American Institute of Mathematical Sciences

Advanced
  • Previous Article
    How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?
  • DCDS-B Home
  • This Issue
  • Next Article
    Hopf bifurcation for a spatially and age structured population dynamics model
August  2015, 20(6): 1759-1783. doi: 10.3934/dcdsb.2015.20.1759

Optimal control of integrodifference equations in a pest-pathogen system

Marco V. Martinez 1, , Suzanne Lenhart 2, and  K. A. Jane White 3,

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.
Keywords: pest-pathogen, Integrodifference equations, optimal control, invasive species..
Mathematics Subject Classification: Primary: 49J21, 92D30; Secondary: 92D4.
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.   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), 1023-1031. doi: 10.4039/ent1251055-6.  Google Scholar

[4]

P. Barbosa, D. K. Letourneau and A. A. Agrawal, Insect Outbreaks Revisited, Wiley-Blackwell, Hoboken, 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), 106-118. 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), 243-257. doi: 10.4169/002557010X521778.  Google Scholar

[7]

N. F. Britton, Essential Mathematical Biology, Springer-Verlag, New York, 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), 204-224. 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, Washington D.C., 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), 105-120. 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), 341-345. 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), 1-11. 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), 387-401. 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), 86-97. 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), 571-596. Google Scholar

[16]

S. P. Ellner and M. Rees, Integral projection models for species with complex demography, The American Naturalist, 167 (2006), 410-428. doi: 10.1086/499438.  Google Scholar

[17]

S. P. Ellner and M. Rees, Stochastic stable population growth in integral projection models: Theory and application, Journal of Mathematical Biology, 54 (2007), 227-256. 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), 528-541. 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), 509-520. 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), 673-686. 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), 264-269. 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), 91-101. 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), 239-250. 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, Utah State University, 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), 549-590. 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), 86-97. 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), 61-75. 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), 417-429. 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), 162-169. 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), 413-436. 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), University of California Press, 2012, 382-384. Google Scholar

[32]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042. doi: 10.2307/2265698.  Google Scholar

[33]

M. Kot and W. M. Schaffer, Discrete time growth dispersal models, Mathematical Biosciences, 80 (1986), 109-136. 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, Modeling, and Computational Science (eds. R. Melnik and I. Kotsireas), University of California Press, 66 (2013), 209-238. 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, New York, 2007.  Google Scholar

[36]

S. Lenhart and P. Zhong, Investigating the order of events in optimal control of integrodifference equations, Systems Theory: Modeling, Analysis and Control Proceedings Volume, (2009), 89-100. 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), 513-520. 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), 67-110. doi: 10.1016/S0065-2504(08)60041-0.  Google Scholar

[39]

R. M. May, Stability and complexity in model ecosystems, IEEE Transactions on Systems, Man and Cybernetics, SMC-6 (1976), p887. 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), 363-371. 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), 2289-2301. doi: 10.1098/rstb.2010.0082.  Google Scholar

[42]

J. D. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[43]

J. Musgrave and F. Lutscher, Integrodifference equations in patchy landscapes, Journal of Mathematical Biology, 69 (2014), 583-615. doi: 10.1007/s00285-013-0714-2.  Google Scholar

[44]

M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613-1628. 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), 7-43. doi: 10.1006/tpbi.1995.1020.  Google Scholar

[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), 1603-1610. doi: 10.1098/rspb.2000.1185.  Google Scholar

[47]

A. J. Nicholson and V. A. Bailey, The balance of animal populations, Proceedings Zoological Society London, 105 (1935), 551-598. doi: 10.1111/j.1096-3642.1935.tb01680.x.  Google Scholar

[48]

A. Okubo and S. Levin, Diffusion and Ecological Problems, Modern Perspectives, Springer, New York 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[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), 1079-1087. doi: 10.1111/ele.12136.  Google Scholar

[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), 2595-2601. doi: 10.1111/j.1420-9101.2010.02118.x.  Google Scholar

[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), 273-288. doi: 10.1016/j.ecolecon.2004.10.002.  Google Scholar

[52]

J. Podgwaite, Gypchek - biological insecticide for the gypsy moth, Journal of Forestry, 97 (1999), 16-19. Google Scholar

[53]

P. Pyšek, V. Jarošik, P. E. Hulme, J. Pergl, M. Hejda, U. Schaffner and M. Vilà, A global assessment of invasive plant impacts on resident species, communities and ecosystems: the interaction of impact measures, invading species' traits and environment, Global Change Biology, 18 (2012), 1725-1737. doi: 10.1111/j.1365-2486.2011.02636.x.  Google Scholar

[54]

R. Reardon, J. Podgwaite and R. Zerillo, Gypchek-The Gypsy Moth Nucleopolyhedrosis Virus Product, USDA Forest Service, FHTET-96-16, 1996. Google Scholar

[55]

R. Reardon, J. Podgwaite and R. Zerillo, Gypchek - Bioinsecticide For The Gypsy Moth, USDA Forest Service, FHTET-2009-01, 2009. Available from: http://www.nrs.fs.fed.us/pubs/gtr/gtr_nrs6.pdf. Google Scholar

[56]

M. Rees and K. E. Rose, Evolution of flowering strategies in Oenothera glazioviana: An integral projection model approach, Proceedings of the Royal Society B, 269 (2002), 1509-1515. doi: 10.1098/rspb.2002.2037.  Google Scholar

[57]

M. Shapiro, H. K. Preisler and J. L. Robertson, Enhancement of baculovirus activity on gypsy moth (Lepidoptera: Lymantriidae) by chitinas, Journal of Economic Entomology, 80 (1987), 1113-1116. doi: 10.1093/jee/80.6.1113.  Google Scholar

[58]

A. A. Sharov, D. Leonard, A. M. Liebhold and N. S. Clemens, Evaluation of preventive treatments in low-density gypsy moth populations using pheromone traps, Journal of Economic Entomology, 95 (2002), 1205-1215. doi: 10.1603/0022-0493-95.6.1205.  Google Scholar

[59]

A. A. Sharov, D. Leonard, A. M. Liebhold, E. A. Roberts and W. Dickerson, "Slow the spread{''}: A national program to contain the gypsy moth, Journal of Forestry, 100 (2002), 30-35. Google Scholar

[60]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, New York, 1997. Google Scholar

[61]

J. B. Skellam, Random dispersal in theoretical population, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[62]

P. Tobin and L. Blackburn, Slow the Spread: A National Program to Manage the Gypsy Moth, USDA Forest Service, Northern Research Station, NRS-6, 2007. Available from: http://www.nrs.fs.fed.us/pubs/gtr/gtr_nrs6.pdf. Google Scholar

[63]

P. Tobin and A. Liebhold, Gypsy moth, in Encyclopedia of Biological Invasions (eds. D. Simberloff and M. Rejmanek), University of California Press, 2011, 298-304. Google Scholar

[64]

R. W. VanKirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137. Google Scholar

[65]

R. R. Viet and M. A. Lewis, Dispersal, population growth, and the allee effect: Dynamics of the house finch invasion of eastern North America, The American Naturalist, 148 (1996), 255-274. doi: 10.1086/285924.  Google Scholar

[66]

M. H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Mathematical Bioscience, 171 (2001), 83-97. doi: 10.1016/S0025-5564(01)00048-7.  Google Scholar

[67]

M. H. Wang, M. Kot and M. G. Neubert, Integrodifference equations, Allee effects, and invasions, Journal of Mathematical Biology, 44 (2002), 150-168. doi: 10.1007/s002850100116.  Google Scholar

[68]

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), 463-475. doi: 10.1016/j.jtbi.2005.01.026.  Google Scholar

[69]

A. Whittle, S. Lenhart and K. A. White, Optimal control of gypsy moth populations, Bulletin of Mathematical Biology, 70 (2008), 398-411. doi: 10.1007/s11538-007-9260-7.  Google Scholar

[70]

M. Williamson, Biological Invasions, Springer, New York, 1996. Google Scholar

[71]

Y. Zhou and M. Kot, Life on the move: Modeling the effects of climate-driven range shifts with integrodifference equations, in Dispersal, Individual Movement and Spatial Ecology (eds. M. A. Lewis, P. K. Maini and S. V. Petrovskii), Lecture Notes in Math., 2071, Springer, Heidelberg, 2013, 263-292. doi: 10.1007/978-3-642-35497-7_9.  Google Scholar

[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), 2281-2298. doi: 10.3934/dcdsb.2012.17.2281.  Google Scholar

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), 1023-1031. doi: 10.4039/ent1251055-6.  Google Scholar

[4]

P. Barbosa, D. K. Letourneau and A. A. Agrawal, Insect Outbreaks Revisited, Wiley-Blackwell, Hoboken, 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), 106-118. 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), 243-257. doi: 10.4169/002557010X521778.  Google Scholar

[7]

N. F. Britton, Essential Mathematical Biology, Springer-Verlag, New York, 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), 204-224. 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, Washington D.C., 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), 105-120. 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), 341-345. 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), 1-11. 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), 387-401. 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), 86-97. 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), 571-596. Google Scholar

[16]

S. P. Ellner and M. Rees, Integral projection models for species with complex demography, The American Naturalist, 167 (2006), 410-428. doi: 10.1086/499438.  Google Scholar

[17]

S. P. Ellner and M. Rees, Stochastic stable population growth in integral projection models: Theory and application, Journal of Mathematical Biology, 54 (2007), 227-256. 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), 528-541. 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), 509-520. 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), 673-686. 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), 264-269. 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), 91-101. 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), 239-250. 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, Utah State University, 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), 549-590. 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), 86-97. 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), 61-75. 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), 417-429. 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), 162-169. 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), 413-436. 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), University of California Press, 2012, 382-384. Google Scholar

[32]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042. doi: 10.2307/2265698.  Google Scholar

[33]

M. Kot and W. M. Schaffer, Discrete time growth dispersal models, Mathematical Biosciences, 80 (1986), 109-136. 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, Modeling, and Computational Science (eds. R. Melnik and I. Kotsireas), University of California Press, 66 (2013), 209-238. 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, New York, 2007.  Google Scholar

[36]

S. Lenhart and P. Zhong, Investigating the order of events in optimal control of integrodifference equations, Systems Theory: Modeling, Analysis and Control Proceedings Volume, (2009), 89-100. 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), 513-520. 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), 67-110. doi: 10.1016/S0065-2504(08)60041-0.  Google Scholar

[39]

R. M. May, Stability and complexity in model ecosystems, IEEE Transactions on Systems, Man and Cybernetics, SMC-6 (1976), p887. 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), 363-371. 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), 2289-2301. doi: 10.1098/rstb.2010.0082.  Google Scholar

[42]

J. D. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[43]

J. Musgrave and F. Lutscher, Integrodifference equations in patchy landscapes, Journal of Mathematical Biology, 69 (2014), 583-615. doi: 10.1007/s00285-013-0714-2.  Google Scholar

[44]

M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613-1628. 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), 7-43. doi: 10.1006/tpbi.1995.1020.  Google Scholar

[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), 1603-1610. doi: 10.1098/rspb.2000.1185.  Google Scholar

[47]

A. J. Nicholson and V. A. Bailey, The balance of animal populations, Proceedings Zoological Society London, 105 (1935), 551-598. doi: 10.1111/j.1096-3642.1935.tb01680.x.  Google Scholar

[48]

A. Okubo and S. Levin, Diffusion and Ecological Problems, Modern Perspectives, Springer, New York 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[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), 1079-1087. doi: 10.1111/ele.12136.  Google Scholar

[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), 2595-2601. doi: 10.1111/j.1420-9101.2010.02118.x.  Google Scholar

[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), 273-288. doi: 10.1016/j.ecolecon.2004.10.002.  Google Scholar

[52]

J. Podgwaite, Gypchek - biological insecticide for the gypsy moth, Journal of Forestry, 97 (1999), 16-19. Google Scholar

[53]

P. Pyšek, V. Jarošik, P. E. Hulme, J. Pergl, M. Hejda, U. Schaffner and M. Vilà, A global assessment of invasive plant impacts on resident species, communities and ecosystems: the interaction of impact measures, invading species' traits and environment, Global Change Biology, 18 (2012), 1725-1737. doi: 10.1111/j.1365-2486.2011.02636.x.  Google Scholar

[54]

R. Reardon, J. Podgwaite and R. Zerillo, Gypchek-The Gypsy Moth Nucleopolyhedrosis Virus Product, USDA Forest Service, FHTET-96-16, 1996. Google Scholar

[55]

R. Reardon, J. Podgwaite and R. Zerillo, Gypchek - Bioinsecticide For The Gypsy Moth, USDA Forest Service, FHTET-2009-01, 2009. Available from: http://www.nrs.fs.fed.us/pubs/gtr/gtr_nrs6.pdf. Google Scholar

[56]

M. Rees and K. E. Rose, Evolution of flowering strategies in Oenothera glazioviana: An integral projection model approach, Proceedings of the Royal Society B, 269 (2002), 1509-1515. doi: 10.1098/rspb.2002.2037.  Google Scholar

[57]

M. Shapiro, H. K. Preisler and J. L. Robertson, Enhancement of baculovirus activity on gypsy moth (Lepidoptera: Lymantriidae) by chitinas, Journal of Economic Entomology, 80 (1987), 1113-1116. doi: 10.1093/jee/80.6.1113.  Google Scholar

[58]

A. A. Sharov, D. Leonard, A. M. Liebhold and N. S. Clemens, Evaluation of preventive treatments in low-density gypsy moth populations using pheromone traps, Journal of Economic Entomology, 95 (2002), 1205-1215. doi: 10.1603/0022-0493-95.6.1205.  Google Scholar

[59]

A. A. Sharov, D. Leonard, A. M. Liebhold, E. A. Roberts and W. Dickerson, "Slow the spread{''}: A national program to contain the gypsy moth, Journal of Forestry, 100 (2002), 30-35. Google Scholar

[60]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, New York, 1997. Google Scholar

[61]

J. B. Skellam, Random dispersal in theoretical population, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[62]

P. Tobin and L. Blackburn, Slow the Spread: A National Program to Manage the Gypsy Moth, USDA Forest Service, Northern Research Station, NRS-6, 2007. Available from: http://www.nrs.fs.fed.us/pubs/gtr/gtr_nrs6.pdf. Google Scholar

[63]

P. Tobin and A. Liebhold, Gypsy moth, in Encyclopedia of Biological Invasions (eds. D. Simberloff and M. Rejmanek), University of California Press, 2011, 298-304. Google Scholar

[64]

R. W. VanKirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137. Google Scholar

[65]

R. R. Viet and M. A. Lewis, Dispersal, population growth, and the allee effect: Dynamics of the house finch invasion of eastern North America, The American Naturalist, 148 (1996), 255-274. doi: 10.1086/285924.  Google Scholar

[66]

M. H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Mathematical Bioscience, 171 (2001), 83-97. doi: 10.1016/S0025-5564(01)00048-7.  Google Scholar

[67]

M. H. Wang, M. Kot and M. G. Neubert, Integrodifference equations, Allee effects, and invasions, Journal of Mathematical Biology, 44 (2002), 150-168. doi: 10.1007/s002850100116.  Google Scholar

[68]

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), 463-475. doi: 10.1016/j.jtbi.2005.01.026.  Google Scholar

[69]

A. Whittle, S. Lenhart and K. A. White, Optimal control of gypsy moth populations, Bulletin of Mathematical Biology, 70 (2008), 398-411. doi: 10.1007/s11538-007-9260-7.  Google Scholar

[70]

M. Williamson, Biological Invasions, Springer, New York, 1996. Google Scholar

[71]

Y. Zhou and M. Kot, Life on the move: Modeling the effects of climate-driven range shifts with integrodifference equations, in Dispersal, Individual Movement and Spatial Ecology (eds. M. A. Lewis, P. K. Maini and S. V. Petrovskii), Lecture Notes in Math., 2071, Springer, Heidelberg, 2013, 263-292. doi: 10.1007/978-3-642-35497-7_9.  Google Scholar

[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), 2281-2298. doi: 10.3934/dcdsb.2012.17.2281.  Google Scholar

[1]

Andrew J. Whittle, Suzanne Lenhart, Louis J. Gross. Optimal control for management of an invasive plant species. Mathematical Biosciences & Engineering, 2007, 4 (1) : 101-112. doi: 10.3934/mbe.2007.4.101
[2]

Peng Zhong, Suzanne Lenhart. Study on the order of events in optimal control of a harvesting problem modeled by integrodifference equations. Evolution Equations & Control Theory, 2013, 2 (4) : 749-769. doi: 10.3934/eect.2013.2.749
[3]

Peng Zhong, Suzanne Lenhart. Optimal control of integrodifference equations with growth-harvesting-dispersal order. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2281-2298. doi: 10.3934/dcdsb.2012.17.2281
[4]

Linhao Xu, Marya Claire Zdechlik, Melissa C. Smith, Min B. Rayamajhi, Don L. DeAngelis, Bo Zhang. Simulation of post-hurricane impact on invasive species with biological control management. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 4059-4071. doi: 10.3934/dcds.2020038
[5]

Linda J. S. Allen, Vrushali A. Bokil. Stochastic models for competing species with a shared pathogen. Mathematical Biosciences & Engineering, 2012, 9 (3) : 461-485. doi: 10.3934/mbe.2012.9.461
[6]

Erin N. Bodine, Louis J. Gross, Suzanne Lenhart. Optimal control applied to a model for species augmentation. Mathematical Biosciences & Engineering, 2008, 5 (4) : 669-680. doi: 10.3934/mbe.2008.5.669
[7]

Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3157-3187. doi: 10.3934/dcdss.2020163
[8]

H. T. Banks, R. A. Everett, Neha Murad, R. D. White, J. E. Banks, Bodil N. Cass, Jay A. Rosenheim. Optimal design for dynamical modeling of pest populations. Mathematical Biosciences & Engineering, 2018, 15 (4) : 993-1010. doi: 10.3934/mbe.2018044
[9]

Chaohong Pan, Hongyong Wang, Chunhua Ou. Invasive speed for a competition-diffusion system with three species. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021194
[10]

Luis F. Gordillo. Optimal sterile insect release for area-wide integrated pest management in a density regulated pest population. Mathematical Biosciences & Engineering, 2014, 11 (3) : 511-521. doi: 10.3934/mbe.2014.11.511
[11]

Adèle Bourgeois, Victor LeBlanc, Frithjof Lutscher. Dynamical stabilization and traveling waves in integrodifference equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3029-3045. doi: 10.3934/dcdss.2020117
[12]

Guirong Jiang, Qishao Lu, Linping Peng. Impulsive Ecological Control Of A Stage-Structured Pest Management System. Mathematical Biosciences & Engineering, 2005, 2 (2) : 329-344. doi: 10.3934/mbe.2005.2.329
[13]

Holly Gaff, Robyn Nadolny. Identifying requirements for the invasion of a tick species and tick-borne pathogen through TICKSIM. Mathematical Biosciences & Engineering, 2013, 10 (3) : 625-635. doi: 10.3934/mbe.2013.10.625
[14]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279
[15]

Eduardo Casas, Konstantinos Chrysafinos. Analysis and optimal control of some quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 607-623. doi: 10.3934/mcrf.2018025
[16]

Gregory Zitelli, Seddik M. Djouadi, Judy D. Day. Combining robust state estimation with nonlinear model predictive control to regulate the acute inflammatory response to pathogen. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1127-1139. doi: 10.3934/mbe.2015.12.1127
[17]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ Ⅱ: Discrete torus bifurcations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1847-1874. doi: 10.3934/cpaa.2020081
[18]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231
[19]

Leszek Gasiński, Nikolaos S. Papageorgiou. Relaxation of optimal control problems driven by nonlinear evolution equations. Evolution Equations & Control Theory, 2020, 9 (4) : 1027-1040. doi: 10.3934/eect.2020050
[20]

Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (101)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences