# Exemplary Discussion Draft 1

Author: Bradley Potter

## Contents

## Model and Hypothesis

The results (shown in the results section) indicate that the larger the initial volume of dead oyster shells, the more sustainable an oyster reef will be. Specifically, there is a critical volume below which the oyster reef will die and above which the oyster reef will go to a stable, living equilibrium. The hypothesis was that a taller, or higher-relief, reef would be more likely to stay alive long term. The volume of the reef is related to the height of the reef by dividing the volume over the area on the sea floor that the oysters inhabit. Therefore, larger volumes (over roughly the same area) translate to higher-relief reefs. Therefore, our model results suggest that our hypothesis was correct: high-relief reefs are more likely to reach a stable, living equilibrium.

## Model Limitations

The model that we have created based on that given by Jordan-Cooley, et al.^{[1]} has some inherent limitations. First of all, the numbers that we have are not directly real-world values. For example, the volumes of sediment, live oysters, and dead oysters show the response of the system for hypothetical values, but to apply this model to a specific, real oyster reef, all of the parameters and initial conditions would have to be measured and scaled to fit the system. What we learn from our model is really more of a qualitative analysis. Adjust a parameter here will have this effect, or changing an initial condition in this way would have this effect. From this knowledge, we could determine some ideal scenarios, such as maximizing the initial reef height, minimizing sediment deposition rate, etc., and then try to impose these optimal conditions on a real system to the best of our abilities in an effort to conserve or reconstruct oyster reefs.

Furthermore, the model that we have does make several assumptions, as detailed in the model description, which are all biologically founded, but as a real system is never ideal, there will inevitably be some loss of accuracy and precision with these assumptions.

Lastly, there are some additional details and factors which the model ignores. Some examples include a more detailed geometric description of each individual reef (something more hill-shaped rather than a rectangular prism perpendicular to the sea floor)as well as the geometry and influence of neighboring reefs. The model treats each reef as a completely independent, free-standing system, which in a bay with a substantial concentration of reefs may not be entirely accurate. There are also many other factors which may influence the oyster prosperity to varying degrees, including the seasonal climate changes, salinity or chemical composition of the water (including pollution), and harvesting for human consumption.

## Discrepancies With Original Paper

As shown and described in the results section, the specific numerical results obtained using the exact equations and parameters of the original paper^{[1]} do not quantitatively agree. After extensive checking of our work and email contact with the corresponding author, we have determined that the likely source of the discrepancy is in the parameter values given in Table 3 of the original paper^{[1]}. Unfortunately, this table of parameter values is used for all other analysis in the original work, making it impossible to duplicate exactly what the authors describe. However, we have results that match quantitatively with everything described in the paper, including both numerical simulations and bifurcation analysis. Since, as described in the Model Limitations above and confirmed by the corresponding author, the purpose of the model is to gain a qualitative understand of the effects of certain conditions, the quantitative discrepancy is not a major concern. We have noted the discrepancy and researched possible causes, and have determined that our model is still adequate for analysis of the system.

## Relationship to Literature

### Hypothesis

As mentioned, our model shows that high-relief reefs should be more stable and sustainable long-term than low-relief reefs, which will tend to die out. This has been confirmed experimentally in some of the restoration efforts of the past decade. It was found in a study of reconstructed reefs, some high-relief and some low-relief, in the Great Wicomico River (Virginia) that "oyster density was fourfold greater on high-relief than on low-relief reefs."^{[2]} This agrees with our findings through the model reconstruction and confirms our hypothesis.

### Model

In the construction of the model, several assumptions were made and each term in the equations was meant to represent a specific biological phenomenon. This is all detailed on the Model Description page. The basis for the model comes from a rough biological understanding and the detailed research that already exists in the field. The logistic growth of live oysters was described by Schulte, et al.^{[2]} The degradation of dead oyster shells, resulting in a decrease in the volume of dead oyster shells, was described by Smith et al.^{[3]} The influence of reef height in the water column and filtration by the live oyster volume on the sediment volume, as well as the influence this sediment volume has on the live oyster population, was described by Lenihan, et al. and Jordan^{[4]}^{[5]} These sources all provide a background in the literature for the assumptions and model construction used in the original paper^{[1]}.

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}“Bistability in a Differential Equation Model of Oyster Reef Height and Sediment Accumulation." Jordan-Cooley, W.C., et al.*Journal of Theoretical Biology.*289 (2011): 1-11. Print. - ↑
^{2.0}^{2.1}Schulte, D. M., et al. "Unprecedented Restoration of a Native Oyster Metapopulation."*Science.*Volume 325. 28 August 2009. 1124-1128. Print. - ↑ Smith, G.F., Bruce, D.G., Roach, E.B., Hansen, A., Newell, R.I.E., McManus, A.M., 2005. "Assessment of Recent Habitat Conditions of Eastern Oyster Crassostrea Virginica Bars in Mesohaline Chesapeake Bay."
*North American Journal of Fisheries Management.*25, 1569–1590. - ↑ Lenihan, H., Micheli, F., Shelton, S., Peterson, C., 1999. "The Influence of Multiple Environmental Stressors on Susceptibility to Parasites: An Experimental Determination With Oysters."
*Limnology and Oceanography.*44(3), 910–924. - ↑ Jordan, S., 1987. "Sedimentation and Remineralization Associated with Biodeposition by the American Oyster Crassostrea Virginica."
*Gmelin.*Ph.D.Thesis, University of Maryland.