# American Institute of Mathematical Sciences

2013, 2(4): 749-769. doi: 10.3934/eect.2013.2.749

## Study on the order of events in optimal control of a harvesting problem modeled by integrodifference equations

 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 37996-1320, United States

Received  October 2012 Revised  June 2013 Published  November 2013

Integrodifference equations are discrete in time and continuous in space, and are used to model populations that are growing at discrete times, and dispersing spatially. A harvesting problem modeled by integrodifference equations involves three events: growth, dispersal and harvesting. The order of arranging the three events affects the optimized harvesting behavior. In this paper we investigate all six possible cases of orders of events, study the equivalences among them under certain conditions, and show how the six cases can be reduced to three cases.
Citation: 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
##### References:
 [1] M. Andersen, Properties of some density-dependent integrodifference equation population models，, Mathematical Biosciences, 104 (1991), 135. doi: 10.1016/0025-5564(91)90034-G. [2] D. A. Andow, P. M. Kareiva, S. A. Levin and A. Okubo, Spread of invading organisms,, Landscape Ecology, 4 (1990), 177. doi: 10.1007/BF00132860. [3] A. J. Bateman, Is gene dispersion normal?, Heredity, 4 (1950), 353. doi: 10.1038/hdy.1950.27. [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. [5] S. Chandrasekhar, Stochastic problems in physics and astronomy,, Rev.Mod. Phys., 15 (1943), 1. 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. doi: 10.1086/286162. [7] M. R. Easterling, S. P. Ellner and P. M. Dixon, Size-specific sensitivity: Applying a new structured population model,, Ecological Society of America, 81 (2000), 694. [8] I. Ekeland and R. Témam, Convex Analysis and Variational Problems,, Society fro Industrial and Applied Mathematics, (1976). [9] E. Fermi, Thermodynamics,, Dover Publications, (1956). [10] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Applications of Mathematics, (1975). [11] H. I. Freedman, J. B. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment,, Math. Biosci., 95 (1989), 111. 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. doi: 10.1017/S1355770X07003828. [13] W. Hackbush, A numerical method for solving parabolic equations with opposite orientations,, Computing, 20 (1978), 229. 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), 17. 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. 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. doi: 10.1016/j.nahs.2006.10.010. [17] 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, 1 (2001), 162. [18] M. Kot, Discrete-time travelling waves: Ecological examples,, Journal of Mathematical. Biology, 30 (1992), 413. doi: 10.1007/BF00173295. [19] M. Kot, Do invading organisms do the wave? (English summary),, Can. Appl. Math. Q., 10 (2002), 139. [20] 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. [21] M. Kot and W. Schaffer, Discrete-time growth-dispersal models,, Math. BiosciMathematical Biosciences, 80 (1986), 109. doi: 10.1016/0025-5564(86)90069-6. [22] S. Lenhart and J. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical and Computational Biology Series. Chapman & Hall/CRC, (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, d. tilman and p. kareiva, editors,, Princetion University Press, (1997), 46. [24] M. A. Lewis and V. Kirk, Integrodifference models for persistence in fragmented habitats,, Bulletin of Mathematical Biology, 59 (1997), 107. [25] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Systems & Control: Foundations & Applications. Birkhäuser Boston, (1995). doi: 10.1007/978-1-4612-4260-4. [26] 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. [27] D. L. Lukes, Differential Equations: Classical to Controlled,, Mathematics in Science and Engineering, (1982). [28] G. M. MacDonald, Fossil pollen analysis and the reconstruction fo plant invasions,, Advances in ecological research, 24 (1993), 67. [29] 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. [30] 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. [31] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E. F. Mishchenko, Modeling Optimal Intervention Strategies for Cholera, The Mathematical Theory of Optimal Processes,, Wiley, (1956). [32] R. Rebarber, B. Tenhumberg and S. Townley, Global asymptotic stability of density dependent integral population projection models,, Theoretical Population Biology, 81 (2012), 81. doi: 10.1016/j.tpb.2011.11.002. [33] C. Reid, The Origin of the British Flora,, Dualu, (1899). [34] S. P. Sethi and G. L. Thompson, Optimal Control Theory; Applications to Management Science and Economics,, Second edition. Kluwer Academic Publishers, (2000). [35] M. Slatkin, Gene flow and selection in a cline,, Genetics, 75 (1973), 733. [36] M. Slatkin, Gene flow and selection in a two-locus system,, Genetics, 81 (1975), 787. [37] H. F. Weinberger, Asymptotic behavior of a model in population genetics,, Nonlinear partial differential equations and applications, 648 (1978), 47. [38] K. Yosida, Functional Analysis, 6th ed,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1980). [39] 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.

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##### References:
 [1] M. Andersen, Properties of some density-dependent integrodifference equation population models，, Mathematical Biosciences, 104 (1991), 135. doi: 10.1016/0025-5564(91)90034-G. [2] D. A. Andow, P. M. Kareiva, S. A. Levin and A. Okubo, Spread of invading organisms,, Landscape Ecology, 4 (1990), 177. doi: 10.1007/BF00132860. [3] A. J. Bateman, Is gene dispersion normal?, Heredity, 4 (1950), 353. doi: 10.1038/hdy.1950.27. [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. [5] S. Chandrasekhar, Stochastic problems in physics and astronomy,, Rev.Mod. Phys., 15 (1943), 1. 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. doi: 10.1086/286162. [7] M. R. Easterling, S. P. Ellner and P. M. Dixon, Size-specific sensitivity: Applying a new structured population model,, Ecological Society of America, 81 (2000), 694. [8] I. Ekeland and R. Témam, Convex Analysis and Variational Problems,, Society fro Industrial and Applied Mathematics, (1976). [9] E. Fermi, Thermodynamics,, Dover Publications, (1956). [10] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Applications of Mathematics, (1975). [11] H. I. Freedman, J. B. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment,, Math. Biosci., 95 (1989), 111. 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. doi: 10.1017/S1355770X07003828. [13] W. Hackbush, A numerical method for solving parabolic equations with opposite orientations,, Computing, 20 (1978), 229. 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), 17. 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. 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. doi: 10.1016/j.nahs.2006.10.010. [17] 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, 1 (2001), 162. [18] M. Kot, Discrete-time travelling waves: Ecological examples,, Journal of Mathematical. Biology, 30 (1992), 413. doi: 10.1007/BF00173295. [19] M. Kot, Do invading organisms do the wave? (English summary),, Can. Appl. Math. Q., 10 (2002), 139. [20] 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. [21] M. Kot and W. Schaffer, Discrete-time growth-dispersal models,, Math. BiosciMathematical Biosciences, 80 (1986), 109. doi: 10.1016/0025-5564(86)90069-6. [22] S. Lenhart and J. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical and Computational Biology Series. Chapman & Hall/CRC, (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, d. tilman and p. kareiva, editors,, Princetion University Press, (1997), 46. [24] M. A. Lewis and V. Kirk, Integrodifference models for persistence in fragmented habitats,, Bulletin of Mathematical Biology, 59 (1997), 107. [25] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Systems & Control: Foundations & Applications. Birkhäuser Boston, (1995). doi: 10.1007/978-1-4612-4260-4. [26] 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. [27] D. L. Lukes, Differential Equations: Classical to Controlled,, Mathematics in Science and Engineering, (1982). [28] G. M. MacDonald, Fossil pollen analysis and the reconstruction fo plant invasions,, Advances in ecological research, 24 (1993), 67. [29] 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. [30] 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. [31] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E. F. Mishchenko, Modeling Optimal Intervention Strategies for Cholera, The Mathematical Theory of Optimal Processes,, Wiley, (1956). [32] R. Rebarber, B. Tenhumberg and S. Townley, Global asymptotic stability of density dependent integral population projection models,, Theoretical Population Biology, 81 (2012), 81. doi: 10.1016/j.tpb.2011.11.002. [33] C. Reid, The Origin of the British Flora,, Dualu, (1899). [34] S. P. Sethi and G. L. Thompson, Optimal Control Theory; Applications to Management Science and Economics,, Second edition. Kluwer Academic Publishers, (2000). [35] M. Slatkin, Gene flow and selection in a cline,, Genetics, 75 (1973), 733. [36] M. Slatkin, Gene flow and selection in a two-locus system,, Genetics, 81 (1975), 787. [37] H. F. Weinberger, Asymptotic behavior of a model in population genetics,, Nonlinear partial differential equations and applications, 648 (1978), 47. [38] K. Yosida, Functional Analysis, 6th ed,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1980). [39] 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.
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