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Optimal control of integrodifference equations in a pest-pathogen system
1. | Department of Mathematics, North Central College, Naperville, IL 60540, United States |
2. | Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300 |
3. | Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom |
References:
[1] |
, Gypsy moth management in the United States: a cooperative approach. Final Environmental Impact Statement Vol. 1-5, USDA-Forest Service and Animal and Plant Health Inspection Service,, NA-MR-01-08, (2008), 01. Google Scholar |
[2] |
, Harmful non-indigenous species in the United States, U.S. Congress, Office of Technology Assessment,, OTA-F-565, (1993). Google Scholar |
[3] |
K. Barber, W. Kaupp and S. Holmes, Specificity testing of the nuclear polyhedrosis virus of the gypsy moth, Lymantria dispar (L.),, The Canadian Entomologist, 125 (1993), 1023.
doi: 10.4039/ent1251055-6. |
[4] |
P. Barbosa, D. K. Letourneau and A. A. Agrawal, Insect Outbreaks Revisited,, Wiley-Blackwell, (2012).
doi: 10.1002/9781118295205. |
[5] |
O. N. Bjørnstad, C. Robinet and A. M. Liebhold, Geographic variation in North-American gypsy moth population cycles: Sub-harmonics, generalist predators and spatial coupling,, Ecology, 91 (2010), 106.
doi: 10.1890/08-1246.1. |
[6] |
J. Briggs, J. Dabbs, M. Holm, J. Lubben, R. Rebarber, B. Tenhumberg and D. Riser-Espinoza, Structured population dynamics: An introduction to integral modeling,, Mathematics Magazine, 83 (2010), 243.
doi: 10.4169/002557010X521778. |
[7] |
N. F. Britton, Essential Mathematical Biology,, Springer-Verlag, (2003).
doi: 10.1007/978-1-4471-0049-2. |
[8] |
J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord,, The American Naturalist, 152 (1998), 204.
doi: 10.1086/286162. |
[9] |
C. Doane and M. McManus, The Gypsy Moth: Research toward Integrated Pest Management,, U.S. Department of Agriculture, (1981). Google Scholar |
[10] |
G. Dwyer, J. Dushoff, J. S. Elkinton and S. A. Levin, Pathogen-driven outbreaks in forest defoliators revisited: Building models from experimental data,, The American Naturalist, 156 (2000), 105.
doi: 10.1086/303379. |
[11] |
G. Dwyer, J. Dushoff, J. S. and S. Yee, The combined effects of pathogens and predators on insect outbreaks,, Nature, 430 (2004), 341.
doi: 10.1038/nature02569. |
[12] |
G. Dwyer and J. S. Elkinton, Using simple models to predict virus epizootics in gypsy moth populations,, Journal of Animal Ecology, 62 (1993), 1.
doi: 10.2307/5477. |
[13] |
E. A. Eager, R. Rebarber and B. Tenhumberg, Choice of density-dependent seedling recruitment function affects predicted transient dynamics: A case study with Platte thistle,, Theoretical Ecology, 5 (2012), 387.
doi: 10.1007/s12080-011-0131-3. |
[14] |
M. R. Easterling, S. P. Ellner and P. M. Dixon, Importance of individual variation to the spread of invasive species: A spatial integral projection model,, Ecology, 92 (2011), 86.
doi: 10.1890/09-2226.1. |
[15] |
J. Elkinton and A. Liebhold, Population dynamics of gypsy moth in North America,, Annual Review of Entomology, 35 (1990), 571. Google Scholar |
[16] |
S. P. Ellner and M. Rees, Integral projection models for species with complex demography,, The American Naturalist, 167 (2006), 410.
doi: 10.1086/499438. |
[17] |
S. P. Ellner and M. Rees, Stochastic stable population growth in integral projection models: Theory and application,, Journal of Mathematical Biology, 54 (2007), 227.
doi: 10.1007/s00285-006-0044-8. |
[18] |
R. S. Epanchin-Niell and A. Hastings, Controlling established invaders: Integrating economics and spread dynamics to determine optimal management,, Ecology Letters, 13 (2010), 528.
doi: 10.1111/j.1461-0248.2010.01440.x. |
[19] |
M. A. Foster, J. C. Schultz and M. D. Hunter, Modelling gypsy moth-virus-leaf chemistry interactions: Implications of plant quality for pest and pathogen dynamics,, Journal of Animal Ecology, 61 (1992), 509.
doi: 10.2307/5606. |
[20] |
H. Gaff, H. R. Joshi and S. Lenhart, Optimal harvesting during an invasion of a sublethal plant pathogen,, Environment and Development Economics, 12 (2007), 673.
doi: 10.1017/S1355770X07003828. |
[21] |
T. Glare, E. Newby and T. Nelson, Safety testing of a nuclear polyhedrosis virus for use against gypsy moth, Lymantria dispar, in New Zealand,, Proceedings of the Forty-Eighth New Zealand Plant Protection Congress, 8 (1995), 264. Google Scholar |
[22] |
A. Hastings, K. Cuddington, K. F. Davies, C. J. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holl , J. Lambrinos, U. Malvadkar, B. A. Melbourne, K. Moore, C. Taylor and D. Thomson, The spatial spread of invasions: New developments in theory and evidence,, Ecology Letters, 8 (2005), 91.
doi: 10.1111/j.1461-0248.2004.00687.x. |
[23] |
K. J. Haynes, A. M. Liebhold and D. M. Johnson, Elevational gradient in the cyclicity of a forest-defoliating insect,, Population Ecology, 54 (2012), 239.
doi: 10.1007/s10144-012-0305-x. |
[24] |
J. Heavilin, The Red Top Model: A Landscape Scale Integrodifference Equation Model of the Mountain Pine Beetle-Lodgepole Pine Forest Interaction,, Ph.D thesis, (2007).
|
[25] |
J. Jacobsen, Y. Jin and M. A. Lewis, Integrodifference models for persistence in temporally varying river environments,, Journal of Mathematical Biology, 70 (2015), 549.
doi: 10.1007/s00285-014-0774-y. |
[26] |
E. Jongejans, K. Shea, O. Skarpaas, D. Kelly and S. P. Ellner, Importance of individual variation to the spread of invasive species: A spatial integral projection model,, Ecology, 92 (2011), 86.
doi: 10.1890/09-2226.1. |
[27] |
H. R. Joshi, S. Lenhart and H. Gaff, Optimal harvesting in an integrodifference population model,, Optimal Control Applications and Methods, 27 (2006), 61.
doi: 10.1002/oca.763. |
[28] |
H. R. Joshi, S. Lenhart, H. Lou and H. Gaff, Harvesting control in an integrodifference population model with concave growth term,, Nonlinear Analysis: Hybrid Systems, 1 (2007), 417.
doi: 10.1016/j.nahs.2006.10.010. |
[29] |
J. M. Kean and N. D. Barlow, A spatial model for the successful biological control of sitona discoideus by microctonus aethiopoides,, The Journal of Applied Ecology, 38 (2001), 162.
doi: 10.1046/j.1365-2664.2001.00579.x. |
[30] |
M. Kot, Discrete-time traveling waves: Ecological examples,, Journal of Mathematical Biology, 30 (1992), 413.
doi: 10.1007/bf00173295. |
[31] |
M. Kot, M. A. Lewis and M. G. Neubert, Integrodifference equations,, in Encyclopedia of Theoretical Ecology (eds. A. Hastings and L. Gross), (2012), 382. Google Scholar |
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M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027.
doi: 10.2307/2265698. |
[33] |
M. Kot and W. M. Schaffer, Discrete time growth dispersal models,, Mathematical Biosciences, 80 (1986), 109.
doi: 10.1016/0025-5564(86)90069-6. |
[34] |
S. Lenhart, E. N. Bodine, P. Zhong and H. Joshi, Illustrating optimal control applications with discrete and continuous features,, in Advances in Applied Mathematics, 66 (2013), 209.
doi: 10.1007/978-1-4614-5389-5_9. |
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S. Lenhart and J. T. Workman, Optimal Control of Biological Models,, Chapman and Hall/CRC, (2007).
|
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A. Liebhold, J. Halverson and G. Elmes, Quantitative analysis of the invasion of gypsy moth in North America,, Journal of Biogeography, 19 (1992), 513.
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J. Musgrave and F. Lutscher, Integrodifference equations in patchy landscapes,, Journal of Mathematical Biology, 69 (2014), 583.
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M. G. Neubert, M. Kot and M. A. Lewis, Invasion speeds in fluctuating environments,, Proceedings of the Royal Society Biological Sciences Series B, 267 (2000), 1603.
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doi: 10.1111/j.1096-3642.1935.tb01680.x. |
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doi: 10.1007/978-1-4757-4978-6. |
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T. A. Perkins, B. L. Phillips, M. L. Baskett and A. Hastings, Evolution of dispersal and life history interact to drive accelerating spread of an invasive species,, Ecology Letters, 16 (2013), 1079.
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show all references
References:
[1] |
, Gypsy moth management in the United States: a cooperative approach. Final Environmental Impact Statement Vol. 1-5, USDA-Forest Service and Animal and Plant Health Inspection Service,, NA-MR-01-08, (2008), 01. Google Scholar |
[2] |
, Harmful non-indigenous species in the United States, U.S. Congress, Office of Technology Assessment,, OTA-F-565, (1993). Google Scholar |
[3] |
K. Barber, W. Kaupp and S. Holmes, Specificity testing of the nuclear polyhedrosis virus of the gypsy moth, Lymantria dispar (L.),, The Canadian Entomologist, 125 (1993), 1023.
doi: 10.4039/ent1251055-6. |
[4] |
P. Barbosa, D. K. Letourneau and A. A. Agrawal, Insect Outbreaks Revisited,, Wiley-Blackwell, (2012).
doi: 10.1002/9781118295205. |
[5] |
O. N. Bjørnstad, C. Robinet and A. M. Liebhold, Geographic variation in North-American gypsy moth population cycles: Sub-harmonics, generalist predators and spatial coupling,, Ecology, 91 (2010), 106.
doi: 10.1890/08-1246.1. |
[6] |
J. Briggs, J. Dabbs, M. Holm, J. Lubben, R. Rebarber, B. Tenhumberg and D. Riser-Espinoza, Structured population dynamics: An introduction to integral modeling,, Mathematics Magazine, 83 (2010), 243.
doi: 10.4169/002557010X521778. |
[7] |
N. F. Britton, Essential Mathematical Biology,, Springer-Verlag, (2003).
doi: 10.1007/978-1-4471-0049-2. |
[8] |
J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord,, The American Naturalist, 152 (1998), 204.
doi: 10.1086/286162. |
[9] |
C. Doane and M. McManus, The Gypsy Moth: Research toward Integrated Pest Management,, U.S. Department of Agriculture, (1981). Google Scholar |
[10] |
G. Dwyer, J. Dushoff, J. S. Elkinton and S. A. Levin, Pathogen-driven outbreaks in forest defoliators revisited: Building models from experimental data,, The American Naturalist, 156 (2000), 105.
doi: 10.1086/303379. |
[11] |
G. Dwyer, J. Dushoff, J. S. and S. Yee, The combined effects of pathogens and predators on insect outbreaks,, Nature, 430 (2004), 341.
doi: 10.1038/nature02569. |
[12] |
G. Dwyer and J. S. Elkinton, Using simple models to predict virus epizootics in gypsy moth populations,, Journal of Animal Ecology, 62 (1993), 1.
doi: 10.2307/5477. |
[13] |
E. A. Eager, R. Rebarber and B. Tenhumberg, Choice of density-dependent seedling recruitment function affects predicted transient dynamics: A case study with Platte thistle,, Theoretical Ecology, 5 (2012), 387.
doi: 10.1007/s12080-011-0131-3. |
[14] |
M. R. Easterling, S. P. Ellner and P. M. Dixon, Importance of individual variation to the spread of invasive species: A spatial integral projection model,, Ecology, 92 (2011), 86.
doi: 10.1890/09-2226.1. |
[15] |
J. Elkinton and A. Liebhold, Population dynamics of gypsy moth in North America,, Annual Review of Entomology, 35 (1990), 571. Google Scholar |
[16] |
S. P. Ellner and M. Rees, Integral projection models for species with complex demography,, The American Naturalist, 167 (2006), 410.
doi: 10.1086/499438. |
[17] |
S. P. Ellner and M. Rees, Stochastic stable population growth in integral projection models: Theory and application,, Journal of Mathematical Biology, 54 (2007), 227.
doi: 10.1007/s00285-006-0044-8. |
[18] |
R. S. Epanchin-Niell and A. Hastings, Controlling established invaders: Integrating economics and spread dynamics to determine optimal management,, Ecology Letters, 13 (2010), 528.
doi: 10.1111/j.1461-0248.2010.01440.x. |
[19] |
M. A. Foster, J. C. Schultz and M. D. Hunter, Modelling gypsy moth-virus-leaf chemistry interactions: Implications of plant quality for pest and pathogen dynamics,, Journal of Animal Ecology, 61 (1992), 509.
doi: 10.2307/5606. |
[20] |
H. Gaff, H. R. Joshi and S. Lenhart, Optimal harvesting during an invasion of a sublethal plant pathogen,, Environment and Development Economics, 12 (2007), 673.
doi: 10.1017/S1355770X07003828. |
[21] |
T. Glare, E. Newby and T. Nelson, Safety testing of a nuclear polyhedrosis virus for use against gypsy moth, Lymantria dispar, in New Zealand,, Proceedings of the Forty-Eighth New Zealand Plant Protection Congress, 8 (1995), 264. Google Scholar |
[22] |
A. Hastings, K. Cuddington, K. F. Davies, C. J. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holl , J. Lambrinos, U. Malvadkar, B. A. Melbourne, K. Moore, C. Taylor and D. Thomson, The spatial spread of invasions: New developments in theory and evidence,, Ecology Letters, 8 (2005), 91.
doi: 10.1111/j.1461-0248.2004.00687.x. |
[23] |
K. J. Haynes, A. M. Liebhold and D. M. Johnson, Elevational gradient in the cyclicity of a forest-defoliating insect,, Population Ecology, 54 (2012), 239.
doi: 10.1007/s10144-012-0305-x. |
[24] |
J. Heavilin, The Red Top Model: A Landscape Scale Integrodifference Equation Model of the Mountain Pine Beetle-Lodgepole Pine Forest Interaction,, Ph.D thesis, (2007).
|
[25] |
J. Jacobsen, Y. Jin and M. A. Lewis, Integrodifference models for persistence in temporally varying river environments,, Journal of Mathematical Biology, 70 (2015), 549.
doi: 10.1007/s00285-014-0774-y. |
[26] |
E. Jongejans, K. Shea, O. Skarpaas, D. Kelly and S. P. Ellner, Importance of individual variation to the spread of invasive species: A spatial integral projection model,, Ecology, 92 (2011), 86.
doi: 10.1890/09-2226.1. |
[27] |
H. R. Joshi, S. Lenhart and H. Gaff, Optimal harvesting in an integrodifference population model,, Optimal Control Applications and Methods, 27 (2006), 61.
doi: 10.1002/oca.763. |
[28] |
H. R. Joshi, S. Lenhart, H. Lou and H. Gaff, Harvesting control in an integrodifference population model with concave growth term,, Nonlinear Analysis: Hybrid Systems, 1 (2007), 417.
doi: 10.1016/j.nahs.2006.10.010. |
[29] |
J. M. Kean and N. D. Barlow, A spatial model for the successful biological control of sitona discoideus by microctonus aethiopoides,, The Journal of Applied Ecology, 38 (2001), 162.
doi: 10.1046/j.1365-2664.2001.00579.x. |
[30] |
M. Kot, Discrete-time traveling waves: Ecological examples,, Journal of Mathematical Biology, 30 (1992), 413.
doi: 10.1007/bf00173295. |
[31] |
M. Kot, M. A. Lewis and M. G. Neubert, Integrodifference equations,, in Encyclopedia of Theoretical Ecology (eds. A. Hastings and L. Gross), (2012), 382. Google Scholar |
[32] |
M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027.
doi: 10.2307/2265698. |
[33] |
M. Kot and W. M. Schaffer, Discrete time growth dispersal models,, Mathematical Biosciences, 80 (1986), 109.
doi: 10.1016/0025-5564(86)90069-6. |
[34] |
S. Lenhart, E. N. Bodine, P. Zhong and H. Joshi, Illustrating optimal control applications with discrete and continuous features,, in Advances in Applied Mathematics, 66 (2013), 209.
doi: 10.1007/978-1-4614-5389-5_9. |
[35] |
S. Lenhart and J. T. Workman, Optimal Control of Biological Models,, Chapman and Hall/CRC, (2007).
|
[36] |
S. Lenhart and P. Zhong, Investigating the order of events in optimal control of integrodifference equations,, Systems Theory: Modeling, (2009), 89. Google Scholar |
[37] |
A. Liebhold, J. Halverson and G. Elmes, Quantitative analysis of the invasion of gypsy moth in North America,, Journal of Biogeography, 19 (1992), 513.
doi: 10.2307/2845770. |
[38] |
G. M. MacDonald, Fossil pollen analysis and the reconstruction of plant invasions,, Advances in Ecological Research, 24 (1993), 67.
doi: 10.1016/S0065-2504(08)60041-0. |
[39] |
R. M. May, Stability and complexity in model ecosystems,, IEEE Transactions on Systems, SMC-6 (1976).
doi: 10.1109/TSMC.1976.4309488. |
[40] |
R. Mendes, W. A. Conde and R. A. Kraenkel, Integrodifference model for blowfly invasion,, Theoretical Ecology, 5 (2012), 363.
doi: 10.1007/s12080-012-0157-1. |
[41] |
J. M. Morales, P. R. Moorcroft, J. Matthiopoulos, J. L. Frair, J. G. Kie, R. A. Powell, E. H. Merrill and D. T. Haydon, Building the bridge between animal movement and population dynamics,, Philosophical Transactions of the Royal Society B, 365 (2010), 2289.
doi: 10.1098/rstb.2010.0082. |
[42] |
J. D. Murray, Mathematical Biology I: An Introduction,, Springer-Verlag, (2002).
doi: 10.1007/b98868. |
[43] |
J. Musgrave and F. Lutscher, Integrodifference equations in patchy landscapes,, Journal of Mathematical Biology, 69 (2014), 583.
doi: 10.1007/s00285-013-0714-2. |
[44] |
M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations,, Ecology, 81 (2000), 1613. Google Scholar |
[45] |
M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern-formation in a discrete-time predator-prey model,, Theoretical Population Biology, 48 (1995), 7.
doi: 10.1006/tpbi.1995.1020. |
[46] |
M. G. Neubert, M. Kot and M. A. Lewis, Invasion speeds in fluctuating environments,, Proceedings of the Royal Society Biological Sciences Series B, 267 (2000), 1603.
doi: 10.1098/rspb.2000.1185. |
[47] |
A. J. Nicholson and V. A. Bailey, The balance of animal populations,, Proceedings Zoological Society London, 105 (1935), 551.
doi: 10.1111/j.1096-3642.1935.tb01680.x. |
[48] |
A. Okubo and S. Levin, Diffusion and Ecological Problems, Modern Perspectives,, Springer, (2001).
doi: 10.1007/978-1-4757-4978-6. |
[49] |
T. A. Perkins, B. L. Phillips, M. L. Baskett and A. Hastings, Evolution of dispersal and life history interact to drive accelerating spread of an invasive species,, Ecology Letters, 16 (2013), 1079.
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