Modeling Oyster Reef Reproductive Sustainability: Analyzing Gamete Viability, Hydrodynamics, and Reef Structure to Facilitate Restoration of \textit{Crassostrea virginica}
MModeling Oyster Reef ReproductiveSustainability: Analyzing Gamete Viability,Hydrodynamics, and Reef Structure to FacilitateRestoration of
Crassostrea virginica
Justin Weissberg ∗ and Vinny Pagano Brandeis University Princeton UniversityJanuary 2021
Abstract
The eastern oyster is a keystone species and ecosystem engineer. How-ever, restoration efforts of wild oysters are often unsuccessful, in that theydo not produce a robust population of oysters that are able to success-fully reproduce. Furthermore, the dynamics of wild oyster fertilizationis not yet well understood. Through conducting an experiment predi-cated on quantifying the influence of elementary aspects of fertilizationkinetics—sperm concentration, gamete age, and success rate—we foundthat, as stochastic as the mating process may seem, there are correla-tions which fundamentally serve as the framework for assessing long-termsustainability, reef structure, and hydrodynamic parameters in relationto fertilization. We then focused on mathematically defining a procedurewhich simulated a concentration distribution of a single sperm and egg re-lease where there existed conditions necessary for breeding to take place.We found a very significant impact of both gamete age and sperm con-centration on fertilization rate ( p < . ∗ The authors would like to thank Dr. James Browne and Cody Onufrock for their expertiseand assistance. a r X i v : . [ q - b i o . Q M ] J a n ontents The eastern oyster (
Crassostrea virginica ) acts as an ecosystem engineer and isa keystone species (Sanjeeva, 2008). Therefore, the eastern oyster is a sourceof invaluable natural capital. Humans rely on earth’s natural capital to carryout vital ecosystem services, including water filtration, storm surge protection,and habitat creation. The oyster provides habitats and nutrition for a vari-ety of organisms, and studies have shown that established reefs enhance thespecies’ richness (Luckenbach et al., 2005). Anecdotally, organisms such ascrabs, amphipods, and some fish rely on oyster reefs for survival (Grabowski,2004). Oyster reefs increase habitat complexity; many organisms depend ontheir shell structure for shelter and refuge from predators (Coen and Lucken-bach, 2000; Rodney and Paynter, 2003). Additionally, oysters help to filterexcessive amounts of phytoplankton present in the water (Lenihan et al., 1996;Jackson et al., 2001). They also play a valuable role in nitrogen cycling in estu-aries. By absorbing excess nitrogen in the water, they help to reduce instancesof eutrophication brought about by human nitrogen pollution (Newell et al.,2005, Sebastiano, 2013). Furthermore, oyster reefs may also act as barriers to2rosion via absorbing wave and storm surge energy (Gereffi et al., 2012). Theyare important, hard barriers which protect Spartina marshes from erosive forces(Meyer et al., 2008). Over long periods of time, oysters sequester carbon fromfossil fuel emissions in the form of calcium carbonate (Grabowski and Peterson,2007).Although the eastern oyster was once numerous and prolific, this species isnow struggling in most major bodies of water (Beck et al., 2011). It is estimatedthat 85% of all-natural reefs around the world have been destroyed (Grabowskiand Peterson, 2007; Beck et al., 2011). Consequently, humans have investedmillions of dollars into restoration efforts, most notably in the Chesapeake Bay,Gulf of Mexico, Delaware Bay, and Hudson River (Hargis and Haven, 1988;Frankenberg, 1995, Kreeger, 2011). There are a myriad of ongoing restorationprojects, particularly in the New York waters, where oysters are both a profitablecommercial enterprise and a natural bio filter (Stark et al., 2011).The conditions which have led to the worldwide ecological decline of wildoysters can be either natural or anthropogenic. Oysters have struggled dueto excessive harvesting, destruction of habitats, and two main diseases:
Has-plosporidium nelsoni (MSX) and
Perkinsus marinus (Dermo) (Bosch and Shab-man, 1990; Horton and Eichbaum, 1991; MacKenzie, 1996; Paynter, 1996). Hu-mans appear to be the primary cause for the decline of oyster reefs because theyhave caused the reefs to become too sparsely populated, thus making sufficientexternal fertilization virtually impossible (Lenihan and Peterson, 1998; Lenihanand Peterson, 2004). As a prominent commercial product, oysters have beenoverharvested for food and their calcium carbonate shells, which have variouscommercial applications. The reduction of oyster populations, insofar as toofew adult oysters are available to sustain a reef, has become nearly pervasive.Therefore, an inadequate number of larvae is produced, thereby jeopardizingreef sustainability (Mann and Powell, 2007). This paradigm is then magnifiedby a poor habitat for recruitment due to human interaction. Naturally, oys-ters grow in vertical reefs, yet humans break up, spread out, and even dredgethese reefs for the attainment of oysters of ideally marketable size and shape(Kennedy, 1995; Gosling, 2003).Due to an insufficient understanding of the influence of gamete quality onfertilization, research pertaining to the reproduction of oyster populations hasproven to be highly variable (Boudry et al., 2002). Although there have beenextensive resources invested into the reproduction process of fresh and marine or-ganisms, little research has been performed on mollusks and especially bivalves.Likewise, restoration efforts have been hampered due to an unsatisfactory un-derstanding of the factors that affect fertilization in the wild. Our findingsare novel in that they demystify oyster fertilization, illuminating the scholarlyconversation and presenting an opportunity to restore a once prevalent species.In order to extract information regarding fertilization success in free-spawningorganisms such as the eastern oyster, two approaches can be used (Denny andShibata, 1991): (1) field data can be collected during a spawning event, or (2)models can be constructed based on both laboratory experiments about fertil-ization and field observations about water flow conditions. Issues related to an3n situ approach includes experimental artifacts that may skew estimates of fer-tilization (Pennington, 1985; Yund, 1990; Levitan, 1991; Levitan et al., 1991, inpress; and Denny and Shibata, 1991). Secondly, that natural spawning eventsare rare makes it especially difficult to collect adequate data (Petersen et al.,2008). Limitations on constructing fertilization models include the difficultyin collecting accurate flow data and the knowledge of how gametes behave inspawn-like conditions. In the laboratory, we investigated the interaction betweengamete age and sperm concentration on overall zygote production. There hasbeen an abundance of literature directed at describing how gamete number, age,and other factors impact fertilization in invertebrates (Lilie, 1915; Rothschildand Swann, 1951; Hulton and Hagstrom, 1956; Brown and Knouse, 1973; Vogelet al., 1982; Pennington, 1985). However, in regard to
Crassostrea virginica ,literature on this topic is limited.We collected data on the interaction of gamete age and sperm concentrationin the lab to accurately model oyster fertilization in the wild. Since zygotesform though external reproduction via broadcast spawning, the egg and spermoriginate from different oysters, making the process random and susceptible toconfounding variables such as swimming speeds and anisotropic conditions, no-tably seawater turbulence. Establishing a robust model for predicting a reef’soverall success rate in a controlled environment may minimize other environ-mental factors from being overlooked. This is a first step toward eliminatingthe problems which currently revolve around wild oyster reintroduction.
The physical structure of an environment is known to have profound implica-tions on population biology and interactions between organisms (Bell et al.,1991). The spread and abundance of a species throughout a habitat is directlyinfluenced by structure, due to the availability of substrate for colonization orrecruitment (Underwood and Denley, 1984). Furthermore, habitat structurecan provide protection from competition (Huffaker, 1958). The physical habi-tat structure has major biological implications since it plays a relevant role inthe coupling of physical and biological variables (Belsky et al., 1989; Mannand Lazier, 1991). The coupling of physical and biological habitat variableshas many ecological consequences that are greatly appreciated, especially in themarine environment. Studies have analyzed how habitats affect flow regimes(Vogel, 1981), salinity (Klinne, 1963), and dissolved oxygen concentration (Diazand Rosenberg, 1985). In terms of oyster reefs, the marine benthic habitatcontributes to the settlement of larvae (Breitburg et al., 1995), recruitment(Eckman, 1993), filtering of food (Muschenheim, 1987; Butman et al., 1994;Sanford et al., 1994), predation (Skilleter and Peterson, 1994), and the compo-sition of a community (Wildish and Kristmanson, 1979; Baynes and Szmant,1989; Leonard et al., 1998). Physical habitat structure has been shown to affectphysical variables such as flow velocity, which has been shown to have a sig-nificant influence on the performance of the species that resides in the habitat(Lenihan, 1999). An improved understanding of the role of physical habitat4tructure on the reproduction of organisms such as the eastern oyster will allowfor an increased ability to manage and replenish destroyed reefs using ecolog-ical engineering (Jones et al., 1994). While successful habitat restoration canpotentially be achieved by recognizing and rebuilding the functional propertiesof a habitat, there is still limited literature explaining how habitats function(Thayer, 1992). The purpose of this research is to set forth the foundation fordetermining the critical mass and reef structure necessary to maintain the long-term integrity of a reef. This exploration targets the influence of gamete ageand sperm dilution on fertilization success. It is likely that, as sessile organisms,oysters depend on gamete age, dilution, hydrodynamics, and distance betweenreefs for successful fertilization (Denny, 1988; Denny and Shibata, 1989).
On June 24, 2014, twenty-five eastern oysters, donated by the Town of Hemp-stead Department of Waterways and Conservation, were used in this spawn-ing experiment. The researchers at the lab fed them two types of microalgae:
Tetraselmis chui and
Isochysis galbana . The oysters were conditioned for eightweeks in tanks with a constant flow of water from Reynolds Channel, as wellas a gradual increase in water temperature up to approximately 18 ◦ C to ensuregamete maturation (Davis and Loosandoff, 1952). The dry stripping methodwas used to extract the gametes (Allen and Bushek, 1992). In order to extractthe gametes, each oyster was shucked open. A pipette was used to puncturethe gonad and subsequently extract the gametes. A light microscope was usedin order to identify the type of gamete (egg or sperm). The sperm were pipet-ted into one stock solution and eggs were pipetted into another. This methodhelped to optimize the precision of dilutions and aging of the gametes.
Once the gametes were separated into two stock solutions, different dilutions ofsperm were created. Dilutions of 1:10 were used in order to produce four dilu-tions (ten percent, one percent, one tenth of one percent, and one hundredthof one percent). The gametes were also mixed at varying intervals in order tomanipulate gamete contact time. These include 25, 45, 65, 125, 185, and 245minutes. Each sample received the same amount of egg (0.5 ml) at a concen-tration of twelve percent. Each sample also received 0.2 ml of sperm with thesperm concentration varying between the aforementioned dilutions. Lastly, 5ml of filtered seawater was added to prevent clumping of gametes. The sampleswere transferred from plastic cups to glass containers for storage in a refriger-ator at 2 − ◦ C. Lugol’s iodine was added to every sample for the purpose ofpreservation. 5 .3 Data Collection
A microscope and a Sedgewick-Rafter counting cell were used to determine thetotal quantity of eggs, fertilized eggs, and unfertilized eggs for each sample. Theaverage egg concentration was also obtained.Figure 1: A sample containing gametes that were aged for 25 minutes and hada sperm dilution of 0 . In order to determine the curve of best fit for the data, exponential regressionswere utilized, and R values were calculated to analyze the strength of thecorrelations. A multi-factor analysis of variance (ANOVA) was performed todetermine the significance of the dilution and gamete age on fertilization. Lastly,a forward step-wise regression was used to create a model for extrapolating thefertilization rate of eggs. Subsequent to our fertilization trials, we began refining a mathematical modelin 2018. The model can be divided into two components. First, the empiri-cal fertilization data was acquired, which allowed for the quantification of theinfluence of gamete age (time after gametic release) and sperm concentration.Next, in an effort to parametrize gamete survival in a fluid dynamics model,the data was analyzed using a forward step-wise regression in both gamete ageand concentration variables. This regression model was necessary to estimatefertilization in the wild, as it was applied to Csanady’s Equation to produce afertilization model in a hydrodynamic context (Csanady, 1973).6 .6 Outline of Simulation
Consider one of the simplest cases in which fertilization can occur: • Two oysters are present within the model – One oyster releases sperm and the otherreleases eggs – The total mass of each release is denotedby M – The difference between swimming speedsof gametes is taken as trivial • Oysters rest on the same leveled platform(two-dimensional model) • The gametic releases from the oysters aremodeled as diffusion equations, with eachfunction being relative to a coordinate axis • There are isotropic conditions (steady-state),meaning turbulence is nonexistent and fluiddynamics will follow basic seawater principles • The centers of both graphs of the gametic re-leases are a distance r apart – In the egg equation, a variable can be ap-pended to the spatial variables as a func-tional phase shift Figure 2: Time evo-lution of elementarydiffusion model.These conditions can easily be generalized to simulate diffusion for three spatialvariables. However, to model wild oyster fertilization, it suffices to use only onediffusion equation for the sperm gametes: c ( r, t ) = M √ πDt exp (cid:16) − r Dt (cid:17) . (1)In other words, given a quantity M of a substance, we are able to compute itsconcentration c ( r, t ) at time t and distance r from the point of release.Naturally, more precise models may involve both classes of gametes to cap-ture more nuanced behavior. For instance, the three-dimensional analogue ofthe previous equation commonly used for anisotropic models is the following: c ( x, y, z, t ) = M ( √ πDt ) exp (cid:16) − x + y + z Dt (cid:17) . (2)It is easy to verify that we satisfy the law of conservation of mass since (cid:90) ∞−∞ c ( r, t ) dr = (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:90) ∞−∞ c ( x, y, z, t ) dzdydz = M. (3)7ore specifically, if ∇ denotes the Laplace operator and we consider an anisotropicmedium such as the ocean, we can obtain a solution corresponding to an instan-taneous ( t = 0) and localized ( x = y = z = 0) release: ∂c∂t = − (cid:126) ∇ · (cid:126)q = (cid:126) ∇ · ( D (cid:126) ∇ c ) = D ∇ c (4) c ( x, y, z, t ) = M ( √ πDt ) (cid:112) D x D y D z exp (cid:16) − x D x t − y D y t − z D z t (cid:17) , (5)where (cid:126)q = (cid:104) q x , q y , q z (cid:105) and D are the flux and the diffusion coefficient of thesubstance, respectively (Diffusion, 2012).Note that time can have arbitrary units and will affect the diffusion coeffi-cient accordingly in the solutions to the partial differential equations. Moreover,the process of fertilization will be modeled using the first diffusion equation.Figure 3: Elementary diffusion model at fixed time.In a marine environment, sperm and ova will be mixed due to turbulence.However, they will significantly dilute over time. The concentration of spermand egg in relation to oceanic turbulence can be estimated. Egg concentrationhas been shown to have an insignificant effect on fertilization (Levitan et al.,1991) and, therefore, egg concentration is not included as a variable in thefertilization regression model.Our mathematical contribution expounds upon Equation 2 in a more real-istic setting. Namely, we adopt an experimental perspective and modify theseminal research of Denny and Shibata, applying their methodology to Cras-sostrea virginica . As defined in Equation 12, we consider F ( S ( x, y, z ) , t ) , (6)where F denotes the probability of fertilization in space-time and S ( x, y, z )represents the sperm concentration as defined by Csanady’s model (Csanady,1973; Lauzon-Guay and Scheibling, 2007): S ( x, y, z ) = Q s ¯ u πα y α z u ∗ x (cid:16) e − y u α z +( z − s )2 ¯ u α y α yα zu ∗ x + e − y u α z +( z + s )2 ¯ u α y α yα zu ∗ x (cid:17) . (7)8ariable Name Symbol ValueHorizontal Diffusion Coefficient α y α z Q s . × Current Velocity ¯ u u ∗ s S − Table 1: Values for simulation.Notably, our simulation is an improvement of Equation 7, both for the purposesof our regression and for future models which build upon correlations such asgamete concentration, because of its efficient multidimensional averaging im-plementation. By excluding unanticipated, emergent outliers arising from thechoice of variables in Table 1, we also allow for a more accurate simulation thatis generalizable to other aquatic species.
In applicative hydrodynamics, oceanic eddy turbulence is caused when a wavetraverses a barrier that is not parallel to its velocity vector. It can create anenvironment that is temporarily more prone to mixing. The conditions underwhich an orbital vortex may take place are not well-defined, making any mathe-matical model of oyster diffusion somewhat unfinished or inconclusive. However,simplifications can confine the reaction to a set number of reacting elements topredict the overall behavior of the chaotic system.Consider an anisotropic system with an intrinsic time scale. The time ittakes for a particle to cover a circular orbit of circumference πd o with a nominalorbital velocity u ∗ is: τ = πd o u ∗ ∼ d o u ∗ . (8)Furthermore, the relationship in a random-walk model for the eddy diffusioncoefficient is: ε ∼ d o u ∗ . (9)The physics of these equations, of course, only hold for perfectly circular orbits.Let ε i and D represent axis turbulent transport and molecular transport in astandard mass advection-diffusion equation respectively. (Note that mass canbe synonymous with heat transfer.) Once expanded to three dimensions andsince ε (cid:29) D , we have from Equation 4 that ∂c∂t + (cid:126)u · (cid:126) ∇ c = ε ∇ c (10) ∂c∂t + u ∂c∂x + v ∂c∂y + w ∂c∂z = ∂∂x (cid:16) ε x ∂c∂x (cid:17) + ∂∂y (cid:16) ε y ∂c∂y (cid:17) + ∂∂z (cid:16) ε z ∂c∂z (cid:17) , (11)9here u , v , and w are the corresponding components of the velocity vector (cid:126)u for an arbitrary current. The representation of the eddy diffusion coefficientis currently abstract because of its natural randomness and variation (Robertsand Webster, 2003). It follows that the general solution corresponding to aninstantaneous and localized release is: c ( x, y, z, t ) = M ( √ πDt ) exp (cid:16) − ( x − ut ) + ( y − vt ) + ( z − wt ) Dt (cid:17) . This concludes all of the mathematically defined relationships for diffusion.
Gamete Age (minutes) Sperm Concentration (decimal)0.1000 0.0100 0.0010 0.000125 0.470 0.270 0.280 0.17945 0.392 0.235 0.223 0.14765 0.328 0.204 0.188 0.12185 0.274 0.178 0.154 0.099145 0.159 0.117 0.0844 0.054205 0.093 0.077 0.046 0.029Table 2: Mean fertilization rate for sperm concentrations and gamete ages.
The highest fertilization was recorded for the 10% sperm dilution. A significantdecrease in fertilization occurred due to sperm dilutions ( p = 2 . × − ). Thelowest fertilization was achieved when the gametes were aged to 205 minutes.Gamete age alone appeared to have a significant effect on fertilization rate ( p =1 . × − ). Lastly, results showed a significant interaction between gameteage and sperm dilution ( p = 0 . Using these fertilization results from Table 2 in a regression, it was found thatboth sperm concentration and gamete age variables were highly significant ( p < . F = 1 . · S − . · A + 0 . , (12)where F, S, and A represent the probability of fertilization, sperm concentra-tion (as a decimal proportion of the starting concentration), and gamete age inminutes, respectively. This regression model provides an estimate fertilization10nder different temporal and concentration parameters for use in the hydrody-namic model. This method differs from others such as Denny and Shibata whomodeled the process of fertilization and suggested only about one percent of anindividual egg’s surface is fertilizable (Denny and Shibata, 1989).Figure 4: Percent fertilization at varying sperm concentrations and gamete ageswith exponential regression trendlines.Concentration Fitted Equation R R . . · e − . · x . . · e − . · x . . · e − . · x . . · e − . · x To the authors’ knowledge, this model is the first of its kind to use empiricaldata as a baseline for modeling external fertilization in bivalves. The empiricaldata indicated higher fertilization rates compared to fertilization models of seaurchins (Denny and Shibata, 1989). Unlike sea urchins, oysters are completelysessile and cannot move toward other oysters to increases chances of success-ful fertilization. Therefore, oysters must rely on the quantity of gametes theyproduce to facilitate fertilization.There are a multitude of other factors which could be mathematically imple-mented into the procedure to make it more specific. The fertilization equationhas been shown to essentially be time-dependent (in many cases, the sperm11igure 5: Fertilization hydrodynamics model produced in Python.concentration is negligible when finding maximum positive time). Turbulencegenerally increases the diffusion coefficient, which could allow the partial dif-ferential equations to yield higher probabilities of fertilization. Furthermore,oysters tend to switch sex halfway through their lifespan, and therefore thelong-term modeling is made significantly difficult. Molecular viscosity appearedto be an impractical method to use for the diffusivity, but this conclusion maybe subject to change as the precision of analysis increases.
The work of Mann and Luckenbach suggests that the sperm swims about atrillion times faster than the egg (Mann and Luckenbach, 2013). However, thevalidity of this experiment could be brought into question. Mann and Luck-enbach only used 25 of the 30 samples—namely, the ones for which the spermtravelled a measurable distance. Even when disregarding the small sample sizeof the resultant data, it seems unreasonable to reject those 5 samples from math-ematical models involving sperm swimming speeds because of how delayed orcircular movement is an acceptable feature of Brownian motion. As such, re-moving the samples would create a pseudo-distribution of values for the spermswimming speeds that is not representative of the measured data. Hence, weadded the 5 events of 0.0 movement back (See Figure 6).This new histogram is a skewed-right distribution and reveals a high outlierin the data that was not accounted for in the original graph. Using only the 25data points, the mean of the estimates is 0.058 mm/sec, which lies within thevalue of 0 . ± .
01 mm/sec that Mann and Luckenbach reported. However,if the non-motile sperm are also included, then a more realistic mean of 0.048mm/sec and median of 0.034 mm/sec are found. Regardless, further analysisand experimentation can be conducted to ensure a more accurate and reliableaverage for sperm swimming speeds. 12igure 6: Adjusted histogram of sperm swimming speeds (Mann and Lucken-bach, 2013).
The results and preliminary hydrodynamic models are particularly promising,as they have direct real-world implications on oyster reef restoration. Prior tothis study, there was very little information available for researchers looking tomost successfully engineer self-sustainable oyster reefs. Having now establishedseveral novel correlations pertinent to fertilization, we present the reader witha clearer picture.The distance between oysters is paramount to permitting interbreeding withinsub-populations of a reef. Moreover, in situ experiments have revealed that in-creasing the distance between spawning organisms of opposite sex also resultsin an exponential decline in fertilization (Denny and Shibata 1989; Pennington1985; Yund 1990; Levitan 1991; Marshall 2002; Metaxas 2002). In these exper-iments, just a few meters of separation between spawning organisms resulted ina significant reduction in fertilization success. This study supports past workshowing that fertilization is considerably dependent on the distance betweenorganisms of the opposite gender. For example, at a distance of 80 meters andgamete age of at least 85 minutes, our model yields an average fertilization valueof only 6.38%. Thus, if a researcher constructs sub-populations that are 80 me-ters apart, they will likely be unable to interbreed with each other—which mayeventually lead to the collapse of the entire reef and potentially the ecosystem.Although this research did not address different gender ratios, this may alsobe a non-trivial factor to consider when constructing reefs. Management effortsmust make sure that reefs are constructed with a sufficient ratio of males tofemales. It is interesting to note that, while the eastern oyster is capable of be-ginning as a male and transitioning to a female throughout their life cycle, theycan even switch sex in response to the environment (Buroker, 1983; Kennedy,1983). Based on combined measurements from Delaware Bay that were pub-13ength (mm) p f
25 0.10035 0.20545 0.24655 0.40065 0.55075 0.56285 0.583Table 4: Data from Powell et al. (2013) showing the combined proportion offemales ( p f ) across several sub-populations by size.lished by Powel et al. (2013), oysters of 25 mm average only 10% female, butincrease to over 70% past the 90 mm size (Table 4). When combined with ourmodel results, this implies that a constant addition of new recruitment is neededto maintain a high ratio of small young individuals that will represent a highermale to female ratio. The sparser the population, the more critical it becomesthat a high ratio of new recruits is maintained. Restorations therefore must besupplied with a growing percentage of new individuals for several years in orderto succeed. Tidal estuaries are dynamic environments in which water velocities are contin-gent upon local bathymetry, sun and moon orbital phases, and meteorologicalactivity. Thus the velocity differs by time and location. Directions of currentflow can also vary depending on tidal cycles. Rates of spawning in the wildare highly variable and seldom predictable. In a mass spawning event, mostoysters may spawn but not necessarily the entire population. Other relevantenvironmental parameters such as predation, temperature, and salinity shouldbe incorporated into the model. Settlement of larvae can also be used to esti-mate the recruitment rate of an oyster population. The model should be testedagainst an actual spawning episode in the wild, and sub-samples should becollected to evaluate its robustness.
This model is based on gamete kinetics and should be compared to field estima-tions of actual fertilization success. Observations regarding natural spawning arealso required in order to effectively model fertilization in the wild (Pennington,1985; Yund, 1990; Grosberg, 1991; Levitan, 1991). This model helps explainhow human overharvesting could have led to the decline of natural oyster reefs.As shown by this model, the reefs are very sensitive to the distance between theindividual oysters. A few meters appear to be critical for fertilization and thus14he failure or success of a reef. Humans should learn and adapt from their mis-takes by using the same principles of population density to help restore oysterbeds. By applying the results of this investigation to restoration efforts, reefscan be more effectively and robustly engineered. Humans have benefited greatlycommercially and environmentally from the oyster. Now there is an opportunitywhere humans can give back by helping to restore this keystone species.
Conducting experiments on sperm swimming speeds and further testing Equa-tion 12 at different temperatures (20 − ◦ C) will increase the precision, ac-curacy, and scope of our results. One compelling implementation of a robustmodel would be toward the development of a user-friendly software designedto assist biologists by mathematically informing them about the practicalityand effectiveness of their wild oyster reef setups. A Monte Carlo simulationfor measuring the sustainability of oyster populations is another interesting ap-proach, since it provides a more numerical and probabilistic interpretation ofthe random-walk model. For example, it could easily deal with the statisticthat 80% of wild oysters in established reefs are female and better calculatethe long-term effects of population growth. Generalizing the partial differentialequations to four spatial dimensions and trying to find similarities, differences,and the potential usefulness of the properties of its solutions could also allow fora sustained investigation. Incorporating more variables such as salinity of sea-water, density, pressure, kinematic viscosity, swimming speeds of gametes, andeddy turbulence will make the equation more realistic and applicable. Furtherexploring this exponential correlation between the percentage of zygotes formedand the distance away that the sperm is from the egg should overall improveour understanding of the fertilization process. Finally, giving results that arein context and citing specific examples which can be empirically verified wouldfurther confirm the efficacy of our model.
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