美国数学建模一棵树的叶子TheLeave.docx

1、美国数学建模一棵树的叶子TheLeaveThe Leaves of a TreeHow 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

2、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 (

3、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

4、 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 nu

5、mber of fish the average adult can safely eat per month. From City Officials research1, 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 consu

6、ming 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

7、 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 situatio

8、n. 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

9、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. As it is known to us, different countries and di

10、stricts 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 eli

11、minating 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 reservo

12、ir 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

13、 997.Keywords: methylmercury discrete dynamical system model variable-elimination modeldiscrete uniform random distribution model random-ingestion model IntroductionWith the development of industry, the degree of environmental pollution is also increasing. Human activities are responsible for most o

14、f 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 c

15、olliding 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 worrie

16、d 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 of methylmercury, if the reservoir is polluted, we can make sure that th

17、e 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

18、 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. A

19、s 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 reser

20、voir 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 Onediscrete dynamical system model The mean value of methylmercury

21、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 th

22、e maximum amount of methylmercury. Therefore, we assume that people consume one fish per month.AssumptionsThe 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

23、at fixed term per month.We assume the half-life of methylmercury in human body is 69.3 days.Solutions Let denote the proportion of eliminating methylmercury per month, denote the accumulation proportion. As we know, methylmercury decays about 50 percent every 65 to 75 days, if no further methylmercu

24、ry is ingested during that time. Consequently,Through calculating, we getLets define the following variables： 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

25、in the month , denotes the amount of methylmercury that people ingest per month and .Moreover, we assumeThough,we getWith 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. Therefo

26、re, with time going by, there will be a balance between absorption and elimination. We can obtain the steady-state value of remaining methylmercury as approaches infinity.The value of is shown by figure 1.Figure 1. merthylmercury completely coming from fish and ingested at fixed term per monthIf the

27、 difference of the remaining methylmercury between the month and is less than five percent of the amount of methylmercury that people ingest per month, that is,Then we can getAt 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 t

28、he 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,As a result, we obtain the maximun amoun

29、t 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

30、 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 methylmercury comes from fish and 24 percent comes from other seafood. In order to make our model mo

31、re 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 methylmercu

32、ry 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 denote the amount of methylmercury at initial time, denote the number of days, denote the remaining