• 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...





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.













In million











Now, scatter plot the above data:


Now we can observe that population data can be checked for fitness in the logistic function.



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.


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).

...read more.


The formula for the logistic function,


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.

...read more.


  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.

...read more.

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

See related essaysSee related essays

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. Math IA type 2. In this task I will be investigating Probabilities and investigating ...

    Therefore Adam wins of the deuce games. Therefore Adam will win Now I will add the value of the probability of Adam winning deuce and non-deuce games to find his total probability of winning and then I will find his odds of winning.

  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. Stopping distances portfolio. In this task, we may develop individual functions that model the ...

    previous equations are added together /the linear equations for thinking and braking distances/, the new combined equation is y = x + x - 27 or y = x - 27 This linear equation is reasonable in the center of the domain but loses its effectiveness as distances are shorter or very large.

  1. 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 ?

  2. Finding Functions to Model Population trends in China

    Therefore, I have the conclusion that linear function is not the model of the graph. Quadratic Function: y=ax2 +bx+c Quadratic function has three parameters, in which I will calculate them out below. I use (1955, 609), (1985, 1070) and (1960, 657.5)

  1. Function that best models the population of China. Some of the functions that ...

    and exponential function (). If we look at graph 1 closely, we can see that most of the data points have aligned themselves in almost a linear form. This informs us that the slope differences will be very small and not drastically big. Apart from the linear function, I also think that the exponential function would work on this particular data set.

  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
    improve your own work