Saving Nemo. algae and it needs water that is basically free of all nutrients and waste in order to survive. So,

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1 Team #5884 Page 1 of 15 Saving Nemo I. Statement of the Problem Coral survives in a very delicately balanced ecosystem. It can easily be overwhelmed by algae and it needs water that is basically free of all nutrients and waste in order to survive. So, when milkfish farms were started in Bolinao, Pangasinan, and the cages were placed where the waste would wash out to the coral reefs, it was not surprising that the coral reefs began to die. This type of situation requires a creative approach, since the people of Bolinao need the food that is provided by the fish farms. Currently, the fish farming technique in Bolinao is known as a monoculture, as it is introducing only one species to the environment. The only solution is to build an ecosystem around the fish farms that would balance out the negative effects the milkfish on the coral. This solution is known as a polyculture. The basic idea is similar to having a perfectly balanced mobile, and then having someone come along and put an extra item on it. If you are not allowed to take it off, then you just add more weight somewhere else to balance it out. The extra weight that throws the ecosystem off is the monoculture and the weight to balance it once again is the polyculture. We will look at our polyculture as if there is no human interaction, then we will examine the current situation, and finally we will observe how it would look if we put our polyculture in an environment with human involvement. II. Description of the Food Web In order to create a balanced polyculture, we pick one creature from each of the following categories: predatory fish, herbivorous fish, mollusk, crustacean, echinoderm, and algae. Since our goal is to sustain fish farming in Bolinao, we choose the milkfish as our

2 Team #5884 Page 2 of 15 predatory fish. Since rabbit fish are such efficient herbivorous fish and Pterocladia capillacea is a slower growing algae, we choose them so the coral will not be overrun with algae. We choose oysters, spiny lobsters, and sea cucumbers from the mollusk, crustacean, and echinoderm families, respectively, because of their economic benefits. Oysters are not only fantastic filter feeders, which are extremely important in keeping the water clean for the coral, but they can also be eaten and harvested for pearls. Spiny lobsters and sea cucumbers can also be harvested for food, and, in some countries, sea cucumbers are considered a delicacy. Figure 1- Food Web Our simple food web is sketched out in Figure 1. The herbivorous rabbit fish, spiny lobster, and sea cucumber feed on algae, the milkfish feed on the rabbit fish, and all of these produce waste that is filtered by the oyster.

3 Team #5884 Page 3 of 15 III. Assumptions A human-influenced ecosystem is an ecosystem directly influenced by humans there are too many indirect factors that could affect the coral, so in order to make the model manageable we need to eliminate these factors. No sediment runoff this is a reasonable assumption because it is an indirect human influence. Ignore bacteria and viruses although they help to filter the water, they would make the model too complicated, but could be added at a later date. Milkfish only eat rabbit fish this is logical because rabbit fish are the only thing among the polyculture s species that milkfish eat. The only method of death is starvation or old age except for the rabbit fish. Rabbit fish can also die by being eaten by milkfish this is a necessary assumption because accounting for disease would make our models much too complex for calculations at this time. Change in human population does not affect harvesting rates this is reasonable because any milkfish that are not needed can be sold or more food can be bought from other countries. Human influence would lower I zy and I yz (or the impact of the milkfish on the rabbit fish and vice versa) - this is sensible since the owners would be spreading feed out for the milkfish and so they would not eat nearly as many rabbit fish. Unbounded algae is the cause of death for coral this is reasonable since too much algae can choke out the coral.

4 Team #5884 Page 4 of 15 IV. Description of the Mathematics After realizing that this model must take into account some very complex predator-prey relationships, we decide to base our model on the Lotka-Volterra equations. The Lotka-Volterra equations model a simple predator-prey situation between just one predator (y) and just one prey (x). dx = ax αxy = x a αy (1) dt dy = cy + γxy = y c + γx (2) dt Where a and c are the growth rate of the prey and the death rate of the predator, respectively, and α and γ are measures of the effect of the interaction between the two species. In equation (1), the first term (ax) represents how the number of prey would grow without the effect of the predator. The second term ( αxy) is the negative impact the predator s interaction has on the prey. Likewise, for equation (2), the first term ( cy) is the decline of the predator population in the absence of prey. The second term (γxy) represents the positive impact of their interaction on the predator. As this system has many more species in the food web than just one predator and one prey, we need to adapt this same idea to fit the needs of our ecosystem. V. Description of the Model Our approach is to create a very simple model, ignoring the oyster, spiny lobster, and sea cucumber. Once we perfect one model, we will add in a new creature. This way, we will be able to diagnose problems for each individual species. Original Bolinao Coral Reef Ecosystem Before Fish Farm Introduction Our first model consists of algae, rabbit fish, and milkfish. The relationships between these three species are represented in the following equations:

5 Team #5884 Page 5 of 15 dx dt = g xx I xy xy = x g x I xy y (3) dy dt = g yy + I yx yx I yz yz = y g y + I yx x I yz z (4) dz dt = g zz + I zy zy = z g z + I zy y (5) Where x is algae population, y is the rabbit fish population, and z is milkfish population. Growth rates are represented by g with the species variable subscript. The effect of the interaction between the species is I with the first subscript denoting who is being acted upon and the second subscript denoting who is doing the acting. It is easy to see the similarities between Lotka-Volterra s original equations and equations (3) and (5). The biggest change is the population model for the rabbit fish represented by (4). The first part ( g y y + I yx yx) looks like the usual predator model, which makes sense since the rabbit fish is eating the algae. There is an added piece because the rabbit fish is also prey to the milkfish. This is represented by the term ( I yz yz).

6 Team #5884 Page 6 of 15 Graph 1 - Stable Ecosystem with No Human Involvement of 3 Species In graph 1, it is clearly seen that the ecosystem oscillates in a stable state. As the rabbit fish eat the algae, the algae dies off, the rabbit fish numbers decrease, which in turn impacts the milkfish. Graph 2 - Ecosystem Cycle with No Human Involvement and 3 Species Graph 2 shows the cyclical nature of our stable system. In most good models of a predator-prey situation, this type of cycle is what we look for. This proves that every species within our food web are dependent on each other.

7 Team #5884 Page 7 of 15 After several more models, we develop our final model, which accounts for all of our species. The following table and six equations represent our six creatures: Table 1 - Variable Definitions x y z s l m Population of algae Population of rabbit fish Population of milkfish Population of sea cucumber Population of spiny lobster Population of oyster dx dt = g xx I xy xy I xs xs I xl xl + W Am xm = x g x I xy y I xs s I xl l + W Am xm (6) dy dt = g yy + I yx yx I yz yz = y g y + I yx x I yz z (7) dz dt = g zz + I zy zy = z g z + I zy y (8) dl dt = g ll + I lx lx = l g l + I lx x (9) ds dt = g ss + I sx sx = s g s + I sx x (10) dm dt = g m m + I my my + I mz mz + I ms ms + I ml ml W Ax mx = m( g m + I my y + I mz z + I ms s + I ml l) (11) The coefficients g and I follow the same subscript rule as the simple model (see above). Algae has terms of negative interaction with rabbit fish, sea cucumbers, and spiny lobsters. However, the algae has an added term (W Am xm) of the nutrients that are not taken by the oysters. The equations for rabbit fish and milkfish stay the same. For the spiny lobster and sea cucumbers,

8 Team #5884 Page 8 of 15 the equations follow the classic predator equation. Since the oyster is a filter feeder, it needs the existence of the rabbit fish, milkfish, sea cucumber, and spiny lobster for their waste, but the oyster has no impact on them. There is also the term (W Ax mx), which represents the amount of nutrients that are taken by the algae. Because it is not really important that we know from which creature the waste comes, we lump all of I my, I mz, I ms, and I ml into one constant, W T, which stands for waste total, giving us the following equation: dm dt = g m m + W T m y + z + s + l W Ax mx = m( g m + W T y + z + s + l W Ax x) (12) Currently, finding appropriate data for the constants proves difficult, so for the sake of time we choose constants that give us graphs that will simulate a normal ecosystem. We know that we are looking for a sinusoidal graph, and we are looking for logical situations; for example, as the algae population declines, the rabbit fish population also declines. Table 2 - Coefficient Values for Original Boliano Ecosystem Coefficient Value Coefficient Value Coefficient Value Coefficient Value g x 2 I xy 0.02 I xs 0.01 I xl 0.01 g y 0.1 I yx I yz 0.03 g z 0.3 I zy g s 0.5 I sx g l 0.5 I lx g m W T W Am W Ax

9 Team #5884 Page 9 of 15 Table 3 - Numbers of Species in original Bolinao Ecosystem Species Number for Stable State Algae 100 Rabbit Fish 25 Milkfish 56 Sea Cucumbers 20 Spiny Lobsters 25 Oysters 20 Graph 3 - Stable Ecosystem with No Human Involvement and 6 Species Red Yellow Green Light Blue Blue Violet Algae Milkfish Rabbit Fish Sea Cucumber Oyster Spiny Lobster

10 Team #5884 Page 10 of 15 Once again graph 3 proves our model to be a stable oscillating system. It also follows our logical reasoning of how the population of the different species would increase/decrease depending on its food source or predator s numbers. This is a healthy state for coral growth because the algae growth is bounded. If left unchecked, the algae would overwhelm the coral. Current Bolinao Monoculture In order to model the current monoculture in Bolinao, we set all of the populations equal to zero except for the milkfish, which the fish farm introduced, and algae, which is inherently in the ocean. Since we essentially got rid of the milkfish s food source, the rabbit fish, we added in a new constant, F, to stand for the food that the farmer would feed his fish. The final equations are as follows: dx dt = g xx (13) dz dt = g zz + Fz = z g z + F (14) Just by looking at the equations, we can tell that the algae will be unbounded. Table 4 - Coefficients for the Current Bolinao State Coefficient Value g x 2 g z 0.4 F 1

11 Team #5884 Page 11 of 15 Graph 4 - Current Bolinao Situation As predicted, the algae is unbounded. In fact the population skyrockets. This suggests that the algae population would be smothering the coral. Not only that, but the problem will continue because the numbers of milkfish are skyrocketing as well. This will just provide more waste, which will in turn provide more nutrients for the algae. It is a vicious cycle that ends up killing the coral reef. This is a little worse than the actual situation in Bolinao where the coral reefs are dying at a slower rate. If we add in a few of our other species, but still have a significantly larger population of algae and milkfish, we should see an effect much like what is happening in Bolinao right now. The coefficients stay the same as Table 2, except for the following: Table 5 - Coefficients for Current Bolinao Situation Coefficient Value I zy I yz 0.003

12 Team #5884 Page 12 of 15 Graph 5 - Current Bolinao Situation Red Yellow Green Light Blue Blue Violet Algae Milkfish Rabbit Fish Sea Cucumber Oyster Spiny Lobster As you can see, the algae is still growing out of control. If left in this state, it would eventually smother the coral. This model is more like the current Bolinao situation since some of the algae is in check because there are some creatures there in the environment, so the algae s population doesn t grow exponentially. Remediation of Bolinao via Polyculture Now we will introduce the idea of a polyculture into the ecosystem. In each pen we would have some algae, along the top of the sides of the pen, oysters would go along the

13 Team #5884 Page 13 of 15 bottom and sides to filter. Then the system of algae, oysters, sea cucumbers, spiny lobsters, rabbit fish, and milkfish would nearly be self-sufficient. For this model, we will use the coefficients in Table 2 and Table 5. We use the same equations (6)-(10) and (12) as our original ecosystem, except that we have adjusted them for the human involvement. We are using the feed factor that we used in equation (14). This has changed I zy and I yz since the milkfish are not eating as many rabbit fish. Graph 6 - Remediation of Bolinao via Polyculture Red Yellow Green Light Blue Blue Violet Algae Milkfish Rabbit Fish Sea Cucumber Oyster Spiny Lobster Although there is once again the problem with the algae, it is a better situation because there are more oysters. Since the oysters are filter feeders, they help to filter waste from the

14 Team #5884 Page 14 of 15 water. If not filtered, the waste would not only kill off the coral, but help the algae grow even faster. Harvesting and Maximizing the Harvest Unfortunately without sufficient data or time, we are unable to predict how much the harvest would benefit the people of Bolinao. We would have liked to harvest a set number of each of the species and find the value of each species on the market per ton. With this kind of information we would be able to estimate the profit of the harvest. Each harvested pound would depend on what creature is being harvested and what its current value is. Once we do this for all of the species, then we could easily subtract the price for the amount of feed from the total gained from sales to get the total profit. The best way to maximize the harvest with our model is to keep the model stable. By taking out too many of one type of creature, you once again through off the balance of the ecosystem. The best thing to do is harvest proportionally to the population. VI. Conclusions (Strengths and Weaknesses) If our model were to be implemented, it would be a very stable system. This is a huge benefit because the fish farm would not require much upkeep and maintenance. Unfortunately, since we did not have sufficient data, we were not able to actually base our model on a real life ecosystem. However, this information could be collected and easily entered into this model. Also, since we were not able to base our coefficients on real data, some of our numbers were off. For example, in Table 2, there are more milkfish than rabbit fish, which does not make much sense since the milkfish will be eating the rabbit fish. However, our model would not work otherwise. So overall it could use some adjustment and useful data, but for the amount of information available and time allotted, this is a good stable base.

15 Team #5884 Page 15 of 15 VII. Call to Action To Whom It May Concern: There is a growing problem of dying coral reefs everywhere in our oceans. Everything from natural disasters to eroding shorelines are slowly driving these beautiful creatures to extinction. Some things we cannot control, but there is one thing we are able to fix. We can fix the way fish farms introduce their fish into an ecosystem. At the moment, a system called a monoculture is being used. This system only introduces one type of fish. This technique throws off the entire ecosystem by weighting one specific branch of the ecosystem. The rest of the ecosystem cannot handle the strain and soon species die off. It is not just a problem for the coral, but for the native fish as well. We propose a new system be put in use, the polyculture. This system introduces a miniecosystem rather than just one type of fish. It is completely balanced and would not throw of the natural ecosystem. Although the fishery would have to farm creatures they would not originally have chosen, they would profit overall with the added value of the other harvested species. It would have the added benefit of clean water and a balanced system where each and every species would thrive, giving a maximum harvest. It is our hope that you would seriously consider this new farming technique for all fisheries. It would not only benefit all of the farmers, but nature as well. Sincerely, Mathematical Consulting Team