doi: 10.3934/dcdss.2021151
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Mathematical modeling of algal blooms due to swine CAFOs in Eastern North Carolina

1. 

St. Mary's College of Maryland, Department of Economics, St. Mary's City, MD 20686, USA

2. 

St. Mary's College of Maryland, Department of Mathematics and Computer Science, St. Mary's City, MD 20686, USA

3. 

Lafayette College, Department of Mathematics, Easton, PA 18042, USA

4. 

Centre College, Department of Mathematics, Danville, KY 40422, USA

* Corresponding author: ekose@smcm.edu

Received  January 2020 Revised  August 2021 Early access December 2021

Dramatic strides have been made in treating human waste to remove pathogens and excess nutrients before discharge into the environment, to the benefit of ground and surface water quality. Yet these advances have been undermined by the dramatic growth of Confined Animal Feeding Operations (CAFOs) which produce voluminous quantities of untreated waste. Industrial swine routinely produce waste streams similar to that of a municipality, yet these wastes are held in open-pit "lagoons" which are at risk of rupture or overflow. Eastern North Carolina is a coastal plain with productive estuaries which are imperiled by more than 2000 permitted swine facilities housing over 9 million hogs; the associated 3,500 permitted manure lagoons pose a risk to sensitive estuarine ecosystems, as breaches or overflows send large plumes of nutrient and pathogen-rich waste into surface waters. Understanding the relationship between nutrient pulses and surface water quality in coastal environments is essential to effective CAFO policy formation. In this work, we develop a system of ODEs to model algae growth in a coastal estuary due to a manure lagoon breach and investigate nutrient thresholds above which algal blooms are unresolvable.

Citation: Amy Henderson, Emek Kose, Allison Lewis, Ellen R. Swanson. Mathematical modeling of algal blooms due to swine CAFOs in Eastern North Carolina. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021151
References:
[1]

D. F. Boesch, D. M. Anderson and R. A. Horner et al., Harmful algal blooms in coastal waters: Options for prevention, control, and mitigation, NOAA Coastal Ocean Program Decision Analysis Series, 10 (1997). Google Scholar

[2]

J. M. Burkholder et al., Impacts to a coastal river and estuary from rupture of a large swine waste holding lagoon, Journal of Environmental Quality, 26, (1997), 1451–1466. Google Scholar

[3]

J. Burkholder, B. Libra and P. Weyer et al., Impacts of waste from concentrated animal feeding operations on water quality, Environ. Health Persp., 115, (2007), 308–312. doi: 10.1289/ehp.8839.  Google Scholar

[4]

E. J. Buskey and D. A. Stockwell, Effects of a persistent "brown tide" on zooplankton populations in the Laguna Madre of south Texas, Toxic Phytoplankton Blooms in the Sea, (1993), 659–666. Google Scholar

[5]

S. J. Du PlooyN. K. Carrasco and R. Perissinotto, Effects of zooplankton grazing on the bloom-forming Cyanothece sp. in a subtropical estuarine lake, J. Plankton Res., 39 (2017), 826-835.  doi: 10.1093/plankt/fbx039.  Google Scholar

[6]

S. P. Epperly and S. W. Ross, Characterization of the North Carolina Pamlico-Albemarle estuarine complex, Estuarine Ecol., 1986. Google Scholar

[7]

J. A. Freund, S. Mieruch and B. Scholze et al., Bloom dynamics in a seasonally forced phytoplankton-zooplankton model: Trigger mechanisms and timing effects, Ecol. Complex., 3 (2006), 129-139. doi: 10.1016/j.ecocom.2005.11.001.  Google Scholar

[8]

R. E. Fuhrman, History of water pollution control, J. Water Pollut. Con. F., 56 (1984), 306-313.   Google Scholar

[9]

J. C. Goldman and E. Carpenter, A kinetic approach to the effect of temperature on algal growth, Limnol. Oceanogr., 19 (1974), 756-766.   Google Scholar

[10]

S. M. Z. Hossain, N. Al-Bastaki and A. M. A. Alnoaimi et al., Mathematical modeling of temperature effect on algal growth for biodiesel application, Renewable Energy and Environ. Sustainability, 4 (2019), 517-528. doi: 10.1007/978-3-030-18488-9_41.  Google Scholar

[11]

C. Hribar, Understanding concentrated animal feeding operations and their impact on communities, The National Assoc. of Local Boards of Health, 2010. Google Scholar

[12]

J. Kravchenko, S. H. Rhew and I. Akushevich et al., Mortality and health outcomes in North Carolina communities located in close proximity to hog concentrated animal feeding operations, NC Med. J., 79 (2018), 278-288. doi: 10.18043/ncm.79.5.278.  Google Scholar

[13]

M. A. Mallin, Impacts of industrial animal production on rivers and estuaries: Animal-waste lagoons and sprayfields near aquatic environments may significantly degrade water quality and endanger health, Am. Sci., 88 (2000), 26-37.   Google Scholar

[14]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing globaluncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar

[15]

D. F. MartinM. T. Doij and C. B. Stackhouse, Biocontrol of the Florida red tide organism, Gymnodinium breve, through predator organisms, Environ. Lett., 4 (1973), 297-301.  doi: 10.1080/00139307309435500.  Google Scholar

[16]

W. D. McBride and N. Key, US hog production from 1992 to 2009: Technology, restructuring, and productivity growth, USDA Econ. Res. Report, 158 (2013). Google Scholar

[17]

A. Shirota, Red tide problem and countermeasures, Int. J. Aquaculture and Fisheries Tech., 1 (1989), 195-293.   Google Scholar

[18]

J. B. ShuklaA. K. Misra and P. Chandra, Modeling and analysis of the algal bloom in a lake caused by discharge of nutrients, Appl. Math. Comput., 196 (2008), 782-790.  doi: 10.1016/j.amc.2007.07.010.  Google Scholar

[19]

V. H. Smith, Responses of estuarine and coastal marine phytoplankton to nitrogen and phosphorus enrichment, Limnol. Oceanogr., 51 (2006), 377-384.  doi: 10.4319/lo.2006.51.1_part_2.0377.  Google Scholar

[20]

K. A. Steidinger, A re-evaluation of toxic dinoflagellate biology and ecology, Prog. Phycol. Res., 2 (1983), 147-188.   Google Scholar

[21]

M. Swinker, Human health effects of hog waste, NC Med. J., 59 (1998), 16-18.   Google Scholar

[22]

J. M. Testa, Y. Li and Y. J. Lee et al., Quantifying the effects of nutrient loading on dissolved O2 cycling and hypoxia in Chesapeake Bay using a couple hydrodynamic-biogeochemical model, J. Marine Syst., 139 (2014), 139-158. Google Scholar

[23]

J. E. Truscott and J. Brindley, Ocean plankton populations as excitable media, Bull. Math. Biol., 56 (1994), 981-998.   Google Scholar

[24]

S. WingD. Cole and G. Grant, Environmental injustice in North Carolina's hog industry, Environ. Health Persp., 108 (2000), 225-231.  doi: 10.1289/ehp.00108225.  Google Scholar

[25]

S. WingS. Freedman and L. Band, The potential impact of flooding on confined animal feeding operations in eastern north carolina, Environ. Health Persp., 110 (2002), 387-391.  doi: 10.1289/ehp.02110387.  Google Scholar

[26]

J. Zhao and Y. Yan, Dynamics of a seasonally forced phytoplankton-zooplankton model with impulsive biological control, Discrete Dyn. Nat. Soc., 2016 (2016). doi: 10.1155/2016/2560195.  Google Scholar

[27]

What Are Phytoplankton?, Available from: https://oceanservice.noaa.gov/facts/phyto.html. Google Scholar

[28]

North Carolina Department of Environmental Quality: MajorHydro, Available from: http://data-ncdenr.opendata.arcgis.com/datasets/majorhydro. Google Scholar

[29]

North Carolina Department of Environmental Quality: List of Permitted Animal Facilities, Available from: https://deq.nc.gov/cafo-map. Google Scholar

[30]

TIGER/Line Shapefiles, Available from: https://www.census.gov/geographies/mapping-files/time-series/geo/tiger-line-file.html. Google Scholar

[31]

Zooplankton Vs. Phytoplankton, Available from: https://sciencing.com/zooplankton-vs-phytoplankton-5432413.html. Google Scholar

show all references

References:
[1]

D. F. Boesch, D. M. Anderson and R. A. Horner et al., Harmful algal blooms in coastal waters: Options for prevention, control, and mitigation, NOAA Coastal Ocean Program Decision Analysis Series, 10 (1997). Google Scholar

[2]

J. M. Burkholder et al., Impacts to a coastal river and estuary from rupture of a large swine waste holding lagoon, Journal of Environmental Quality, 26, (1997), 1451–1466. Google Scholar

[3]

J. Burkholder, B. Libra and P. Weyer et al., Impacts of waste from concentrated animal feeding operations on water quality, Environ. Health Persp., 115, (2007), 308–312. doi: 10.1289/ehp.8839.  Google Scholar

[4]

E. J. Buskey and D. A. Stockwell, Effects of a persistent "brown tide" on zooplankton populations in the Laguna Madre of south Texas, Toxic Phytoplankton Blooms in the Sea, (1993), 659–666. Google Scholar

[5]

S. J. Du PlooyN. K. Carrasco and R. Perissinotto, Effects of zooplankton grazing on the bloom-forming Cyanothece sp. in a subtropical estuarine lake, J. Plankton Res., 39 (2017), 826-835.  doi: 10.1093/plankt/fbx039.  Google Scholar

[6]

S. P. Epperly and S. W. Ross, Characterization of the North Carolina Pamlico-Albemarle estuarine complex, Estuarine Ecol., 1986. Google Scholar

[7]

J. A. Freund, S. Mieruch and B. Scholze et al., Bloom dynamics in a seasonally forced phytoplankton-zooplankton model: Trigger mechanisms and timing effects, Ecol. Complex., 3 (2006), 129-139. doi: 10.1016/j.ecocom.2005.11.001.  Google Scholar

[8]

R. E. Fuhrman, History of water pollution control, J. Water Pollut. Con. F., 56 (1984), 306-313.   Google Scholar

[9]

J. C. Goldman and E. Carpenter, A kinetic approach to the effect of temperature on algal growth, Limnol. Oceanogr., 19 (1974), 756-766.   Google Scholar

[10]

S. M. Z. Hossain, N. Al-Bastaki and A. M. A. Alnoaimi et al., Mathematical modeling of temperature effect on algal growth for biodiesel application, Renewable Energy and Environ. Sustainability, 4 (2019), 517-528. doi: 10.1007/978-3-030-18488-9_41.  Google Scholar

[11]

C. Hribar, Understanding concentrated animal feeding operations and their impact on communities, The National Assoc. of Local Boards of Health, 2010. Google Scholar

[12]

J. Kravchenko, S. H. Rhew and I. Akushevich et al., Mortality and health outcomes in North Carolina communities located in close proximity to hog concentrated animal feeding operations, NC Med. J., 79 (2018), 278-288. doi: 10.18043/ncm.79.5.278.  Google Scholar

[13]

M. A. Mallin, Impacts of industrial animal production on rivers and estuaries: Animal-waste lagoons and sprayfields near aquatic environments may significantly degrade water quality and endanger health, Am. Sci., 88 (2000), 26-37.   Google Scholar

[14]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing globaluncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar

[15]

D. F. MartinM. T. Doij and C. B. Stackhouse, Biocontrol of the Florida red tide organism, Gymnodinium breve, through predator organisms, Environ. Lett., 4 (1973), 297-301.  doi: 10.1080/00139307309435500.  Google Scholar

[16]

W. D. McBride and N. Key, US hog production from 1992 to 2009: Technology, restructuring, and productivity growth, USDA Econ. Res. Report, 158 (2013). Google Scholar

[17]

A. Shirota, Red tide problem and countermeasures, Int. J. Aquaculture and Fisheries Tech., 1 (1989), 195-293.   Google Scholar

[18]

J. B. ShuklaA. K. Misra and P. Chandra, Modeling and analysis of the algal bloom in a lake caused by discharge of nutrients, Appl. Math. Comput., 196 (2008), 782-790.  doi: 10.1016/j.amc.2007.07.010.  Google Scholar

[19]

V. H. Smith, Responses of estuarine and coastal marine phytoplankton to nitrogen and phosphorus enrichment, Limnol. Oceanogr., 51 (2006), 377-384.  doi: 10.4319/lo.2006.51.1_part_2.0377.  Google Scholar

[20]

K. A. Steidinger, A re-evaluation of toxic dinoflagellate biology and ecology, Prog. Phycol. Res., 2 (1983), 147-188.   Google Scholar

[21]

M. Swinker, Human health effects of hog waste, NC Med. J., 59 (1998), 16-18.   Google Scholar

[22]

J. M. Testa, Y. Li and Y. J. Lee et al., Quantifying the effects of nutrient loading on dissolved O2 cycling and hypoxia in Chesapeake Bay using a couple hydrodynamic-biogeochemical model, J. Marine Syst., 139 (2014), 139-158. Google Scholar

[23]

J. E. Truscott and J. Brindley, Ocean plankton populations as excitable media, Bull. Math. Biol., 56 (1994), 981-998.   Google Scholar

[24]

S. WingD. Cole and G. Grant, Environmental injustice in North Carolina's hog industry, Environ. Health Persp., 108 (2000), 225-231.  doi: 10.1289/ehp.00108225.  Google Scholar

[25]

S. WingS. Freedman and L. Band, The potential impact of flooding on confined animal feeding operations in eastern north carolina, Environ. Health Persp., 110 (2002), 387-391.  doi: 10.1289/ehp.02110387.  Google Scholar

[26]

J. Zhao and Y. Yan, Dynamics of a seasonally forced phytoplankton-zooplankton model with impulsive biological control, Discrete Dyn. Nat. Soc., 2016 (2016). doi: 10.1155/2016/2560195.  Google Scholar

[27]

What Are Phytoplankton?, Available from: https://oceanservice.noaa.gov/facts/phyto.html. Google Scholar

[28]

North Carolina Department of Environmental Quality: MajorHydro, Available from: http://data-ncdenr.opendata.arcgis.com/datasets/majorhydro. Google Scholar

[29]

North Carolina Department of Environmental Quality: List of Permitted Animal Facilities, Available from: https://deq.nc.gov/cafo-map. Google Scholar

[30]

TIGER/Line Shapefiles, Available from: https://www.census.gov/geographies/mapping-files/time-series/geo/tiger-line-file.html. Google Scholar

[31]

Zooplankton Vs. Phytoplankton, Available from: https://sciencing.com/zooplankton-vs-phytoplankton-5432413.html. Google Scholar

30] and the N.C. Department of Environmental Quality [28]">Figure 1.  North Carolina Major River Systems Map produced using QGIS. Data obtained from the U.S. Census Bureau [30] and the N.C. Department of Environmental Quality [28]
30] and the N.C. Department of Environmental Quality [28,29]">Figure 2.  Locations of swine CAFOs (brown dots) relative to major river basins which drain into the Pamlico-Albemarle Sound estuary. Map produced using QGIS. Data obtained from the U.S. Census Bureau [30] and the N.C. Department of Environmental Quality [28,29]
Figure 3.  System dynamics of Model 1 with no additional nutrients added over a 30-day time period. With no additional nutrient influx, any current algal presence quickly resolves itself. As the algae dies out, the amount of dissolved oxygen in the system flourishes
Figure 4.  System dynamics of Model 2 with no additional nutrients added over a 30-day time period. The presence of zooplankton in the system results in a quicker decline in the algae population (Day 5 comparison: $ A $ = 0.6127 $ \mu $g/L in Model 2, as opposed to $ A $ = 1.642 $ \mu $g/L in Model 1 - see Figure 3)
Figure 5.  PRCC sensitivity scores for (a) Model 1 and (b) Model 2
Figure 6.  Changes in the eigenvalue $ \lambda_2 $ corresponding to the equilibrium point $ (A,O,N) = (200, 1,150.0171) $, depending on $ \beta_{N} \text{ and } \mu_{AN} $
Figure 7.  Bifurcation diagram of Model 1 relating the steady nitrogen levels to the average temperature at varying values of $ K_N $, the half-saturation constant for nutrient uptake
Figure 8.  System dynamics of Model 1 with constant nutrient flow at $ 19.6 $ mg/L over a 60-day time period. The algal bloom is resolvable in this case
Figure 9.  System dynamics of Model 1 with constant nutrient flow at $ 19.8 $ mg/L over a 60-day time period. The algal bloom is unresolvable in this case and we find that $ \lambda = 19.7 $ is a bifurcation value for Model 1
Figure 10.  System dynamics of Model 2 with constant nutrient flow at $ 300 $ mg/L over a one year time period. The algal bloom is resolvable in this case
Figure 11.  Model 1 under variable nutrient flow $ \lambda(t) = 20te^{-t/5} $ in a 60-day period
Figure 12.  Long-term dynamics of Model 2 with variable nutrient flow term, $ \lambda(t) = 20te^{-t/5} $. Note that the dissolved oxygen population does recover from the initial hypoxia when the algal population eventually reaches zero
Figure 13.  Long-term dynamics of Model 1 with two breaches, 3 months apart from each other, under variable nutrient flow, $ \lambda(t) = 20te^{-t/5} $
Figure 14.  Long-term dynamics of Model 2 with two breaches, 3 months apart from each other, under variable nutrient flow, $ \lambda(t) = 20te^{-t/5} $
Table 1.  Table of parameter descriptions and values. Where literature values are unavailable, parameters are estimated manually to produce behavior consistent with that which would be expected in model simulations
Name Description Estimate Units Reference
$A_0$ Arrhenius equation constant 5.35$\times 10^9$ days$^{-1}$ [9]
$E/R$ Activation energy/universal gas constant 6472 $^{\circ}$K [9]
$T$ Average air temperature 305.3722 $^{\circ}$K Estimated
$K_N$ Half-saturation constant for nutrient uptake 50.5226 mg/L [10]
$\delta_1$ Natural algal death rate 0.5 days$^{-1}$ [18]
$\delta_2$ Algal death rate due to overcrowding 0.01 L/$(\mu$g$\cdot$days) Estimated
$R_M$ Maximum specific predation rate 0.7 days$^{-1}$ [23]
$\alpha$ Governing rate for predation maximum achievement 5.7 $\mu$g$^2$/L$^2$ [23]
$q_0$ Constant influx of dissolved oxygen 6 days$^{-1}$ Estimated
$\delta_0$ Natural depletion rate of dissolved oxygen 1 days$^{-1}$ [18]
$\alpha_0$ Depletion rate of dissolved oxygen due to algae consumption 0.01 mg/$\mu$g Estimated
$\lambda(t)$ Nutrient flow due to spill event Variable mg/(L$\cdot$ days)
$\beta_N$ Influx rate of nutrients due to death of algae 0.2 mg/$\mu$g Estimated
$\mu_{AN}$ Consumption rate of nutrients by algae 0.5 mg/$\mu$g Estimated
$\gamma$ Production rate of zooplankton 0.05 Unitless [23]
$\delta_Z$ Natural death rate of zooplankton 0.017 days$^{-1}$ [23]
Name Description Estimate Units Reference
$A_0$ Arrhenius equation constant 5.35$\times 10^9$ days$^{-1}$ [9]
$E/R$ Activation energy/universal gas constant 6472 $^{\circ}$K [9]
$T$ Average air temperature 305.3722 $^{\circ}$K Estimated
$K_N$ Half-saturation constant for nutrient uptake 50.5226 mg/L [10]
$\delta_1$ Natural algal death rate 0.5 days$^{-1}$ [18]
$\delta_2$ Algal death rate due to overcrowding 0.01 L/$(\mu$g$\cdot$days) Estimated
$R_M$ Maximum specific predation rate 0.7 days$^{-1}$ [23]
$\alpha$ Governing rate for predation maximum achievement 5.7 $\mu$g$^2$/L$^2$ [23]
$q_0$ Constant influx of dissolved oxygen 6 days$^{-1}$ Estimated
$\delta_0$ Natural depletion rate of dissolved oxygen 1 days$^{-1}$ [18]
$\alpha_0$ Depletion rate of dissolved oxygen due to algae consumption 0.01 mg/$\mu$g Estimated
$\lambda(t)$ Nutrient flow due to spill event Variable mg/(L$\cdot$ days)
$\beta_N$ Influx rate of nutrients due to death of algae 0.2 mg/$\mu$g Estimated
$\mu_{AN}$ Consumption rate of nutrients by algae 0.5 mg/$\mu$g Estimated
$\gamma$ Production rate of zooplankton 0.05 Unitless [23]
$\delta_Z$ Natural death rate of zooplankton 0.017 days$^{-1}$ [23]
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