Figure 1.
Different resource functions $m(x)$
-
Previous Article
Eigenvectors of homogeneous order-bounded order-preserving maps
- DCDS-B Home
- This Issue
-
Next Article
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
![]()
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. |
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 $v$
| [1] |
Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879 |
| [2] |
Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013 |
| [3] |
Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control & Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020 |
| [4] |
Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019 |
| [5] |
Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231 |
| [6] |
Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221 |
| [7] |
Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345 |
| [8] |
Eduardo Casas, Konstantinos Chrysafinos. Analysis and optimal control of some quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 607-623. doi: 10.3934/mcrf.2018025 |
| [9] |
Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021020 |
| [10] |
Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131 |
| [11] |
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 |
| [12] |
Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014 |
| [13] |
Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279 |
| [14] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
| [15] |
Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761 |
| [16] |
Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737 |
| [17] |
Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001 |
| [18] |
Serge Nicaise, Fredi Tröltzsch. Optimal control of some quasilinear Maxwell equations of parabolic type. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1375-1391. doi: 10.3934/dcdss.2017073 |
| [19] |
Sébastien Court, Karl Kunisch, Laurent Pfeiffer. Hybrid optimal control problems for a class of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1031-1060. doi: 10.3934/dcdss.2018060 |
| [20] |
Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]