# American Institute of Mathematical Sciences

March  2022, 15(3): 555-572. doi: 10.3934/dcdss.2021151

## 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 Published  March 2022 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, 2022, 15 (3) : 555-572. doi: 10.3934/dcdss.2021151
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##### References:
] 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]
] 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]
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
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)
PRCC sensitivity scores for (a) Model 1 and (b) Model 2
Changes in the eigenvalue $\lambda_2$ corresponding to the equilibrium point $(A,O,N) = (200, 1,150.0171)$, depending on $\beta_{N} \text{ and } \mu_{AN}$
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
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
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
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
Model 1 under variable nutrient flow $\lambda(t) = 20te^{-t/5}$ in a 60-day period
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
Long-term dynamics of Model 1 with two breaches, 3 months apart from each other, under variable nutrient flow, $\lambda(t) = 20te^{-t/5}$
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 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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