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.

[2]

H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Universitext, (2011).

[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. doi: 10.1017/S030821050001876X.

[4]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics,, J. Math. Biol., 29 (1991), 315. doi: 10.1007/BF00167155.

[5]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinb. Sect A, 137 (2007), 497.

[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. doi: 10.1016/j.jde.2008.07.024.

[7]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution,, Math. Bios. Eng., 7 (2010), 17. doi: 10.3934/mbe.2010.7.17.

[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. doi: 10.1512/iumj.2008.57.3204.

[9]

C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?,, J. Math. Anal. Appl., 277 (2003), 489. doi: 10.1016/S0022-247X(02)00575-9.

[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. doi: 10.1016/j.nonrwa.2009.01.015.

[11]

L. C. Evans, "Partial Differential Equations," 2nd edition,, Graduate Studies in Mathematics, 19 (2010).

[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. doi: 10.2307/1939378.

[13]

P. Kareiva, Population dynamics in spatially complex environments: Theory and data,, Phil. Trans. Riy. Soc. London Ser. B, 330 (1987), 175. doi: 10.1098/rstb.1990.0191.

[14]

M. Kot, "Elements of Mathematical Ecology,", Cambridge University Press, (2001).

[15]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model,, J. Diff. Eqns., 250 (2011), 161. doi: 10.1016/j.jde.2010.08.028.

[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. doi: 10.3934/dcds.2010.28.1051.

[17]

S. Lenhart and T. J. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007).

[18]

X. J. Li and J. M. Yong, "Optimal Control Theory for Infinite-Dimensional Systems,", Systems & Control: Foundations & Applications, (1995).

[19]

J.-L. Lions, "Optimal Control Systems Governed by Partial Differential Equations,", Die Grundlehren der mathematischen Wissenschaften, (1971).

[20]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translation of Mathematical Monographs, 23 (1967).

[21]

J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications,", Third edition, 18 (2003).

[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. doi: 10.1007/BF00280665.

[23]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives,", Second edition, 14 (2001).

[24]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.

show all references

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.

[2]

H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Universitext, (2011).

[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. doi: 10.1017/S030821050001876X.

[4]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics,, J. Math. Biol., 29 (1991), 315. doi: 10.1007/BF00167155.

[5]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinb. Sect A, 137 (2007), 497.

[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. doi: 10.1016/j.jde.2008.07.024.

[7]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution,, Math. Bios. Eng., 7 (2010), 17. doi: 10.3934/mbe.2010.7.17.

[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. doi: 10.1512/iumj.2008.57.3204.

[9]

C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?,, J. Math. Anal. Appl., 277 (2003), 489. doi: 10.1016/S0022-247X(02)00575-9.

[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. doi: 10.1016/j.nonrwa.2009.01.015.

[11]

L. C. Evans, "Partial Differential Equations," 2nd edition,, Graduate Studies in Mathematics, 19 (2010).

[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. doi: 10.2307/1939378.

[13]

P. Kareiva, Population dynamics in spatially complex environments: Theory and data,, Phil. Trans. Riy. Soc. London Ser. B, 330 (1987), 175. doi: 10.1098/rstb.1990.0191.

[14]

M. Kot, "Elements of Mathematical Ecology,", Cambridge University Press, (2001).

[15]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model,, J. Diff. Eqns., 250 (2011), 161. doi: 10.1016/j.jde.2010.08.028.

[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. doi: 10.3934/dcds.2010.28.1051.

[17]

S. Lenhart and T. J. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007).

[18]

X. J. Li and J. M. Yong, "Optimal Control Theory for Infinite-Dimensional Systems,", Systems & Control: Foundations & Applications, (1995).

[19]

J.-L. Lions, "Optimal Control Systems Governed by Partial Differential Equations,", Die Grundlehren der mathematischen Wissenschaften, (1971).

[20]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translation of Mathematical Monographs, 23 (1967).

[21]

J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications,", Third edition, 18 (2003).

[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. doi: 10.1007/BF00280665.

[23]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives,", Second edition, 14 (2001).

[24]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.

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