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August  2015, 20(6): 1759-1783. doi: 10.3934/dcdsb.2015.20.1759

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

Received  October 2013 Revised  November 2013 Published  June 2015

We develop the theory of optimal control for a system of integrodifference equations modelling a pest-pathogen system. Integrodifference equations incorporate continuous space into a system of discrete time equations. We design an objective functional to minimize the damaged cost generated by an invasive species and the cost of controlling the population with a pathogen. Existence, characterization, and uniqueness results for the optimal control and corresponding states have been completed. We use a forward-backward sweep numerical method to implement our optimization which produces spatio-temporal control strategies for the gypsy moth case study.
Citation: Marco V. Martinez, Suzanne Lenhart, K. A. Jane White. Optimal control of integrodifference equations in a pest-pathogen system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1759-1783. doi: 10.3934/dcdsb.2015.20.1759
References:
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, 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

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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.  Google Scholar

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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.  Google Scholar

[6]

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N. F. Britton, Essential Mathematical Biology,, Springer-Verlag, (2003).  doi: 10.1007/978-1-4471-0049-2.  Google Scholar

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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.  Google Scholar

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C. Doane and M. McManus, The Gypsy Moth: Research toward Integrated Pest Management,, U.S. Department of Agriculture, (1981).   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. P. Ellner and M. Rees, Integral projection models for species with complex demography,, The American Naturalist, 167 (2006), 410.  doi: 10.1086/499438.  Google Scholar

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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.  Google Scholar

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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).   Google Scholar

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[27]

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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.  Google Scholar

[4]

P. Barbosa, D. K. Letourneau and A. A. Agrawal, Insect Outbreaks Revisited,, Wiley-Blackwell, (2012).  doi: 10.1002/9781118295205.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. F. Britton, Essential Mathematical Biology,, Springer-Verlag, (2003).  doi: 10.1007/978-1-4471-0049-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Elkinton and A. Liebhold, Population dynamics of gypsy moth in North America,, Annual Review of Entomology, 35 (1990), 571.   Google Scholar

[16]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Kot, Discrete-time traveling waves: Ecological examples,, Journal of Mathematical Biology, 30 (1992), 413.  doi: 10.1007/bf00173295.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

S. Lenhart and J. T. Workman, Optimal Control of Biological Models,, Chapman and Hall/CRC, (2007).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

R. M. May, Stability and complexity in model ecosystems,, IEEE Transactions on Systems, SMC-6 (1976).  doi: 10.1109/TSMC.1976.4309488.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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