Backman's Growth Function

This is a term paper on Backman's growth function submitted by the author to his course coordinator of Advanced Silviculture as per the fulfillment of assignment assign to him. This is not an accepted article from any renowned institution/Science Journal. However, it can be an important piece of literature for the learner and reader.

Author: Binaya Adhikari
M.Sc. Forestry, IOF, Pokhara Campus, TU, Nepal.
Email: [email protected]

Plant Growth

Growth refers to the irreversible changes in the size of a cell, organ or whole plant. It involves both the cell division and enlargement. The plant growth can be visualized in terms of increase in length or plant height, stem diameter, the volume of tissue, increase in cell numbers, increase in fresh weight and dry weight, increase in leaf area, leaf weight, etc.(Pandey 2017).
Backman's Growth Function -- growth rate
Plant growth rates

The increment in volume or mass requires the deposition of mass in the cytoplasm and cell walls. However, increment in dry mass may not coincide with the changes in size. For example, during starch accumulation, a potato tuber may gain dry weight without a concomitant change in volume. Thus, the increase in dry matter is the most important parameter for quantitative analysis of plant growth. Plant growth analysis is required to explain the differences in plant growth in terms of differences between species growing under the same environmental condition or differences within a species growing in different environments.

The quantitative analysis of plant growth is a branch of plant physiology to which adequate attention has not as yet been paid, but which should be able nevertheless to yield results of much theoretical interest and economic importance. Methods for obtaining data for the analysis of plant growth under ordinary cultural conditions are in general simple, consisting principally of periodic dry-weight and leaf-area measurements, and a quantity of excellent data of this nature has already been collected and exists in the literature. As yet thorough analysis of these results has not been presented. Attempts have been made to fit in a few isolated results with various empirical laws without a wide examination of existing data (Briggs 1920).

The simplest expression of plant growth is presented as the average growth rate (AGR). It is defined as an increase in dry weight per unit time. The calculation of the average growth rate assumes a linear increase in plant growth. However, the plants show a sigmoidal pattern of growth where the initial plant growth (e.g. in terms of weight) is exponential which later becomes reduced and reaches a maximum level finally. This pattern is applicable to the growth of plant organs in terms of size, volume, weight, length, etc. Also, the Richards function is often inappropriate to describe growth for crop stands, in which dry matter characteristically increases at an almost constant rate during the main growth stage (Goudriaan 1990).

Backman's Growth Function

Based on the plant growth studies, Blackman (1919) mentioned the rate of production of new material would be proportional to the size of the plant, i.e. the increase in plant weight would follow the compound interest law”. The proportional rate of increase was termed as efficiency index by Blackman (1919) and he further stated that “clearly the efficiency of the plant is greatest at first and then falls somewhat but the fall is only slightly until the formation of the inflorescence when there is a marked diminution of the efficiency index.

The conceptual basis of Backman's function of growth is the postulate that the logarithm of growth rate is negatively proportional to the square of time's logarithm. (Levich 1995)

It is clear that in the case of an ordinary plant the leaf area will increase as growth proceeds, and with increasing leaf area the rate of production of material by assimilation will also increase; this again will lead to a still, more rapid growth, and thus to a greater leaf area and a greater production of assimilating material, and so on. If the rate of assimilation per unit area of leaf surface and the rate of respiration remain constant, and the size of the leaf system bears a constant relation to the dry weight of the whole plant, then the rate of production of new material, as measured by the dry weight, will be proportional to the size of the plant, i. e. the plant in its increase of dry weight will follow the compound interest law. When money accumulates at compound interest, the final amount reached depends on:

  1. the capital originally employed, 
  2. the rate of interest, 
  3. the time during which the money accumulates. 

In the case of an annual plant, the ultimate dry weight attained will depend on

  1. the weight of the seed, since that determines the size of the seedling at the time that accumulation of new material begins; 
  2. on the rate at which the material present is employed to produce new material, i. e. the percentage increase of dry weight per day or week or another period  
  3. the time during which the plant is increasing in weight.

Growth Equation

During the daylight period, the plant is adding new material continuously, and during rapid growth, the plant is continuously, or nearly continuously, unfolding its leaves and increasing its assimilating area. The plant's increase is thus comparable rather to money accumulating at compound interest, in which the interest is added to ' the principal not daily or weekly, but continuously. (Blackman 1919)

The simple equation which best applies to the growth of active annual plants is thus:
where, as before,
Wx = the final weight,
Wo = the initial weight,
r = the rate of interest, and
t = time and
e is the base of natural logarithms.

The rate of interest required to give the same final dry weight is naturally less when it is added continuously than when it is assumed to be added discontinuously. The final weight attained will depend on the initial weight, the rate of interest (r), and the time. The differences in the dry weight attained by two plants may thus depend on simply the initial dry weights of the seedlings; if the rate of interest is the same the final weights will then vary directly as the initial weights.

Equation of straight line,
 W0 = Initial weight,
 r   = interest rate,
t   = time period

r determines the slope of the line, rate of interest  and represents the plant’s capacity to add to its own dry weight

lnWt=lnW0+rtlne        (lne=1, e=2.7182)

The rate of interest (r of the equation) is clearly a very important physiological constant. It represents the efficiency of the plant as a producer of new material and gives a measure of the plant's economy is working. The rate of interest, r, may be termed the efficiency index of dry weight production, since not only does it indicate the plant's growth efficiency as measured by increase of dry material, but it also appears as an exponential term in the equation which expresses the relation between the initial dry weight, the final dry weight, and the period of growth. It may also be termed the 'economy constant of the plant; it is, of course, comparable to the velocity constant of a chemical reaction.

Nature of Plant Growth

The growth is naturally affected by external conditions, being higher when conditions are favorable, but even under the same conditions, there are large variations in the economy of working of different plants, so the efficiency index is certainly to a large extent a characteristic of different species and varieties. It would be of great interest to determine what these differences in efficiency are due. They may be the result of differences in the rate of assimilation per unit area of leaf surface, of differences in the rate of respiration, of differences in the thickness of the leaves, or of differences in the distribution of material to leaves on the one hand and to the axis on the other. The larger the proportion of new material that the plant can utilize in leaf production the greater, other things being equal, should be its efficiency.

The fall in efficiency after the first few weeks of growth may perhaps be correlated with the mechanical relations connected with the larger size. A doubling of the leaf area would require a stem of more than twice the weight to attain equal strength. With larger leaf areas the ratio of stem weight to leaf weight goes up. As the efficiency of the plant is highest in its early stages favorable conditions at that time should have a marked effect on growth. If a plant, owing to such conditions, should double its size as compared with another plant, then there is no reason why that advantage should not be retained until the end of the growing season. A ' good start' means, among other things, a larger capital to work with throughout the growing season. (Blackman 1919)

The growth of an annual plant, at least in its early stages, follows approximately the ' compound interest law. The dry weight attained by such a plant at the end of any period will depend on

  1. the weight of the seed (or the seedling at its start), representing the initial capital with which the plant starts
  2. the average rate at which the plant makes use of the material already present to build up new material: this represents the rate of interest on the capital (material) employee
  3. the period of growth
The plant is continually unfolding its leaves and increasing its assimilating power. New material is added continuously during daylight, and during rapid growth, the plant is continuously, or nearly continuously, unfolding its leaves and increasing its assimilating rate. The growth of the plant more nearly approximates' to money accumulating at compound interest where the interest is added continuously. The simple equation which best expresses the growing relations of active, annual plants is:
Wi= Woerl
Wi = the final weight,
Wo = the initial weight
r = the rate at which the material already present is used to produce new material
t = time

The term r is an important physiological constant, for it is a measure of the efficiency of the plant in the production of new material; the greater r is, the higher the return which the plant obtains for its outlay of material. r. may thus be termed the ' efficiency index' of dry weight production, for not only is it a measure of the plant's efficiency but it is also an exponential term in the equation expressing the growth of the plant.
A small difference in the 'efficiency indices' of two plants (resulting, for example, from a slightly greater rate of assimilation or a more economical distribution of material between leaves and axis) may lead to a large difference in the final weight.
It is suggested that in all experiments (such as water cultures, pot experiments) dealing with the production of vegetative material the efficiency index be calculated. The relative efficiency of different plants and of the same plant at different stages can thus be determined; also the effect on the efficiency index of various external conditions.


  • BLACKMAN, V. H. (1919). The Compound Interest Law and Plant Growth. Annals of Botany, os-33(3), 353–360. doi:10.1093/oxfordjournals.aob.a089727
  • BRIGGS, G. E., KIDD, F., & WEST, C. (1920). A QUANTITATIVE ANALYSIS OF PLANT GROWTH: PART I. Annals of Applied Biology, 7(1), 103–123. doi:10.1111/j.1744-7348.1920.tb05107.x 
  • Goudriaan, J., & Monteith, J. L. (1990). A mathematical function for crop growth based on light interception and leaf area expansion. Annals of botany, 66(6), 695-701.
  • Levich, A.P. On the Way to Understanding the Time Phenomenon: the Constructions of Time in Natural Science. Pt. 1. Interdisciplinary Time Studies. World Scientific, 1995.
  • Pandey, Rakesh & Paul, Vijay & Das, Madurima & Meena, Mahesh & Meena, Ramavatar. (2017). Plant growth analysis. 10.13140/RG.2.2.21657.72808.