7 Measurement of Growth of Geographical Phenomena

Prof Surendra Singh

1.0 Introduction

Phenomenal changes on the earth over time is a ‘law of nature’. Natural as well as human phenomena over earth are always changeable which creates variations in geographical properties and changes its characteristics over time. In real sense, earth is neither flat nor stable over time. Therefore, space – time concept of geography integrates its subject matter. Changes in biological phenomena (plants as well as human beings) follow certain rules and regulations. The concept of natural growth of an organism is associated with a natural regulation of biological growth with reference to physical factors of landscape. Before starting discussion on growth measurement and mechanism, a concept note on phenomenal growth would help in understanding the insights of growth measurement.

2.0 What is Phenomenal Growth?

One must be clear in concept of phenomenal growth. Growth refers to temporal changes of an attribute. It means time is the dimension of measuring the growth of any organism. Secondly, growth is associated with the ‘volumetric’ measurement of geographic phenomena rather than unitary changes. For instance, the population of main workers in a village and its share as given in the following Table-1 has two dimensions of change, volumetric as well as unitary, which would depict the measurement of workers growth over time.

Table-1: Main Workers and Total Population in a Village in 2001 and 2011

In the last raw of the above table, change in the share of workers to total population is not ‘growth’. It is simply a difference in share of workers over time rather than growth. It diminishes over time while population of main workers increased by 3,000 i.e. 20% during the decade. The volumetric change given in the second row in the table reflects growth (either absolute or relative in %) of main workers in the village. Registrar General, census of India operations, New Delhi measures this change in absolute and relative manner for the population of India during the decade called ‘decadal growth’.

Thirdly, since time of phenomenal change is considered as unitary variable which is generally expressed on X-axis on the graph, it is used as independent variable in measuring the growth; the volumetric change of a particular attribute is considered as dependent variable and is shown by trend graph rather than scatter diagram.

Table-2: wheat production in India (Production Million tones)

Source: At A Glance, Bulletin, Directorate of Economics and Statistics, Ministry of Agriculture, New Delhi

Fig-1: Growth Trend of Wheat Production in India (1950-1 to 2005-06)

In this figure, trend line shows a significant fluctuation in the changing nature of wheat production. However, a general trend line may be fixed by using moving average, straight line, curve fitting and so to infer the growth results.

A detail discussion on these techniques of generalisation of trend line has been given in preceding concerned modules. However, the measurements of the growth of biological phenomena are to be given in present discussion.

3.0 Growth of Biological Phenomena and its Regulation

The facts of natural growth of a plant are associated with the specific environment created for plant-growth. Kormondy (1996) states that the inherent capacity of each species in ecosystem is enable to produce at given rate called ‘reproductive potential rate (r)’ in specific environment. If this rate is multiplied by total number of producers, N, that are number of plaints in specific area, rN, it refers to ‘biotic potential’. It is biological concept of natural growth of plant. On the other hand, the force of physical and biological environment in which organism exists opposes or restricts the growth called ‘environmental resistance’ is dynamic phenomena and limited to its varying degree. It determines carrying capacity of plant growth (K). Further, K is considered as the maximum limit of growth. Therefore, total population of plants can not go beyond K. It means K regulates the growth rate at different points of time (t). In a given ecosystem, the difference between maximum population and the already present (K-N) is unoccupied places of plant at a piece of land. It is hidden capacity or ‘environmental resistance’ as population approaches the maximum, more resistance is encountered in the system. This resistance effect is expressed quantitatively as

(K-N)/ K.                               … …   ……(1)

When N is small, the resistance value is very low near zero. But as N increases and approaches to K, then (K-N)/K comes to zero, there is fewer opportunities to population growth (Fig.-1).

Fig.-1: Regulation of Biotic Potential and Environmental Resistance

Since this resistance (K-N)/N acts against the increased potential (rN) of the population in the product of two, it would indicate the rate of change in the population (Kormondy 1996) as

dN/dt = rN *{(K-N)/K} .    ……….. (2)

The logistic equation of plant growth is regulated by two components of environmental forces. Simmons (1981) modified the above equation in the form of exponent as

N = [ K/(1+ e rt ) .               …………..(3)

This equation is used by many demographers for the regulation of urban population growth by using exponent, e, in the equation that would be discussed in the next part of the discussion in detail. However, the measurement of growth of agricultural production was tried to adopt the same concept of ecosystem and land water management to fix the reciprocity law of production increase (Singh 1980, 2000).

4.0 Mathematical Concept of Phenomenal Growth

In fact, growth refers to the change in the value of an attribute between two points of time. For example, population of a particular area/region, Po, at particular time called base year, t0 changes to P1 at current year, t1. The difference of population (P1 – Po) at a particular duration t (i.e, tn–t0 = t) is the total change in population that is called absolute change in the total volume of population. If this amount is calculated in proportion to the population of base year [(P1-Po)/Po], it becomes the relative change called growth rate of population over the specific period of time, t. Thus, there are following four types of growth measurements that can be used for the analysis of given distribution of an attribute.

  4.1  Simple Growth Rate:

It is simply a proportionate difference of the values of an attribute within given time. If period of time (t = tn – to) is considered as unit of time for measuring the phenomenal change, the simple growth rate, r for a unit of time, t, is

r = {(P1 – Po) / Po t} . …………. (4)

  Simple growth rate follows straight line equation in the distribution of attribute over time, P(t). As absolute change, i.e.,

P1- Po = K, then

P1 = Po + K,   …………. (5)

where K = rt because, rate r multiplied by time duration becomes total change, K. Replacing rt for K, we find

rt = (P1 – Po)

So,  P1 = Po + rt .     ………….(6)

It is linear growth equation and shows specific characteristics of geographical phenomena.

4.1.1 Example-1:

Total population in India was 1071 million and 1210 million in the years 2001 and 2011 respectively. Calculate average annual growth of population as per given simple growth rate of formulae?

P1 = 1210 million, Po = 1071 million and t = 10 years, calculate r=?

r (%) = 100*(1210-1071)/(1071*10) = 13,900/10710, that is 1.307 percent average annually growth of population in India during the year 2001 to 2011.

4.2 Compound Growth Rate:

If the whole duration of growth period is considered as a unit of time, then the equation-4 can be written ignoring t because t=1

(P1 – Po)/Po = r;  P1=Po + rPo

= Po (1+r) . ……….. (7)

This simple growth rate for specific period of time say between t1 and to is compounded for the next unit of time, t2 – t1 and so on. It is mathematically written to calculate the compound growth for whole span of time (tn – to) as

P1 = Po (1+r)

P2 = P1 (1+r) =Po(1+r) (1+r) =Po (1+r)2

P3 = P2 (1+r) =Po (1+r) ( 1+r) (1+r) =Po (1+r)3

Pn = Pn-1 (1+r) (1+r) ( 1+r) … n = Po ( 1+r) n , n = 1,2,3, …, t

where n is number of time units In present context, it is translated as

P (t) = Po (1+r)t ………. (8)

Note that r is in its unitary value not in percent

4.3 Is Compound Growth Semi – logarithmic?

The answer to this equation may only be given to transform the growth equation into its logarithmic form. The log – form of the above equation is

log Pt = log Po + t (log (1+r))              ………..(9)

It means compound growth equation follow semi-log rule when time is shown at simple scale of the growth – graph.

Fig.-2: Graphical Representation of Different Measurements of Phenomenal Growth

Utility of this growth rate is in variety of ways. According to Malthus, population increases in ‘multiplicity’ manner. Adding new population of a year in the previous one is also capable of produce population. Therefore, population increases fast by time. So, compound growth gives a correct assessment of population increase and actual rate of population growth.

Secondly, now-a-days, bank interest rates are calculated compounded by using the interest rate software based on the compound growth formula. As far as calculation of various components of this formula of compound growth is concerned, only of its four components, they are: Pt = current years population, Po = base years population, t = time unit and r = growth rate in its unit term, if any three are given, the fourth one can be calculated easily. It is exemplified below.

4.3.1 Example-2:

The population in India in 2011 was 1210 million which is increased at annual compound rate of 1.307% during the decade 2011-2021. What would be projected population in India in 2021? In this question, Po = 1210 million, r (%) = 1.307 t = 10 are given and P1 = ?

Solving compound growth equation (8) for P1, one must calculate

P (t) = Po (1+r)t .

Putting numerical values in the equation as

P1 = 1210 million (1.0+0.01307) = 1377.77, we find answer that approximately, 1378 million population in India is estimated for the year 2021.

4.3.2 Example-3:

Population of India was 1071 million and 1210 million in 2001 and 2011 respectively. What is annual compound growth rate of population during the decade?

In this question, P1 = 1210 million, Po = 1071 million, t = 10 year and r = ?

As per given formula of compound growth rate, it is simplified for r, because r is to be calculated. It is P1 = Po (1+r) t

There are two methods of calculating r as given below.

4.3.2.1 First method (Simple calculation)

If  Po (1+r)t = P1

Then (1+r)t = (P1/Po)

1+r = (P1/Po)1/t

r = [{(P1/Po) 1/t } – 1]…………… (10)

4.3.2.2 Second method (log – transformation)

If  log Po + t log (1+r) = log P1

Then  log (1+r) = (log P1 – log Po) / t

1 + r = [Antilog {(log P1 – log Po)/t }

r = [Antilog {(log P1 – Log Po)/t}-1] .

When given values of components are inserted either in equations ( 10) and (11), we get the results as

First method:            r     = [{(1210/1071)0.1 }-1] = .0122775 = 1.2277%

Second method:      r          =[ Antilog {(3.0827- 3.02979/10}-1] = 1.2275%

Annual compound rate of population growth in India during the decade 2001 – 2011 is 1.277%

4.4 Exponential Growth:

In the equation of geometric growth time is generally measured in a long duration say a Year. If it is measured in months it will be Multiplied by 12 and Similarly if we measure time in days the time in years will be multiplied by 365. Theoretically time can be measured for every instance and growth can be considered as a continuous process compounded not annually or monthly but instantaneously. This process of instantaneous growth is known as exponential growth and is captured by a mathematical constant e =2.718282…….. by the equation:

P1 = Po (e)rt ……….(12)

Its logarithmic transformation in natural logarithms ( Logarithms to the base e =2.718282 Log e ) is;

Ln ( P1/ Po )=rt ………..    (13)

4.4.2 Example-4:

Population in India was 1071 million and 1210 million in 2001 and 2011 respectively. It increased exponentially. What is annual growth of Indian population?

P1 = 1210 million, Po = 1071 million, t = 10 years, r = ?

4.4.2.1 Answer:

Simplifying equation ( 13) for r, we obtain

Ln ( P1 / Po ) = Log e  ( P1 / Po ) = Log e ( 1210/1071)  = = Log e  ( 1.129785 ) = 0.122027578 = r t

The value of : r  = 0.012203

It is to be noted that exponential growth rate (r = 0.012203 will be less than the geometric growth rate r =0.0122775 as given above )

4.5 Allometric Growth( Mainly used in Biological Sciences )

When all the parts of an organism grow proportionally, it is called allometric growth. In the equation representing allometric growth in which time factor, t, is a variable. It is kept at base and r, which is constant, is used as power function. So the equation of allometric growth is written as

P1= Po (t )r ………….(15)

Logarithmic form of this equation is

log P1 = log Po + r ( log t )    ………………… (16)

So, population and time variables are represented in logarithmically transformed manner. As a result, allometric growth is double- log linear growth measurement of an organism (Fig.-2d).

4.6 Logistic ‘S-Type’ Curve for Urban Growth – An Example

As described in preceding section, the exponential growth changes its rate in different stages of phenomenal increase. A suitable example for this statement may be given by taking into account the growth of urban population with the condition that urban area is limited and boundaries of this area is fixed over space, while changing functional nature of a town and increasing population pressure within the town changes its growth rate. Such urban growth processes are analyzed by using ‘logistic’ curve which shows solver increasing rate at low population level in the initial stage of urban growth. Second is the take – off stage of growth when rural to urban migration takes place with very fast rate of urban growth. Increasing function intensity and diversity in activities are major characteristics of town growth. Finally, in the third stage of growth when there is limited space to grow, the space thinning processes start. People move outside town and develop satellite towns in the peripheries of town. Population growth rate becomes stagnant in this final stage of town population (Fig.-3).

Fig.-3: Stages of Urban Growth

 4.6.1 Mathematical Explanation:

In order to obtain more natural curve fit in distribution, we need a curve that drops off towards 0 and ascends towards 1 more gradually if population on Y-axis is shown as probability function of population distribution, p(t). The formula that produces the same native of curve increase is logistic formula given below.

P(x)= ex/(1+ex) .                 ……………………(17 )

This formula is simple in form and may be used in respect to time t instead of X in its extended form:

P(t)= e (a+bt)/( 1+e a+bt)      ,      ……………….. (18)

Where a and b determine the logistic intercept and slope. When t = 0, we have

p = ea/(1+ea). Since the relationship between p and t is non-linear, the slope varies as t varies. At a point where p = 0.5, the slope is b/4. a and b have same qualitative effect on the logistic trend of an attribute.

5.0 Summary:

Measurement of growth of an organism and understanding the growth mechanism are main dimensions of geographical studies. Growth of natural vegetation (plant-growth) and human population growth have different processes. As a result, their measurements are different. Agricultural production generally follows simple growth trend while population is correctly measured by compound growth. However, growth models for agricultural production have wider application of cumulative effects of input, applications which may be described in separate module.

References

• Kormondy E.J. (1996): Concepts of Ecology, IV Edn, Prentice Hall of India Pvt. Ltd., New Delhi (East Economy Edition)
• Singh Surendra (1980): Reciprocity model, agricultural Development and Regional Planning – A case study of Uttar Pradesh I India,Transactions, Institute of Indian Geographers, Vol 2 (No 2) : 37 – 47
• WWW stat, ubc. ca/ rollin / teach / 643wo4 / lec / node 46.html
• Ceuom.weebly. com / uploads / 1/ … / group – 8, logistic curve methodfinal. pptx.
• Singh Surendra (1994): Agricultural Development in India – A Regional Analysis, Kaushal Publication Shillong.
• Simmons, I.G. (1981): The Ecology of Natural Resources, II Edn, ELBS and Edward Arnold Publications.
• Singh Surendra (2000): Modelling the Structure and Functions of Agricultural Yield Potentials, Transactions, Institute of Indian Geographers, Vol 22: 1-15