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

by amitbahl (student)

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.

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).

It is easy to find the explicit solution of the logistic equation, since it is a first-order separable differential equation.

The growing solutions are all time-translated versions of the logistic function

ys= eses +1=11+e-s

which looks like this:

This function goes from 0 to 1 as t goes from −∞ to +∞. All other growing solutions have the same limiting behaviour and are time-translated versions of this one. After rescaling back to the original variables, we have

xt= K1+e-r(t-to)

There are also decreasing solutions where x>1 and solutions (irrelevant to population biology) where x<0 decreases explosively to −∞.

Logistic functions are good models of biological population growth in species which have grown so large that they are near to saturating their ecosystems, or of the spread of information within societies. They are also common in marketing, where they chart the sales of new products over time; in a different context, they can also describe demand curves: the decline of demand for a product as a function of increasing price can be modelled by a logistic function, as in the figure below.

The formula for the logistic function,

Y=CAe-Bx

involves three parameters A, B, C. (Compare ...