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 and 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.

[2]

K. R. De SilvaT. V. PhanT. 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.

[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.

[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.

[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.

[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.

[7]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.

[8]

FAO, Fisheries Management, Marine Protected Areas and Fisheries, Food and Agriculture Organization (FAO) of the United Nations, Rome, (2011).

[9]

W. J. FletcherJ. ShawS. 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.

[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.

[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.

[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.

[13]

C. R. GwaltneyM. 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.

[14]

W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.  doi: 10.1007/BF02251947.

[15]

G. E. Herrera and S. Lenhart, Spatial optimal control of renewable resource stocks, in Spatial Ecology, CRC Press, 2009,343-357.

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R. Hilborn and C. J. Walters, Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty, Springer, 1992. doi: 10.1007/978-1-4615-3598-0.

[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.

[18]

H. R. JoshiG. E. HerreraS. 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.

[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.

[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.

[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.

[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.

[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.

[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/.

[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.

[26]

E. A. MobergE. ShyuG. E. HerreraS. LenhartY. 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.

[27]

M. G. Neubert, Marine reserves and optimal harvesting, Ecology Lett., 6 (2003), 843-849.  doi: 10.1046/j.1461-0248.2003.00493.x.

[28]

T. Oguz, Controls of multiple stressors on the Black Sea fishery, Frontiers in Marine Sci., 4 (2017). doi: 10.3389/fmars.2017.00110.

[29]

T. OguzE. 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.

[30]

T. OguzB. 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.

[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.

[32]

E. K. PikitchC. SantoraE. A. BabcockA. BakunR. Bonfil and et al., Ecosystem-based fishery management, Science, 305 (2004), 346-347.  doi: 10.1126/science.1098222.

[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.

[34]

M. Skern-MauritzenG. OttersenN. O. HandegardG. HuseG. 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.

[35]

STECF, Scientific, Technical and Economic Committee for Fisheries (STECF) Black Sea Assessments, Publications Office of the European Union, Luxembourg, EU, 2017,284pp.

[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.

[37]

A. W. TritesV. 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.

[38]

J. E. Wilen, Spatial management of fisheries, Marine Resource Economics, 19 (2004), 7-19.  doi: 10.1086/mre.19.1.42629416.

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.

[2]

K. R. De SilvaT. V. PhanT. 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.

[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.

[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.

[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.

[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.

[7]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.

[8]

FAO, Fisheries Management, Marine Protected Areas and Fisheries, Food and Agriculture Organization (FAO) of the United Nations, Rome, (2011).

[9]

W. J. FletcherJ. ShawS. 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.

[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.

[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.

[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.

[13]

C. R. GwaltneyM. 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.

[14]

W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.  doi: 10.1007/BF02251947.

[15]

G. E. Herrera and S. Lenhart, Spatial optimal control of renewable resource stocks, in Spatial Ecology, CRC Press, 2009,343-357.

[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.

[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.

[18]

H. R. JoshiG. E. HerreraS. 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.

[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.

[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.

[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.

[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.

[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.

[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/.

[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.

[26]

E. A. MobergE. ShyuG. E. HerreraS. LenhartY. 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.

[27]

M. G. Neubert, Marine reserves and optimal harvesting, Ecology Lett., 6 (2003), 843-849.  doi: 10.1046/j.1461-0248.2003.00493.x.

[28]

T. Oguz, Controls of multiple stressors on the Black Sea fishery, Frontiers in Marine Sci., 4 (2017). doi: 10.3389/fmars.2017.00110.

[29]

T. OguzE. 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.

[30]

T. OguzB. 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.

[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.

[32]

E. K. PikitchC. SantoraE. A. BabcockA. BakunR. Bonfil and et al., Ecosystem-based fishery management, Science, 305 (2004), 346-347.  doi: 10.1126/science.1098222.

[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.

[34]

M. Skern-MauritzenG. OttersenN. O. HandegardG. HuseG. 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.

[35]

STECF, Scientific, Technical and Economic Committee for Fisheries (STECF) Black Sea Assessments, Publications Office of the European Union, Luxembourg, EU, 2017,284pp.

[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.

[37]

A. W. TritesV. 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.

[38]

J. E. Wilen, Spatial management of fisheries, Marine Resource Economics, 19 (2004), 7-19.  doi: 10.1086/mre.19.1.42629416.

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.
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
Figure 3.  Initial Biomass of anchovy (blue), jellyfish (red), and zooplankton (green).
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
Figure 6.  Seasonal optimal harvesting rate applied for 3 months with different advection rates, $ b_i $ for $ i = 1,2,3 $
Figure 7.  Seasonal optimal harvesting rate applied for 3 months with different diffusion rates, $ D_i $ for $ i = 1,2,3 $
Table 1.  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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