Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 49J22, 92D25; Secondary: 39A05.


    \begin{equation} \\ \end{equation}
  • [1]

    D. A. Andow, P. M. Kareiva, Simon A. Levin and Akira Okubo, Spread of invading organisms, Landscape Ecology, 4 (1990), 177-188.


    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.


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


    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.


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


    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.


    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.


    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.


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


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


    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.


    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.


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


    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.


    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.


    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.


    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.


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


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


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


    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.


    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.


    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.


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


    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.


    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.


    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.


    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.


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


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


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


    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.


    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.


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


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


    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.


    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.


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

  • 加载中

Article Metrics

HTML views() PDF downloads(47) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint