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, 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, 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, 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.

[5]

R. S. Cantrell, C. Cosner, 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, 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, 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.

[9]

H. Finotti, S. Lenhart, 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, 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.

[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, 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, 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, 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.

[5]

R. S. Cantrell, C. Cosner, 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, 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, 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.

[9]

H. Finotti, S. Lenhart, 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, 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.

[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 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 $v$
[1]

Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

[2]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with Lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[3]

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

[4]

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

[5]

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

[6]

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

[7]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations . Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279

[8]

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

[9]

Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737

[10]

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

[11]

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

[12]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115

[13]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001

[14]

Marco V. Martinez, Suzanne Lenhart, K. A. Jane White. Optimal control of integrodifference equations in a pest-pathogen system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1759-1783. doi: 10.3934/dcdsb.2015.20.1759

[15]

B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463

[16]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[17]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455

[18]

Kareem T. Elgindy. Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted Gegenbauer integral pseudospectral method. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1-24. doi: 10.3934/jimo.2017056

[19]

Shuhong Chen, Zhong Tan. Optimal interior partial regularity for nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 981-993. doi: 10.3934/dcds.2010.27.981

[20]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (1)
  • HTML views (1)
  • Cited by (0)

[Back to Top]