Figure 1.
Different resource functions $m(x)$
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Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications
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 |
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
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References:
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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.
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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. |
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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.
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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. 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. |
| [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. |
| [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. |
| [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. |
| [10] |
W. Hackbusch,
A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.
doi: 10.1007/BF02251947. |
| [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. |
| [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. |
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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
W. Hackbusch,
A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.
doi: 10.1007/BF02251947. |
| [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. |
| [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. |
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 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 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 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 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 2a and larger resources $m=6x/5$ as in Figure 1c; (5a) Optimal control $h_1$ , (5b) Population distribution of $u$ ">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 2a and the resource function $m=x/5$ as given in Figure 1a; (6a) Optimal control $h_1$ , (6b) 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 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 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 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 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 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 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 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 $v$ ">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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