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

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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

Population

In million

554.8

609.0

657.5

729.2

830.7

927.8

998.9

1070.0

1155.3

1220.5

Now, scatter plot the above data:

image00.png

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).

...read more.

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.

...read more.

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.

...read more.

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