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Allee Effect and Control of Lake System Invasion

Abstract

We consider the model of invasion prevention in a system of lakes that are connected via traffic of recreational boats. It is shown that in presence of an Allee effect, the general optimal control problem can be reduced to a significantly simpler stationary optimization problem of optimal invasion stopping. We consider possible values of model parameters for zebra mussels. The general N-lake control problem has to be solved numerically, and we show a number of typical features of solutions: distribution of control efforts in space and optimal stopping configurations related with the clusters in lake connection structure.

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References

  • Bossenbroek, J.M., Kraft, C.E., Nekola, J.C., 2001. Prediction of long-distance dispersal using gravity models: zebra mussel invasion of inland lakes. Ecol. Appl. 11(6), 1778–788.

    Article  Google Scholar 

  • Brock, W.A., Xepapadeas, A., 2003. Valuing biodiversity from an economic perspective: a unified economic, ecological, and genetic approach. Am. Econ. Rev. 93(5), 1597–614.

    Article  Google Scholar 

  • Buchan, L.A.J., Padilla, D.K., 1999. Estimating the probability of long-distance overland dispersal of invading aquatic species. Ecol. Appl. 9, 254–65.

    Article  Google Scholar 

  • Carlson, D.A., Haurie, A.B., Leizarowitz, A., 1991. Infinite Horizon Optimal Control. Springer, Berlin.

    MATH  Google Scholar 

  • Clark, C.W., 1990. Mathematical Bioeconomics. The Optimal Management of Renewable Resources. Wiley, New York.

    MATH  Google Scholar 

  • Courchamp, F., Clutton-Brock, T., Grenfell, B., 1999. Inverse density dependence and the Allee effect. Trends Ecol. Evol. 14(10), 405–10.

    Article  Google Scholar 

  • Gregory, J., Lin, C., 1992. Constrained Optimization in the Calculus of Variations and Optimal Control Theory. Van Nostrand Reinhold, New York.

    MATH  Google Scholar 

  • Hastings, A., Cuddington, K., Davies, K.F., Dugaw, C.J., Elmendorf, S., Freestone, A., Harrison, S., Holland, M., Lambrinos, J., Malvadkar, U., Melbourne, B.A., Moore, K., Taylor, C., Thomson, D., 2005. The spatial spread of invasions: new developments in theory and evidence. Ecol. Lett. 8, 91–01.

    Article  Google Scholar 

  • Horvath, T.G., Lamberti, G.A., 1997. Drifting macrophytes as a mechanism for zebra mussel (Dreissena polymorpha) invasion of lake-outlet streams. Am. Midl. Nat. 138, 29–6.

    Article  Google Scholar 

  • Johnson, L.E., Carlton, J.T., 1996. Post-establishment spread in large-scale invasions: dispersal mechanisms of the zebra mussel dreissena polymorpha. Ecology 77(6), 1686–690.

    Article  Google Scholar 

  • Johnson, L.E., Riccardi, A., Carlton, J.T., 2001. Overland dispersal of aquatic invasive species: a risk assessment of transient recreational boating. Ecol. Appl. 11(6), 1789–799.

    Article  Google Scholar 

  • Kamien, M.I., Schwartz, N.L., 1991. Dynamic Optimization. North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Keitt, T.H., Lewis, M.A., Holt, R.D., 2001. Allee effect, invasion pinning, and species’ borders. Am. Nat. 157(2), 203–16.

    Article  Google Scholar 

  • Kolar, C., Lodge, D.M., 2002. Ecological Predictions and Risk Assessment for Alien Fishes in North America. Science 298, 1233–236.

    Article  Google Scholar 

  • Leung, B., Drake, J.M., Lodge, D.M., 2004. Predicting invasions: propagule pressure and the gravity of Allee effects. Ecology 85, 1651–660.

    Article  Google Scholar 

  • Lewis, M.A., Kareiva, P., 1993. Allee dynamics and the spread of invading organisms. Theor. Pop. Biol. 43, 141–58.

    MATH  Article  Google Scholar 

  • MacIsaac, H.J., Borbely, J.V.M., Muirhead, J.R., Graniero, P.A., 2004. Backcasting and forecasting biological invasions of inland lakes. Ecol. Appl. 14, 773–83.

    Article  Google Scholar 

  • Mangin, S., 2001. The 100th Meridian Initiative: A Strategic Approach to Prevent the Westward Spread of Zebra Mussels and Other Aquatic Nuisance Species. U.S. Fish and Wildlife Service, Washington.

    Google Scholar 

  • O’Neill, C.R. Jr., 1997. Economic impact of zebra mussels. Grt. Lakes Res. Rev. 3(1), 3–1.

    Google Scholar 

  • Nicholis, S.J., 1996. Variations in the reproductive cycle of Dreissena Polymorpha in Europe, Russia, and North America. Am. Zool. 36, 311–25.

    Google Scholar 

  • Olson, L.J., Roy, S., 2002. The economics of controlling a stochastic biological invasion. Am. J. Agron. Econ. 84(5), 1311–316.

    Article  Google Scholar 

  • Ontario Ministry of Natural Resources web site, 2003. http://www.mnr.gov.on.ca/MNR/fishing/threat.html.

  • Pennington, J.T., 1985. The ecology of fertilization of echinoid eggs: the consequences of sperm dilution, adult aggregation, and synchronous spawning. Biol. Bull. 169(4), 17–30.

    Google Scholar 

  • Perrings, C., Williamson, M., Barbier, E., Delfino, D., Dalmazzone, S., Shogren, J., Simmons, P., Watkinson, A., 2002. Biological invasion risks and the public good: an economic perspective. Conserv. Ecol. 6(1), 1. URL: http://www.consecol.org/vol6/iss1/art1/.

    Google Scholar 

  • Pontryagin, L.S., Boltyanskii, V.G., Gamkrelize, R.V., Mishchenko, E.F., 1962. The Mathematical Theory of Optimal Processes. Wiley, New York.

    MATH  Google Scholar 

  • Potapov, A., Lewis, M.A., Finnoff, D., 2007. Optimal control of biological invasions in lake networks. Nat. Res. Mod. 20(3), 351–79.

    MATH  MathSciNet  Google Scholar 

  • Sen, A., Smith, T.E., 1995. Gravity Models of Spatial Interaction Behavior. Springer, Berlin.

    Google Scholar 

  • Taylor, C.M., Hastings, A., 2005. Allee effects in biological invasions. Ecol. Lett. 8, 895–08.

    Article  Google Scholar 

  • Veit, R.R., Lewis, M.A., 1996. Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America. Am. Nat. 148(2), 255–74.

    Article  Google Scholar 

  • Williamson, M., 1996. Biological Invasions. Chapman & Hall, London.

    Google Scholar 

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Correspondence to A. B. Potapov.

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Potapov, A.B., Lewis, M.A. Allee Effect and Control of Lake System Invasion. Bull. Math. Biol. 70, 1371 (2008). https://doi.org/10.1007/s11538-008-9303-8

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  • DOI: https://doi.org/10.1007/s11538-008-9303-8

Keywords

  • Biological invasions
  • Spatial control
  • Optimal control
  • Aquatic invaders
  • Zebra mussels