• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# In this task, you will investigate different functions that best model the population of China from 1950 to 1995.

Extracts from this document...

Introduction

MATHS

PORTFOLIO

SL TYPE-II

Aim: In this task, you will investigate different functions that best model the population of China from 1950 to 1995.

The following table1 shows the population of China from 1950 to 1995.

 Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 PopulationIn million 554.8 609 657.5 729.2 830.7 927.8 998.9 1070 1155.3 1220.5

Now, scatter plot the above data: Now we can observe that population data can be checked for fitness in the logistic function.

LOGISTIC MODEL

Idea

The logistic equation is a simple model of population growth in conditions where there are limited resources. When the population is low it grows in an approximately exponential way. Then, as the effects of limited resources become important, the growth slows, and approaches a limiting value, the equilibrium population or carrying capacity.

Details

The logistic growth model is

dxdt=rK  K-xx.

Here x is the population, which is a function of time tK is the equilibrium population, and r is the growth rate.

Note that in the limit K→∞, we get the simpler model:

dxdt= rx

describing exponential population growth:

x (t)=x0ert.

When K is finite and positive, the logistic model describes population growth that is approximately exponential when the population is much less than K, but levels off as the population approaches K. If the population is larger than K, it will decrease. Every positive solution has

limt→+∞ x (t) =K.

The logistic model can be normalised by rescaling the units of population and time. Define y := x/K and s := rt. The result is

dyds = y (1−y).

Middle

The formula for the logistic function,

Y=CAe-Bx

involves three parameters A, B, C. (Compare with the case of a quadratic function y = ax2+bx+c which also has three parameters.) We will now investigate the meaning of these parameters. First we will assume that the parameters represent positive constants. As the input x grows in size, the term –Bx that appears in the exponent in the denominator of the formula becomes a larger and larger negative value. As a result, the term e–Bx becomes smaller and smaller.

To identify the exact meaning of the parameter A, set x = 0 in the formula; we find that

Y=C1+Ae-Bx0 =C1+A

Clearing the denominator gives the equation (1 + A) y(0) = C. One way to interpret this last equation is to say that the limiting value C is 1 + A times larger than the initial output y(0) An equivalent interpretation is that A is the number of times that the initial population must grow to reach C .The parameter B is much harder to interpret exactly. We will be content to simply mention that if B is positive, the logistic function will always increase, while if B is negative, the function will always decrease.

A look at a scatterplot of these data makes clear that a logistic function provides a good model. It also appears that data seems to be quickly approaching its limiting value. It makes sense for us to estimate this limiting value as C = 1250.

Conclusion

1. Chaotic: The population will eventually visit every neighbourhood in a subinterval of (0, 1). Nested among the points it does visit, there is a countable infinite set of fixed points and periodic points of every period. The points are equivalent to Cantor middle thirds set and are wildly unstable. It is highly likely that any real population would ever begin with one of these values. In addition, chaotic orbits exhibit sensitive dependence on initial conditions such that any two nearby points will eventually diverge in their orbits to any arbitrary separation one chooses.

The behaviour of the logistic equation is more complex than that of the simple harmonic oscillator. The type of orbit depends on the growth rate parameter, but in a manner that does not lend itself to "less than", "greater than", "equal to" statements. The best way to visualize the behaviour of the orbits as a function of the growth rate is with a bifurcation diagram. Pick a convenient seed value, generate a large number of iterations, discard the first few and plot the rest as a function of the growth factor.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## Math IA -Modelling Population Growth in China.

This will make your screen look like this: LinReg(ax+b) L1 , L2 Hit the comma (,) button again. Then your screen will look like this: LinReg(ax+b) L1 , L2 , Next hit the VARS button. Scroll over to the field labeled Y-VARS. Scroll down to where it says Function.

2. ## A logistic model

? c 1.5 ? (?1?10?5 )(1?104 ) ? c ? c ? 1.5 ? (?1? 10?5 )(1? 104 ) ? 1.6 Hence the equation of the linear growth factor is: r ? ?1? 10?5 u ? 1.6 {2} n n 2. Find the logistic function model for un+1 Using equations {1} and {2}, one can find the equation for un+1: un?1 ?

1. ## Maths Portfolio - Population trends in China

The average systematic error percentage is 1.5 %. By taking these two approaches, we are able to see the difference, evaluate and find the proper function that most likely suited the model I was given, and got the points presented with quite an accurate precision.

2. ## An investigation of different functions that best model the population of China.

An exponential model would not make sense as it would not represent the slowing down of the population growth and the linear model would not allow the rate of population increase to change as it does on the graph. However polynomials can be integrated to have a higher precision and

1. ## Math SL Circle Portfolio. The aim of this task is to investigate positions ...

In the following graph, 2, r = . Through the same method of calculation as the sample calculation, = , which validates the general statement. The following graph is a close-up of the graph above, showing the value of = = 0.18. In the following graph, 2, r = .

2. ## Math IA Type 1 Circles. The aim of this task is to investigate ...

be require to provide further evidence of the trend that has developed, see Graph 1, below. Graph 1 Refer to Table 4 for the equations Excel provided. Values of r r = 1 r =2 r =3 r =4 r =5 Values of OP? Table 4.

1. ## Infinite Summation- The Aim of this task is to investigate the sum of infinite ...

S10= S9+t10=2.718281 803 n Sn 0 1 1 2 2 2.5 3 2.666667 4 2.708333 5 2.716667 6 2.718056 7 2.718254 8 2.718279 9 2.718281 10 2.718281 This diagram shows the relation between Sn and n using the gained results.

2. ## In this Internal Assessment, functions that best model the population of China from 1950-1995 ...

A quadratic fit applied to the data points via Graphical Analysis 3 follows: Cubic Fit: From the graph, the data points also seem to show trends apparent in graphs of cubic functions, as the rate at which the population increases proliferates as the years go by. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 