September  2012, 17(6): 2281-2298. doi: 10.3934/dcdsb.2012.17.2281

Optimal control of integrodifference equations with growth-harvesting-dispersal order

1. 

Department of Ecology, Evolution, and Natural Resources, Rutgers University, New Brunswick, NJ 08901, United States

2. 

Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300

Received  September 2011 Revised  November 2011 Published  May 2012

Integrodifference equations are discrete in time and continuous in space, and are used to model the spread of populations that are growing in discrete generations, or at discrete times, and dispersing spatially. We investigate optimal harvesting strategies, in order to maximize the profit and minimize the cost of harvesting. Theoretical results on the existence, uniqueness and characterization, as well as numerical results of optimized harvesting rates are obtained. The order of how the three events, growth, dispersal and harvesting, are arranged affects the harvesting behavior.
Citation: 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
References:
[1]

D. A. Andow, P. M. Kareiva, Simon A. Levin and Akira Okubo, Spread of invading organisms,, Landscape Ecology, 4 (1990), 177.   Google Scholar

[2]

M. Andersen, Properties of some density-dependent integrodifference equation population models,, Mathematical Biosciences, 104 (1991), 135.  doi: 10.1016/0025-5564(91)90034-G.  Google Scholar

[3]

A. J. Bateman, Is gene dispersion normal,, Heredity, 4 (1950), 353.   Google Scholar

[4]

M. G. Bhat, K. R. Fister and S. Lenhart, An optimal control model for surface runoff contamination of a large river basin,, Natural Resource Modeling Journal, 12 (1999), 175.  doi: 10.1111/j.1939-7445.1999.tb00009.x.  Google Scholar

[5]

S. Chandrasekhar, Stochastic problems in physics and astronomy,, Reviews of Modern Physics, 15 (1943), 1.  doi: 10.1103/RevModPhys.15.1.  Google Scholar

[6]

J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord,, American Naturalist, 152 (1998), 204.  doi: 10.1086/286162.  Google Scholar

[7]

Michael R. Easterling, Stephen P. Ellner and Philip M. Dixon, Size-specific sensitivity: Applying a new structured population model,, Ecological Society of America, 81 (2000), 694.   Google Scholar

[8]

Ivar Ekeland and Roger Témam, "Convex Analysis and Variational Problems,", Studies in Mathematics and its Applications, (1976).   Google Scholar

[9]

Enrico Fermi, "Thermodynamics,'', Dover Publications, (1956).   Google Scholar

[10]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,'', Applications of Mathematics, (1975).   Google Scholar

[11]

H. I. Freedman, J. B. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment,, Mathematical Biosciences, 95 (1989), 111.  doi: 10.1016/0025-5564(89)90055-2.  Google Scholar

[12]

H. Gaff, H. R. Joshi and S. Lenhart, Optimal harvesting during an invasion of a sublethal plant pathogen,, Environment and Development Economics Journal, 12 (2007), 673.   Google Scholar

[13]

W. Hackbush, A numerical method for solving parabolic equations with opposite orientations,, Computing, 20 (1978), 229.  doi: 10.1007/BF02251947.  Google Scholar

[14]

E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit, Partial differential equations in ecology spatial interactions and population dynamics,, Ecology, 75 (1994), 18.  doi: 10.2307/1939378.  Google Scholar

[15]

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

[16]

H. R. Joshi, S. Lenhart, H. Lou and H. Gaff, Harvesting control in an integrodifference population model with concave growth term,, Nonlinear Anal. Hybrid Syst, 1 (2007), 417.   Google Scholar

[17]

John M. Kean and Nigel D. Barlow, A spatial model for the successful biological control of sitona discoideus by microctonus aethiopoides,, The Journal of Applied Ecology, 1 (2001), 162.   Google Scholar

[18]

M. Kot and W. Schaffer, Discrete-time growth-dispersal models,, Math. Biosci., 80 (1986), 109.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar

[19]

M. Kot, Discrete-time travelling waves: Ecological examples,, Journal of Mathematical Biology, 30 (1992), 413.  doi: 10.1007/BF00173295.  Google Scholar

[20]

M. Kot, Do invading organisms do the wave,, Canadian Applied Mathematics Quarterly, 10 (2002), 139.   Google Scholar

[21]

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

[22]

Suzanne Lenhart and John Workman, "Optimal Control Applied to Biological Models,'', Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007).   Google Scholar

[23]

M. A. Lewis, Variability, patchiness, and jump dispersal in the spread of an invading population,, in, (1997), 46.   Google Scholar

[24]

Xun Jing Li and Jiong Min Yong, "Optimal Control Theory for Infinite Dimensional Systems,'', Systems & Control: Foundations & Applications, (1995).   Google Scholar

[25]

D. L. Lukes, "Differential Equations. Classical to Controlled,'', Mathematics in Science and Engineering, 162 (1982).   Google Scholar

[26]

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

[27]

J. D. Murray, E. A. Stanley and D. L. Brown, On the spread of rabies among foxes,, Proc. Roy. Soc. London Ser., 229 (1986), 111.  doi: 10.1098/rspb.1986.0078.  Google Scholar

[28]

M. 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.  Google Scholar

[29]

L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'', Wiley, (1956).   Google Scholar

[30]

M. Slatkin, Gene flow and selection in a cline,, Genetics, 75 (1973), 733.   Google Scholar

[31]

M. Slatkin, Gene flow and selection in a two-locus system,, Genetics, 81 (1975), 787.   Google Scholar

[32]

J. Lubben, D. Boeckner, R. Rebarber, S. Townley and B. Tenhumberg, Parameterizing the growth-decline boundary for uncertain population projection models,, Theoretical Population Biology, 75 (2009), 85.  doi: 10.1016/j.tpb.2008.11.004.  Google Scholar

[33]

Richard Rebarber, Brigitte Tenhumberg and Stuart Townley, Global asymptotic stability of density dependent integral population projection models,, Theoretical Population Biology, 81 (2002), 81.  doi: 10.1016/j.tpb.2011.11.002.  Google Scholar

[34]

C. Reid, "The origin of the British Flora,'', Dualu, (1899).   Google Scholar

[35]

S. P. Sethi and G. L. Thompson, "Optimal Control Theory. Applications to Management Science and Economics,'', Second edition, (2000).   Google Scholar

[36]

M. A. Lewis and R. W. Van Kirk, Integrodifference models for persistence in fragmented habitats,, Bulletin of Mathematical Biology, 59 (1997), 107.  doi: 10.1016/S0092-8240(96)00060-2.  Google Scholar

[37]

H. F. Weinberger, Asymptotic behavior of a model in population genetics,, in, 648 (1978), 1976.   Google Scholar

[38]

K. Yosida, "Functional Analysis,'', 6th edition, 123 (1980).   Google Scholar

show all references

References:
[1]

D. A. Andow, P. M. Kareiva, Simon A. Levin and Akira Okubo, Spread of invading organisms,, Landscape Ecology, 4 (1990), 177.   Google Scholar

[2]

M. Andersen, Properties of some density-dependent integrodifference equation population models,, Mathematical Biosciences, 104 (1991), 135.  doi: 10.1016/0025-5564(91)90034-G.  Google Scholar

[3]

A. J. Bateman, Is gene dispersion normal,, Heredity, 4 (1950), 353.   Google Scholar

[4]

M. G. Bhat, K. R. Fister and S. Lenhart, An optimal control model for surface runoff contamination of a large river basin,, Natural Resource Modeling Journal, 12 (1999), 175.  doi: 10.1111/j.1939-7445.1999.tb00009.x.  Google Scholar

[5]

S. Chandrasekhar, Stochastic problems in physics and astronomy,, Reviews of Modern Physics, 15 (1943), 1.  doi: 10.1103/RevModPhys.15.1.  Google Scholar

[6]

J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord,, American Naturalist, 152 (1998), 204.  doi: 10.1086/286162.  Google Scholar

[7]

Michael R. Easterling, Stephen P. Ellner and Philip M. Dixon, Size-specific sensitivity: Applying a new structured population model,, Ecological Society of America, 81 (2000), 694.   Google Scholar

[8]

Ivar Ekeland and Roger Témam, "Convex Analysis and Variational Problems,", Studies in Mathematics and its Applications, (1976).   Google Scholar

[9]

Enrico Fermi, "Thermodynamics,'', Dover Publications, (1956).   Google Scholar

[10]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,'', Applications of Mathematics, (1975).   Google Scholar

[11]

H. I. Freedman, J. B. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment,, Mathematical Biosciences, 95 (1989), 111.  doi: 10.1016/0025-5564(89)90055-2.  Google Scholar

[12]

H. Gaff, H. R. Joshi and S. Lenhart, Optimal harvesting during an invasion of a sublethal plant pathogen,, Environment and Development Economics Journal, 12 (2007), 673.   Google Scholar

[13]

W. Hackbush, A numerical method for solving parabolic equations with opposite orientations,, Computing, 20 (1978), 229.  doi: 10.1007/BF02251947.  Google Scholar

[14]

E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit, Partial differential equations in ecology spatial interactions and population dynamics,, Ecology, 75 (1994), 18.  doi: 10.2307/1939378.  Google Scholar

[15]

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

[16]

H. R. Joshi, S. Lenhart, H. Lou and H. Gaff, Harvesting control in an integrodifference population model with concave growth term,, Nonlinear Anal. Hybrid Syst, 1 (2007), 417.   Google Scholar

[17]

John M. Kean and Nigel D. Barlow, A spatial model for the successful biological control of sitona discoideus by microctonus aethiopoides,, The Journal of Applied Ecology, 1 (2001), 162.   Google Scholar

[18]

M. Kot and W. Schaffer, Discrete-time growth-dispersal models,, Math. Biosci., 80 (1986), 109.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar

[19]

M. Kot, Discrete-time travelling waves: Ecological examples,, Journal of Mathematical Biology, 30 (1992), 413.  doi: 10.1007/BF00173295.  Google Scholar

[20]

M. Kot, Do invading organisms do the wave,, Canadian Applied Mathematics Quarterly, 10 (2002), 139.   Google Scholar

[21]

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

[22]

Suzanne Lenhart and John Workman, "Optimal Control Applied to Biological Models,'', Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007).   Google Scholar

[23]

M. A. Lewis, Variability, patchiness, and jump dispersal in the spread of an invading population,, in, (1997), 46.   Google Scholar

[24]

Xun Jing Li and Jiong Min Yong, "Optimal Control Theory for Infinite Dimensional Systems,'', Systems & Control: Foundations & Applications, (1995).   Google Scholar

[25]

D. L. Lukes, "Differential Equations. Classical to Controlled,'', Mathematics in Science and Engineering, 162 (1982).   Google Scholar

[26]

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

[27]

J. D. Murray, E. A. Stanley and D. L. Brown, On the spread of rabies among foxes,, Proc. Roy. Soc. London Ser., 229 (1986), 111.  doi: 10.1098/rspb.1986.0078.  Google Scholar

[28]

M. 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.  Google Scholar

[29]

L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'', Wiley, (1956).   Google Scholar

[30]

M. Slatkin, Gene flow and selection in a cline,, Genetics, 75 (1973), 733.   Google Scholar

[31]

M. Slatkin, Gene flow and selection in a two-locus system,, Genetics, 81 (1975), 787.   Google Scholar

[32]

J. Lubben, D. Boeckner, R. Rebarber, S. Townley and B. Tenhumberg, Parameterizing the growth-decline boundary for uncertain population projection models,, Theoretical Population Biology, 75 (2009), 85.  doi: 10.1016/j.tpb.2008.11.004.  Google Scholar

[33]

Richard Rebarber, Brigitte Tenhumberg and Stuart Townley, Global asymptotic stability of density dependent integral population projection models,, Theoretical Population Biology, 81 (2002), 81.  doi: 10.1016/j.tpb.2011.11.002.  Google Scholar

[34]

C. Reid, "The origin of the British Flora,'', Dualu, (1899).   Google Scholar

[35]

S. P. Sethi and G. L. Thompson, "Optimal Control Theory. Applications to Management Science and Economics,'', Second edition, (2000).   Google Scholar

[36]

M. A. Lewis and R. W. Van Kirk, Integrodifference models for persistence in fragmented habitats,, Bulletin of Mathematical Biology, 59 (1997), 107.  doi: 10.1016/S0092-8240(96)00060-2.  Google Scholar

[37]

H. F. Weinberger, Asymptotic behavior of a model in population genetics,, in, 648 (1978), 1976.   Google Scholar

[38]

K. Yosida, "Functional Analysis,'', 6th edition, 123 (1980).   Google Scholar

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