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Optimal control of fish-feeding in a three-dimensional calm freshwater pond considering environmental concern
1. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong |
2. | School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa |
This paper describes the optimal fish-feeding in a three-dimensional calm freshwater pond based on the concentrations of seven water quality variables. A certain number of baby fishes are inserted into the pond simultaneously. They are then taken out of the pond simultaneously for harvest after having gone through a feeding program. This feeding program creates additional loads of water quality variables in the pond, which becomes pollutants. Thus, an optimal fish-feeding problem is formulated to maximize the final weight of the fishes, subject to the restrictions that the fishes are not under-fed and over-fed and the concentrations of the pollutants created by the fish-feeding program are not too large. A computational scheme using the finite element Galerkin scheme for the three-dimensional cubic domain and the control parameterization method is developed for solving the problem. Finally, a numerical example is solved.
References:
[1] |
V. S. Amundsen and T. C. Osmundsen,
Sustainability indicators for salmon aquaculture, Data Brief, 20 (2018), 20-29.
doi: 10.1016/j.dib.2018.07.043. |
[2] |
F. A. Bohnes and A. Laurent,
Environmental impacts of existing and future aquaculture production: Comparison of technologies and feed options in singapore, Aquaculture, 532 (2021), 736001.
doi: 10.1016/j.aquaculture.2020.736001. |
[3] |
C. B. C. K. Cerbule, P. Senff and I. K. Stolz, Towards environmental sustainability in marine finfish aquaculture, Frontier in Marine Science, 2021. |
[4] |
R. A. Falconer and M. Hartnett,
Mathematical modeling of flow, pesticide and nutrient transport for fish-farm planning and management, Ocean Coastal Management, 19 (1993), 37-57.
|
[5] |
M. Kumlu and M. Aktas,
Optimal feeding rates for european sea bass dicentrarchus labrax L. reared in seawater and freshwater, Aquaculture, 231 (2004), 501-515.
|
[6] |
GERSAMP, Monitoring the ecological effects of coastal aquaculture wastes food and agriculture, IMO/FAO/UNESCO/WMO/WHO/IAEA/UN/UNEP joint group of experts on the scientific aspects of marine pollution, GESAMP Rep Stud., (1996), 57. |
[7] |
S. Goyal, D. Ott, J. Liebscher, J. Dautz, H. O. Gutzeit, D. Schmidt and R. Reuss,
Sustainability analysis of fish feed derived from aquatic plant and insect, Sustainability, 13 (2021), 7371.
doi: 10.3390/su13137371. |
[8] |
H. W. J. Lee, C. K. Chan, K. Yau, K. H. Wong and C. Myburgh,
Control parametrization and finite element method for controlling multi-species reactive transport in a circular Pool, J. Ind. Manag. Optim., 9 (2013), 505-524.
doi: 10.3934/jimo.2013.9.505. |
[9] |
Q. Lin, R. Loxton and K. L. Teo,
The control parametrization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10 (2014), 275-309.
doi: 10.3934/jimo.2014.10.275. |
[10] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equation, Springer-Verlag, Berlin, Germany, 1971. |
[11] |
C. Liu, Z. Gong, E. Feng and H. Yin,
Optimal switching control of a fed-batch fermentation process, J. Global Optim., 52 (2012), 265-280.
doi: 10.1007/s10898-011-9663-8. |
[12] |
A. I. Stamou, M. Karamanoli, N. Vasiliadou, E. Douka, A. Bergamasco and L. Genovese,
Mathematical modelling of the interactions between aquacultures and the sea environment, Desalination, 248 (2009), 826-835.
|
[13] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991. |
[14] |
K. H. Wong, L. S. Jennings and F. Benyah,
Control parametrization method for free planning time optimal control problems with time-delayed argument, Nonlinear Anal., 47 (2001), 5679-5689.
doi: 10.1016/S0362-546X(01)00669-1. |
[15] |
K. H. Wong, H. W. J. Lee and C. K. Chan,
Control parametrization and finite element method for controlling multi-species reactive transport in a rectangular diffuser unit, J. Optim. Theory Appl., 150 (2011), 118-141.
doi: 10.1007/s10957-011-9826-2. |
[16] |
K. H. Wong, H. W. J. Lee, C. K. Chan and C. Myburgh,
Control parametrization and finite element method for controlling multi-species reactive transport in an underground tunnel, J. Optim. Theory Appl., 157 (2013), 168-187.
doi: 10.1007/s10957-012-0148-9. |
[17] |
D. Wu, Y. Bai and C. Yu,
A new computational approach for optimal control problems with multiple time-delay, Automatica, 101 (2019), 388-395.
doi: 10.1016/j.automatica.2018.12.036. |
[18] |
R. S. S. Wu,
The environmental impact of marine fish culture: Towards a sustainable future, Marine Pollution Bulletin, 31 (1995), 159-166.
doi: 10.1016/0025-326X(95)00100-2. |
[19] |
R. S. S. Wu, P. K. S. Shin, D. W. MacKay, M. Mollowney and D. Johnson,
Management of marine fish farming in the sub-tropical environment: A modelling approach, Aquaculture, 174 (1999), 279-298.
doi: 10.1016/S0044-8486(99)00024-1. |
[20] |
Z. S. Wu and K. L. Teo,
Optimal control problems involving second boundary value problems of parabolic type, SIAM J. Control Optim., 21 (1983), 729-756.
doi: 10.1137/0321045. |
[21] |
F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings,
Visual MISER: An efficient user-friendly visual program for solving optimal control problems, J. Ind. Manag. Optim., 12 (2016), 781-810.
doi: 10.3934/jimo.2016.12.781. |
[22] |
C. Yu, Q. Lin, R. Loxton, K. L. Teo and G. Wang,
A hybrid time-scaling transformation for time-delay optimal control problems, J. Optim. Theory Appl., 169 (2016), 867-901.
doi: 10.1007/s10957-015-0783-z. |
[23] |
X. Zeng, T. C. Rasmussen, M. B. Beck, A. K. Parker and Z. Lin,
A biogeochemical model for metabolism and nutrient cycling in a southeastern piedmont impoundment, Environmental Modelling and Software, 21 (2006), 1073-1095.
doi: 10.1016/j.envsoft.2005.05.009. |
show all references
References:
[1] |
V. S. Amundsen and T. C. Osmundsen,
Sustainability indicators for salmon aquaculture, Data Brief, 20 (2018), 20-29.
doi: 10.1016/j.dib.2018.07.043. |
[2] |
F. A. Bohnes and A. Laurent,
Environmental impacts of existing and future aquaculture production: Comparison of technologies and feed options in singapore, Aquaculture, 532 (2021), 736001.
doi: 10.1016/j.aquaculture.2020.736001. |
[3] |
C. B. C. K. Cerbule, P. Senff and I. K. Stolz, Towards environmental sustainability in marine finfish aquaculture, Frontier in Marine Science, 2021. |
[4] |
R. A. Falconer and M. Hartnett,
Mathematical modeling of flow, pesticide and nutrient transport for fish-farm planning and management, Ocean Coastal Management, 19 (1993), 37-57.
|
[5] |
M. Kumlu and M. Aktas,
Optimal feeding rates for european sea bass dicentrarchus labrax L. reared in seawater and freshwater, Aquaculture, 231 (2004), 501-515.
|
[6] |
GERSAMP, Monitoring the ecological effects of coastal aquaculture wastes food and agriculture, IMO/FAO/UNESCO/WMO/WHO/IAEA/UN/UNEP joint group of experts on the scientific aspects of marine pollution, GESAMP Rep Stud., (1996), 57. |
[7] |
S. Goyal, D. Ott, J. Liebscher, J. Dautz, H. O. Gutzeit, D. Schmidt and R. Reuss,
Sustainability analysis of fish feed derived from aquatic plant and insect, Sustainability, 13 (2021), 7371.
doi: 10.3390/su13137371. |
[8] |
H. W. J. Lee, C. K. Chan, K. Yau, K. H. Wong and C. Myburgh,
Control parametrization and finite element method for controlling multi-species reactive transport in a circular Pool, J. Ind. Manag. Optim., 9 (2013), 505-524.
doi: 10.3934/jimo.2013.9.505. |
[9] |
Q. Lin, R. Loxton and K. L. Teo,
The control parametrization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10 (2014), 275-309.
doi: 10.3934/jimo.2014.10.275. |
[10] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equation, Springer-Verlag, Berlin, Germany, 1971. |
[11] |
C. Liu, Z. Gong, E. Feng and H. Yin,
Optimal switching control of a fed-batch fermentation process, J. Global Optim., 52 (2012), 265-280.
doi: 10.1007/s10898-011-9663-8. |
[12] |
A. I. Stamou, M. Karamanoli, N. Vasiliadou, E. Douka, A. Bergamasco and L. Genovese,
Mathematical modelling of the interactions between aquacultures and the sea environment, Desalination, 248 (2009), 826-835.
|
[13] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991. |
[14] |
K. H. Wong, L. S. Jennings and F. Benyah,
Control parametrization method for free planning time optimal control problems with time-delayed argument, Nonlinear Anal., 47 (2001), 5679-5689.
doi: 10.1016/S0362-546X(01)00669-1. |
[15] |
K. H. Wong, H. W. J. Lee and C. K. Chan,
Control parametrization and finite element method for controlling multi-species reactive transport in a rectangular diffuser unit, J. Optim. Theory Appl., 150 (2011), 118-141.
doi: 10.1007/s10957-011-9826-2. |
[16] |
K. H. Wong, H. W. J. Lee, C. K. Chan and C. Myburgh,
Control parametrization and finite element method for controlling multi-species reactive transport in an underground tunnel, J. Optim. Theory Appl., 157 (2013), 168-187.
doi: 10.1007/s10957-012-0148-9. |
[17] |
D. Wu, Y. Bai and C. Yu,
A new computational approach for optimal control problems with multiple time-delay, Automatica, 101 (2019), 388-395.
doi: 10.1016/j.automatica.2018.12.036. |
[18] |
R. S. S. Wu,
The environmental impact of marine fish culture: Towards a sustainable future, Marine Pollution Bulletin, 31 (1995), 159-166.
doi: 10.1016/0025-326X(95)00100-2. |
[19] |
R. S. S. Wu, P. K. S. Shin, D. W. MacKay, M. Mollowney and D. Johnson,
Management of marine fish farming in the sub-tropical environment: A modelling approach, Aquaculture, 174 (1999), 279-298.
doi: 10.1016/S0044-8486(99)00024-1. |
[20] |
Z. S. Wu and K. L. Teo,
Optimal control problems involving second boundary value problems of parabolic type, SIAM J. Control Optim., 21 (1983), 729-756.
doi: 10.1137/0321045. |
[21] |
F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings,
Visual MISER: An efficient user-friendly visual program for solving optimal control problems, J. Ind. Manag. Optim., 12 (2016), 781-810.
doi: 10.3934/jimo.2016.12.781. |
[22] |
C. Yu, Q. Lin, R. Loxton, K. L. Teo and G. Wang,
A hybrid time-scaling transformation for time-delay optimal control problems, J. Optim. Theory Appl., 169 (2016), 867-901.
doi: 10.1007/s10957-015-0783-z. |
[23] |
X. Zeng, T. C. Rasmussen, M. B. Beck, A. K. Parker and Z. Lin,
A biogeochemical model for metabolism and nutrient cycling in a southeastern piedmont impoundment, Environmental Modelling and Software, 21 (2006), 1073-1095.
doi: 10.1016/j.envsoft.2005.05.009. |




(a) Phytoplankton (PHY) (b) Organic Nitrogen (NOR) (c) Ammonia (NAM) (d) Nitrates (NIT) (e) Organic Phosphorous (POR) (f) Inorganic Phosphorous (PIN) (g) Suspended Solid (SS)

(a) Organic Nitrogen (NOR) (b) Ammonia (NAM) (c) Nitrates (NIT) (d)Organic Phosphorous(POR) (e)Inorganic Phosphorous(PIN) (f)Suspended Solid(SS) (The purple and the brown curve in Figure 5(e) almost coincide with each other.)


(a) Ammonia (NAM) (b) Nitrates (NIT) (c) Inorganic Phosphorous (PIN) (In Figure 7(a)-Figure 7(c), the black curves represent the instantaneous concentrations of the water quality variables of the No-Fish-Feeding Water Pollution (NFFWP) sub-model, and the brown curves represent the instantaneous Fish-Feeding Water Pollution (FFWP) sub-model obtained by using the optimal control
![]() |
![]() |
D. Decay of PHY |
where |
(ⅰ) |
(ⅱ) |
G. Photosynthesis (Growth of PHY, |
where |
(ⅰ) |
is the growth rate of PHY; |
is the effect of the temperature on the growth of PHY at temperature |
with |
with |
(ⅱ) |
A. Adsorption of NAM |
where |
(ⅰ) |
(ⅱ) |
Set. Settling of PHY |
where |
(ⅰ) |
(ii) |
Am. Ammonification - Mineralization of NOR |
(ⅰ) |
(ⅱ) |
is the saturation constant for Nitrogen mineralization at |
(ⅲ) |
N. Nitrification |
where |
(ⅰ) |
(ⅱ) |
(ⅲ) |
R. Endogenous respiration of PHY |
where |
|
(ⅱ) |
where the formula for |
(ⅲ) |
P. Mineralization of POR |
where |
(ⅰ) |
(ⅱ) |
is the mineralization rate of |
(ⅲ) |
D. Decay of PHY |
where |
(ⅰ) |
(ⅱ) |
G. Photosynthesis (Growth of PHY, |
where |
(ⅰ) |
is the growth rate of PHY; |
is the effect of the temperature on the growth of PHY at temperature |
with |
with |
(ⅱ) |
A. Adsorption of NAM |
where |
(ⅰ) |
(ⅱ) |
Set. Settling of PHY |
where |
(ⅰ) |
(ii) |
Am. Ammonification - Mineralization of NOR |
(ⅰ) |
(ⅱ) |
is the saturation constant for Nitrogen mineralization at |
(ⅲ) |
N. Nitrification |
where |
(ⅰ) |
(ⅱ) |
(ⅲ) |
R. Endogenous respiration of PHY |
where |
|
(ⅱ) |
where the formula for |
(ⅲ) |
P. Mineralization of POR |
where |
(ⅰ) |
(ⅱ) |
is the mineralization rate of |
(ⅲ) |
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