Advertisement

Mathematical Models Can Predict the Spread of an Invasive Species

  • John G. AlfordEmail author
Living reference work entry

Abstract

Invasive species are nonindigenous plants and animals that have the potential to cause great harm to both the environment and native species. If an invasive species is able to survive and spread throughout the environment, there may be large financial losses to the public. Public policy makers and scientists are responsible for developing and funding programs to control and eradicate invasive species. These programs are founded on the science of invasion ecology and the biological characteristics of the invader. Conclusions drawn from mathematical modeling have contributed to this knowledge base. This chapter presents some of the fundamental mathematical models in population ecology that have been used to predict how an invasive species population can grow and disperse after its introduction. These predictions are shown to be consistent with experimental data. The chapter concludes with a brief discussion of how the basic modeling principles and results that are presented here have contributed to developing strategies for control and eradication efforts.

Keywords

Invasive species Mathematical model Exponential Logistic Allee effect Reaction-diffusion equation 

References

  1. Anderson JL, Albergotti L, Proulx S, Peden C, Huey RB, Phillips PC (2007) Thermal preference of Caenorhabditis elegans: a null model and empirical tests. J Exp Biol 210:3107–3116CrossRefGoogle Scholar
  2. Andow DA, Kareiva PM, Levin SA, Okubo A (1990) Spread of invading organisms. Landsc Ecol 4:177–188CrossRefGoogle Scholar
  3. Bacaër N (2011) A short history of mathematical population dynamics. Springer, LondonCrossRefGoogle Scholar
  4. Beck KG, Zimmerman K, Schardt JD, Stone J, Lukens RR, Reichard S, Randall J, Cangelosi AA, Cooper D, Thompson JP (2008) Invasive species defined in a policy context: recommendations from the federal invasive species advisory committee. Invasive Plant Sci Manage 1:414–421CrossRefGoogle Scholar
  5. Becker K (1972) Muskrats in central Europe and their control. In: Proceedings of the 5th vertebrate pest conference. University of Nebraska, LincolnGoogle Scholar
  6. Bertignac M, Lehodey P, Hampton J (1998) A spatial population dynamics simulation model of tropical tunas using a habitat index based on environmental parameters. Fish Oceanogr 7:326–334CrossRefGoogle Scholar
  7. Boyce WE, DiPrima RC (2012) Elementary differential equations and boundary value problems, 10th edn. Wiley, New YorkzbMATHGoogle Scholar
  8. Brassil CE (2001) Mean time to extinction of a metapopulation with an Allee effect. Ecol Model 143:9–16CrossRefGoogle Scholar
  9. Dennis B (2002) Allee effects in stochastic populations. Oikos 96:389–401CrossRefGoogle Scholar
  10. Edelstein-Keshet L (2005) Mathematical models in biology. SIAM, New YorkCrossRefGoogle Scholar
  11. Fujisaki I, Pearlstine EV, Mazzotti FJ (2010) The rapid spread of invasive Eurasian collared doves Streptopelia decaocto in the continental USA follows human-altered habitats. Ibis 152:622–632CrossRefGoogle Scholar
  12. Hastings A (1996) Models of spatial spread: a synthesis. Biol Conserv 78:143–148CrossRefGoogle Scholar
  13. Hengeveld R (1992) Potential and limitations of predicting invasion rates. Fla Entomol 75:60–72CrossRefGoogle Scholar
  14. Karieva P (1983) Local movement in herbivorous insects: applying a passive diffusion model to mark-recapture field experiments. Oecologia 57:322–327CrossRefGoogle Scholar
  15. Keller RP, Lodge DM, Lewis MA, Shogren JF (2009) Bioeconomics of invasive species: integrating ecology, economics, policy and management. Oxford University Press, New YorkGoogle Scholar
  16. Kot M (2001) Elements of mathematical ecology. Cambridge University Press, Cambridge, UKCrossRefGoogle Scholar
  17. Kramer AM, Dennis B, Liebhold AM, Drake JM (2009) The evidence for Allee effects. Popul Ecol 51:341–354CrossRefGoogle Scholar
  18. Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman & Hall, LondonzbMATHGoogle Scholar
  19. Lewis MA, Petrovskii SV, Potts JR (2016) The mathematics behind biological invasions, vol 44. Springer, BerlinzbMATHGoogle Scholar
  20. Liebhold A, Bascompte J (2003) The Allee effect, stochastic dynamics and the eradication of alien species. Ecol Lett 6:133–140CrossRefGoogle Scholar
  21. Liebhold AM, Tobin PC (2006) Growth of newly established alien populations: comparison of North American gypsy moth colonies with invasion theory. Popul Ecol 48:253–262CrossRefGoogle Scholar
  22. Liebhold AM, Work TT, McCullough DG, Cavey JF (2006) Airline baggage as a pathway for alien insect species entering the United States. Am Entomol 52:48–54CrossRefGoogle Scholar
  23. Liebhold AM et al (2016) Eradication of invading insect populations: from concepts to applications. Annu Rev Entomol 61:335–352CrossRefGoogle Scholar
  24. Lockwood JL, Hoopes MF, Marchetti MP (2007) Invasion ecology. Black-well, MaldenGoogle Scholar
  25. Logan JD (2004) Applied partial differential equations, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  26. Lonsdale WM (1993) Rates of spread of an invading species Mimosa pigra in northern Australia. J Ecol 81:513–521CrossRefGoogle Scholar
  27. Madsen JD, Owens CS (2000) Factors contributing to the spread of Hydrilla in lakes and reservoirs. Aquatic plant control technical notes collection (ERDC TN-APCRP-EA-01). US Army Engineer Research and Development Center, Vicksburg, pp 1–11Google Scholar
  28. McJunkin JW, Zelmer DA, Applegate RD (2005) Population dynamics of wild turkeys in Kansas (Meleagris gallopavo): theoretical considerations and implications of rural mail carrier survey (RMCS) data. Am Midl Nat 154:178–187CrossRefGoogle Scholar
  29. Mills EL, Leach JH, Carlton JT, Secor CL (1994) Exotic species and the integrity of the Great Lakes. Bioscience 44:666–676CrossRefGoogle Scholar
  30. Neubert MG, Parker IM (2004) Projecting rates of spread for invasive species. Risk Anal 24:817–831CrossRefGoogle Scholar
  31. Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives. Springer, New YorkCrossRefGoogle Scholar
  32. Petrovska BB (2012) Historical review of medicinal plants usage. Pharm Rev 6:1–5CrossRefGoogle Scholar
  33. Picart D, Milner FA, Thiery D (2015) Optimal treatments schedule in insect pest control in viticulture. Math Popul Stud 22:172–181MathSciNetCrossRefGoogle Scholar
  34. Pimentel D, Zuniga R, Morrison D (2005) Update on the environmental and economic costs associated with alien-invasive species in the United States. Ecol Econ 52:273–288CrossRefGoogle Scholar
  35. Ricklefs RE, Miller GL (1999) Ecology, 4th edn. W.H Freeman, New YorkGoogle Scholar
  36. Rodriguez-Brenes IA, Komarova NL, Wodarz D (2013) Tumor growth dynamics: in-sights into somatic evolutionary processes. Trends Ecol Evol 28:597–604CrossRefGoogle Scholar
  37. Santos VB, Yoshihara E, Freitas RTF, Reis Neto RV (2008) Exponential growth model of Nile tilapia (Oreochromis niloticus) strains considering heteroscedastic variance. Aquaculture 274:96–100CrossRefGoogle Scholar
  38. Seidl I, Tisdell CA (1999) Carrying capacity reconsidered: from Malthus’ population theory to cultural carrying capacity. Ecol Econ 31(3):395–408CrossRefGoogle Scholar
  39. Shigesada N, Kawasaki K (1997) Biological invasions: theory and practice. Oxford series in ecology and evolution. Oxford University Press, OxfordGoogle Scholar
  40. Skellam JG (1951) Random dispersal in theoretical populations. Biometrika 38(1–2):196–218MathSciNetCrossRefGoogle Scholar
  41. Stephens PA, Sutherland WJ, Freckleton RP (1999) What is the Allee effect? Oikos 87:185–190CrossRefGoogle Scholar
  42. Suckling DM, Tobin PC, McCullough DG, Herms DA (2012) Combining tactics to exploit Allee effects for eradication of alien insect populations. J Econ Entomol 105:1–13CrossRefGoogle Scholar
  43. Taylor CM, Hastings A (2005) Allee effects in biological invasions. Ecol Lett 8:895–908CrossRefGoogle Scholar
  44. Tobin PC, Liebhold AM, Roberts EA, Blackburn LM (2015) Estimating spread rates of non-native species: the gypsy moth as a case study. In: Venette RC (ed) Pest risk modelling and mapping for invasive alien species. CABI Press, Wallingford, pp 131–144Google Scholar
  45. Turchin P (2001) Does population ecology have general laws? Oikos 94:17–26CrossRefGoogle Scholar
  46. Volpert V, Petrovskii S (2009) Reaction-diffusion waves in biology. Phys Life Rev 6:267–310CrossRefGoogle Scholar
  47. Walker MS (2018) Spotted lanternfly: states urge citizens to report sightings of invasive insect hitchhiker. In: Entomology Today. Entomological Society of America. https://entomologytoday.org/2018/02/26/spotted-lanternfly-states-urge-citizens-report-sightings-invasive-insect-hitchhiker/. Accessed 9 March, 2019

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics HuntsvilleSam Houston State UniversityHuntsvilleUSA

Section editors and affiliations

  • Torsten Lindström
    • 1
  • Bharath Sriraman
    • 2
  1. 1.Linneaeus UniversityVäxjöSweden
  2. 2.Department of Mathematical SciencesThe University of MontanaMissoulaUSA

Personalised recommendations