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

Advection control in parabolic PDE systems for competitive populations

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.

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, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, 2003. Google Scholar

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R. S. CantrellC. 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

[7]

W. DingH. FinottiS. LenhartY. 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. FinottiS. 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. KellyJr. 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

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. CantrellC. 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. DingH. FinottiS. LenhartY. 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. FinottiS. 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. KellyJr. 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 $m(x)$
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
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
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
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 $u$
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 $u$
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 $u$
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 $v$
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 $v$
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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