Chaphuis JL (1995) Alien mammals in the French Subantarctic Islands. In: Dingwall PR (ed) Progress in conservation of the subantarctic islands, conservation of the southern polar region, vol 2. The World Conservation Union, Paimpont, pp 127–132Google Scholar
  7. Cleaveland S, Thirgood S, Laurenson K (1999) Pathogens as allies in island conservation? Trends Ecol Evol 14:83–84CrossRefGoogle Scholar
  8. Courchamp F, Pontier P (1994) Feline immunodeficiency virus: an epidemiological review. C R Acad Sci Paris Sér III Sci Vie 317:1123–1134Google Scholar
  9. Courchamp F, Sugihara G (1999) Modeling the biological control of an alien predator to protect island species from extinction. Ecol Appl 9:112–123CrossRefGoogle Scholar
  10. Courchamp F, Pontier D, Langlais M (1995) Population dynamics of feline immunodeficiency virus within cat populations. J Theor Biol 175:553–560CrossRefGoogle Scholar
  11. Courchamp F, Yoccoz NG, Artois M, Pontier D (1998) At-risk individuals in feline immunodeficiency virus epidemiology: evidence from multivariate approach in a natural population of domestic cats (feline catus). Epidemiol Infect 121:227–236CrossRefGoogle Scholar
  12. Courchamp F, Langlais M, Sugihara G (1999) Cats protecting birds: modelling the mesopredator release effect. J Anim Ecol 68:282–292CrossRefGoogle Scholar
  13. Courchamp F, Say L, Pontier D (2000) Transmission of feline immunodeficiency virus in a population of cats. Wildl Res 27:227–236CrossRefGoogle Scholar
  14. Courchamp F, Chapuis JL, Pascal M (2003) Mammal invaders on islands: impact, control and control impact. Biol Rev 78:347–383CrossRefGoogle Scholar
  15. Diamond J (1989) Overview of recent extinctions. In: Western D, Pearl MC (eds) Conservation for the twenty-first century. Oxford University Press, New York, pp 37–41Google Scholar
  16. Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley, New YorkMATHGoogle Scholar
  17. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations. J Math Biol 28:365–382MathSciNetCrossRefMATHGoogle Scholar
  18. Diekmann O, Dietz K, Heesterbeek JAP (1991) The basic reproduction ratio for sexually transmitted diseases: theoretical considerations. Math Biosci 107:325–339CrossRefMATHGoogle Scholar
  19. Diekmann O, Heesterbeek JAP, Robert MG (2010) The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface 9:873–885CrossRefGoogle Scholar
  20. Diekmann O, Heesterbeek H, Britton T (2013) Mathematical tools for understanding infectious disease dynamics. Princeton University Press, New JerseyMATHGoogle Scholar
  21. Dobson AP (1988) Restoring island ecosystems: the potential of parasites to control introduced mammals. Conserv Biol 3:31–38CrossRefGoogle Scholar
  22. Driessche PVD, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48MathSciNetCrossRefMATHGoogle Scholar
  23. Fan M, Kuang Y, Feng Z (2005) Cats protecting birds revisited. Bull Math Biol 167:1081–1106MathSciNetCrossRefMATHGoogle Scholar
  24. Fitzgerald BM, Turner DC (2000) Hunting behaviour of domestic cats and their impact on prey populations. In: Turner DC, Bateson P (eds) The domestic cat: the biology of its behaviour, 2nd edn. Cambridge University Press, Cambridge, pp 151–175Google Scholar
  25. Fromont E, Pontier D, Langlais M (1998) Dynamics of a feline retrovirus (FeLV) in host populations with variable spatial structure. Proc R Soc Lond B 265:1097–1104CrossRefGoogle Scholar
  26. Gaff H, Schaefer E (2009) Optimal control applied to vaccination and treatment strategies for various epidemiological models. Math Biosci Eng 6:469–492MathSciNetCrossRefMATHGoogle Scholar
  27. Hackbusch W (1978) A numerical method for solving parabolic equations with opposite orientations. Computing 20:229–240MathSciNetCrossRefMATHGoogle Scholar
  28. Hadeler KP, Freedman HI (1989) Predator–prey populations with parasite infection. J Math Biol 27:609–631MathSciNetCrossRefMATHGoogle Scholar
  29. Hartmann K (1998) Feline immunodeficiency virus infection: an overview. Vet J 155:123–137CrossRefGoogle Scholar
  30. Hastings A (2004) Transients: the key to long-term ecological understanding? Trends Ecol Evol 19:39–45CrossRefGoogle Scholar
  31. Hilker FM, Schmitz K (2008) Disease-induced stabilization of predator–prey oscillations. J Theor Biol 255:299–306CrossRefGoogle Scholar
  32. Kooi BW, van Voorn GAK, pada Das K (2011) Stabilization and complex dynamics in a predator–prey model with predator suffering from an infectious disease. Ecol Complex 8:113–122CrossRefGoogle Scholar
  33. Kot M (2001) Elements of mathematical ecology. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  34. Lavers JL, Wilcox C, Donlan CJ (2010) Bird demographic responses to predator removal programs. Biol Invasions 12:3839–3859CrossRefGoogle Scholar
  35. Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman and Hall, Boca RatonMATHGoogle Scholar
  36. Loyd KA, Miller CA (2010) Influence of demographics, experience and value orientation on preferences for lethal management of feral cats. Hum Dimens Wildl 15:262–273CrossRefGoogle Scholar
  37. Moors PJ, Atkinson IAE (1984) Predation on seabirds by introduced animals, and factors affecting its severity. In: Croxall PJ, Evans PGH, Schreiber RW (eds) Status and conservation of the world’s seabirds, vol 2. ICBP Technical Publications, Cambridge, pp 667–690Google Scholar
  38. Murray JD (1993) Mathematical biology, 2nd edn. Springer, HeidelbergCrossRefMATHGoogle Scholar
  39. Nogales M, Martin A, Tershey BR, Donlan CJ, Veitch D, Puerta N, Wood B, Alonso J (2004) A review of feral cat eradication on islands. Conserv Biol 18:310–319CrossRefGoogle Scholar
  40. Nutter FB, Levine JF, Stoskopf MK (2004) Reproductive capacity of free-roaming domestic cats and kitten survival rate. J Am Vet Med Assoc 225:1399–1402CrossRefGoogle Scholar
  41. Oliveira NM, Hilker FM (2010) Modeling disease introduction as biological control of invasive predators to preserve endangered prey. Bull Math Biol 72:444–468MathSciNetCrossRefMATHGoogle Scholar
  42. Pascal M (1980) Structure et dynamique la population de chats harets de l’archipel del kerguelen. Mammalia 44:161–182CrossRefGoogle Scholar
  43. Pimm SL (1982) Food webs. Chapman and Hall, LondonCrossRefGoogle Scholar
  44. Pontryagin Boltyanskii V, Gamkrelize R, Mishchenko E (1967) The mathematical theory of optimal processes. Wiley, New YorkGoogle Scholar
  45. Rauzon MJ (1985) Feral cats on Jarvis Island: their effects and their eradication. Atoll Res Bull 282:1–32CrossRefGoogle Scholar
  46. Rayner MJ, Hauber ME, Imber MJ, Stamp RK, Clout MN (2007) Spatial heterogeneity of mesopredator release within an oceanic island system. Proc Natl Acad Sci USA 104:20,862–20,865CrossRefGoogle Scholar
  47. Robertson SA (2008) A review of feral cat control. J Feline Med Surg 10:366–375CrossRefGoogle Scholar
  48. Rounsevel DE, Copson GR (1982) Growth rate and recovery of a king penguin aptenodytes patagonicus population after exploitation. Aust Wildl Res 9:519–525CrossRefGoogle Scholar
  49. Russell JC, Lecomte V, Dumont Y, Le Corre M (2009) Intraguild predation and mesopredator release effect on long-lived prey. Ecol Model 220:1098–1104CrossRefGoogle Scholar
  50. van Aarde RJ (1980) The diet and feeding behaviour of feral cats, Felis catus at Marion Island. S Afr J Wild Res 10(3/4):123–128Google Scholar
  51. van Rensburg PJJ, Bester MN (1988) The effect of cat, Felis catus, predation on three breeding Procellariidae species on Marion Island. S Afr J Zool 23(4):301–305CrossRefGoogle Scholar
  52. Xiao Y, Van Den Bosch F (2003) The dynamics of an eco-epidemic model with biological control. Ecol Model 168:203–214CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2016

Authors and Affiliations

  • Eric Numfor
    • 1
  • Frank M. Hilker
    • 2
  • Suzanne Lenhart
    • 3
  1. 1.Department of MathematicsAugusta UniversityAugustaUSA
  2. 2.Institute of Environmental Systems Research, School of Mathematics and Computer ScienceOsnabrück UniversityOsnabrückGermany
  3. 3.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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