# American Institute of Mathematical Sciences

June  2012, 1(1): 81-107. doi: 10.3934/eect.2012.1.81

## Optimal control of advective direction in reaction-diffusion population models

 1 1400 Kenesaw Ave, 31F, Knoxville, TN 37132, United States 2 Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, United States

Received  November 2011 Revised  February 2012 Published  March 2012

We investigate optimal control of the advective coefficient in a class of parabolic partial differential equations, modeling a population with nonlinear growth. This work is motivated by the question: Does movement toward a better resource environment benefit a population? Our objective functional is formulated with interpreting "benefit" as the total population size integrated over our finite time interval. Results on existence, uniqueness, and characterization of the optimal control are established. Our numerical illustrations for several growth functions and resource functions indicate that movement along the resource spatial gradient benefits the population, meaning that the optimal control is close to the spatial gradient of the resource function.
Citation: Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations & Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81
##### References:
 [1] F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Canad. Appl. Math. Quart., 3 (1995), 379-397.  Google Scholar [2] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011.  Google Scholar [3] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in a disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X.  Google Scholar [4] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155.  Google Scholar [5] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinb. Sect A, 137 (2007), 497-518.  Google Scholar [6] R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024.  Google Scholar [7] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Bios. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17.  Google Scholar [8] X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunction of elliptic operator with large convection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.  Google Scholar [9] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl., 277 (2003), 489-503. doi: 10.1016/S0022-247X(02)00575-9.  Google Scholar [10] W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Anal. Real World Appl., 11 (2010), 688-704. doi: 10.1016/j.nonrwa.2009.01.015.  Google Scholar [11] L. C. Evans, "Partial Differential Equations," 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar [12] 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), 17-29. doi: 10.2307/1939378.  Google Scholar [13] P. Kareiva, Population dynamics in spatially complex environments: Theory and data, Phil. Trans. Riy. Soc. London Ser. B, 330 (1987), 175-190. doi: 10.1098/rstb.1990.0191.  Google Scholar [14] M. Kot, "Elements of Mathematical Ecology," Cambridge University Press, Cambridge, 2001.  Google Scholar [15] K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Diff. Eqns., 250 (2011), 161-181. doi: 10.1016/j.jde.2010.08.028.  Google Scholar [16] K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. Series A, 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051.  Google Scholar [17] S. Lenhart and T. J. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.  Google Scholar [18] X. J. Li and J. M. Yong, "Optimal Control Theory for Infinite-Dimensional Systems," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995.  Google Scholar [19] J.-L. Lions, "Optimal Control Systems Governed by Partial Differential Equations," Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971.  Google Scholar [20] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translation of Mathematical Monographs, 23, AMS, Providence, RI, 1967.  Google Scholar [21] J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.  Google Scholar [22] J. D. Murray and R. P. Sperb, Minimum domains for spatial patterns in a class of reaction-diffusion equations, J. Math. Biol., 18 (1983), 169-184. doi: 10.1007/BF00280665.  Google Scholar [23] A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.  Google Scholar [24] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  Google Scholar

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##### References:
 [1] F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Canad. Appl. Math. Quart., 3 (1995), 379-397.  Google Scholar [2] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011.  Google Scholar [3] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in a disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X.  Google Scholar [4] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155.  Google Scholar [5] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinb. Sect A, 137 (2007), 497-518.  Google Scholar [6] R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024.  Google Scholar [7] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Bios. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17.  Google Scholar [8] X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunction of elliptic operator with large convection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.  Google Scholar [9] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl., 277 (2003), 489-503. doi: 10.1016/S0022-247X(02)00575-9.  Google Scholar [10] W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Anal. Real World Appl., 11 (2010), 688-704. doi: 10.1016/j.nonrwa.2009.01.015.  Google Scholar [11] L. C. Evans, "Partial Differential Equations," 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar [12] 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), 17-29. doi: 10.2307/1939378.  Google Scholar [13] P. Kareiva, Population dynamics in spatially complex environments: Theory and data, Phil. Trans. Riy. Soc. London Ser. B, 330 (1987), 175-190. doi: 10.1098/rstb.1990.0191.  Google Scholar [14] M. Kot, "Elements of Mathematical Ecology," Cambridge University Press, Cambridge, 2001.  Google Scholar [15] K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Diff. Eqns., 250 (2011), 161-181. doi: 10.1016/j.jde.2010.08.028.  Google Scholar [16] K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. Series A, 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051.  Google Scholar [17] S. Lenhart and T. J. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.  Google Scholar [18] X. J. Li and J. M. Yong, "Optimal Control Theory for Infinite-Dimensional Systems," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995.  Google Scholar [19] J.-L. Lions, "Optimal Control Systems Governed by Partial Differential Equations," Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971.  Google Scholar [20] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translation of Mathematical Monographs, 23, AMS, Providence, RI, 1967.  Google Scholar [21] J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.  Google Scholar [22] J. D. Murray and R. P. Sperb, Minimum domains for spatial patterns in a class of reaction-diffusion equations, J. Math. Biol., 18 (1983), 169-184. doi: 10.1007/BF00280665.  Google Scholar [23] A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.  Google Scholar [24] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  Google Scholar
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