# American Institute of Mathematical Sciences

January  2021, 26(1): 155-171. doi: 10.3934/dcdsb.2020373

## A spatial food chain model for the Black Sea Anchovy, and its optimal fishery

 1 Department of Fisheries and Wildlife, Michigan State University, East Lansing, MI 48824, USA 2 Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USA

* Corresponding author: Mahir Demir

Received  June 2020 Revised  November 2020 Published  January 2021 Early access  December 2020

We present a spatial food chain model on a bounded domain coupled with optimal control theory to examine harvesting strategies. Motivated by the fishery industry in the Black Sea, the anchovy stock and two more trophic levels are modeled using nonlinear parabolic partial differential equations with logistic growth, movement by diffusion and advection, and Neumann boundary conditions. Necessary conditions for the optimal harvesting control are established. The objective for the problem is to find the spatial optimal harvesting strategy that maximizes the discounted net value of the anchovy population. Numerical simulations using data from the Black Sea are presented. We discuss spatial features of harvesting and the effects of diffusion and advection (migration speed) on the anchovy population. We also present the landing of anchovy and its net profit by applying two different harvesting strategies.

Citation: Mahir Demir, Suzanne Lenhart. A spatial food chain model for the Black Sea Anchovy, and its optimal fishery. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 155-171. doi: 10.3934/dcdsb.2020373
##### References:
 [1] K. S. Chaudhuri and S. S. Ray, On the combined harvesting of a prey predator system, Biol. Syst., 4 (1996), 373-389.  doi: 10.1142/S0218339096000259.  Google Scholar [2] K. R. De Silva, T. V. Phan, T. and S. Lenhart, Advection control in parabolic PDE systems for competitive populations, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1049-1072.  doi: 10.3934/dcdsb.2017052.  Google Scholar [3] M. Demir, Optimal Control Strategies in Ecosystem-Based Fishery Models, Ph.D dissertation, University of Tennessee in Knoxville, 2019. Available from: https://trace.tennessee.edu/utk_graddiss/5421. Google Scholar [4] M. Demir and S. Lenhart, Optimal sustainable fishery management of the Black Sea anchovy with food chain modeling framework, Nat. Resour. Model., 33 (2020), 29pp. doi: 10.1111/nrm.12253.  Google Scholar [5] W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Nat. Resour. Model., 22 (2009), 173-211.  doi: 10.1111/j.1939-7445.2008.00033.x.  Google Scholar [6] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar [7] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.  Google Scholar [8] FAO, Fisheries Management, Marine Protected Areas and Fisheries, Food and Agriculture Organization (FAO) of the United Nations, Rome, (2011). Google Scholar [9] W. J. Fletcher, J. Shaw, S. J. Metcalf and D. J. Gaughan, An ecosystem based fisheries management framework: The efficient, regional-level planning tool for management agencies, Marine Policy, 34 (2010), 1226-1238.  doi: 10.1016/j.marpol.2010.04.007.  Google Scholar [10] E. A. Fulton, A. D. M. Smith, D. C. Smith and P. Johnson, An integrated approach is needed for ecosystem based fisheries management: Insights from ecosystem-level management strategy evaluation, PLoS One, 9 (2014). doi: 10.1371/journal.pone.0084242.  Google Scholar [11] A. Grishin, G. Daskalov, V. Shlyakhov and V. Mihneva, Influence of gelatinous zoo-plankton on fish stocks in the Black Sea: Analysis of biological time-series, Marine Ecological J., 6 (2007), 5–24. Google Scholar [12] A. C. Gücü, Y. Genç, M. Dağtekin, S. Sakınan and O. Ak, et al., On Black Sea anchovy and its fishery, Rev. Fisheries Sci. Aquaculture, 25 (2017), 230-244. doi: 10.1080/23308249.2016.1276152.  Google Scholar [13] C. R. Gwaltney, M. P. Styczynski and M. A. Stadtherr, Reliable computation of equilibrium states and bifurcations in food chain models, Comput. Chemical Engrg., 28 (2004), 1981-1996.  doi: 10.1016/j.compchemeng.2004.03.012.  Google Scholar [14] W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.  doi: 10.1007/BF02251947.  Google Scholar [15] G. E. Herrera and S. Lenhart, Spatial optimal control of renewable resource stocks, in Spatial Ecology, CRC Press, 2009,343-357. Google Scholar [16] R. Hilborn and C. J. Walters, Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty, Springer, 1992. doi: 10.1007/978-1-4615-3598-0.  Google Scholar [17] J. Hoekstra and J. C. J. M. van den Bergh, Harvesting and conservation in a predator-prey system, J. Econom. Dynam. Control, 29 (2005), 1097-1120.  doi: 10.1016/j.jedc.2004.03.006.  Google Scholar [18] H. R. Joshi, G. E. Herrera, S. Lenhart and M. G. Neubert, Optimal dynamic harvest of a mobile renewable resource, Nat. Resour. Model., 22 (2009), 322-343.  doi: 10.1111/j.1939-7445.2008.00038.x.  Google Scholar [19] T. K. Kar and K. S. Chaudhuri, Harvesting in a two-prey one-predator fishery: A bioeconomic model, ANZIAM J., 45 (2004), 443-456.  doi: 10.1017/S144618110001347X.  Google Scholar [20] M. R. Kelly Jr., M. G. Neubert and S. Lenhart, Marine reserves and optimal dynamic harvesting when fishing damages habitat, Theoretical Ecology, 12 (2019), 131-144.  doi: 10.1007/s12080-018-0399-7.  Google Scholar [21] M. R. Kelly Jr., Y. Xing and S. Lenhart, Optimal fish harvesting for a population modeled by a nonlinear parabolic partial differential equation, Nat. Resour. Model., 29 (2016), 36-70.  doi: 10.1111/nrm.12073.  Google Scholar [22] T. Lauck, C. W. Clark, M. Mangel and G. R. Munro, Implementing the precautionary principle in fisheries management through marine reserves, Ecological Appl., 8 (1998), S72–S78. doi: 10.1890/1051-0761(1998)8[S72:ITPPIF]2.0.CO;2.  Google Scholar [23] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007. doi: 10.1201/9781420011418.  Google Scholar [24] R. Miller Neilan, Optimal Control Applied to Population and Disease Models, Ph.D. dissertation, University of Tennessee in Knoxville, 2009. Available from: https://trace.tennessee.edu/utk_graddiss/74/. Google Scholar [25] R. Miller Neilan and S. Lenhart, Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons, J. Math. Anal. Appl., 378 (2011), 603-619.  doi: 10.1016/j.jmaa.2010.12.035.  Google Scholar [26] E. A. Moberg, E. Shyu, G. E. Herrera, S. Lenhart, Y. Lou and M. G. Neubert, On the bioeconomics of marine reserves when dispersal evolves, Nat. Resour. Model., 28 (2015), 456-474.  doi: 10.1111/nrm.12075.  Google Scholar [27] M. G. Neubert, Marine reserves and optimal harvesting, Ecology Lett., 6 (2003), 843-849.  doi: 10.1046/j.1461-0248.2003.00493.x.  Google Scholar [28] T. Oguz, Controls of multiple stressors on the Black Sea fishery, Frontiers in Marine Sci., 4 (2017). doi: 10.3389/fmars.2017.00110.  Google Scholar [29] T. Oguz, E. Akoglu and B. Salihoglu, Current state of overfishing and its regional differences in the Black Sea, Ocean & Coastal Mgmt., 58 (2012), 47-56.  doi: 10.1016/j.ocecoaman.2011.12.013.  Google Scholar [30] T. Oguz, B. Fach and B. Salihoglu, Invasion dynamics of the alien ctenophore Mnemiopsis leidyi and its impact on anchovy collapse in the Black Sea, J. Plankton Res., 30 (2008), 1385-1397.  doi: 10.1093/plankt/fbn094.  Google Scholar [31] B. Öztürk, B. A. Fach, Ç. Keskin, S. Arkin and B. Topaloğlu, et al., Prospects for marine protected areas in the Turkish Black Sea, in Management of Marine Protected Areas: A Network Perspective, John Wiley & Sons Ltd., 2017,247–262. Google Scholar [32] E. K. Pikitch, C. Santora, E. A. Babcock, A. Bakun, R. Bonfil and et al., Ecosystem-based fishery management, Science, 305 (2004), 346-347.  doi: 10.1126/science.1098222.  Google Scholar [33] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [34] M. Skern-Mauritzen, G. Ottersen, N. O. Handegard, G. Huse, G. E. Dingsor and et al., Ecosystem processes are rarely included in tactical fisheries management, Fish and Fisheries, 17 (2016), 165-175.  doi: 10.1111/faf.12111.  Google Scholar [35] STECF, Scientific, Technical and Economic Committee for Fisheries (STECF) Black Sea Assessments, Publications Office of the European Union, Luxembourg, EU, 2017,284pp. Google Scholar [36] W. J. Ströbele and H. Wacker, The economics of harvesting predator-prey systems, Zeitschr. f. Nationalökonomie, 61 (1995), 65-81.  doi: 10.1007/BF01231484.  Google Scholar [37] A. W. Trites, V. Christensen and D. Pauly, Competition between fisheries and marine mammals for prey and primary production in the pacific ocean, J. Northwest Atlantic Fishery Sci., 22 (1997), 173-187.  doi: 10.2960/J.v22.a14.  Google Scholar [38] J. E. Wilen, Spatial management of fisheries, Marine Resource Economics, 19 (2004), 7-19.  doi: 10.1086/mre.19.1.42629416.  Google Scholar

show all references

##### References:
 [1] K. S. Chaudhuri and S. S. Ray, On the combined harvesting of a prey predator system, Biol. Syst., 4 (1996), 373-389.  doi: 10.1142/S0218339096000259.  Google Scholar [2] K. R. De Silva, T. V. Phan, T. and S. Lenhart, Advection control in parabolic PDE systems for competitive populations, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1049-1072.  doi: 10.3934/dcdsb.2017052.  Google Scholar [3] M. Demir, Optimal Control Strategies in Ecosystem-Based Fishery Models, Ph.D dissertation, University of Tennessee in Knoxville, 2019. Available from: https://trace.tennessee.edu/utk_graddiss/5421. Google Scholar [4] M. Demir and S. Lenhart, Optimal sustainable fishery management of the Black Sea anchovy with food chain modeling framework, Nat. Resour. Model., 33 (2020), 29pp. doi: 10.1111/nrm.12253.  Google Scholar [5] W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Nat. Resour. Model., 22 (2009), 173-211.  doi: 10.1111/j.1939-7445.2008.00033.x.  Google Scholar [6] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar [7] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.  Google Scholar [8] FAO, Fisheries Management, Marine Protected Areas and Fisheries, Food and Agriculture Organization (FAO) of the United Nations, Rome, (2011). Google Scholar [9] W. J. Fletcher, J. Shaw, S. J. Metcalf and D. J. Gaughan, An ecosystem based fisheries management framework: The efficient, regional-level planning tool for management agencies, Marine Policy, 34 (2010), 1226-1238.  doi: 10.1016/j.marpol.2010.04.007.  Google Scholar [10] E. A. Fulton, A. D. M. Smith, D. C. Smith and P. Johnson, An integrated approach is needed for ecosystem based fisheries management: Insights from ecosystem-level management strategy evaluation, PLoS One, 9 (2014). doi: 10.1371/journal.pone.0084242.  Google Scholar [11] A. Grishin, G. Daskalov, V. Shlyakhov and V. Mihneva, Influence of gelatinous zoo-plankton on fish stocks in the Black Sea: Analysis of biological time-series, Marine Ecological J., 6 (2007), 5–24. Google Scholar [12] A. C. Gücü, Y. Genç, M. Dağtekin, S. Sakınan and O. Ak, et al., On Black Sea anchovy and its fishery, Rev. Fisheries Sci. Aquaculture, 25 (2017), 230-244. doi: 10.1080/23308249.2016.1276152.  Google Scholar [13] C. R. Gwaltney, M. P. Styczynski and M. A. Stadtherr, Reliable computation of equilibrium states and bifurcations in food chain models, Comput. Chemical Engrg., 28 (2004), 1981-1996.  doi: 10.1016/j.compchemeng.2004.03.012.  Google Scholar [14] W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.  doi: 10.1007/BF02251947.  Google Scholar [15] G. E. Herrera and S. Lenhart, Spatial optimal control of renewable resource stocks, in Spatial Ecology, CRC Press, 2009,343-357. Google Scholar [16] R. Hilborn and C. J. Walters, Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty, Springer, 1992. doi: 10.1007/978-1-4615-3598-0.  Google Scholar [17] J. Hoekstra and J. C. J. M. van den Bergh, Harvesting and conservation in a predator-prey system, J. Econom. Dynam. Control, 29 (2005), 1097-1120.  doi: 10.1016/j.jedc.2004.03.006.  Google Scholar [18] H. R. Joshi, G. E. Herrera, S. Lenhart and M. G. Neubert, Optimal dynamic harvest of a mobile renewable resource, Nat. Resour. Model., 22 (2009), 322-343.  doi: 10.1111/j.1939-7445.2008.00038.x.  Google Scholar [19] T. K. Kar and K. S. Chaudhuri, Harvesting in a two-prey one-predator fishery: A bioeconomic model, ANZIAM J., 45 (2004), 443-456.  doi: 10.1017/S144618110001347X.  Google Scholar [20] M. R. Kelly Jr., M. G. Neubert and S. Lenhart, Marine reserves and optimal dynamic harvesting when fishing damages habitat, Theoretical Ecology, 12 (2019), 131-144.  doi: 10.1007/s12080-018-0399-7.  Google Scholar [21] M. R. Kelly Jr., Y. Xing and S. Lenhart, Optimal fish harvesting for a population modeled by a nonlinear parabolic partial differential equation, Nat. Resour. Model., 29 (2016), 36-70.  doi: 10.1111/nrm.12073.  Google Scholar [22] T. Lauck, C. W. Clark, M. Mangel and G. R. Munro, Implementing the precautionary principle in fisheries management through marine reserves, Ecological Appl., 8 (1998), S72–S78. doi: 10.1890/1051-0761(1998)8[S72:ITPPIF]2.0.CO;2.  Google Scholar [23] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007. doi: 10.1201/9781420011418.  Google Scholar [24] R. Miller Neilan, Optimal Control Applied to Population and Disease Models, Ph.D. dissertation, University of Tennessee in Knoxville, 2009. Available from: https://trace.tennessee.edu/utk_graddiss/74/. Google Scholar [25] R. Miller Neilan and S. Lenhart, Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons, J. Math. Anal. Appl., 378 (2011), 603-619.  doi: 10.1016/j.jmaa.2010.12.035.  Google Scholar [26] E. A. Moberg, E. Shyu, G. E. Herrera, S. Lenhart, Y. Lou and M. G. Neubert, On the bioeconomics of marine reserves when dispersal evolves, Nat. Resour. Model., 28 (2015), 456-474.  doi: 10.1111/nrm.12075.  Google Scholar [27] M. G. Neubert, Marine reserves and optimal harvesting, Ecology Lett., 6 (2003), 843-849.  doi: 10.1046/j.1461-0248.2003.00493.x.  Google Scholar [28] T. Oguz, Controls of multiple stressors on the Black Sea fishery, Frontiers in Marine Sci., 4 (2017). doi: 10.3389/fmars.2017.00110.  Google Scholar [29] T. Oguz, E. Akoglu and B. Salihoglu, Current state of overfishing and its regional differences in the Black Sea, Ocean & Coastal Mgmt., 58 (2012), 47-56.  doi: 10.1016/j.ocecoaman.2011.12.013.  Google Scholar [30] T. Oguz, B. Fach and B. Salihoglu, Invasion dynamics of the alien ctenophore Mnemiopsis leidyi and its impact on anchovy collapse in the Black Sea, J. Plankton Res., 30 (2008), 1385-1397.  doi: 10.1093/plankt/fbn094.  Google Scholar [31] B. Öztürk, B. A. Fach, Ç. Keskin, S. Arkin and B. Topaloğlu, et al., Prospects for marine protected areas in the Turkish Black Sea, in Management of Marine Protected Areas: A Network Perspective, John Wiley & Sons Ltd., 2017,247–262. Google Scholar [32] E. K. Pikitch, C. Santora, E. A. Babcock, A. Bakun, R. Bonfil and et al., Ecosystem-based fishery management, Science, 305 (2004), 346-347.  doi: 10.1126/science.1098222.  Google Scholar [33] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [34] M. Skern-Mauritzen, G. Ottersen, N. O. Handegard, G. Huse, G. E. Dingsor and et al., Ecosystem processes are rarely included in tactical fisheries management, Fish and Fisheries, 17 (2016), 165-175.  doi: 10.1111/faf.12111.  Google Scholar [35] STECF, Scientific, Technical and Economic Committee for Fisheries (STECF) Black Sea Assessments, Publications Office of the European Union, Luxembourg, EU, 2017,284pp. Google Scholar [36] W. J. Ströbele and H. Wacker, The economics of harvesting predator-prey systems, Zeitschr. f. Nationalökonomie, 61 (1995), 65-81.  doi: 10.1007/BF01231484.  Google Scholar [37] A. W. Trites, V. Christensen and D. Pauly, Competition between fisheries and marine mammals for prey and primary production in the pacific ocean, J. Northwest Atlantic Fishery Sci., 22 (1997), 173-187.  doi: 10.2960/J.v22.a14.  Google Scholar [38] J. E. Wilen, Spatial management of fisheries, Marine Resource Economics, 19 (2004), 7-19.  doi: 10.1086/mre.19.1.42629416.  Google Scholar
and right graph obtained with different cost of anchovy fishing ($\mu_1$) and the other parameters from Table 1.">Figure 1.  Optimal harvesting rate applied for 9 months. Left graph obtained with baseline parameters given in Table 1 and right graph obtained with different cost of anchovy fishing ($\mu_1$) and the other parameters from Table 1.
and right graph obtained with a different initial condition and the other parameters from Table 1">Figure 2.  Optimal harvesting rate applied for 9 months. Left graph obtained with baseline parameters given in Table 1 and right graph obtained with a different initial condition and the other parameters from Table 1
), jellyfish (red), and zooplankton (green).">Figure 3.  Initial Biomass of anchovy (blue), jellyfish (red), and zooplankton (green).
. The left plots show the populations with the optimal harvesting strategy with maximum harvest rate $0.35$, and the right plots show the populations with constant harvesting strategy $h = 0.35$.">Figure 4.  Comparison of seasonal optimal harvesting strategy and the constant harvesting strategy applied for three months with the baseline parameters given in Table 1. The left plots show the populations with the optimal harvesting strategy with maximum harvest rate $0.35$, and the right plots show the populations with constant harvesting strategy $h = 0.35$.
">Figure 5.  3D plot of optimal harvesting strategy given in Figure 4
Seasonal optimal harvesting rate applied for 3 months with different advection rates, $b_i$ for $i = 1,2,3$
Seasonal optimal harvesting rate applied for 3 months with different diffusion rates, $D_i$ for $i = 1,2,3$
Parameter descriptions, units and numerical values
 Parameters Descriptions Unit Value $r_1$ Intrinsic growth rate of anchovy, $A$ days$^{-1}$ 0.3 $r_2$ Intrinsic growth rate of jelly fish, $P$ days$^{-1}$ 0.75 $r_3$ Intrinsic growth rate of zoo-plankton, $Z$ days$^{-1}$ 0.9 $K_1$ Carrying capacity of anchovy, $A$ Tonnes 3.5$e^{+5}$ $K_2$ Carrying capacity of jelly fish, $P$ Tonnes 7.5$e^{+3}$ $K_3$ Carrying capacity of zoo-plankton, $Z$ Tonnes 4$e^{+4}$ $m_0$ Growth rate of $A$ due to predation of $Z$ (days x Tonnes)$^{-1}$ 1.4$e^{-6}$ $m_1$ Consumption rate of $A$ by its predator, $P$ (days x Tonnes)$^{-1}$ $0.66e^{-5}$ $m_2$ Growth rate of $P$ due to predation of $A$ (days x Tonnes)$^{-1}$ $4.95e^{-6}$ $m_3$ Growth rate of $P$ due to predation of $Z$ (days x Tonnes)$^{-1}$ $5.7e^{-6}$ $m_4$ Consumption rate of $Z$ due to its predator $A$ (days x Tonnes)$^{-1}$ $0.2e^{-5}$ $m_5$ Consumption rate of $Z$ due to its predator $P$ (days x Tonnes)$^{-1}$ $1e^{-5}$ $m_6$ Consumption rate of $P$ due to its predators days$^{-1}$ 0.2 $\mu_1$ Coefficient of linear part of the cost function US Dollar ＄44,750,000 $\mu_2$ Coefficient of quadratic part of the cost function US Dollar x days 0.1 $p$ Price of anchovy per tonnes US Dollar x (Tonnes)$^{-1}$ ＄1000 $\alpha$ The interest rate of the discount rate year$^{-1}$ 0.01 $b_i$ Advection Coefficient, ($b_i > 0$ for $i=1,2,3$) km x days$^{-1}$ 0.18 $D_i$ Diffusion Coefficient, ($D_i > \theta > 0$ for $i=1,2,3$) km$^{2}$ x days$^{-1}$ 0.05
 Parameters Descriptions Unit Value $r_1$ Intrinsic growth rate of anchovy, $A$ days$^{-1}$ 0.3 $r_2$ Intrinsic growth rate of jelly fish, $P$ days$^{-1}$ 0.75 $r_3$ Intrinsic growth rate of zoo-plankton, $Z$ days$^{-1}$ 0.9 $K_1$ Carrying capacity of anchovy, $A$ Tonnes 3.5$e^{+5}$ $K_2$ Carrying capacity of jelly fish, $P$ Tonnes 7.5$e^{+3}$ $K_3$ Carrying capacity of zoo-plankton, $Z$ Tonnes 4$e^{+4}$ $m_0$ Growth rate of $A$ due to predation of $Z$ (days x Tonnes)$^{-1}$ 1.4$e^{-6}$ $m_1$ Consumption rate of $A$ by its predator, $P$ (days x Tonnes)$^{-1}$ $0.66e^{-5}$ $m_2$ Growth rate of $P$ due to predation of $A$ (days x Tonnes)$^{-1}$ $4.95e^{-6}$ $m_3$ Growth rate of $P$ due to predation of $Z$ (days x Tonnes)$^{-1}$ $5.7e^{-6}$ $m_4$ Consumption rate of $Z$ due to its predator $A$ (days x Tonnes)$^{-1}$ $0.2e^{-5}$ $m_5$ Consumption rate of $Z$ due to its predator $P$ (days x Tonnes)$^{-1}$ $1e^{-5}$ $m_6$ Consumption rate of $P$ due to its predators days$^{-1}$ 0.2 $\mu_1$ Coefficient of linear part of the cost function US Dollar ＄44,750,000 $\mu_2$ Coefficient of quadratic part of the cost function US Dollar x days 0.1 $p$ Price of anchovy per tonnes US Dollar x (Tonnes)$^{-1}$ ＄1000 $\alpha$ The interest rate of the discount rate year$^{-1}$ 0.01 $b_i$ Advection Coefficient, ($b_i > 0$ for $i=1,2,3$) km x days$^{-1}$ 0.18 $D_i$ Diffusion Coefficient, ($D_i > \theta > 0$ for $i=1,2,3$) km$^{2}$ x days$^{-1}$ 0.05
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