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Bilinear equations in Hilbert space driven by paths of low regularity
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 |
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.
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 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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-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. |
[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. 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. |
[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 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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-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. |
[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. 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. |
[38] |
J. E. Wilen,
Spatial management of fisheries, Marine Resource Economics, 19 (2004), 7-19.
doi: 10.1086/mre.19.1.42629416. |



Parameters | Descriptions | Unit | Value |
Intrinsic growth rate of anchovy, |
days |
0.3 | |
Intrinsic growth rate of jelly fish, |
days |
0.75 | |
Intrinsic growth rate of zoo-plankton, |
days |
0.9 | |
Carrying capacity of anchovy, |
Tonnes | 3.5 |
|
Carrying capacity of jelly fish, |
Tonnes | 7.5 |
|
Carrying capacity of zoo-plankton, |
Tonnes | 4 |
|
Growth rate of |
(days x Tonnes) |
1.4 |
|
Consumption rate of |
(days x Tonnes) |
||
Growth rate of |
(days x Tonnes) |
||
Growth rate of |
(days x Tonnes) |
||
Consumption rate of |
(days x Tonnes) |
||
Consumption rate of |
(days x Tonnes) |
||
Consumption rate of |
days |
0.2 | |
Coefficient of linear part of the cost function | US Dollar | $44,750,000 | |
Coefficient of quadratic part of the cost function | US Dollar x days | 0.1 | |
Price of anchovy per tonnes | US Dollar x (Tonnes) |
$1000 | |
The interest rate of the discount rate | year |
0.01 | |
Advection Coefficient, ( |
km x days |
0.18 | |
Diffusion Coefficient, ( |
km |
0.05 |
Parameters | Descriptions | Unit | Value |
Intrinsic growth rate of anchovy, |
days |
0.3 | |
Intrinsic growth rate of jelly fish, |
days |
0.75 | |
Intrinsic growth rate of zoo-plankton, |
days |
0.9 | |
Carrying capacity of anchovy, |
Tonnes | 3.5 |
|
Carrying capacity of jelly fish, |
Tonnes | 7.5 |
|
Carrying capacity of zoo-plankton, |
Tonnes | 4 |
|
Growth rate of |
(days x Tonnes) |
1.4 |
|
Consumption rate of |
(days x Tonnes) |
||
Growth rate of |
(days x Tonnes) |
||
Growth rate of |
(days x Tonnes) |
||
Consumption rate of |
(days x Tonnes) |
||
Consumption rate of |
(days x Tonnes) |
||
Consumption rate of |
days |
0.2 | |
Coefficient of linear part of the cost function | US Dollar | $44,750,000 | |
Coefficient of quadratic part of the cost function | US Dollar x days | 0.1 | |
Price of anchovy per tonnes | US Dollar x (Tonnes) |
$1000 | |
The interest rate of the discount rate | year |
0.01 | |
Advection Coefficient, ( |
km x days |
0.18 | |
Diffusion Coefficient, ( |
km |
0.05 |
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