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Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |







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