In Task 2a, we discussed the independent farming of the milkfish; but actually in the pen, there are more than just milkfish and algae. So here we have to introduce the removed species as the middle strata, and according to the requirements of the problem, adjust the numbers of the species in the middle strata to simulate the water quality in the Bolinao area until the water quality matches the one currently observed.
The concrete practices are as follows: Simulate the water quality (in Site D, for example) and solve the problem according to the model in Task 1.
It is easier to find out the water quality from the initial values of algae, milkfish, and other species than vice versa.
We adopt brute-force random search:
• Set the initial values of algae, other species, and milkfish to 100 ×103, 10×103, and 1.3×103.
• According to the introductory ratio between the milkfish and the algae, and the requirements for the capacity of the pen obtained from Task 2a, we introduce the algae and the milkfish respectively as 72 × 103 and 1.3 × 103, and at the same time have the introductory numbers of other species come from a random distribution between8 × 103 and10 × 103, with the aim of searching for the theoretical value matching the observed water quality.
quality in steady state that is consistent with the actually observed value.
We set out the criteria for judging water quality:
• Chlorophyll a≡ (0.0001x1 − 1.2785)/0.7568.
• Total concentration of organics =
x2× 0.2438 × 6.9 × [0.2, 11.5] + x1 × [242, 493].
• Percentage of different elements in the excrement: C 10%, N 0.4%, P 0.6%.
• C meets|c(1) − c1(1)| ≤ 100and|c(2) − c1(2)| ≤ 100.
• N meets|n(1) − n1(1)| ≤ 10and|n(2) − n1(2)| ≤ 10.
• Chlorophyll meets|ca − 4.5| ≤ 0.15.
We sort out results meeting the above requirements, that is, the numbers of three species when the water quality obtained through simulation similar to the observed one, and show the result in Table 5.2
Table 5.
Simulation results.
Pop. 1 Pop. 2 Pop. 3 Simulation results Initial number ×103 70.0 [8.01,9.00] 1.10
Number in steady state ×103 46.1 9.0 1.04 Estimated from data Number in steady state ×103 45.7 9.3 0.9
To make the numbers of the species close to those predicted in the model, we compare the numbers of existing species with those observed in Bolinao area. Here we take into account that the added feedstuff for milkfish can revise the model in Task 1, that is, we can add a constantλto the the third equation of the model in Task 1 to express the influence of feedstuff on the numbers of the species. The revised model is:
˙x1(t) = x1r1
We set initial values (70000, [8008,8995], 1100), and calculate the steady-state numbers of all the species: (46062, 8989, 1051), as shown in Figure 6.
2EDITOR’S NOTE: The accompanying Matlab code does not impose the constraints indicated above on N and C.
0 5 10 15 20 25 30 35 40 45 50
Figure 6. Comparison between observed values and simulated values.
Task 3
Task 3a: Develop a commercial polyculture to remediate Bolinao We start from the model of Task 1 (the Bolinao coral reef ecosystem model before farming), introduce filter feeders, and revise the model. We renumber the species, with algae as 1, filter feeders as 2, herbivores as 3, and milkfish as 4. Following the same modeling principles as earlier, we arrive at the system:
where we now usek for the constant of feedstuff.
We solve this system in Matlab to obtain the numbers of algae, filter feeders, herbivorous fish, and milkfish: (14314, 6092, 6129, 6979). Figure 7 shows the system tending toward equilibrium.
0 1 2 3 4 5 6 7 8 9 10 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5
Figure 7. The changes in the numbers of algae, filter feeders, herbivorous fish, and milkfish.
Report on the outputs of the model Based on (6), we find:
• This model optimize the water quality, since only when the water quality reaches a certain standard, can it provide the ideal growing environment for a species, and only in the viable environment, is it meaningful to talk about the number of each species.
• We establish a newly-born coral reef habitat without the help of man, that is, without feedstuff casting, with least leftover nutriment and particles (foodstuff and excrement) sediment.
• According to Task 3a, we get the steady-state numbers of algae, filter feeders, herbivorous fish, and milkfish. We regard those as the initial values and determine the concentration of chlorophyll as 0.202 µg/L.
Based on the information about the elements percentage given in prob-lem, we calculate the content of different elements, as shown in Table 6.
• Assume that the total income isK = P
xivi, whereviis the market value of a unit of speciesi.
• Based on market investigation and relevant online data, we get the
aver-Element Concentration (µg/L)
C (10%) 35 –72
N (0.4%) 1.4– 2.9 P (0.6%) 2.1– 4.3
age weight and price of each species, and finally figure out the income:
K = $114 × 103/pen.
• To calculate the cost of improving water quality, assume that we intro-duce 1,000 mussels into the pen. We investigate such factors as weight and market price of mussels, and put them into the model in Task 1 to figure out all the indexes.
Table 7.
Steady-state numbers (×104) of species before adjustment.
Algae Molluscs Herbivorous Milkfish (mussels) Fish
Before adjustment 1.43 0.61 0.61 0.70
After adjustment 1.37 0.62 0.61 0.70
Table 8.
Concentrations of elements (µg/L) before and after adjustment.
Chlorophyll C N P
Before adjustment 0.202 35–71 1.4–2.9 2.1–4.3 After adjustment 0.125 33–69 1.3–2.8 2.0–4.1
From Table 8, it is easy to see that the water quality has improved.
For one thing, the introduced mussels feed on the algae for one thing, and for another they decompose the organic particles.
• The 1,000 introduced mussels cost $361 or so, scarcely making a dent in the income.
Task 4
From Task 3a, we know the numbers of algae, filter feeders, herbivo-rous fish, and milkfish: (14314, 6092, 6129, 6979). The algae are the most numerous, and the numbers of the other species are roughly equal. In such a steady state:
fish is higher that that of seaweed. In addition, although the amount of seaweed is large, it is light, so we cannot pursue maximizing weight.
• Measuring harvest with the price of each species harvested, we have to differentiate the values of the species. Since it costs to feed the milkfish, we should take these costs into consideration when calculating the values of each species. We define the value of edible biomass as the sum of the values of each species harvested, minus the cost of milkfish feed.
Task 5
When evaluating a commercial polyculture scheme, we usually consider not only the economic benefits of farming, but also try to ensure reaching a win-win between economy and environment under the premiss of keeping the ecological environment and water quality in good condition.
Hence, we establish the following optimal model to pursue the maxi-mum commercial benefits, with the premiss of not having water quality worsen. Combined with the previous polyculture system model, we es-tablish the following nonlinear optimization model of balance to maximize the total values of harvest. It is a complex nonlinear single-objective opti-mization model, since nonlinear differential equations are embedded into the constraints:
Objective function: max f = ax1 + bx2+ cx3+ dx4 − µ,
wherea, b, c, dare the unit market prices of the species andµis the feedstuff price.
The constraints on water quality are:
• concentration of of chlorophyll≤0.28 mg/mL,
• concentration of C≤196µg/L, and
• concentration of N≤39µg/L.
We can express these conditions in the equations involving thexias follows:
0.0001x1 − 1.2785
0.756 ≤ 0.28, 1.68222x2[0.2, 11.5] + 0.1x4[242, 493] ≤ 196, 1.68222x2[0.2, 11.5] + 0.004x4[242, 493] ≤ 39.
In addition, we have the equality relations among thexiin (6).
Such a complex optimization problem cannot be solved directly with any software, so first we make a cycle simulation search (actually still a brute-force search) to find enough solutions meeting water quality conditions, and obtain intervals for the steady-state numbers of the species that meet the demands of water quality, as shown in Table 9.
Algae Molluscs Herbivorous Milkfish (mussels) Fish
Maximum 1.3922 0.6249 0.6233 0.7061
Minimum 1.3286 0.6152 0.6174 0.7018
Therefore, we can replace the equality conditions among thexi by in-tervals for the steady-state numbers:
1.3286 ≤ x1 ≤ 1.3922, 0.6152 ≤ x2 ≤ 0.6249, 0.6174 ≤ x3 ≤ 0.6233, 0.7108 ≤ x4 ≤ 0.7061.
We can now use Lingo to solve the equivalent model, with the results of Table 10.
Table 10.
Optimal steady-state numbers (×104) of species.
Algae Molluscs Herbivorous Milkfish (mussels) Fish
Optimal 1.39 0.62 0.62 0.71
The corresponding the maximum harvest value is $115×103, and the corresponding water quality is shown in Table 11.
Table 11.
Concentrations of elements (µg/L) after optimization.
Chlorophyll C N P
At optimal 0.15 17–36 0.7–1.4 1.0–2.2
Compared to the water quality required by coral growth, the water quality obtained here is obviously satisfactory, and we reap relatively high economic benefits at the same time.
In order to prove the results of our model are correct, we define:
fishing/harvest index= feed cost net income
Then the result we obtained is: fishing/harvest index = 0.06%. The actual result is: fishing/harvest index = 2.8%.
feeding cost for one unit of net income is obviously less than the current cost, so our feeding strategy can produce better harvest.