Abstract
This study investigates an optimal strategy for the cost effective control of gypsy moth populations. Gypsy moth populations cycle between low sparse numbers to high outbreak levels and it is during the outbreak levels that the moths cause extensive damage to plant foliage which can lead to deforestation. Deforestation can result in significant economic damage to infested areas, and consequently, there have been many efforts to control moth populations. One effective method of control is the use of the biocontrol agent, Gypchek, but its production is costly. We develop a mathematical model which combines population dynamics and optimal control of the moth population to explore strategies by which the total cost of the gypsy moth problem (economic damage and cost of Gypchek) can be minimized.
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References
Anderson, J.F., Kaya, H.K. (Eds.) 1976. Perspectives in forest entomology. In: Insect Pathogens in Natural and Microbial Control of Forest Defoliators, pp. 251–263. Academic, New York.
Bess, H.A., Spurr, S.H., Littlefield, E.W., 1947. Forest site conditions and the gypsy moth. Harv. For. Bull. 22, 56.
Campbell, R.W., 1969. Studies on gypsy moth population dynamics. For. Insect. Popul. Dyn. USDA For. Serv. Res. Pap. 125, 29–34.
Campbell, R.W., 1975. The bimodality of gypsy moth porthetria dispar (l.) populations. J. NY Entomol. Soc. Am. 183, 287–288.
Campbell, R.W., 1978. Numerical bimodality among North American gypsy moth populations. Environ. Entomol. 7, 641–646.
Campbell, R.W., 1981. The gypsy moth. Technical report, USDA.
Campbell, R.W., Sloan, R.J., 1977. Forest stand responses to defoliation by the gypsy moth. For. Sci. Monogr. 19, 1–34.
Campbell, R.W., Sloan, R.J., 1978. Natural maintenance and decline of gypsy moth outbreaks. Environ. Entomol. 7, 389–395.
Clark, C., 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. Wiley, New York.
Doane, C.C., 1970. Primary pathogens and their role in the development of an epizootic in the gypsy moth. J. Invertebr. Pathol. 15, 21–33.
Edel’man, N.M., 1963. Effect of the biochemical composition of food on growth changes in the physiological conditions of insects. questions of ecology. Vys. Shkola 7, 211–213.
Elkington, J.S., Leibhold, A.M., 1990. Population dynamics of the gypsy moth in North America. Annu. Rev. Entomol. 35, 571–596.
USDA Facesheet, April 2003. Usda facesheet. USDA Facesheet.
Fister, R., 1997. Optimal control of harvesting in a predator-prey parabolic system. Houst. J. Math. 23(2), 341–355.
Foster, M.A., Schultz, J.C., Hunter, M.D., 1992. Modelling gypsy moth-virus-leaf chemistry interactions: implications of plant quality for pest and pathogen dynamics. J. Animal. Ecol. 61, 509–520.
Hackbusch, W., 1978. A numerical method for solving parabolic equations with opposite orientations. Computing 20(3), 229–240.
Hough, J.A., Pimental, D., 1978. Influence of host foliage on development, survival, and fecundity of gypsy moth. Environ. Entomol. 7, 97–102.
Jung, E., Lenhart, S., Protopopescu, V., Babbs, C.F., 2005. Optimal control theory applied to a difference equation model for cardiopulmonary resuscitation. Math. Models Methods Appl. Sci. 15, 1519–1531.
Keating, S.T., Yendol, W.G., 1987. Influence of selected host plants on gypsy moth larval mortality caused by a baculovirus. Environ. Entomol. 16, 459–462.
Keating, S.T., Yendol, W.G., Schultz, J.C., 1988. Relationship between susceptibility of gypsy moth (lepidoptera: Lymantriidae) to a baculovirus and host-plant foliage constituents. Environ. Entomol. 17, 952–958.
Keating, S.T., Hunter, M.D., Schultz, J.C., 1990. Leaf phenolic inhibition of the gypsy moth nuclear polyhedrosis virus: the role of the polyhedral inclusion body aggregation. J. Chem. Ecol. 16, 1445.
Lenhart, S., Liang, M., 2000. Bilinear optimal control for a wave equation with viscous damping. Houst. J. Math. 26, 575–595.
Lenhart, S., Workman, J.T., 2007. Optimal Control Applied to Biological Models. Chapman & Hall/CRC, London/Boca Raton.
Lenhart, S.M., Bhat, M.G., 1992. Application of distributed parameter control model in wildlife damage management. Math. Models Methods Appl. Sci. 2(4), 423–439.
Liebhold, S., 2003. Gypchek—the gypsy moth nucleopolyhedrosis virus product [online]. USDA Forest Service, available from: http://www.fs.fed.us/ne/morgantown/4557/gmoth/manag/gypchek.html, accessed 12/01/06.
Macki, J., Strauss, A., 1982. Introduction to Optimal Control Theory. Springer, Berlin.
Murray, K.D., Elkinton, J.S., 1989. Environmental contamination of egg mass as major component of transgenerational transmission of gypsy moth nuclear polyhedrosis virus (ldmnpv). J. Invertebr. Pathol. 53, 324–334.
Nicholson, A.J., Bailey, V.A., 1935. The balance of animal populations. Proc. Zool. Soc. Lond. 3, 551–598.
Rockafellar, R.T., 1970. Convex Analysis. Princeton University Press, Princeton.
Schultz, J.C., Foster, M.A., Montgomery, M.E., 1990. Host-plant-mediated impacts of baculovirus on gypsy moth populations. In: Population Dynamics of Forest Insects. Intercept Ltd, Andover.
Sethi, S.P., Thompson, G.L., 2000. Optimal Control Theory: Application to Management Science and Economics. Kluwer Academic, Dordrecht.
Webb, R.E., White, G.B., Sukontarak, T., Podgwaite, J.D., Schumacher, D., Schumacher, D., Diss, A., Reardon, R.C., 2004. Biological efficacy of gypchek against a low-density leading edge gypsy moth population. Nort. J. Appl. For. 21(3), 144–149.
Woods, S.A., Elkinton, J.S., Podgewaite, J.D., 1989. Acquisition of nuclear polyhedrosis virus from tree stems by newly emerged gypsy moth (lepidoptera: Lymantriidae) larvae. Environ. Entomol. 18, 298–301.
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Whittle, A., Lenhart, S. & White, K.A.J. Optimal Control of Gypsy Moth Populations. Bull. Math. Biol. 70, 398–411 (2008). https://doi.org/10.1007/s11538-007-9260-7
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DOI: https://doi.org/10.1007/s11538-007-9260-7