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May  2017, 22(3): 1049-1072. doi: 10.3934/dcdsb.2017052

Advection control in parabolic PDE systems for competitive populations

Kokum R. De Silva 1, , Tuoc V. Phan 2, and  Suzanne Lenhart 2,,

1. 

Department of Mathematics, University of Peradeniya, Peradeniya, KY 20400, Sri Lanka
2. 

Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USA

* Corresponding author: lenhart@math.utk.edu

Received  May 2016 Revised  December 2016 Published  January 2017

This paper investigates the response of two competing species to a given resource using optimal control techniques. We explore the choices of directed movement through controlling the advective coefficients in a system of parabolic partial differential equations. The objective is to maximize the abundance represented by a weighted combination of the two populations while minimizing the 'risk' costs caused by the movements.

Keywords: Optimal control, advection direction, competitive systems, partial differential equations, parabolic system.
Mathematics Subject Classification: Primary: 49K20, 92B05; Secondary: 35K5.
Citation: Kokum R. De Silva, Tuoc V. Phan, Suzanne Lenhart. Advection control in parabolic PDE systems for competitive populations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1049-1072. doi: 10.3934/dcdsb.2017052
References:
[1]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canadian Applied Mathematics Quarterly, 3 (1995), 379-397.   Google Scholar

[2]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments Ⅱ, SIAM J. Math. Anal., 22 (1991), 1043-1064.  doi: 10.1137/0522068.  Google Scholar

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R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, Journal of Mathematical Biology, 29 (1991), 315-338.  doi: 10.1007/BF00167155.  Google Scholar

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R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, 2003. Google Scholar

[5]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proceedings of the Royal Society of Edinburgh Section A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.  Google Scholar

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C. Cosner and Y. Lou, Does movement towards better environments always benefit a population?, Journal of Mathematical analysis and Applications, 277 (2003), 489-503.  doi: 10.1016/S0022-247X(02)00575-9.  Google Scholar

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W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Analysis: Real World Applications, 11 (2010), 688-704.  doi: 10.1016/j.nonrwa.2009.01.015.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations, 2nd edition, American Mathematical Society, Providence, RI, 2010. Google Scholar

[9]

H. Finotti, S. Lenhart and T. V. Phan, Optimal control of advective direction in reaction-diffusion population models, Evolution equations and control theory, 1 (2012), 81-107.  doi: 10.3934/eect.2012.1.81.  Google Scholar

[10]

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

[11]

M. R. Kelly, Jr. Y. Xing and S. Lenhart, Optimal fish harvesting for a population modeled by a nonlinear parabolic partial differential equation, Natural Resources Modeling Journal, 29 (2016), 36-70.  doi: 10.1111/nrm.12073.  Google Scholar

[12]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC Mathematical and Computational Biology Series, 2007. Google Scholar

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show all references

References:
[1]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canadian Applied Mathematics Quarterly, 3 (1995), 379-397.   Google Scholar

[2]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments Ⅱ, SIAM J. Math. Anal., 22 (1991), 1043-1064.  doi: 10.1137/0522068.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, Journal of Mathematical Biology, 29 (1991), 315-338.  doi: 10.1007/BF00167155.  Google Scholar

[4]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, 2003. Google Scholar

[5]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proceedings of the Royal Society of Edinburgh Section A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.  Google Scholar

[6]

C. Cosner and Y. Lou, Does movement towards better environments always benefit a population?, Journal of Mathematical analysis and Applications, 277 (2003), 489-503.  doi: 10.1016/S0022-247X(02)00575-9.  Google Scholar

[7]

W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Analysis: Real World Applications, 11 (2010), 688-704.  doi: 10.1016/j.nonrwa.2009.01.015.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations, 2nd edition, American Mathematical Society, Providence, RI, 2010. Google Scholar

[9]

H. Finotti, S. Lenhart and T. V. Phan, Optimal control of advective direction in reaction-diffusion population models, Evolution equations and control theory, 1 (2012), 81-107.  doi: 10.3934/eect.2012.1.81.  Google Scholar

[10]

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

[11]

M. R. Kelly, Jr. Y. Xing and S. Lenhart, Optimal fish harvesting for a population modeled by a nonlinear parabolic partial differential equation, Natural Resources Modeling Journal, 29 (2016), 36-70.  doi: 10.1111/nrm.12073.  Google Scholar

[12]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC Mathematical and Computational Biology Series, 2007. Google Scholar

[13]

J. Simon, Compact sets in the $L^p(0,T;B)$, Annali di Matematica Pure ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

Figure 1.  Different resource functions $m(x)$
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Figure 2.  Different initial conditions: (2a) Smaller initial population at middle, (2b) Larger initial population at middle, (2c) Two smaller initial populations overlapping in the middle
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Figure 3.  One population only: Population dynamics and optimal control for $u$ population with the resource function $m=x/5$ as in Figure 1a; (3a) Optimal control $h_1$ with respect to the IC in Figure 2a, (3b) Population distribution of $u$ with respect to the IC in Figure 2a, (3c) Optimal control $h_1$ with respect to the IC in Figure 2b, (3d) Population distribution of $u$ with respect to the IC in Figure 2b
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Figure 4.  One population only: Population dynamics and optimal control for $u$ population with the resource function $m=sin(\pi x/5)$ as in Figure 1b; (4a) Optimal control $h_1$ with respect to the IC in Figure 2a, (4b) Population distribution of $u$ with respect to the IC in Figure 2a, (4c) Optimal control $h_1$ with respect to the IC in Figure 2b, (4d) Population distribution of $u$ with respect to the IC in Figure 2b
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Figure 5.  One population only: Population dynamics and optimal control with a smaller IC as in Figure 2a and larger resources $m=6x/5$ as in Figure 1c; (5a) Optimal control $h_1$, (5b) Population distribution of $u$
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Figure 6.  Two populations with competition: Population dynamics and optimal control with a smaller IC as in Figure 2a and the resource function $m=x/5$ as given in Figure 1a; (6a) Optimal control $h_1$, (6b) Population distribution of $u$
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Figure 7.  Two populations with competition: Population dynamics and optimal control with a smaller IC as in Figure 2a and the resource function $m=sin(\pi x/5)$ as given in Figure 1b; (7a) Optimal control $h_1$, (7b) Population distribution of $u$
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Figure 8.  Two populations with competition: Population dynamics and optimal control with a smaller IC as in Figure 2c and the resource function $m=x/5$ as given in Figure 1a; (8a) Optimal control $h_1$, (8b) Population distribution of $u$, (8c) Optimal control $h_2$, (8d) Population distribution of $v$
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Figure 9.  Two populations with competition: Population dynamics and optimal control with a smaller IC as in Figure 2c and the resource function $m=sin(\pi x/5)$ as given in Figure 1b; (9a) Optimal control $h_1$, (9b) Population distribution of $u$, (9c) Optimal control $h_2$, (9d) Population distribution of $v$
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Figure 10.  Two populations with different competition rates: Population dynamics and optimal control with a smaller IC as in Figure 2a and the resource function $m=sin(\pi x/5)$ as given in Figure 1b and $b_1 = 4$, $b_2 = 0.5$; (10a) Optimal control $h_1$, (10b) Population distribution of $u$, (10c) Optimal control $h_2$, (10d) Population distribution of $v$
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