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 and 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), 177-188.

[2]

M. Andersen, Properties of some density-dependent integrodifference equation population models, Mathematical Biosciences, 104 (1991), 135-157. doi: 10.1016/0025-5564(91)90034-G.

[3]

A. J. Bateman, Is gene dispersion normal, Heredity, 4 (1950), 353-363.

[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), 175-195. doi: 10.1111/j.1939-7445.1999.tb00009.x.

[5]

S. Chandrasekhar, Stochastic problems in physics and astronomy, Reviews of Modern Physics, 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1.

[6]

J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, American Naturalist, 152 (1998), 204-224. doi: 10.1086/286162.

[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), 694-708.

[8]

Ivar Ekeland and Roger Témam, "Convex Analysis and Variational Problems," Studies in Mathematics and its Applications, Vol. 1, North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1976.

[9]

Enrico Fermi, "Thermodynamics,'' Dover Publications, New York, 1956.

[10]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,'' Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975.

[11]

H. I. Freedman, J. B. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment, Mathematical Biosciences, 95 (1989), 111-123. doi: 10.1016/0025-5564(89)90055-2.

[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), 673-686.

[13]

W. Hackbush, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240. doi: 10.1007/BF02251947.

[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), 18-29. doi: 10.2307/1939378.

[15]

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.

[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), 417-429.

[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), 162-169.

[18]

M. Kot and W. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136. doi: 10.1016/0025-5564(86)90069-6.

[19]

M. Kot, Discrete-time travelling waves: Ecological examples, Journal of Mathematical Biology, 30 (1992), 413-436. doi: 10.1007/BF00173295.

[20]

M. Kot, Do invading organisms do the wave, Canadian Applied Mathematics Quarterly, 10 (2002), 139-170.

[21]

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

[22]

Suzanne Lenhart and John Workman, "Optimal Control Applied to Biological Models,'' Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.

[23]

M. A. Lewis, Variability, patchiness, and jump dispersal in the spread of an invading population, in "Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions" (eds. D. Tilman and P. Kareiva), Princeton University Press, (1997), 46-74.

[24]

Xun Jing Li and Jiong Min Yong, "Optimal Control Theory for Infinite Dimensional Systems,'' Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995.

[25]

D. L. Lukes, "Differential Equations. Classical to Controlled,'' Mathematics in Science and Engineering, 162, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982.

[26]

G. M. MacDonald, Fossil pollen analysis and the reconstruction of plant invasions, Advances in Ecological Research, 24 (1993), 67-109. doi: 10.1016/S0065-2504(08)60041-0.

[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), 111-150. doi: 10.1098/rspb.1986.0078.

[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), 7-43. doi: 10.1006/tpbi.1995.1020.

[29]

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

[30]

M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756.

[31]

M. Slatkin, Gene flow and selection in a two-locus system, Genetics, 81 (1975), 787-802.

[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), 85-97. doi: 10.1016/j.tpb.2008.11.004.

[33]

Richard Rebarber, Brigitte Tenhumberg and Stuart Townley, Global asymptotic stability of density dependent integral population projection models, Theoretical Population Biology, 81 (2002), 81-87. doi: 10.1016/j.tpb.2011.11.002.

[34]

C. Reid, "The origin of the British Flora,'' Dualu, London, 1899.

[35]

S. P. Sethi and G. L. Thompson, "Optimal Control Theory. Applications to Management Science and Economics,'' Second edition, Kluwer Academic Publishers, Boston, MA, 2000.

[36]

M. A. Lewis and R. W. Van Kirk, Integrodifference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137. doi: 10.1016/S0092-8240(96)00060-2.

[37]

H. F. Weinberger, Asymptotic behavior of a model in population genetics, in "Nonlinear Partial Differential Equations and Applications" (Proc. Special Sem., Indiana Univ., Bloomington, Ind., 1976-1977), Lecture Notes in Mathematics, 648, Springer, Berlin, (1978), 47-96.

[38]

K. Yosida, "Functional Analysis,'' 6th edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123, Springer-Verlag, Berlin-New York, 1980.

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), 177-188.

[2]

M. Andersen, Properties of some density-dependent integrodifference equation population models, Mathematical Biosciences, 104 (1991), 135-157. doi: 10.1016/0025-5564(91)90034-G.

[3]

A. J. Bateman, Is gene dispersion normal, Heredity, 4 (1950), 353-363.

[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), 175-195. doi: 10.1111/j.1939-7445.1999.tb00009.x.

[5]

S. Chandrasekhar, Stochastic problems in physics and astronomy, Reviews of Modern Physics, 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1.

[6]

J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, American Naturalist, 152 (1998), 204-224. doi: 10.1086/286162.

[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), 694-708.

[8]

Ivar Ekeland and Roger Témam, "Convex Analysis and Variational Problems," Studies in Mathematics and its Applications, Vol. 1, North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1976.

[9]

Enrico Fermi, "Thermodynamics,'' Dover Publications, New York, 1956.

[10]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,'' Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975.

[11]

H. I. Freedman, J. B. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment, Mathematical Biosciences, 95 (1989), 111-123. doi: 10.1016/0025-5564(89)90055-2.

[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), 673-686.

[13]

W. Hackbush, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240. doi: 10.1007/BF02251947.

[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), 18-29. doi: 10.2307/1939378.

[15]

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.

[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), 417-429.

[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), 162-169.

[18]

M. Kot and W. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136. doi: 10.1016/0025-5564(86)90069-6.

[19]

M. Kot, Discrete-time travelling waves: Ecological examples, Journal of Mathematical Biology, 30 (1992), 413-436. doi: 10.1007/BF00173295.

[20]

M. Kot, Do invading organisms do the wave, Canadian Applied Mathematics Quarterly, 10 (2002), 139-170.

[21]

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

[22]

Suzanne Lenhart and John Workman, "Optimal Control Applied to Biological Models,'' Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.

[23]

M. A. Lewis, Variability, patchiness, and jump dispersal in the spread of an invading population, in "Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions" (eds. D. Tilman and P. Kareiva), Princeton University Press, (1997), 46-74.

[24]

Xun Jing Li and Jiong Min Yong, "Optimal Control Theory for Infinite Dimensional Systems,'' Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995.

[25]

D. L. Lukes, "Differential Equations. Classical to Controlled,'' Mathematics in Science and Engineering, 162, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982.

[26]

G. M. MacDonald, Fossil pollen analysis and the reconstruction of plant invasions, Advances in Ecological Research, 24 (1993), 67-109. doi: 10.1016/S0065-2504(08)60041-0.

[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), 111-150. doi: 10.1098/rspb.1986.0078.

[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), 7-43. doi: 10.1006/tpbi.1995.1020.

[29]

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

[30]

M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756.

[31]

M. Slatkin, Gene flow and selection in a two-locus system, Genetics, 81 (1975), 787-802.

[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), 85-97. doi: 10.1016/j.tpb.2008.11.004.

[33]

Richard Rebarber, Brigitte Tenhumberg and Stuart Townley, Global asymptotic stability of density dependent integral population projection models, Theoretical Population Biology, 81 (2002), 81-87. doi: 10.1016/j.tpb.2011.11.002.

[34]

C. Reid, "The origin of the British Flora,'' Dualu, London, 1899.

[35]

S. P. Sethi and G. L. Thompson, "Optimal Control Theory. Applications to Management Science and Economics,'' Second edition, Kluwer Academic Publishers, Boston, MA, 2000.

[36]

M. A. Lewis and R. W. Van Kirk, Integrodifference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137. doi: 10.1016/S0092-8240(96)00060-2.

[37]

H. F. Weinberger, Asymptotic behavior of a model in population genetics, in "Nonlinear Partial Differential Equations and Applications" (Proc. Special Sem., Indiana Univ., Bloomington, Ind., 1976-1977), Lecture Notes in Mathematics, 648, Springer, Berlin, (1978), 47-96.

[38]

K. Yosida, "Functional Analysis,'' 6th edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123, Springer-Verlag, Berlin-New York, 1980.

[1]

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

[2]

Hiroaki Morimoto. Optimal harvesting and planting control in stochastic logistic population models. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2545-2559. doi: 10.3934/dcdsb.2012.17.2545

[3]

Marco V. Martinez, Suzanne Lenhart, K. A. Jane White. Optimal control of integrodifference equations in a pest-pathogen system. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1759-1783. doi: 10.3934/dcdsb.2015.20.1759

[4]

Sebastian Aniţa, Ana-Maria Moşsneagu. Optimal harvesting for age-structured population dynamics with size-dependent control. Mathematical Control and Related Fields, 2019, 9 (4) : 607-621. doi: 10.3934/mcrf.2019043

[5]

Miaomiao Chen, Rong Yuan. Maximum principle for the optimal harvesting problem of a size-stage-structured population model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4619-4648. doi: 10.3934/dcdsb.2021245

[6]

Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112

[7]

Manoj Kumar, Syed Abbas, Rathinasamy Sakthivel. Analysis of diffusive size-structured population model and optimal birth control. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022036

[8]

Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014

[9]

Meng Liu, Chuanzhi Bai. Optimal harvesting of a stochastic delay competitive model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1493-1508. doi: 10.3934/dcdsb.2017071

[10]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233

[11]

Dongmei Xiao. Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 699-719. doi: 10.3934/dcdsb.2016.21.699

[12]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001

[13]

Jon Jacobsen, Taylor McAdam. A boundary value problem for integrodifference population models with cyclic kernels. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3191-3207. doi: 10.3934/dcdsb.2014.19.3191

[14]

Judith R. Miller, Huihui Zeng. Multidimensional stability of planar traveling waves for an integrodifference model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 741-751. doi: 10.3934/dcdsb.2013.18.741

[15]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[16]

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

[17]

Xiaoling Zou, Yuting Zheng. Stochastic modelling and analysis of harvesting model: Application to "summer fishing moratorium" by intermittent control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5047-5066. doi: 10.3934/dcdsb.2020332

[18]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[19]

Y. Gong, X. Xiang. A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. Journal of Industrial and Management Optimization, 2009, 5 (1) : 1-10. doi: 10.3934/jimo.2009.5.1

[20]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control and Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]