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Logistic Curve Method for Population Forecasting

Logistic curve method is also called mathematical curve fitting method.

Logistic curve/S-curve is the S-shaped curve obtained by plotting population in normal growth situation against time.

Logistic Curve Method for Population Forecasting

The growth in logistic curve method follows some specific logical mathematical relationship.

The population follows the logistic curve, which has an initial low growth rate followed by high rate. The rate again decreases as the population moves towards saturation.

Assumption in Logistic Curve Method: “The population growth is a function of time.”

Logistic Curve Method Formula

P=\frac{P_{s}}{1+m.log_{e}^{-1}(nt)}

Saturation Population for Logistic Curve Method

P_{s}=-\frac{2P_{0}P_{1}P_{2}-P_{1}^{2}(P_{0}+P_{2})}{P_{0}P_{2}-P_{1}^{2}}

Constant m: m=\frac{P_{s}-P_{0}}{P_{0}}

Constant c:

c=\frac{2.3}{t_{1}}\:log_{10}\,[\frac{P_{0}(P_{s}-P_{1})}{P_{1}(P_{s}-P_{0}}]

When do we Use Logistic Curve Method?

When the population growth rate is normal, i.e., birth, death, and migration of population take place in normal situations.

In the absence of war, epidemic, or natural disaster, the population growth curve considers characteristics of living things within limited space and economic opportunity.

Steps of Logistic Curve Method

1. Population data of the last three census are noted.

2. Saturation population (Ps) is then calculated by the following formula:

P_{s}=-\frac{2P_{0}P_{1}P_{2}-P_{1}^{2}(P_{0}+P_{2})}{P_{0}P_{2}-P_{1}^{2}}

Ps = saturation population

P2 = population of the last decade

P1 – population before two decades

P0 = population before three decades

3. Values of constant m and c are to be found.

Constant m – m=\frac{P_{s}-P_{0}}{P_{0}}

Constant c – c=\frac{2.3}{t_{1}}\:log_{10}\,[\frac{P_{0}(P_{s}-P_{1})}{P_{1}(P_{s}-P_{0}}]

Here, n = t2 – t1

4. Population is estimated by the following formula:

P=\frac{P_{s}}{1+m.log_{e}^{-1}(nt)}

Derivation of Logistic Curve Method

The curve JN  in the above figure is called logistic curve.

1. Early Growth Curve (JK)

As this part represents early growth phase of a city, it exhibits geometric growth (log growth).

Here, the rate of population growth will be directly proportional to the present population.

\frac{dP}{dt}\propto P

2. Transitional Growth Curve (KM)

This part represent the city is already developed. Hence, the growth rate decreases.

The growth rate now exhibits arithmetic growth.

Here, the rate of population growth is constant.

\frac{dP}{dt}=constant

3. Late Growth Curve (MN)

This part represents the phase when the population is moving towards its saturation.

Here, the rate of change in population is proportional to the difference in saturation population and existing population.

\frac{dP}{dt}\propto(Ps-P)

Verhaulst has put forth a mathematical solution for this logistic curve (JN) via first order equation given below:

log_{e}(\frac{P_{s}-P}{P})-log_{e}(\frac{P_{s}-P_{0}}{P_{0}})=KP_{s}.t

where,

P = population at any time t from origin J

Ps = saturation population

P0 = population at the origin point J

K = constant

t = time in years

Solving the above equation, we get,

P=\frac{P_{s}}{1\:+\:\frac{P_{s}-P_{0}}{P_{0}}\,.\,log_{e}^{-1}(-KP_{s}.t)}

Constants:

m=\frac{P_{s}-P_{0}}{P_{0}}\\\\c=\frac{2.3}{t_{1}}\:log_{10}\,[\frac{P_{0}(P_{s}-P_{1})}{P_{1}(P_{s}-P_{0}}]

Let us substitute the above constant in the above equation.

Thus, the population P is

P=\frac{P_{s}}{1+m.log_{e}^{-1}(nt)}

 The above equation of logistic curve used for forecasting the population.

McLean simplified the saturation population equation for values of P0, P1, and P2 at fixed time intervals of t, t1, and t2.

He took these values as –

t0 = 0

t1 = 1

t2 = 2 t1

P_{s}=-\frac{2P_{0}P_{1}P_{2}-P_{1}^{2}(P_{0}+P_{2})}{P_{0}P_{2}-P_{1}^{2}}

Logistic Curve Method Example

Find the population of Delhi, India for the year 2011 from the population census data:

Year19711981199120012011             
Population40,65,69862,20,40694,20,6441,38,50,507 

Step1: Given Data:

P2 = 1,38,50,507

P1 = 94,20,644

P0 = 62,20,406

Step2: Find saturation population (S):

P_{s}=-\frac{2P_{0}P_{1}P_{2}-P_{1}^{2}(P_{0}+P_{2})}{P_{0}P_{2}-P_{1}^{2}}\\\\P_{s}=-\frac{2(62,20,406)(94,20,644)(1,38,50,507)-(94,20,644)^{2}(62,20,406+1,38,50,507)}{(62,20,406\times 1,38,50,507)-(94,20,644)^{2}}

Ps = 60930631.1642 = 6,09,30,631

Step3: Find values of constant m & c:

m=\frac{P_{s}-P_{0}}{P_{0}}\\\\m=\frac{6,09,30,631-62,20,406}{62,20,406}

m = 8.8

c=\frac{2.3}{t_{1}}\:log_{10}\,[\frac{P_{0}(P_{s}-P_{1})}{P_{1}(P_{s}-P_{0}}]\\\\c=\frac{2.3}{t_{1}}\:log_{10}\,[\frac{(62,20,406)(6,09,30,631-94,20,644)}{94,20,644(6,09,30,631-)}]

c = – 0.047

Step4: Find population Pn:

P=\frac{P_{s}}{1+m.log_{e}^{-1}(ct)}\\\\P=\frac{6,09,30,631}{1+(8.8)(log_{e}^{-1})(-0.047\times 10)}

Summary

Logical curve method is based on logical and mathematical interpretation of population growth behaviour. The population growth rate is constant to time initially, followed by exponential growth. When the population reaches saturation, growth rate decreases.

FAQ

Problems in logistic curve method?

Logistic curve method applies to a city/town showing normal growth rate only.

For the situations of war, pandemic, and disaster which do not show typical growth rates cannot be considered for in this logistic curve method.

What is the formula for logistic growth method?
P=\frac{P_{s}}{1+m.log_{e}^{-1}(nt)}