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Abstract
The reefs of Bolinao are a precious natural resource. They support a large percentage of the
population's resources. Like all coral reefs, the reefs of Bolinao are vital to the ecosystem, but their
existence is in extreme danger. The reefs of Bolinao are being damaged by the nearby milkfish pens,
which are creating excess bacteria and light-blocking algae. This excess buries the slow-growing coral
reefs beneath feet of debris.
Working to reestablish the reefs of Bolinao creates a fascinating mathematical problem because,
despite the difference in the penned and open ocean systems, each system displays a long-term stable
population. While the population of any species may vary slightly over time, the populations of each
given species must remain roughly stable relative to the other populations in order to keep the system
in equilibrium. Even in the fish pens, a constant population of milkfish produces a relatively constant
level of excretion.
We concluded that the best way to determine the relationships between species was with a
system of linear equations because the systems have long-term stability. All population growths are
directly proportional to each other. In our model, there exists only one predator, allowing us to quantify
its consumption rate. The same type of linear relationship exists throughout the entire ecosystem.
Our basic model allowed us to predict population of each species, the pollution rates, and the
harvestable value of the system as a whole in order to maximize production. Implementation of our
model on the Bolinao reef system creates less pollution and maximizes financial profit from the harvest.
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Modeling Milkfish Excretions
In a channel between the Luzon and Santiago Islands in Bolinao, Pangasinan, there is a fourteen
mile long coral reef [1]. Before the introduction of commercial milkfish, a healthy reef was present.
However, when the local merchants began harvesting the milkfish, the natural habitat became
interrupted. The increase in milkfish led to an increase in milkfish excretion, encouraging a massive
amount of algae growth which in turn kills the reef. While we want to maximize financial and nutritional
benefits for the people, it is also vital to maximize water quality by reducing nutrient input that creates
algae and kills the coral so that the reef can continue to flourish.
Part 1: Modeling the Monoculture before Farming
For this system, we look at a cross-section of ocean 50 meters deep (the depth at which coral
grows) by 100 meters by 100 meters, totaling 500,000 m3, or 500,000,000 liters.
Assumptions
All systems are stable
All population sizes are constant
All organisms are of average weight and lifespan
All algae is phytoplankton
Because all organisms are average, they consume the same amount as other organisms in their
species
Given that average growth of herbivorous and predatory fish is between 10 and 20% of their
ingested matter, we assume 15% for our calculations, leaving 85% waste from food intake
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Algae eat 100% of the edible milkfish sediment. In order to simplify the system, we assume that
the portion which can be used (11% of milkfish excretions) is used
All growth is linear
To create our model, we chose one species from each of the following categories: herbivorous fish,
crustacean, mollusk, echinoderm, and algae, in addition to the milkfish. We selected the following:
rabbitfish for herbivorous fish, giant tiger prawn as our crustaceans, scallops as the mollusks, urchins as
the echinoderms, and phytoplankton as our algae.
Assumptions
An average milkfish weighs 10,000 grams and has an 8 year lifespan [2]
With a given range of milkfish excretion, we selected 350 grams of waste per square meter per
day
All rabbitfish weigh 71 grams and have a lifespan of 5 years [3]
The rabbitfish population per square meter, when greater than zero, is constant
Urchins weigh 7 grams
Mollusks weigh 150 grams [4]
With a large range of the time it takes for algae to double its mass given in the problem, we
assume that the algae mass will double in 5 days. To remain constant this means that algae eat
one-fifth of their weight per day.
Given data tells us that tiger prawns eat 10 to 20 milligrams wet weight of food per individual
per day, and that they eat both algae and bacteria. We assume that the crustaceans eat 10
milligrams of algae and 5 milligrams of bacteria per individual per day
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We found a conversion that wet algae is five times heavier than dry algae [5]. Also, a healthy reef
can live and reproduce in water with one-half million to one million bacteria per milliliter and .25ug
chlorophyll per liter.
Calculating the Amount of Each Species
In order for a healthy reef to exist, the ratios between species need to remain constant, so that
the populations are balanced and the water quality is consistently healthy. In our research, we found
that 1.5% of plankton biomass is chlorophyll [6]. Using this and the fact that a healthy reef system
contains 0.25 micrograms of chlorophyll per liter, we calculated that there are 16.67 micrograms of
phytoplankton per liter of water. Concentrating on our cross-section, we find that there are 8,335
grams of dry algae. Using the conversion between wet and dry algae, this equals 41,675 grams of wet
algae:
Assume from this point forward, all algae measurements are in wet weight. Now that we know
the total amount of algae in the system, we can derive the other populations. Our first calculation finds
the total amount of algae that rabbitfish consume. Given that they eat an average of 20 grams per
month per meter squared of reef and multiplying by the area of reef,
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Given a growth rate of 15% of total consumption, we determine that the rabbitfish will grow at
a rate of 1,000 grams per day among the total rabbitfish.
Comparably, with milkfish eating rabbitfish, the milkfish will grow at a rate of 149 grams per day
spread throughout the total population. To find the milkfish population, we take the total growth of all
the milkfish and multiply it by individual weight over lifespan in days to find the total growth per day.
This yields that 510 milkfish live in our cross-section.
The rabbitfish population is found in the same manner. Rabbitfish weigh 71 grams, live for 5 years, and
the total population grows at a rate of 1,000 grams per day. So we have:
Urchins eat the algae, so to find the number of urchins, we subtract the amount of algae that is eaten
per day by the rabbitfish from the total amount of algae, which leaves us with 1,668 grams of wet algae
remaining to be eaten per day. Based on the given information that urchins eat .05 grams of wet algae
per each gram of urchin per hour and that the urchins each weigh 7 grams, we calculate the total
number of urchins that the system can support:
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Thus, the system can support a maximum of 199 urchins. Because our system also needs to support
tiger prawns, which eat algae, we choose to have 180 urchins in our cross-section. So, 180 urchins
consume 1,512 grams of algae per day, leaving 156 grams per day for the tiger prawn:
We calculate the number of tiger prawns by dividing the grams of algae left by the amount consumed by
the tiger prawn (10 mg of algae per individual per day). This equation tells us that our system, with 180
urchins, supports 15,600 tiger prawns:
If 11% of milkfish excretion is sediment, this means that there is 89% organic material left over. Taking
89% of our total milkfish excretion value from above (15,155 grams), we arrive at a value of 13,487
grams of organic excretion for the mollusks to eat:
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Each 150 gram scallop eats 10 milligrams algae each hour. With this information, we can find out how
much organic material is eaten per day:
We find the total number of scallops by dividing the grams of milkfish excretion the scallops must
consume by the amount that a scallop consumes per day to find the total number of scallops, which
comes is 375.
Organism Interaction and a Steady State of Water Quality
The stable system proves to be circular. Milkfish eat rabbitfish and milkfish excretion feeds the
algae. The algae are, in turn, eaten by the rabbitfish, crustaceans, and urchins. The mollusks eat the left
over organic excretion from the milkfish. So, the water quality remains at a stable and healthy level
because we based our calculations on the amount of algae a healthy system can support before the
coral begins to die off. As long as the algae are eaten at the rate they grow, the water maintains its
quality and the coral continues to flourish.
Part 2: The Current Milkfish Monoculture
Assumptions
Assume an increase of scallops by a factor of 10, making 3750 scallops present in the milkfish
pen
Assume the population of rabbitfish is zero and the milkfish are fed solely by feed that the
farmers introduce into the system
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The milkfish excretions are removed from the system at a rate of 15% per day, either by export
or sedimentation. Thus, 85% remains in the system to be controlled by the population of
mollusks, algae, tiger prawn, and urchins
A linear relationship between bacteria and excretions
Assume a stable system, meaning the total input equals the total output
Relationship between Water Quality and Excretions
Because all variables in a stable system will remain constant in the long term, we know that the fecal
matter concentration (and by extension, the total fecal matter in the system) must remain constant. [By
extension, the sum of all inputs must equal the sum of all outputs, so since our only significant input is
milkfish excretion, and our only outputs are the sedimentation and ‘export’ (or currents carrying water
out of the system) and the rate at which mollusks ingest the feces]:
The input into the system is equal to the portion of the milkfish excretions left unconsumed by the algae
minus the amount consumed by the mollusks. Thus, input equals output and the total change in
excretion in the system is zero:
The input into the system per day is 15% of the total amount of excretions in the system.
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The total excretions divided by the bacterial estimate gives the constant linear relationship between the
grams of excretions and the bacteria:
Part 2A
Assume zero mollusks in system
Excretions and Algae Growth
Based on the previous relationship between bacteria and excretions, the bacteria count in the water is:
The cap on algae growth occurs when the amount leaving the system per day is equal to the amount
that can be added to the system per day because it is a stable system. So, the maximum amount of algae
is 10 times the amount of excretion eaten by the algae per day.
When the algae are at this maximum amount, the amount of chlorophyll in the water is:
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This water quality will not support coral life and is also significantly worse than the water quality
observed in Bolinao during fish farming because the milkfish do not completely suppress all other
aquatic life in the area.
Part 2B
Assumptions
Algae doubles its weight in 5 days
10% of the algae leaves the system per day by way of ocean current
Algae have a cap on growth due to levels of sunlight that can permeate the upper layers of algae
to reach the algae below
Ratio between crustaceans and urchins is the same as in the original model
Relationship between Algae Growth and the Species that Consume Algae
Since algae doubles in 5 days, one-fifth, or 20%, of the total algae [A] must be removed from the system
daily to keep population constant.
Thus,
The amount of algae leaving the system per day is 11% of the milkfish excretion added to the system per
day. So, the total algae in the system is:
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According to equation 1,
Based on the previous model in step one, 15,600 tiger prawns and 180 urchins eat 1668g of algae per
day. The current amount of algae is 115.4 times that amount. Therefore, 115.4 times the previous
populations, or 1,800,359 tiger prawns and 20,773 urchins, would consume one-fifth of the total algae
in the system (found in equation 2).
Calculating the Number of Milkfish
We find the total milkfish contained in the system by
The amount of feed given is 6.58kg/m² of pen/5 months, which converts to 43,700g/day. Thus,
And,
Part 3: Remediation by Polyculture
Assumptions
2,000 milkfish are present in the system
25,704 rabbitfish are present in the system
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The diet of the milkfish consists of rabbitfish and supplementary feed.
Number of Each Species Present in System
Based on the current number of milkfish, 364,337g of excretions are added to the system per day. 89%
of those excretions are consumed by scallops:
Since scallops consume 36g of excretions each, we find the number of scallops present in the system:
The total algae growth per day is equal to 11% of the excretions added to the system per day:
As previously calculated, 25,704 rabbitfish eat 6667g of algae per day. So, the remaining algae to
be consumed by urchins and tiger prawn is 33,410g. This amount of algae will be completely consumed
by 3,000 urchins and 821,000 tiger prawns.
In this model, the number of milkfish increases from the model in part one by a factor of 4. By
extension, the algae, and therefore the chlorophyll, each increase by a factor of 4. So, the water has a
chlorophyll level of about 1 microgram per liter.
The bacteria concentration in the water remains at a healthy and relatively constant level if the
scallops and tiger prawns are allowed to reproduce freely. If the milkfish excretion concentration rises to
an unhealthy level, the scallops will multiply in the presence of the excess food. If the excretion level
decreases, the scallops will have less food and the population will decrease. In the long term, the
excretion level and related bacteria concentration will be maintained.
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In this model, we balance the needs of each species and the water quality. We constrain the
algae growth per day to maintain water quality and avoid overpopulation. We also provide for the
mollusks to consume every gram of excretion that the algae cannot consume.
The market value of sea urchins is $15.34/kg [7], of scallops is $6.50/kg [8], of giant tiger prawn
is $7/kg [9], and of milkfish is $7.10/kg. The total economic value of the system is the combined value of
the organisms in the system:
Species
Number
Weight/organism
Value per 1000g
Total value
Sea urchin
3,000
7g
$15.34
Scallop
9,007
150g
$6.50
$8,781.83
Giant tiger prawn
821,000
150g
$7.00
$862,050.00
Milkfish
2,000
10,000g
$1.28
$25,560.00
$322.14
Total:
$896,713.97
The harvest value of this model is equal to the grams of algae consumed multiplied by the 15% growth
rate multiplied by the value per kilogram.
Species
Algae directly
consumed
per day
Value per kg
Amount of
feed
consumed
per day
Percentage
of diet
consisting of
algae
Harvest
value per
day
$15.34
Amount of
algae
consumed
per day
through
rabbitfish
consumption
0g
Sea urchin
25,200g
0g
100%
$57.99
Scallop
324,260g
$6.50
0g
0g
100%
$316.15
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Giant tiger
prawn
Milkfish
14
8,210g
$7.00
0g
0g
67%
$12.93
0g
$1.28
1,000.1g
36,058g
100%
$7.10
This polyculture is much more efficient because it protects the environment and creates greater
revenue for the fish farms.
The Cost of Improving Water Quality by One Unit
By removing one milkfish:
Excretion lost per one milkfish is
To find the loss of revenue for other species,
By algebra and substitution,
Species
Urchin and
Giant tiger
prawn
Scallop
Calculation for cost in dollars per day by removing one milkfish
Total loss in $
$0.030
$0.020
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Milkfish
(Negligible)
$0.003
Total loss per day:
$0.05
So, the change in total algae is:
So, $0.05 per day improves water quality by: the chlorophyll in the loss of algae per liters of water:
Part 4: Science
We want to use our model to maximize benefit to the local human population. We defined maximum
value in terms of dollars. Thus, we determine the human benefit by the dollar value of the harvest.
Part 5: Maximizing the Harvest
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Daily Harvest in Dollars vs Algae
Concentration
1200
1000
800
600
Harvest daily in dollars
400
200
0
0
41675
1925000 2284333
For maximization of a given level of pollution, we begin by ignoring the bacterial effect of the
giant tiger prawn, as it is difficult to quantify. Then, we hold pollution levels constant. This, by extension,
fixes numbers of mollusks, algae, and milkfish, and the rabbitfish are held constant by the model itself.
So, the question becomes, “Given a fixed amount of algae per day, which is more efficient to raise and
harvest: crustaceans or urchins?”
Since each invertebrate uses the same growth efficiency rate in our model equations, the
question is further simplified to, “Which is greater: the value of an urchin per gram or the value of a
giant tiger prawn per gram?” There is, however, one caveat: giant tiger prawn supplements their algae
intake with bacteria consumption. So, the algae that crustaceans eat is only 2/3 of their diet.
Given our data of the values per kg of each organism ($15.34 for urchin and $7 for giant tiger
prawn), we get a conversion of
And
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Thus, we know that urchins are much more profitable. So, conversion to an entirely urchin-based model,
rather than an urchin and prawn-based model, costs $12.93 per day in crustacean harvest, but returns
$18.90 per day in urchin harvest. Thus, the increase in total returns is $5.96 per day.
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Letter to the Director of the Pacific Marine Fisheries Council
Many years ago, off the shores of Bolinao, healthy coral reefs grew in abundance. Today, much
of that coral is dead due to milkfish farming. The farmers breed the milkfish in pens without other
species to control its excretions. The excretions build up and create a layer of sediment on the ocean
floor, covering the coral reefs. The excess excretions also feed the algae that are already present in the
water and the algae multiply to an unhealthy level. The water quality, therefore, decreases to a level
that is unhealthy for coral growth and the coral in the area dies out.
In order to maintain healthy water quality and reestablish the coral reefs, a polyculture system
of farming must be established. We have created a model that, when in effect, will allow the coral to
reestablish itself without further human intervention. The model includes the following: one particular
predatory fish, the milkfish; one particular herbivorous fish, the rabbitfish; one mollusk species, the
scallop; one crustacean species, the giant tiger prawn; one echinoderm species, the sea urchin; and one
algae species, phytoplankton. We have created a system that not only allows the coral to reproduce, but
creates a valuable harvest for the farmers.
In order to maximize water quality and marine harvest, we chose a mean between the two. It is
not possible to maintain the levels of production necessary to sustain the seaside economy just with a
reef-based ecosystem; instead, we have to take the broad ocean system and supplement it with feed.
However, a more efficient utilization of even the waste products of the system will allow for the
production of much more material, with less cost, in the same area, while doing significantly less
damage.
Our model indicates that it is entirely possible to raise sufficient mollusk populations to return
bacteria levels to normal, by absorbing the fecal content of the water, and it is also very possible to
breed sufficient rabbitfish and other similar creatures to drastically reduce algae levels, although
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decreasing the algae levels below a certain point is environmentally unfeasible. The first step, which will
take years, is to introduce crustaceans, urchins and rabbitfish in large numbers into the pens, while
decreasing the numbers of milkfish. This will allow for the improvement of the environment, and with
harvesting of the milkfish kept high, allow for the populations to grow. After five to ten years, it seems
likely that the populations will reach levels which will sustain the more moderate system we have
entailed. For a relatively low level of pollution, this model will allow the economy to draw nearly the
same income, even disregarding the cost of feed, by drawing upon the profit potential of the waste.
Through algae and mollusks, it can be sieved from the water and turned into profit.
This model will help to sustain the coral reefs, clear the water of pollution, and increase profits
from the harvesting, relative to invested feed. The reefs, once stable, and clean, can provide external
profits from tourism, resulting in a significant net increase in profit to the nearby economy. In short, for
a very low commitment in time and cost, profits can be increased dramatically.
The model will allow us to meet the needs of the people of Bolinao and preserve the beautiful
coral reefs for future generations; it will meet the needs of the present, without selling the future.
Sincerely,
Math team #5889
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References
1.
14 Mile Reef. http://philangler.tripod.com/14%20reef.htm. 6 February 2009.
2. Encyclopedia of Life. http://www.eol.org/pages/224731?category_id=232. 7 February 2009.
3. Fishlore.com. http://fishlore.com/profile_blotchedfoxface.htm. 7 February 2009.
4. Aquaculture Research. Lion’s Paw Scallop.
http://www.fieldstudies.org/download/247_volker_et_al_2005_.pdf. 7 February 2009.
5. CRESP. http://www.cresp.org/Amchitka/Final_WW_DW_3_13_06.pdf. 7 February 2009.
6. NCSU Water Quality Group. Watersheds Algae.
http://www.water.ncsu.edu/watersheds/info/algae.html. 7 February 2009.
7. Freshnews.com. http://www.freshnews.com/news/other-tech-areas/article_27723.html. 8
February 2009.
8. Fisheries and Aquatic Department.
http://www.fao.org/fishery/culturedspecies/Patinopecten_yessoensis. 8 February 2009.
9. Introduction. http://www.fao.org/docrep/009/a0086e/A0086E06.htm. 8 February 2009.