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1、Elephants:When Is Enough, Enough?Jin Tan Zhuotao Zhang Duan Liu South China University of Technology,Guang Dong,China Advisor: Yongan Qin2013/01/221Statement to the Park ManagementWe develop a system that uses contraceptive darts as the primary method for elephant population control. This method pro
2、vides a practical alternative to expensive relocation and unpopular culling. Using a statistical model to simulate the changes in the elephant population from year to year, we deter-mine a darting plan that effectively brings the elephant population down to a stable total population of about 11000,
3、the parks desired target. Theoretically, this model should accurately predict the structure and size of the elephant population based on the information provided to us about the elephants, such as birthrates, reproductive activity, and life span. Although we had to determine the elephants survival r
4、ates from a rather small sample of data, the survival rates that we determined matched the general informa- tion provided. If more accurate survival rates can be found, the model can be adjusted easily by changing a few parameters. Additionally, we generalize our model to an adaptive darting method
5、that ac-counts for random fluctuations due to varying survival rates and birth rates,as well as such external influences as immigration, emigration, and poach-ing. Thus, despite lack of conclusive data, the darting method will effectively control the population even with the random variations introd
6、uced by nature.This method involves the following basic procedure:11) From a survey of the population, determine the approximate population size, age structure, and survival rates. We estimate these from the sample data provided. 2) Feed these data into the mathematical model and from it obtain the
7、initial percentage of females to be darted.2Assumptions1) The number of elephants relocated in the past two years is representative if the actual age structure if the current elephant population. One common practice is to relocate entire family units of elephants at once, which would be generally co
8、nsistent with this assumption.2) The population in park never differs greatly from the stable state of the population. 3) Elephants mate and give birth at a uniform rate throughout the year. 4) The survive rate of new born elephant is 0.75. 5) The survive rate within ten-year-wide age groups is roug
9、hly uniform. 6) Population of elephants over 60-year-old will decline linearly and eventu- ally decline to none. (No elephant be older than 70.)3Analysis of the ProblemThe nature of this problem suggests that the population should be modeled by a system of equations. The data provide by the park are
10、 presented interms of a discrete age distribution(As is shown in figure 1).Since the duration of the darts effectiveness is given in terms of years rather than a fraction thereof, it is appropriate to approach the problem in terms of a discrete time step, namely dT = 1 year. Given that all elephants
11、 die by the age of 70, the problem is reduces to 71 equations, one for each age cohort. Such a system is most naturally represented in terms of a matrix equationPn+1= T PnWhere Pn+1and Pnare column vectors with 71 rows, in which the ith element represents the number of elephants of age i. The matrix
12、 T is 71 71, and2Figure 1: a discrete age distributioneach of the elements in the ith row is a coefficient in the ith equation. T can be manipulated (e.g. by darting) so that it has an eigenvalue of 1, which corresponds to a stable population and age structure. A matrix A with eigenvalue (a scalar)
13、and eigenvector x has the property that x = Ax. For a general population vector P , as n arrow to + we have AnP approaches x or some scalar multiple of x.The convergence is especially fast if P is initially somewhat similar to x, although small variations of x from can cause P to converge to a scala
14、r multiple of X instead of to x itself.This relationship suggests the solution to the dilemma of stabilizing the elephant population: If T is manipulated through darting so that it has an eigenvalue 1, then as it is applied to the population of elephants over time, P will converge to the eigenvector
15、; that is, the population will stabilize.4Determining the Transition MatrixTo determine the elements of the matrix equations, consider the structure ofthe structure of the difference equations. The first reduction in the magnitude of the problem is to consider only female elephants.Given that the se
16、x ratio is very close to 1:1 for adults as well as for newborns, we can consider3only females, knowing that the full population can be determined simply by multiplying by two. Hence, the sum of the elements of the P vector is the newborn elephants, age 0. The size of this stratum at iteration i+1 depends on only the number of reproducing females.The linear equation for the newborn elephants is then(P0)n+1=60Xi=1Pi (pi)nWhere Piis the probability that an elephant in the ith age group ha
