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The Leaves of a Tree

"How much do the leaves on a tree weigh?" How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following:

Why do leaves have the various shapes that they have?

Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape?

Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure?

How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?

In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.

Nowadays the heavy metal pollution is so common that people pay more and more attention to it. The aim of this paper is to calculate the maximum of methylmercury in human body during their lifetime and the maximum number of fish the average adult can safely eat per month. From City Officials research[1], we get information that the mean value of methylmercury in bass samples of the Neversink Reservoir is 1300 ug/kg and the average weight of bass people consume per month is 0.7 kg. According to the different consuming time in every month, we construct a discrete dynamical system model for the amount of methylmercury that will be bioaccumulated in the average adult body. In ideal conditions, we assume people consume bass at fixed term per month. Based on it, we construct fixed-ingestion model and we reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3505 ug. As methylmercury ingested is not only coming from bass but also from other food, hence, we make further revise to our model so that the model is closer to the actual situation. As a result, we figure out the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3679 ug. As a matter of fact, although we assume people consume one fish per month, the consuming time has great randomness. Taking the randomness into consideration, we construct a random-ingestion model at the basis of the first model. Through computer simulations, we obtain the maximum of methylmercury in human body is 4261 ug. We also calculate the maximum amount is 4420 ug after random-ingestion model is revised.

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As it is known to us, different countries and districts have different criterions for mercury toxicity. In our case, we adopt LD50 as the toxic criterions(LD50 is the dosage at which 50% of the humans exposed to a particular chemical will die. The LD50 for methylmercury is 50 mg/kg.). We speculate mercury toxicity has effect on the ability of eliminating mercury, therefore, we set up variable-elimination model at the basis of the first model. According to the first model, the amount of methylmercury in human body is 50 ug/kg, far less than 50 mg/kg, so we reach the conclusion that the fish consumption restrictions put forward by the reservoir advisories can protect the average adult. If the amount of methylmercury ingested increases, the amount of bioaccumulation will go up correspondingly. If 50 mg/kg is the maximum amount of methylmercury in human body, we can obtain the maximum number of fish that people consume safely per month is 997.

Keywords: methylmercury discrete dynamical system model variable-elimination model

discrete uniform random distribution model random-ingestion model

Introduction

With the development of industry, the degree of environmental pollution is also increasing. Human activities are responsible for most of the mercury emitted into the environment. Mercury, a byproduct of coal, comes from acid rain from the smokestack emissions of old, coal-fired power plants in the Midwest and South. Its particles rise on the smokestack plumes and hitch a ride on prevailing winds, which often blow northeast. After colliding with the Catskill mountain range, the particles drop to the earth. Once in the ecosystem, micro-organisms in the soil and reservoir sediment break down the mercury and produce a very toxic chemical form known as methylmercury. It has great effect on human health.

Public officials are worried about the elevated levels of toxic mercury pollution in reservoirs providing drinking water to the New York City. They have asked for our assistance in analyzing the severity of the problem. As a result of the bioaccumulation

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of methylmercury, if the reservoir is polluted, we can make sure that the amount of methylmercury in fish is also increasing. If each person adheres to the fish consumption restrictions as published in the Neversink Reservoir advisory and consumes no more than one fish per month, through analyzing, we construct a discrete dynamical system model of time for the amount of methylmercury that will bioaccumulate in the average adult person. Then we can obtain the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime. At the same time, we can also get the time that people have taken to achieve the maximum amount of methylmercury. As we know, different countries and districts have different criterions for the mercury toxicity. In our case, we adopt the criterion of Keller Army Community Hospital. If the maximum amount of methylmercury in human body is far less than the safe criterion, we can reach the conclusion that the reservoir is not polluted by mercury or the polluted degree is very low, otherwise we can say the reservoir is great polluted by mercury. Finally, the degree of pollution is determined by the amount of methylmercury in human body.

Problem One

discrete dynamical system model

The mean value of methylmercury in bass samples of the Neversink Reservoir is 1300 ug/kg and the average weight of bass is 0.7 kg. According to the subject, people consume no more than one fish per month. For the safety of people, we must consider the bioaccumulation of methylmercury under the worst condition that people absorb the maximum amount of methylmercury. Therefore, we assume that people consume one fish per month. Assumptions

The amount of methylmercury in fish is absorbed completely and instantly by

people.

The elimination of mercury is proportional to the amount remaining. People absorb fixed amount of methylmercury at fixed term per month. We assume the half-life of methylmercury in human body is 69.3 days. Solutions

Let 1 denote the proportion of eliminating methylmercury per month, 1 denote the accumulation proportion. As we know, methylmercury decays about 50 percent every 65 to 75 days, if no further methylmercury is ingested during that time. Consequently,

1 1 1, 1

69.3/30

0.5.

Through calculating, we get

1 0.7408.

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Let’s define the following variables：

0

n

denotes the amount of methylmercury at initial time,

denotes the number of month,

denotes the amount of methylmercury in human body at the moment people have just ingested the methylmercury in the month n,

denotes the amount of methylmercury that people ingest per month and

.

n

x1

x1 1300 0.7ug 910ug

Moreover, we assume

0=0.

Though,

n n 1 1 x1,

we get

1 0 1 x1 2 0 1 x1 1 x1

2

n 0 1 1

nn 1

x1 1 x1 x1 1 1) x1

n 1

n ( 1

n 1

1

n 2

n

1 1

1 1

x1.

With the remaining amount of methylmercury increasing, the elimination of methylmercury is also going up. We know the amount of ingested methylmercury per mouth is a constant. Therefore, with time going by, there will be a balance between absorption and elimination. We can obtain the steady-state value of remaining methylmercury as n approaches infinity.

n lim

*

1 1

n 1

n

1 1

x1

11 1

x1 3505ug.

The value of n is shown by figure 1.

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Figure 1. merthylmercury completely coming from fish and ingested at fixed term per month

If the difference of the remaining methylmercury between the month n and n 1 is less than five percent of the amount of methylmercury that people ingest per month, that is,

n n 1 x1 5%.

Then we can get

11=3380ug.

At the same time, we can work out the time that people have taken to achieve 3380 ug is 11 months.

From our model, we reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3505 ug.

If people ingest methylmercury every half of a month, however, the sum of methylmercury ingested per month is constant, consequently,

x1

9102

405ug, 1 0.86.

As a result, we obtain the maximun amount of methylmercury in human body is 3270ug. When the difference is within 5 %, we get the time people have taken to achieve it is 11 months.

Similarly, if people ingest methylmercury per day, we get the maximum amount is 3050ug, and the time is 10 months. Revising Model

As a matter of fact, the amount of methylmercury in human body is not completely coming from fish. According to the research of Hong Kong SAR Food and Environmental Hygiene Department [1], under normal condition, about 76 percent of

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methylmercury comes from fish and 24 percent comes from other seafood. In order to make our model more and more in line with the actual situation, it is necessary for us to revise it. The U.S. environmental Protection Agency (USEPA) set the safe monthly dose for methylmercury at 3 microgram per kilogram (ug/kg) of body weight. If we adopt USEPA criterion, we can calculate the amount of methylmercury that the average adult ingest from seafood is 50.4 ug per month. Assumptions

The amount of methylmercury in the seafood is absorbed completely and

instantly by people.

The elimination of methylmercury is proportional to the amount remaining. People ingest fixed amount of methylmercury from other seafood every day. We assume the half-life of methylmercury in human body is 69.3 days. Solutions

Let 0denote the amount of methylmercury at initial time, t denote the number of days, t denote the remaining amount on the day t, and x2 denote the amount of methylmercury that people ingest per day. Moreover, we assume

0=0.

In addition, we work out

x2=50.4/30=1.68 ug.

The proportion of remaining methylmercury each day is 2, then

2

69.3

0.5.

Through calculating, we get

2 0.99.

Because of

t

1 2

t 1

1 2

x2,

we obtain steady-state value of methylmercury

t lim

*

1 2

t 1

t

1 2

x2

11 2

x2 168ug.

If the difference of remaining methylmercury between the day t and t 1 is less than five percent of the amount of methylmercury that people ingest every day, that is,

t t 1 x2 5%.

We have

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301= 160 ug.

So we can reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate from seafood is 160 ug and the time that people take to achieve the maximum is 301 days.

Let x1 denote the amount of methylmercury people ingest through bass at fixed term per month, so the amount of methylmercury an average adult accumulate on the day t is

t t 1 2 x2 if t is a positive integer and not divisible by 30

t t 1 2 x2 x1 if t is a positive integer and divisible by 30.

The value of t

is shown by figure 2.

Figure 2. merthylmercury coming from fish and other seafood and ingested at fixed term per day

The change of t reflects the change of the amount of methylmercury in human body. Through revising model, we can figure out the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3679 ug.

Problem Two

Random-ingestion model

Although people consume one fish per month, the consuming time has great randomness. We speculate the randomness has effect on the bioaccumulation of methylmercury, therefore, we construct a new model. Assumptions

The amount of methylmercury in fish is absorbed completely and instantly by

people.

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The elimination of methylmercury is proportional to the amount remaining. People consume one fish per month, but the consuming time has randomness. We assume the half-life of methylmercury in human body is 69.3 days.

Let L0denote the amount of methylmercury at initial time, Ln denote the amount of methylmercury at the moment people have just ingested methylmercury in the month n, and x denote the amount of methylmercury that people absorb each time.

We assume

L0=0.

We have

x 910ug.

We define 1 the proportion of remaining methylmercury every day. Through

1

69.3

0.5,

we can get

1 0.99.

Let i obey discrete uniform random distribution with maximum 30 and minimum 1 and tn denote the number of days between the day day in of the month n, then we have

tn in 30-in-1,

in 1

of the month n 1 and the

Ln L(n 1) 1n x.

t

The value of Ln is shown by figure 3.

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Figure 3. merthylmercury completely coming from fish and ingested at random per month

Figure 3 shows the amount of methylmercury in human body has a great change due to the randomness of consuming time. Through the computer simulation, if we have numberless samples, Ln will achieve the maximum value. That is,

Ln 4261ug.

Revising model

In order to make our model more accurate, we need to make further revise. We take methylmercury coming from other seafood into consideration. We know the amount of methylmercury that people ingest from other seafood every day is 1.68 ug.

In that situation, we have

Ln Ln 1 x2 if n 30 (n-1) in

Ln Ln 1 x2 x if n 30 (n-1) in.

Through the computer simulation, we can get a set of data about Ln shown by figure 4.

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Figure 4. remaining merthylmercury coming from fish consumed at random per month and other

food consumed at fixed term per day

Though the revised model, we reach the conclusion that if we have numberless samples, Ln will achieve the maximum value. That is,

Ln 4420ug.

Variable-eliminateion model

As a matter of fact, the state of human health can affect metabolice rate so that the ability of eliminating methylmercury is not constant. We have koown the amount of methylmercury in human body will affect human health. So we can draw the conclusion that the amount of methylmercury in human body will affect the abilitity of eliminating methylmercury. Assumptions

The amount of methylmercury in fish is absorbed completely and instantly by

people.

the elimination of methylmercury is not only proportional to the amount

remaining, but also affected by the change of human health which are caused by the amount of methylmercury.

People absorb fixed amount of methylmercury at fixed term per month through

consuming bass.

We assume the half-life of methylmercury in human body is 69.3 days.

In condition that no further methylmercury is ingested during a period of time, we

let denote the eliminating proportion per month. We have known methylmercury decays about 50 percent every other day 5 to a turn 5 days, so we determine the half-life of methylmercury in human body is 69.3 days. Then we

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have

1 (1 6)9.3/3 00. .5

By calculating, we get

=0.2592.

We adopt LD50 as the toxic criterions, then we get the maximum amount of methylmercury in human body is 3.5 106 ug. Let’s define the following variables：

0

n

denotes the amount of methylmercury at initial time,

denotes the number of month,

denotes the amount of methylmercury in human body at the moment people have just ingested the methylmercury in the month n,

n

n denotes the ability of eliminating methylmercury in the month n. denotes the effect on human health caused by methylmercury toxicity.

r n 1 n 1

3.5 106

n n 1 (1 n)

Hence, we have

1 0 (1 1)

2 0 (1 2) (1 1) (1 2)

n 0 (1 n)...(1 1) (1 2) (1 3)...(1 n) (1 3)...(1 n) ... (1 n) 1

We define the value of is 0.5, then we get the maximum amount of maximum in human body is 3567 ug, that is,

n=3567 ug

*

Not taking the effect on the ability of eliminating maximum caused by methylmercury toxicity into account in model one,we obtain the maximum amount is 3510 ug. The difffference proves methylmercury toxicity has effect on eliminating methylmercury. We find out through calculating when r increases, the amount of

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methylmercury go up correspondingly. The reason for it is that methylmercury toxicity rises as a result of r increasing. Correspondingly, the effect on human health will increase, which is in accordance with fact.

Problem Three

According to the first model revised, we can get the maximum amount of bioaccumulation methylmercury is 3679 ug. We assume the average weight of an adult is 70 kg and the amount of methylmercury in human body is 53 ug/kg, far less than 50 mg/kg. Therefore, according to our model, the fish consumption restrictions put forward by the reservoir advisories can protect the average adult from reaching the LD50(LD50 is the dosage at which 50% of the humans exposed to a particular chemical will die. The LD50 for methylmercury is 50 mg/kg).

We assume the lethal dosage of methylmercury is not gradually increasing. If the amount of methylmercury people ingests goes up rapidly, the bioaccumulation amount will reach to a higher value. Moreover, the value probably endangers human safety. Let LD50 be the maximum amount of methylmercury in human body, that is,

n=50 mg/kg 70 kg=3500 mg.

*

Let x1 denote the amount of methylmercury people ingest per month. According to the first model,

n lim

*

1 1

n 1

n

1 1

x1

11 1

x1.

We can figure out

x1=907.2 mg.

We know the mean value of methylmercury in bass samples is 1.3 mg/kg, hence, we can obtain the maximum amount of fish that people consume safely per month is

Mmax

x11.3

698kg.

The maximum number of fish is 698/0.7=997.

Conclusion

In problem one, the paper calculates the final steady-state value at the same time interval per month, per half a month and per day. Through comparing the results, we get the final bioaccumulation amount of methylmercury is less, when discrete time unit is smaller. It shows when the interval of consuming fish is smaller and the sum of methylmercury ingested is constant for a period of time, the possibility of poisoning is lower.

In problem two, we analyze the change of the amount of methylmercury under the condition that consuming time is random. We find out the amount of

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methylmercury in human body is changing constantly in fixed range, when people have just consumed fish. Moreover, the maximum is 4261 ug, which is far bigger than 3505 ug. So we can reach the conclusion that people are more endangered when the consuming time is irregular.

In order to closer to the actual situation, we construct a model in which the half-life of methylmercury in human body is not constant. Through analyzing the data of computer simulation, the maximum amount of methylmercury will increase, that is, the risk of poisoning will be higher.

References

[1] Dr.D.N.Rahni, PHD. Airborne Mercury Contamination and the NeversinkReservoir.

http://webpage.pace.edu/dnabirahni/rahnidocs/Envsc/Airborne%20Mercury%20Contamination%20and%20the%20Neversink%20Reservoir.doc [2] Hu Dong Bai Ke. Bass. http:///wiki%E9%B2%88%E9%B1%BC.

[3] Centre for Food Safety Food and Environmental Hygiene Department The Government of the Hong Kong Special Administrative Region. Mercury in Fish and Food Safety.

http://www.cfs.gov.hk/english/Programmme/programme_rafs/Programme_rafs_fc_01_19_mercury_in_fish.html.