Chapter 1. Dynamics of change21.1 Constant rate of change

Application of mathematics in agriculture often requires mathematical functions to describe and predict a phenomenon that is continuously changing under varying rates, and this requires the use of calculus. Calculus is a branch of mathematics that deals with instantaneous rates of change. And there are two broad applications of calculus: differential calculus (or simply known as differentiation) and the other integral calculus (also known as integration or anti-differentiation).

Calculus, unfortunately, has a reputation of being difficult and abstract to most people (agriculturists included). Most of us learn by rote the various expressions for differentiation and integration, including

and its counterpart

where both n and C are constants. But what is the point of calculus? More specifically, how do we use or apply calculus in agriculture?

Consider a hypothetical growing crop. Table 2.1 shows that the plants growth rate remained constant at 10 g day-1 from day 10 to 20. This means that in this period the plant would consistently gain weight by 10 g per day. So after ten days, the plant must gain (10 g day-1 ( 10 days) or 100 g. Hence, provided the plant is growing at a constant rate, there is a simple relationship between the cumulative (total) weight change ((W) and the rate of growth (wr):

where (t is the time interval. In Table 2.1, we can confirm that in ten days from day 10 to 20 ((t = 10) the plant weight increased by 100 g.

And if we plot the growth rate (which is constant) against time, we will get a horizontal line at 10 g day-1 (Fig. 2.1a). Furthermore, the area under this line is actually the cumulative weight change. As mentioned earlier, the cumulative weight change is the growth rate multiplied by the time interval. The area under the line in Fig. 2.1a is a rectangle with a height of 10 g day-1 (i.e., the growth rate) and a width of (20 10) days (i.e., the time interval). Consequently, this gives the area under the line from day 10 to 20 as the area of the rectangle:

This cumulative change in weight of 100 g from day 10 to 20 agrees with our previous calculation and the data in Table 2.1. In short, the cumulative change in weight is the area under the line (or curve) for the graph of rate of weight change against time.Table 2.1. Measured plant growth (constant growth rate)DayGrowth rate (g day-1)Plant weight (g)

1010300

1110310

1210320

1310330

1410340

1510350

1610360

1710370

1810380

1910390

2010400

(a)(b)

Fig. 2.1. Plotted constant growth rate and plant weight against time

Since the growth rate remained constant, the plant weight must increase linearly with time (Fig. 2.1b). And if we take the slope of this linear line, we will obtain 10 g day-1, which is the rate of weight change or growth. Recall that for a linear line, its slope (m) is calculated as

where (x1, y1) and (x2, y2) are two pairs of points along the linear line. From Table 2.1, let us take the two pairs of points as (15, 350) and (20, 400), so that we determine the slope of the line as

In other words, the instantaneous rate of weight change is the slope of the line (or curve) for the graph of plant weight against time.

To generalise: the slope of a curve gives us the instantaneous rate of change, whereas the area under a rate of change curve gives us the cumulative change. As we will later discuss, the slope of a curve and the area under a curve are concerns of differential calculus and integral calculus, respectively.1.2 Variable rate of change

In the previous section, calculations were simple because the rate of plant growth was constant. But when the rates of change are not constant, the use of calculus then becomes essential. Consider a second example using the same hypothetical crop but it grew instead at a variable (non-constant) rate.

Table 2.2 shows the measured daily plant weight from day 10 to 20. Using the data in Table 2.2, we can show that the plant weight is related to time by the following quadratic function:

where f (t) is the weight of the plant (g) at time t (day). For example:at day 10:

at day 11:

at day 20:

all of which agrees with the tabulated data in Table 2.2.

Plotting the plant weight against time shows that the plant weight increased with time in a non-linear manner (Fig. 2.2). This shows that the growth rate of the plant is not constant, in contrast to the first example (Fig. 2.1b). As mentioned in the previous section, finding the slope of a curve will give us the instantaneous rate of change. When the growth rate is constant, the plant weight will increase linearly with time. This increase must be linear because a linear line has a constant slope at all its points (hence, a constant rate of change).

Table 2.2. Measured plant weight which increases at a variable rate

DayPlant weight (g)

10300

11322

12346

13372

14400

15430

16462

17496

18532

19570

20610

Fig. 2.2. Plotted plant weight against time (variable growth rate)

But when the growth rate is variable, as in this second example, the weight of the plant will increase in a non-linear manner with time. Because this curve is non-linear, its slope will vary from point-to-point along the curve, in contrast to a linear curve. Finding the slope of a curve at, say, point (a, b) is, by definition, finding the slope of the tangent line at (a, b). But what exactly is a tangent line?

Fig. 2.3 shows point P on two curves. We have drawn an enlarged version of the box around each point P. Notice that the portion of each curve within the boxed region looks almost straight. With increasing magnification, the portion of the curve near P becomes increasingly more exactly like a straight line. This straight line is called the tangent line to the curve at point P.

Fig. 2.3. Tangent line to the curve at point P

The fundamental idea to determine the slope of the tangent line at a point P is to approximate the tangent line very closely by secant lines. A secant line at P is a straight line passing through P and a nearby point Q on the curve (Fig. 2.4). Now, suppose that point P is (x, f (x)), and that point Q is h horizontal units away from point P so that point Q is located at (x + h, f (x + h)). Consequently, the slope of the secant line through points P and Q is

To let the slope of secant line approach the slope of tangent line, we move point Q increasingly closer to point P, thereby h becomes increasingly smaller (but h is never zero). In other words, by taking h sufficiently small, the slope of the secant line can be taken as the slope of the tangent line with the desired accuracy. Mathematically, we write this as

where h(0 means h approaches but never reaches zero, and f (a) is the derivative of the function f (x) at x = a. In other words, f (a) is the slope of the tangent line at x = a. As mentioned in the previous chapter, the derivative of the function f (x) is sometimes written as dy/dx, df (x)/dx or D(x).

Fig. 2.4. Tangent line to the curve at point P approximated by the secant line through points P and Q

Let us return to the example where the plant growth rate is variable. What is the rate of plant growth at, say, day 15? To do this, we need to find the slope of the tangent line at t = 15. In other words, we differentiate the plant weight function f (t) = t2 + t + 190 at t = 15:

where we see that as h(0, f (15) approaches 31. Consequently, we can take 31 g day-1 as the plant growth rate at day 15. Likewise, to determine the growth rate at day 20 is

where the growth rate at day 20 is 41 g day-1. To generalise, the growth rate of the plant at any given time t is:

So the derivative of the plant weight function f (t) = t2 + t + 190 gives the growth rate function as f (t) = 2t + 1. Table 2.3 expands on the earlier Table 2.2 by showing the measured daily plant weight as well as the calculated daily growth rates (i.e., 2t + 1) from day 10 to 20.Table 2.3. Measured plant weight and its variable (calculated) rate of increase

DayGrowth rate (g day-1)Plant weight (g)

1021300

1123322

1225346

1327372

1429400

1531430

1633462

1735496

1837532

1939570

2041610

Fig. 2.5 shows that the growth rate increased linearly with time, where the line is described by the function 2t + 1. We saw in the previous section that if we took the area under the rate of change curve, we would obtain the cumulative change. In Fig. 2.5, the area under the curve from day 10 to 20 is the combined area of the triangle (Area A) and rectangle (Area B):

which agrees with our measured weight gain of 310 g from day 10 to 20 (Table 2.3).

Fig. 2.5. Plotted variable growth rate against time

Finding the area under the curve is the same as integration, and integration is the converse of differentiation. When we take the slope of the function f, we interpret it as its rate of change. But if we wish to determine the cumulative change (that is, how much change had occurred), we take the anti-derivative of the rate of change function f; that is, we integrate the rate function f. In short, by integrating the function that describes the rate of change curve, it gives us the cumulative change; that is,

In our first example, the instantaneous rate of change was constant at 10 g day-1. Hence, integrating

gives us the cumulative change of 100 g from day 10 to 20 (Table 2.1). But in our second example, the rate of growth varied according to the function 2t + 1 so integrating this function

gives us a cumulative change of 310 g in the same ten-day period (Table 2.3).

1.3 Integration as a summation and the mean value of a function

In Fig. 2.6, the area under the curve ABCD can be approximated by dividing this area into several rectangular strips, where each strip has a width of (x (the symbol ( is interpreted as very small). The area of each strip is approximately equal to f (x)((x (i.e., height ( width for a rectangle area). And the sum area for all the strips gives the approximate area of ABCD, or

[2.1]

Notice that the last term f (b)((x is the area of the last rectangle which lies just outside the region ABCD, and that, in Fig. 2.6, the areas of the rectangles tend to underestimate the area under the curve. The area of the first rectangle, for example, underestimates the area APRD by the area APQ.

Fig. 2.6. Approximating the area under the curve by a series of equal width rectangles

Eq. 2.1, however, becomes increasingly more accurate if we take the width of each strip to be increasingly smaller and so increasing the number of strips to cover ABCD. In other words, as (x approaches zero ((x ( 0), the accuracy of Eq. 2.1 increases. Mathematically, we write this as

We have learnt that the area under a curve ABCD is determined exactly by integrating the function f(x) from x = a to x = b:

so this means that

[2.2]

Eq. 2.2 is important as it tells us that an integral can be approximated by a summation, provided that the area under the curve can be divided into a large number of narrow strips to give the desired accuracy.

Let us further determine the mean (average) of the function f(x) from x = a to x = b (Fig. 2.6). Recall that the mean of n quantities (y1, y2, , yn) is:

In the same way, the mean of the function from a to b is given by

Since (b a) = (n - 1)((x, and as n ( (, (x ( 0:

where earlier in Eq. 2.2 we showed that .

1.4 Numerical integration

There are integrals that are impossible to evaluate analytically such as

There are no exact solutions to such integrals, and we must resort to numerical integration to obtain their approximate solutions. There are many numerical methods available, but in this book, we will discuss three common methods: midpoint, trapezoidal and Gauss.1.4.1 Midpoint method

We have actually discussed a slight variation of this method in the previous section (Eq. 2.2). As described previously, we approximate the area under the curve by rectangles. The area under curve is divided into n number of rectangular strips, each having an interval width of (x (Fig. 2.7). What is different, however, we select the midpoint for each interval, ck, which it is determined by

so that the area for a rectangle strip is determined by f (ck) ( (x (note: k = 1, 2, , n). The sum of all the rectangle areas is then the area under the curve, or

Since (b a) = n ( (x,

[2.3]

Fig. 2.7. The midpoint method to approximate the area under the curve

The more rectangle strips we use (i.e., larger n and smaller (x), the accuracy of the midpoint method increases.1.4.2 Trapezoidal method

Instead of using rectangles to approximate the area under the curve, trapezoids are used instead in this method. We again divide the area under the curve into n equal intervals with width (x (Fig. 2.8).

Fig. 2.8. The trapezoidal method to approximate the area under the curve

We see that the sum of all the trapezoidal areas gives us the area under the curve. From Fig. 2.9, the area of a trapezoid is

where w is the width of the interval (x; and h1 and h2 are f (xi) and f (xi+1), respectively (i = 0, 1, , n). Consequently, the area under the curve is determined by

Note that x0 = a and xn = b. As before, (b a) = n ( (x so that

[2.4]

Like the previous method, the accuracy of the trapezoidal method increases by increasing the number of trapezoidal strips and decreasing the interval width (x for each strip.

Fig. 2.9. Area of a trapezoid

1.4.3 Gauss method

One of the most accurate numerical method is the Gauss-Legendre (or simply known as Gauss) method. This method is also one the most common methods used in agriculture work.

Consider the function y = f (t) which we wish to integrate over the interval t = -1 to 1. We will approximate its solution as

[2.5]

where the coefficients w and t are the weight and abscissa, respectively. We wish to determine all the weights wi and abscissas ti in such a way so that the linear additions of wi(f(ti) for i = 1 to N is the exact solution of the integral of f (t) over [-1, 1]. A two-point Gaussian integration means N = 2, and Eq. 2.5 is expanded to

[2.6]

Let us further approximate the function f (t) as a cubic polynomial which has a polynomial degree (or order) of three:

Substitution into Eq. 2.6 produces

and collecting the terms

means the following must be true:

We now have four unknowns (the weights w1, w2, and abscissas t1, t2) in four independent equations which means they can be solved. Thus, it can be shown that for a two-point Gaussian integration method:

At hindsight, approximating the function f (t) as a cubic polynomial is exactly right for a two-point Gaussian integration. This is because it produces four unknowns in four independent equations. So for a three-point Gaussian integration the function f (t) must be approximated by a polynomial curve having a degree of five. This would produce six unknowns (weights w1, w2, w3, and abscissas t1, t2, t3) in six independent equations.

To generalize this rule, an N-point Gaussian integration requires that the function f (t) be approximated by a polynomial curve with 2N-1 degrees. As expected, the higher the N points of integration the higher the accuracy of solution. This is because we are approximating the function f (t) by an increasingly more flexible polynomial curve (i.e., increasingly higher polynomial degree). Nonetheless, the higher the N points of integration, the higher the number of required computations as well. In agriculture, the three- or five-point Gaussian integration are often used. The abscissa xN,i and weights wN,i up to eight points are tabulated in Table 2.4.

If we had the integral over an interval [a, b] instead of [-1, 1] we must normalize the interval [a, b] to [-1, 1] so that if a ( x ( b and that -1 ( t ( 1 this means, using linear transformation (Fig. 2.10), x is related to t by

[2.7]

where (a = (b + a) / 2 and (b = (b a) / 2, and differentiating x with respect to t means

Thus, the integral over the interval [a, b] is made equivalent to that over the interval [1, 1] by

[2.8]

Table 2.4. Abscissas and weights for the N-point Gaussian integration

NAbscissas, tN,iWeights, wN,iNAbscissas, tN,iWeights, wN,i

2 0.57735027 1.00000000 6 0.93246951 0.17132449

0.57735027 1.00000000 0.66120939 0.36076157

3 0.77459667 0.55555556 0.23861919 0.46791393

0.00000000 0.88888888 7 0.94910791 0.12948497

4 0.86113631 0.34785485 0.74153119 0.27970539

0.33998104 0.65214515 0.40584515 0.38183005

5 0.90617985 0.23692689 0.00000000 0.41795918

0.53846931 0.47862867 8 0.96028986 0.10122854

0.00000000 0.56888888 0.79666648 0.22238103

0.52553241 0.31370665

0.18343464 0.36268378

Fig. 2.10. Normalization of [a, b] integration interval to [-1, 1]Example

Use the midpoint and trapezoidal methods (with n = 4), as well as the Gauss method (integration with 3 points), to approximate the following integral:

Compare their results against the analytical solution.

Solution

Midpoint method:

The interval width for each rectangle strip is given by

Consequently, the midpoints ck for the four rectangle strips are:

Interval, (xk-1, xk)Midpoint, ckf (ck) = ck2

(0, 0.25)0.1250.015625

(0.25, 0.5)0.3750.140625

(0.5, 0.75)0.6250.390625

(0.75, 1.0)0.8750.765625

TOTAL1.312500

Applying Eq. 2.3 to approximate the integral gives:

Trapezoidal method:

As before, the interval width for each strip (trapezoidal) is 0.25, and applying Eq. 2.4 to obtain the integral approximation as:

Gauss method:

Using Eq. 2.7 to normalize the interval from [1, 0] to [-1, 1] means

so, from Eq. 2.8 and Table 2.4, the approximate solution is

Analytical solution:

The exact solution of the integral is

The error for the midpoint method is an underestimation by 1.562% of the exact solution, whereas the trapezoidal method overestimates by 3.125%. The highest accuracy obtained is by the Gauss method: a very small overestimation by 0.02% of the exact solution.

Example

Leaf photosynthesis can be described by the following equation, as given by Goudriaan and van Laar (1994):

[2.9]

where AL is the leaf photosynthesis (that is, the assimilation rate of CO2 by a single leaf; (g CO2 m-2 leaf area s-1); Am is the maximum leaf photosynthesis rate ((g CO2 m-2 leaf area s-1); ( is the solar radiation conversion factor ((g CO2 J-1); k is the canopy extinction coefficient for solar radiation (unitless); I0 is the solar irradiance above canopy (W m2 ground area or J m-2 ground area s-1); and L is the cumulative leaf area index from the canopy top to the leaf being considered (m2 leaf area m-2 ground area).

Determine the canopy photosynthesis (that is, determine the cumulative photosynthesis for all the leaves together). Use the following values: k = 0.5, total leaf area index (LAI) = 3 m2 leaf area m-2 ground area, I0 = 200 W m-2 ground area, Am = 1500 (g CO2 m-2 leaf area s-1, and ( = 12 (g CO2 J-1.

Solution

Eq. 2.9 gives the photosynthetic rate of a single leaf. To determine the canopy photosynthesis (all leaves), we need to integrate Eq. 2.9 over the entire canopy; that is, integration is over the interval [0, LAI], where LAI is the total leaf area index. To integrate complex equations like Eq. 2.9, it is convenient to use mathematical software to perform the integration. The analytical solution to integrating Eq. 2.9 is

[2.10]

Substituting the given parameter values into Eq. 2.10 gives the canopy photosynthesis as 1270.6220 (g CO2 m-2 leaf area s-1.

Let us try the 3-point Gauss integration method to integrate Eq. 2.9. We first need to normalise the integration interval [0, LAI] to [-1, 1]. Using Eq. 2.7, we obtain

so that

[2.11]

where

From Table 2.4, we obtain the values for the three abscissas t3,1, t3,2 and t3,3, and for each abscissa, we determine the leaf photosynthetic rate AL (t):Abscissa, t3,iAL (t3,i)Weight, w3,iw3,i ( AL (t3,i)

-0.7746604.7840.5556336.018

0.0000411.38160.8889365.6771

0.7746261.74370.5556145.4248

TOTAL847.1199

From Eq. 2.11, the canopy photosynthesis is therefore

which is very close the analytical solution (an error less than 0.01%).

1.5 Mathematical analysis of plant growth: application of calculus

In this section, we will discuss how calculus can be applied to describe the growth of a generic plant. We will begin by assuming that the plant has adequate supply of water and nutrients (i.e., no water stress or nutrient deficiency), and the plant is free from pests and diseases. The only limiting factor to plant growth is solar radiation which is the main source of energy for various plant processes including photosynthesis.

Photosynthesis, aided with solar radiation energy, converts carbon dioxide and water into carbohydrates (sugars). These carbohydrates are plant food; that is, substrates which the plant will use for two purposes: maintenance and growth. Maintenance is a process whereby the plant uses the substrates for its continual survival. In other words, the substrates are used to maintain or sustain existing plant parts. Growth, on the other hand, is a process whereby the substrates are used to build or synthesise new materials; thus, enlarging existing plant parts or creating new ones. Consequently, the plant becomes bigger and heavier.

The plant growth model is as shown in Fig. 2.11. The plant weight consists of two components: the storage weight and the plant structural weight. The storage weight, Ws, is supported by the addition of newly produced substrates via the photosynthesis process. A portion of the plants storage, kgWs, will be utilised for maintenance and growth, where of this total (kgWs), Yg of it will be utilised for growth, and the remainder (1 Yg) for maintenance. Of the total plant structural weight, Wg, kd of it will degrade and the substrates returned to the storage, and ks of it will senesce (i.e., loss of materials due to increasing age).

In plant growth studies we are often interested in the plant dry weight (without its water content) per unit ground area. A large portion of the dry matter of the plant is in the form of organic compounds whose primary constituent is carbon which is assimilated in the form of carbon dioxide through the plant leaves. Consequently, we need to express the plant weight with respect to its carbon content (g C m-2 ground area). On average, the carbon content in a plant is about 45% of the plant dry weight. So in the absence of carbon analysis, plant dry weight equivalent to its carbon weight is 45% of the total dry weight.

Moreover, as shown earlier, Eq. 2.10 gives us the gross canopy photosynthesis, AT, in units of (g CO2 m-2 leaf area s-1 which should be expressed with respect to carbon; that is, in units (g C m-2 leaf area s-1. This conversion is done by multiplying AT by 12/44 (or about 0.273) since the atomic weight of carbon and molecular weight of carbon dioxide are 12 and 44 g, respectively. In other words, every 44 g of CO2 has 12 g of C, so AT (g CO2 will have (12/44)(AT (g of C.

Fig. 2.11. A simple plant growth model (adapted from Thornley, 1977)

From Fig. 2.11, the rates of change in the plant storage and structure dry weight at time t (in unit hour) are

[2.12]

[2.13]

where Ws and Wg are the dry weights for the substrates storage and plant structure, respectively (g C m-2 ground area). Since time t is in unit hour, we need to multiply AT by 3600 (i.e., 60 s ( 60 mins) to express AT in units (g CO2 m-2 leaf area hour-1. Further multiplication by LAI is to convert AT into units (g CO2 m-2 ground area hour-1; hence, making it compatible with the dry weight units (which are expressed as g C per unit ground area):

Adding both Eq. 2.12 and 2.13 gives the rate of change in the total plant dry weight as

where W is the total plant dry weight (g C m-2 ground area). The cumulative increase in the total plant dry weight, (W, from time t1 to t2 is then obtained by integrating this rate of change function over the period [t1, t2]:

[2.14]

where (t is the time interval (t2 - t1), and kg ks and Yg are taken as constants. Calculations are straightforward with the exception of the integration of the canopy photosynthesis AT. From Eq. 2.10, canopy photosynthesis is

where both the leaf area index, LAI, and the solar irradiance above canopy, I0, are not constants but both vary with time. For short periods (such as a day, or (t = 24 hours), it is reasonable to assume LAI remains constant. Consequently, we will determine the daily increment in the total dry weight W, with t1 = 0 and t2 = 24.

The instantaneous solar irradiance I0 (J m-2 ground area s-1) is taken to vary with time t (hour) according to a simple equation adapted from France and Thornley (1984) as

[2.15]

where It,d is the daily total solar irradiance (J m-2 ground area day-1); and tsr and tss are the times of sunrise and sunset, respectively (hour). Since there is no photosynthesis (AL = 0) for periods before sunrise and after sunset, integration of canopy photosynthesis over the whole day is equivalent to its integration over the period between sunrise and sunset; that is,

Hence, substituting Eq. 2.15 into Eq. 2.10 and its integration gives the daily canopy photosynthesis, AT,d, as

[2.16]

recalling that LAI is unchanged during the day (thus, LAI is not a function of t for that period). Unfortunately, Eq. 2.16 cannot be integrated analytically, and we must rely on numerical integration methods.

For this case, we will choose the 3-point Gauss integration method to determine AT,d, and Table 2.5 lists the parameters and their values to be used in the plant growth model. The first step in the Gauss method is to normalise the integration interval [tsr = 6, tss = 18] to [-1, 1]. Using Eq. 2.7, we obtain

so that

[2.17]

where

From Table 2.4, we obtain the values for the three abscissas t3,1, t3,2 and t3,3, and for each abscissa, we determine A(t) using Eq. 2.15 with its relevant parameter values from Table 2.5:

Abscissa, t3,iA (t3,i)Weight, w3,iw3,i ( A (t3,i)

-0.77460.12230.55560.0679

0.00000.52840.88890.4697

0.77460.12230.55560.0679

TOTAL0.6055

So the daily canopy photosynthesis AT,d is

[2.18]

From Eq. 2.14, the daily increment ((t = 1) in the total plant dry weight is

This means that the plant has gain about 5 g to give the total plant dry weight after one day of growth as

[2.19]

Table 2.5. Parameter values for the simple plant growth model

SymbolDescriptionUnitsValue

Ammaximum leaf photosynthesis rate(g CO2 m-2 leaf area s-11500

( solar radiation conversion factor(g CO2 J-112

kcanopy extinction coefficient-0.5

LAIleaf area indexm2 leaf area m-2 ground area2

kg growth rate constantday-11.98

kddegradation rate constantday-10.10

kssenescence rate constantday-10.05

Ygefficiency of structural synthesis-0.75

Wssubstrates dry weightg C m-2 ground area20

Wgstructural dry weightg C m-2 ground area130

tsrtime of sunrisehour6

tsstime of sunsethour18

It,ddaily total solar irradianceJ m-2 ground area day-110000000

A more rigorous modeling approach to plant growth would be to include the death rates of leaves (whether due to self shading or age) and roots, as well as the partitioning of substrates to the various plant organs such as roots, stem, leaves and storage organs. In particular, the partitioning of substrates to the leaves is important as it would allow us to work out weight of the leaves, and in turn, the leaf area index, LAI; that is,

where Wleaves is the weight of green leaves (g m-2 ground area); and SLA is the specific leaf area which is the leaf area per unit leaf weight (m2 leaf area g-1). The specific leaf area varies with the plant growth stage and it typically ranges between 0.005 to 0.03 m2 leaf area g-1.

In depth details about plant growth modeling are discussed in Goudriaan and van Laar (1994), Teh (2006), and Thornley and France (2006).

1.5.1 Growth functions

We saw in the previous section that the rate of weight change, dW/dt, is often used to indicate growth rate. dW/dt is known as the absolute growth rate (AGR). A more useful measure of growth rate, however, is the relative or specific growth rate (SGR), which is given by

which shows that SGR is the increase in weight per unit time per unit weight (or the absolute growth rate per unit weight). SGR is a better indicator of growth rate than AGR because SGR expresses growth rate as the rate of weight increase per unit weight. Hence, SGR is a measure of the efficiency of the plant or animal to produce new mass material.

If SGR remains constant throughout the plant growth period, it means

where u is a growth constant; in this case, u is also SGR. When SGR is constant, the rate of weight increase, dW/dt, is proportional only to the weight W. Integrating the above equation, we obtain

[2.20]

where W0 is the initial plant weight at time t = 0. Eq. 2.20 is the exponential function for growth (Fig. 2.12).

Exponential growth functions are seldom used to describe long periods of growth because as plants become larger, their specific growth rates tend to decrease until zero. In other words, SGR is not a constant value. So let us formulate SGR as a function of the plant weight in such a way so that as the plant weight increases, SGR will decrease. SGR is now

where Wm is the maximum plant weight. We can see that as W approaches Wm, the quotient W/Wm approaches 1, so SGR approaches zero. As done previously, we will integrate the above equation to give

which is known as the logistic or sigmoid function for growth (Fig. 2.12). Logistic growth functions are typified by S-shaped curves.

Fig. 2.12. Plant growth as described by three types of growth functions: exponential (u = 0.2, W0 = 1), logistic (u = 0.3, W0 = 1, Wm = 100), and Gompertz (u0 = 0.5, W0 = 1, D = 0.1086)

Another frequently used function for growth is the Gompertz function (Fig. 2.12). For this function, SGR also decreases with increasing plant age, but the growth coefficient u is no longer a constant but it varies exponentially according to:

where u0 is the initial u at time t = 0; and D is the decay constant rate of u. Hence, SGR is now

Integrating, we obtain the Gompertz function for growth:

Fig. 2.12 shows the increase in plant weight as described by the exponential, logistic and Gompertz functions. Of the three functions, the exponential function is the least representative of plant growth. It often shows a very slow initial growth rate, followed by an increasingly rapid growth rate. As mentioned in the previous chapter, exponential growth functions are typically limited to describe the growth of very young seedlings or the growth of embryos. The logistic growth function typically shows an S-shaped curve: growth rate is initially slow, then increases for long periods before slowing down again towards the end of the growth period. Compared to the logistic growth, Gompertz growth typically shows a faster early growth rate, followed by a slower, almost linear approach to the asymptote.1.6 Multivariable calculus

When a function depends on two independent variables, say, x and y, we express the function as f (x, y). Suppose we want to know how the function f (x, y) changes when x and y change. Instead of changing both variables simultaneously, we could change one variable (x) while keeping the other (y) constant. In turn, we could now change the y variable, while keeping the x variable constant. In this way, we will have a better understanding on how the function f (x, y) depends on x and y. This is the idea behind the concept of partial derivatives.

The function f (x, y) has two partial derivatives (as it has two independent variables). The partial derivative of f (x, y) with respect to x, written as , is the derivative of f (x, y) where y is kept constant, and the function f(x, y) depends solely on x. In the same way, the partial derivative of f (x, y) with respect to y, written as , is the derivative of f (x, y) where x is kept constant, and the function f(x, y) depends solely on y. Note that for partial derivatives, we use the symbol ( instead of d.

We can extend these principles to three or more independent variables. The function f(x, y, z) depends on three variables: x, y, z, and the partial derivative means the partial derivative of the function f(x, y, z) with respect to x depends solely on x, with both variables y and z kept constant. The other two partial derivatives: and are interpreted using the same principles.

Finding partial derivatives is no different from finding the derivatives of functions of one variable, since by keeping all but one variable constant, determining a partial derivative is the same as finding the derivative of one variable.Example

Let f(x, y) = 5x3y2. Determine the two partial derivatives.Solution

To determine , we keep the y variable constant. In other words, 5y2 is a constant. So

To determine , we keep the x variable constant instead; hence, 5x3 is now a constant. Finally,

Recall that the converse of differentiation is integration. When a function f (x) is integrated over an interval x = a and x = b, we obtained the cumulative change in that period. The same principle applies for a multivariable function like f (x, y). To determine the cumulative change, we need to perform a double integration over intervals [x = a, x = b] and [y = p, y = q]. Mathematically, this is expressed as

The integrals are called iterated integrals because they are evaluated in more than a single step, from inside out; that is,

where the inside integral is evaluated first by treating the y variable as constant. The result of this first integration is a function of the y variable, and in the second step, this resulting function is integrated over the limits as indicated by the outside integral sign (i.e., over y = p and y = q). The order of the differentials dx and dy indicates which integration is performed first.Example

Evaluate .

Solution

The order of the differentials is dx then dy, indicating that the integration is over the xinterval first [1, 3] (the inside integral) then over the y-interval [2, 4] (the outside integral). In other words,

where the first integration step is with respect to x (the y variable is treated as a constant):

and the resulting function is for the next integration step which is now with respect to y:

Hence, .

We have actually used double integration in the previous section (1.5) on plant growth modeling. We learned that to determine the daily canopy photosynthesis, we first need to integrate the leaf photosynthesis function (Eq. 2.9) over the entire total leaf area [0, LAI] which will give us the canopy photosynthesis function (Eq. 2.10). Eq. 2.10, however, will only give us the canopy photosynthesis at an instantaneous moment. To determine the canopy photosynthesis for the entire day, we must integrate Eq. 2.10 over the whole day, which is equivalent to integrating the function from the time of sunrise to sunset (since there is no photosynthesis for periods before sunrise and after sunset). The daily canopy photosynthesis function is as shown in Eq. 2.16 which can be expressed in units (g CO2 m-2 leaf area day1 as

where I0(t) is from Eq. 2.15, and the multiplication by 3600 is because time t is in unit hour._1225726065.unknown

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