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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:
[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.

[2]

, Harmful non-indigenous species in the United States, U.S. Congress, Office of Technology Assessment,, OTA-F-565, (1993).

[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).

[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.

[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.

[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.

[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.

[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.

[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. doi: 10.1111/ele.12136.

[50]

B. L. Phillips, G. P. Brown and R. Shine, Evolutionarily accelerated invasions: The rate of dispersal evolves upwards during the range advance of cane toads,, Journal of Evolutionary Biology, 23 (2010), 2595. doi: 10.1111/j.1420-9101.2010.02118.x.

[51]

D. Pimentel, R. Zuniga and D. Morrison, Update on the environmental and economic costs associated with alien-invasive species in the United States,, Ecological Economics, 52 (2005), 273. doi: 10.1016/j.ecolecon.2004.10.002.

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M. H. Wang, M. Kot and M. G. Neubert, Integrodifference equations, Allee effects, and invasions,, Journal of Mathematical Biology, 44 (2002), 150. doi: 10.1007/s002850100116.

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S. M. White and K. A. J. White, Relating coupled map lattices to integro-difference equations: Dispersal-driven instabilities in coupled map lattices,, Journal of Theoretical Biology, 235 (2005), 463. doi: 10.1016/j.jtbi.2005.01.026.

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Y. Zhou and M. Kot, Life on the move: Modeling the effects of climate-driven range shifts with integrodifference equations,, in Dispersal, (2071), 263. doi: 10.1007/978-3-642-35497-7_9.

[72]

P. Zhong and S. Lenhart, Optimal control of integrodifference equations with growth-harvesting-dispersal order,, Discrete and Continuous Dynamical Systems - Series B, 17 (2012), 2281. doi: 10.3934/dcdsb.2012.17.2281.

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.

[2]

, Harmful non-indigenous species in the United States, U.S. Congress, Office of Technology Assessment,, OTA-F-565, (1993).

[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).

[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.

[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.

[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.

[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.

[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.

[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. doi: 10.1111/ele.12136.

[50]

B. L. Phillips, G. P. Brown and R. Shine, Evolutionarily accelerated invasions: The rate of dispersal evolves upwards during the range advance of cane toads,, Journal of Evolutionary Biology, 23 (2010), 2595. doi: 10.1111/j.1420-9101.2010.02118.x.

[51]

D. Pimentel, R. Zuniga and D. Morrison, Update on the environmental and economic costs associated with alien-invasive species in the United States,, Ecological Economics, 52 (2005), 273. doi: 10.1016/j.ecolecon.2004.10.002.

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