Growth formula of crops

1. 1 | P a g e
I. Introduction:
Plant growth analysis is now a widely used tool in such different fields as plant breeding
(Wilson and Cooper, 1970; Spitters and Kramer, 1986), plant physiology (Clarkson et ai,
1986; Rodgers and Barneix, 1988) and plant ecology (Grime and Hunt, 1975; Tilman, 1988).
Its methodology started to evolve in the 1920s (Blackman, 1919; West et ai, 1920) with what
is now called the 'classical' approach. In this method, the relative growth rate (RGR) is
calculated by dividing the difference in In-transformed plant weight between two harvests by
the time interval between those harvests. Compound rates, like the net assimilation rate
(NAR; increase in weight per unit of leaf area and time), are computed in a similar, discrete,
way (Evans, 1972). The classical method was challenged in the 1960s, when increased
computational power enabled curve-fitting procedures with polynomial equations on
progressions of plant weight and leaf area with time (Vernon and Allison, 1963; Hughes and
Freeman, 1967)
2. Objectives:
i. To identify spatial and temporal integration of all plant processes
ii. To know the Rate of dry matter accumulation varies across the life cycle of a crop and
dry matter
iii. To quantify the effects of environmental influences or to analyze genotypic
differences between crop cultivars.
iv. To learn what are the most relevant methodologies to measure the daily performance
of crop canopy
v. To know Why does plant-to-plant variability make the measurement of plant
productivity so difficult

2. 2 | P a g e
3. Review of Literature:
Sharief et al. (1997) in Egypt, observed that root length and diameter, LAI, root fresh and dry
weights, foliage fresh and dry weights, root and sugar yields/fad were tended to increase due
to potassium fertilizer at the rate of 36 kg K2O/fad as compared with control (without
potassium fertilization). On the other hand, TSS, sucrose and purity percentages had inverse
effect. EL-Zayat (2000) in Egypt, stated that increasing potassium fertilizer rates from 0 to 24
kg K2O/fad brought out significant increases in root length and diameter, dry matter
accumulation, LAI, CGR, root, top and sugar yields/fad. On the other side, potassium
fertilization failed to exhibit significant differences in RGR, NAR and quality parameters.
EL-Harriri and Mirvat, Gobarh (2001) in Egypt, indicated that high level of potassium
fertilizer (48 kg K2O/fad) exhibited a significant increase on LAI, root/top ratio, root length
and diameter, root and top yields/fad, TSS, sucrose and purity percentages as compared with
control treatment. Kandil et al. (2002 a) in Egypt, reported that potassium fertilizer
significantly affected root and foliage fresh weights/plant, LAI and CGR. Vice versa with
connection RGR and NAR. The highest values of root, top and sugar yields/fad were
obtained from application of 36 kg K2O/fad. Whereas, increasing K2O level up to 48 kg
K2O/fad did not exhibit any significant increase. With respect to quality parameters (TSS %,
sucrose % and purity %), it is worthy to note that potassium fertilizer levels did not
significantly affect these traits.

3. 3 | P a g e
4. Discussion:
i. Leaf Area
This is the area of photosynthetic surface produced by the individual plant over a period of
interval of time and expressed in cm2 plant-1. The importance of leaf area in relation to basic
plant metabolic processes, such as photosynthesis and respiration, is generally recognized.
Furthermore, the quantification of several growth analysis parameters requires the
measurement of leaf area at several stages during the life cycle of the plant.
ii. Leaf area index (LAI):
Leaf area index (LAI) is a dimensionless quantity that characterizes plant canopies. It is
defined as the one-sided green leaf area per unit ground surface area in broadleaf canopies.
The interaction between vegetation surface and the atmosphere, e.g. radiation uptake,
precipitation interception, energy conversion, momentum and gas exchange, is substantially
determined by the vegetation surface (Monteith and Unsworth, 1990).
During vegetation period of deciduous trees, total vegetation surface itself is mainly
composed of leaf area, and by lesser part of twigs, branches and stem surface. Whereas
during times of absent foliage (i.e. winter in temperate climates, dry season in tropical areas)
woody parts determine vegetation surface area.
Various destructive and non-destructive methodologies to measure or derive LAI do exist.
Non-destructive methods include hemispherical photography, sunfleck ceptometers, and
other optical instruments like TRAC, LAI-2000 or LI-COR (detailed description of
techniques are presented by Chen et al. 1997).
It has to be taken into account that all methods do have advantages as well as disadvantages
in estimating LAI and data are not always directly comparable. Espacially older literature
tends to contain higher values of LAI than later references (see Technical Memorandum of
Scurlock et al. 2001). The annual course of LAI for deciduous trees peaks during the height
of growing season, whereas LAI of coniferous stands vary far less over the year. Some

4. 4 | P a g e
deciduous trees keep old leaves till next seasons budding whilst other species completely
shed their branches. LAI of plants, especially grasses, consists of photo synthetically active
green and senescent leaves. Even though old leaves do not influence photosynthesis, they still
play an important role in intercepting precipitation. Therefore, as in the case of modeling
water interception, a LAI greater than zero has to be maintained throughout the year for
forests and pasture species in contrast to agricultural sites, which start with a LAI of zero
after ploughing. Information on this lower bound of LAI, here referred to as minimum LAI or
LAImin are relatively scarce. Published LAImin for grassland species are in a range between
0.3 to 2.0. Measurements of LAI in winter times for Fagus sylvatica or assessments of area
indices of branches and stems for Populus tremoluides reveal a highest LAImin of 1.1. Area
index of woody parts of coniferous trees can be assumed to be around 0.5, covering a range
of 0.2 to 0.9. (Breuer et al. 2002). As some eco-hydrological models need information on LAI
of whole stands, a surcharge for understory and litter has to be added to the LAI of trees to
obtain the integrated total LAI of a given forest stand. We therefore listed values for
understory as well as shrubs and woodland so modellers can sum up the needed LAI for their
purpose. On average we assume that a surcharge for understory vegetation and litter of
approx. 2.0 should be added to total LAI of deciduous and coniferous forests stands.
Treatments like fertilization of pasture or thinning of trees might have a strong effect on LAI.
Narrow plant spacing compared to broad plant spacing in crop stands leads to rising LAI,
whereas thinning in deciduous forest reduced LAI.
LAI = leaf area / ground area, m2
/ m2
iii. Crop Growth Rate (CGR):
The crop growth rate simply indicates the change in dry weight over a period’ of time. This
expression, as well as RGR, can be used without any assumption about the form of the
growth curve. Both ’formulae can be used to compare treatments between and within
experiments. They can even be used when it is possible to make only two harvests. Given
plots of species with differing temperature requirements for growth, the student will be able
to rank crops according to their growth rates and to compare the fall growth habits of species.

5. 5 | P a g e
CGR = (W2-Wl) / (t2-tl) = g m-2/time unit
Wl and W2 are plant dry weights m-2
land area while A1 and A2 are leaf area indices at times tl
and t2
iv. Relative Growth Rate (RGR):
Crop growth rate is a measure of the increase in size, mass or number of crops over a period
of time. The increase can be plotted as a logarithmic or exponential curve in many cases. The
absolute growth rate is the slope of the curve. Relative growth rate is the slope of a curve that
represents logarithmic growth over a period of time. An exponential growth rate is not
sustainable over time. The curve typically flattens out, representing saturation in growth at a
certain point in time. The crop growth rate calculation is dependent on the values of NAR
(Net Assimilation Rate) and LAI (Leaf Area Index) of the crop.
In plant physiology, RGR is a measure used to quantify the speed of plant growth. It is
measured as the mass increase per aboveground biomass per day, for example as g g−1
d−1
. It
is considered to be the most widely used way of estimating plant growth, but has been
criticised as calculations typically involve the destructive harvest of plants. Another problem
is that RGR nearly always decreases over time as the biomass of a plant increases, but
traditionally this has been ignored when modeling plant growth. The RGR decreases for
several reasons - non-photosynthetic biomass (roots and stems) increases, the top leaves of a
plant begin to shade lower leaves and soil nutrients can become limiting. Overall, respiration
scales with total biomass, but photosynthesis only scales with photosynthetic biomass and as
a result biomass accumulates more slowly as total biomass increases. The RGR of trees in
particular slow with increasing size due in part to the large allocation to structural material of
the trunk required to hold photosynthetic material up in the canopy. A novel approach to
separate size effects from intrinsic growth differences is implemented and described in detail
in Philipson et al. (2012).
Relative growth rate, RGR (d–1
), can be expressed in terms of differential calculus as RGR =
(1/W)(dW/dt) (compare Equation 6.2.) so that RGR is increment in dry mass (dW) per
increment in time (dt) divided by existing biomass (W). Averaged over a time interval t1 to t2
during which time biomass increases from W1 to W2, RGR (d–1
) can be calculated from

6. 6 | P a g e
Where:
= natural logarithm
= time one (in days)
= time two (in days)
= Dry weight of plant at time one (in grams)
= Dry weight of plant at time two (in grams)
v. Leaf Weight Ratio (LWR)
It was coined by (Kvet et al., 1971) Leaf weight ratio is expressed as the dry weight of leaves
to whole plant dry weight and is expressed in g g –1.
LWR= Leaf dry weight / Plant dry weight Plant dry weight
vi. Leaf Area Duration (LAD)
To correlate dry matter yield with LAI, Power et al. (1967) integrated the LAI with time and
called as Leaf Area Duration. LAD takes into account, both the duration and extent of
photosynthetic tissue of the crop canopy. The LAD is expressed in days.
LAD = (L1 + L2) X (t2 – t1)
L1 = LAI at the first stage
L2 = LAI at the second stage,
(t 2 - t 1) = Time interval in days
vii. Specific Leaf Area (SLA)
Specific leaf area is a measure of the leaf area of the plant to leaf dry weight and expressed
in cm2g-1 as proposed by Kvet et al. (1971).
SLA = Leaf area / Leaf weight
Hence, if the SLA is high, the photosynthesizing surface will be high. However no
relationship with yield could be expected.
viii. Specific Leaf Weight (SLW)

7. 7 | P a g e
It is a measure of leaf weight per unit leaf area. Hence, it is a ratio expressed as g cm-2 and
the term was suggested by Pearce et al. (1968). More SLW/unit leaf area indicates more
biomass and a positive relationship with yield can be expected.
SLW = Leaf weight / Leaf area
ix. Absolute Growth Rate (AGR)
AGR is the function of amount of growing material present and is influenced by the
environment. It gives Absolute values of biomass between two intervals. It is mainly used
for a single plant or single plant organ e.g. Leaf growth, plant weight etc.
AGR = h2 – h1/ t2 – t1 cm day-1
Where, h1 and h2 are the plant height at t1 and t2 times respectively
x. Net Assimilation Rate (NAR):
Various equations are used to estimate mean net assimilation rate NAR mean. NAR is the
ratio of rate of dry matter accumulation and leaf area index and a mean ratio should take into
account the rate of change of each of its components. During exponential dry matter
accumulation, and assuming an equal exponential rate of increase for LAI and dry matter,
mean NAR can be estimated as follows:
NAR mean = [(W2 - W1) / (t2 - t1)] ÷ [(LAI2 ÷LAI1) / (LAI2 - LAI1)] where NAR mean is
the mean net assimilation rate during a period from t = t1 to t = t2. The second part of
Equation expresses the inverse of mean LAI from t = t1 to t = t2. In contrast to mean NAR,
instantaneous NAR can be estimated by calculating rate of dry matter accumulation at time t
(i.e., by differentiating the "growth curve" at time = t) and measuring LAI at time = t.
Instantaneous NAR at time = t is rate of dry matter accumulation divided by LAI. The NAR
will decline once mutual shading among leaves in the canopy will occur. Rate of dry matter
accumulation will become "constant" when a change in LAI will not influence absorptance of
incident irradiance: the canopy has attained the phase of "constant" growth. Similarly, a crop
will have attained the phase of "constant" growth when leaf-area expansion has been
completed, even if PAR absorptance is less than 100%. Dry matter accumulation during this
period is relatively unimportant in the context of dry matter accumulation during the growing

8. 8 | P a g e
season. For instance, a maize crop in Ontario will accumulate less than 15% of dry matter at
maturity during this period.
Net gain in biomass (W) is clearly an outcome of CO2 assimilation by leaves minus
respiratory loss by the entire plant. Leaf area can therefore be viewed as a driving variable,
and biomass increment (dW) per unit time (dt) can then be divided by leaf area (A) to yield
the net assimilation rate, NAR (g m–2
d–1
), where
Averaged over a short time interval (t1 to t2 days) and provided whole-plant biomass and leaf
area are linearly related (see Radford 1967 and literature cited),
NAR thus represents a plant’s net photosynthetic effectiveness in capturing light, assimilating
CO2 and storing photo assimilate. Variation in NAR can derive from differences in canopy
architecture and light interception, photosynthetic activity of leaves, respiration, transport of
photo assimilate and storage capacity of sinks, or even the chemical nature of stored products.
xi. Leaf Area Ratio (LAR):
Since leaf area is a driving variable for whole-plant growth, the proportion of plant biomass
invested in leaf area or ‘leafiness’ will have an important bearing on RGR, and can be
conveniently defined as leaf area ratio, LAR (m2
g–1
), where
At any instant, or in practice at any harvest, LAR can be taken as A/W and can be factored
into two components, namely specific leaf area (SLA) and leaf weight ratio (LWR). SLA is
simply a ‘ratio’ of leaf area (A) to leaf mass (WL) (dimensions m2
g–1
) and LWR is a true ratio
of leaf mass (WL) to total plant mass (W) (dimensionless). Thus,
Alternatively and as commonly employed for growth analysis, average LAR over the growth
interval t1 to t2 is simply

9. 9 | P a g e
Expressed this way, LAR becomes a more meaningful growth index than A/W (Equation 6.8)
and can help resolve sources of variation in RGR. If both A and W are increasing
exponentially so that W is proportional to A, it follows that
or, summarized in terms of now familiar growth indices,
or more explicitly,
5. Conclusion:
Classic plant growth analysis continues to find application in resolving sources of variation in
RGR but suffers from statistical deficiencies and strict prerequisites for valid application of
the formulae discussed above. Functional growth analysis was developed during the 1960s to
overcome these limitations and was made feasible with the advent of computer-based data
analysis at about that time. In this technique (see Hunt 1982) curves generated by
mathematical functions are fitted to both A and W (either original values or ln-transformed
data). RGR at any particular point in time is then calculated as the slope of ln W versus time.
Other indices can be calculated once an adequate relationship between ln A and time is
established. In effect, an adequate relationship between ln W and ln A versus time allows
calculation of instantaneous values for RGR, NAR and LAR. As mentioned above, the slope
of ln W versus time yields RGR, and at that same instant A can be derived from the ln A
versus time relationship, allowing LAR (A/W) to be calculated. With RGR already derived,
NAR is then RGR/LAR.
Functional growth analysis enables experimenters to follow a time-course in growth indices
and to derive instantaneous values. In practical terms, large harvests at weekly intervals are
no longer needed. Instead, smaller harvests of two to four plants every 3–4 d are sufficient.
However, data analysis remains critical, and especially important is choice of a mathematical

10. 10 | P a g e
function with biologically meaningful parameters that best fits ln-transformed values (see
Hunt 1978, 1982 for further details).
6. References
 Wilson D, Cooper JP. 1970. Effect of selection for mesophyll cell size on growth and
assimilation in Lolium perenne L.New Phytologist 69, 233-45.
 Spitters CJT, Kramer T. 1986. Differences between spring wheat cultivars in early
growth. Euphytica 35, 273-92.
 Rodgers CO, Barneix AJ. 1988. Cultivar differences in the rate of nitrate intake by
intact wheat plants as related to growth rate. Physiologia Plantarum 72, 121-6.
 Grime JP, Hunt R. 1975. Relative growth rate: its range and adaptive significance in a
local flora. Journal of Ecology 63, 393-422.
 Evans GC. 1972. The quantitative analysis of plant growth. Oxford: Blackwell
Scientific Publications.
 Vernon AJ, Allison JCS. 1963. A method of calculating net assimilation rate. Nature
200, 814.
 Hughes AP, Freeman PR. 1967. Growth analysis using frequent small harvests.
Journal of Applied Ecology 4, 553-60.
 Abd EL-Gawad, A.A. ; H.K. Hassan and W.H. Hassany (2000). Transplanting
technique to adjust plant stand of sugar beet under saline conditions: I- Response of
sugar beet yield to transplanting missed hill by different transplant ages. Proc. 9th of
Agron., 1-2 Sept. 2000, Minufiya Univ., II: 533-548.
 Aitchison J, Brown JAC. 1966. The lognormal distribution with special reference to
its use in economics. Cambridge University Press.
 Barnett V, Lewis T. 1978. Outliers in statistical data. Chichester: Wiley and Sons.
 Biere A. 1987. Ecological significance of size variation within populations. In: van
Andel J, Bakker JP, Snaydon RW, eds. Disturbance in grasslands. Dordrecht: Junk
Publishers, 253-63.
 Blackman VH. 1919. The compound interest law and plant growth. Annals of Botany
33, 353-60.

11. 11 | P a g e
 Breuer L, Eckhardt K, Frede H-G, 2003. Plant parameter values for models in
temperate climates. Ecol Model 169, 237-293.
 Chen JM, Rich PM, Gower ST, Norman JM, Plummer S, 1997. Leaf area index of
boreal forests: Theory, techniques, and measurements. J Geophys Res 102, 29429-
29443.
 Echarte, L., Rothstein, S, and Tollenaar, M. 2008. The response of leaf
photosynthesis and dry matter accumulation to nitrogen supply in an older and a
newer maize hybrid. Crop Sci. 48:656-665.
 Monteith JL, Unsworth MH, 1990. Principles of environmental physics. Edward
Arnold, London, 291 pp.
 Scurlock JMO, Asner GP, Gower ST, 2001. Worldwide Historical Estimates and
Bibliography of Leaf Area Index, 1932-2000. ORNL Technical Memorandum TM-
2001/268, Oak Ridge National Laboratory, Oak Ridge, Tennessee, U.S.A.
 Sinclair, T.R. and Muchow, R.C. 1999. Radiation use efficiency. Advances in
Agronomy 65: 215-265.