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Article Contents

# Optimal control of fish-feeding in a three-dimensional calm freshwater pond considering environmental concern

• * Corresponding author: Kar Hung Wong

© 2021 The Author(s). Published by AIMS, LLC. This is an Open Access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/ ).

• 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.

Mathematics Subject Classification: Primary: 49M15, 65M60; Secondary: 35Q92.

 Citation:

• Figure 1.  Boundary conditions of the water pollution model

Figure 2.  The Front Part of the Fish Pond

Figure 3.  The global node point of the fish pond

Figure 4.  Instantaneous average concentration in the No-Fish-Feeding Water Pollution (NFFWP) sub-model

Figure 5.  Comparison of the instantaneous average concentration between the NFFWP sub-model, the FFWP sub-model with $u(t) = 1.25$ for all $t\in[180, 550]$, and the FFWP sub-model with $u(t) = 1.5$ for all $t\in[180, 550]$

Figure 6.  The instantaneous optimal control (i.e. the instantaneous optimal fishes' feeding rate)

Figure 7.  Comparison of the instantaneous average concentration between the NFFWP sub-model and the FFWP sub-model obtained by using the optimal control $u^*(t)$

Table 1.  Weight of the fishes at the final time

Table 2.  Maximum concentration at various places of the pond in the fish-feeding water pollution (FFWP) model obtained by using optimal fish-feeding rate

Table A1.  Processes described in the Three-Dimensional Water Pollution Model and their equations

 D. Decay of PHY $\left(D_{1}\right)$ with releases of NOR $\left(D_{2}\right)$, POR $\left(D_{5}\right)$, and SS $\left(D_{7}\right).$ $D_{1} = k_{d 1} \times X_{P H Y}, D_{2} = k_{d 2} \times X_{P H Y}, D_{5} = k_{d 5} \times X_{P H Y}, D_{7} = k_{d 7} \times X_{P H Y},$ where (ⅰ) $k_{d 1} = -k_{d}, k_{d 2} = 0.1761 \times k_{d}, k_{d 5} = 0.1761 \times k_{d}, k_{d 7} = 0.379 \times k_{d},$ (ⅱ) $k_{d}$ is a parameter whose value is given in Table 2. G. Photosynthesis (Growth of PHY, $G_1$), with uptakes of NAM ($G_3$), NIT($G_4$), and PIN($G_6$). $G_1 = k_{g1}(t)\times X_{PHY}$, $G_3 = k_{g3}(t)\times X_{PHY}$, $G_4 = k_{g4}(t)\times X_{PHY}$, $G_6 = k_{g6}(t)\times X_{PHY}$, where (ⅰ) $k_{g 1}(t) = k_{g \max }(t) \times e f(L(t)) \times[$ $e f(N)+e f(P)]$ is the growth rate of PHY; $k_{g \max }(t) = k_{g \max 20} \times 1.047^{Tenp(t)-20}$ is the effect of the temperature on the growth of PHY at temperature $Temp(t){ }^{\circ} C$, where $Temp(t)$ is the water temperature in at time $t$ given by $Temp(t) = Temp_{\min }+\frac{\left(Temp_{\max }-Temp_{\min }\right)}{2}\left(1-\cos \frac{360(t-76)}{365} \times \frac{\pi}{180}\right)$, with $Tem p_{\min } = 13.7$ and $Tem p_{\max } = 25.9$; $e f $$(L(t)) = \frac{L(t)}{L_{S}} \exp \left(1-\frac{L(t)}{L_{S}}\right) is the effect of light on the growth rate of PHY, where L(t) is the incident solar radiation (expressed in cal/ \text{cm}^{2} ) given by L(t) = L_{\min }+\frac{\left(L_{\max }-L_{\min }\right)}{2}\left(1-\cos \frac{(t-15) \times 360}{365} \times \frac{\pi}{180}\right) with L_{\min } = 120 \mathrm{cal} / \mathrm{cm}^{2} , and L_{\max } = 192 \mathrm{cal} / \mathrm{cm}^{2} ; ef$$ (N)$ is the effect of nutrients due to the uptake of NAM and NIT, $ef$$(P)$ is the effect of nutrients due to the uptake of PIN; their average values are given in Table $A2$; $k_{g \max 20}$ and $L_{S}$ are parameters, whose values are given in Table A2, (ⅱ) $k_{g 3}(t) = -0.1761 \times P_{N A M} \times k_{g 1}(t)$, $k_{g 4}(t) = -0.1761 \times\left(1-P_{N A M}\right) k_{g 1}(t)$, $k_{g 6}(t) = -0.1761 \times k_{g 1}(t)$, $P_{N A M}$ is the preference term for NAM whose approximated value is also given in Table A2. A. Adsorption of NAM $\left(A_{3}\right)$ and PIN $\left(A_{6}\right)$. $A_{3} = k_{A 3} \times X_{S S}, \ A_{6} = k_{A 6} \times X_{S S},$ where (ⅰ) $k_{A 3} = -\frac{S V_{S S}}{H} \times a_{NAM}, k_{A 6} = -\frac{S V_{S S}}{H} \times a_{P I N}$, (ⅱ) $S V_{S S}, a_{N A M}, H, a_{P I N}$ are parameters whose values are given in Table 2. Set. Settling of PHY $\left(S_{1}\right)$, NOR $\left(S_{2}\right)$, POR $\left(S_{5}\right)$, and SS $\left(S_{7}\right)$ \begin{align*} &Set_{1} = k_{S e t 1} \times X_{P H Y}, \ Set_{2} = k_{S e t 2} \times X_{N O R}, \ \quad Set_{5} = k_{S e t 5} \times X_{P O R}, \\ &Set_{7} = k_{Set7} \times X_{S S}, \end{align*} where (ⅰ) $k_{Set1} = -\frac{S V_{P H Y}}{H}, \ k_{Set2} = -\frac{S V_{N O R}}{H}\left(1-C_{N O R}\right)$, $k_{Set5} = -\frac{S V_{P O R}}{H}\left(1-C_{P O R}\right), \ k_{Set7} = -\frac{S V_{S S}}{H},$ (ii) $S V_{P H Y}, \ S V_{S S}, \ S V_{N O R}, \ S V_{P O R}, \ C_{N O R}, \ C_{P O R}$ and $H$ are parameters whose values are given in Table 2. Am. Ammonification - Mineralization of NOR $A m_{2} = k_{A m 2}(t) \times X_{N O R}, A m_{3} = k_{A m 3}(t) \times X_{N O R}$ where (ⅰ) $k_{A m 2}(t) = -k_{N min }(t) \times c_{N O R}, k_{A m 3}(t) = k_{N min }(t) \times c_{N O R}$, (ⅱ) $k_{N min }(t) = k_{N min } \times 1.047^{Temp(t)-20}$ is the saturation constant for Nitrogen mineralization at $Temp(t){ }^{o} C$, where the formula for $Temp(t)$ is as given in the Photosynthesis process, (ⅲ) $c_{N O R}$ is a parameter whose value is given in Table 2. N. Nitrification $\left(N i t_{3}\right.$ and $\left.N i t_{4}\right)$. $N i t_{3} = k_{N i t 3}(t) \times X_{N A M}, N i t_{4} = k_{N i t 4}(t) \times X_{N A M},$ where (ⅰ) $k_{N i t 3}(t) = -k_{N i t}(t), k_{N i t 4}(t) = k_{N i t}(t),$ (ⅱ) $k_{N i t}(t) = k_{N i t 20} \times 1.047^{Temp(t)-20}$ is the nitrification rate at $Temp(t)^{o} C$, where the formula for $Temp(t)$ is as given in the Photosynthesis process. (ⅲ) $k_{N i t 20}$ is a parameter whose value is given in Table A2. R. Endogenous respiration of PHY $\left(R_{1}\right)$ with the release of NOR $\left(R_{2}\right)$, NAM $\left(R_{3}\right)$, POR $\left(R_{5}\right)$ and $\operatorname{PIN}\left(R_{6}\right)$. \begin{align*} &R_{1} = k_{r 1}(t) \times X_{P H Y}, \ R_{2} = k_{r 2}(t) \times X_{P H Y}, \ R_{3} = k_{r 3}(t) \times X_{P H Y}, \\ &R_{5} = k_{r 5}(t) \times X_{P H Y}, \ R_{6} = k_{r 6}(t) \times X_{P H Y}, \end{align*} where $\text {(ⅰ) } k_{r 1}(t) = -k_{r}(t), \\ k_{r 2}(t) = 0.1761 \times f_{N O R} \times k_{r}(t), \\ k_{r 3}(t) = 0.1761 \times\left(1-f_{N O R}\right)\times k_{r}(t)\\ k_{r 5}(t) = 0.1761 \times f_{P O R} \times k_{r}(t), \\ k_{r 6}(t) = 0.1761 \times\left(1-f_{P O R}\right) \times k_{r}(t),$ (ⅱ) $k_{r}(t) = k_{r 20} \times 1.047^{Temp(t)-20}$ is the respiration rate at $Temp(t)^{o} C,$ where the formula for $Temp(t)$ is as given in the Photosynthesis process, (ⅲ) $k_{r 20}, f_{N O R}$ are parameters whose values are given in Table 2. P. Mineralization of POR $\left(P_{5}\right.$ and $\left.P_{6}\right)$. $Min_{5} = k_{Min 5}(t) \times X_{P O R}, M i n_{6} = k_{Min6}(t) \times X_{P O R}$ where (ⅰ) $k_{min 5}(t) = -k_{P min }(t) \times c_{P O R}, k_{Min 6}(t) = k_{P min }(t) \times c_{P O R}$, (ⅱ) $k_{P min } = k_{P min 20} \times 1.047^{Temp(t)-20}$ is the mineralization rate of $\mathrm{NOR}$ at $Temp(t)^{o} C$, where the formula for $Temp(t)$ is as given in the Photosynthesis process, (ⅲ) $c_{P O R}$ is a parameter whose value is given in Table A2.

Table A2.  Parameters of the three-dimensional water pollution model and their values

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Figures(7)

Tables(4)