Sigmoidal Curve in Geometric Growth

Question

Here is an excerpt from my textbook:

Let us now see what happens in geometrical growth. In most systems, the initial growth is slow (lag phase), and it increases rapidly thereafter – at an exponential rate (log or exponential phase). Here, both the progeny cells following mitotic cell division retain the ability to divide and continue to do so. However, with limited nutrient supply, the growth slows down leading to a stationary phase. If we plot the parameter of growth against time, we get a typical sigmoid or S-curve.

The exponential growth can be expressed as $W_1 = W_oe^r{^t}$

$W_1$ = final size (weight, height, number etc.)
$W_0$ = initial size at the beginning of the period
$r$ = growth rate
$t$ = time of growth
$e$ = base of natural logarithms

The first paragraph makes perfect sense to me, but what bothers me is the formula given. When I plot the given equation $W_1 = W_oe^r{^t}$, this is what I get;

Which is clearly not sigmoidal (s-shaped) as my textbook says. What I think should be the correct equation is: $$W_1 = \frac{W_o}{1+e^-{^r{^t}}}$$

When I plot this equation, this is what I get:

Which is slightly better than the previous plot. Though this formula might also be erroneous as I just made it up from the general formula of a sigmoid function $S(x) = \frac {1}{1+e^-{^r{^t}}}$

Why is there a discrepancy between the in-text statement and the plot of the mentioned formula and is there any better formula to depict this growth?

Also, I have referred to various sites but could not find any satisfactory answer. Any help is appreciated. Thanks!

Answer 1

I discussed this question with my biology teacher and he confirmed that the equation $W_1 = W_oe^r{^t}$ just represents the exponential phase of the sigmoidal growth, but not the entire growth period. This was also helpfully pointed out in the comments.

Though I did find an article talking about the mathematical functions of sigmoidal growth which includes both the exponential and linear phases of growth.

Within the life cycle of an organ, a plant or a crop, the total growth duration can be divided into three sub‐phases: an early accelerating phase; a linear phase; and a saturation phase for ripening (Goudriaan and van Laar, 1994). Therefore, the growth pattern typically follows a sigmoid curve, and the growth rate a bell‐shaped curve. While the sigmoid pattern can be represented piecewise using an exponential, a linear and a convex equation sequentially (e.g. Lieth et al., 1996), a more elegant way is to use a curvilinear equation which gives a gradual transition from one phase to the next. For example, based on principles of light interception and leaf area expansion, Goudriaan and Monteith (1990) derived a single equation, the expolinear equation, for both the exponential and linear phases of crop growth: $$w=\frac{c_m}{r_m} ln [1+e^{r_m(t-t_o)}] \tag{1}$$ where $w$ is mass, $t$ is time, to is the moment at which the linear phase effectively begins, $c_m$ and $r_m$ are maximum growth rate in the ‘linear phase’ and maximum relative growth rate (RGR) in the ‘exponential phase’, respectively.

YIN, XINYOU, GOUDRIAAN, JAN, LANTINGA, EGBERT A., VOS, JAN, and HUUB J. SPIERTZ. "A Flexible Sigmoid Function of Determinate Growth." Annals of Botany 91, no. 3 (2003): 361-371. Accessed February 26, 2023. https://doi.org/10.1093/aob/mcg029.

But, this equation too does not achieve the smooth sigmoid curve I wanted to see. With this equation you get a plot like the following:

The equation I suggested is almost close to an equation mentioned in the paper and is not entirely incorrect. The equation uses the logistics function and gets the smooth sigmoidal curve I was hoping to see on the plot.

Alternative, but simpler, functions that can produce two smooth transitions in a single formula are the classical growth functions. The first is the well‐known logistic function (Verhulst, 1838): $$w = \frac {w_{max}}{1+e^{-k(t-t_m)}} \tag{3}$$ where $k$ is a constant that determines the curvature of the growth pattern, and $t_m$ is the inflection point at which the growth rate reaches its maximum value. At time $t_m$, the RGR is $\dfrac{k}{2}$. As can be seen from eqn (3), the weight at $t_m$ is half of its maximum value, $w_{max}$.

Now, when I plot equation $(3)$, I get what I was hoping for! The smooth sigmoidal growth curve:

NOTE: All the graphs in this Q&A take the growth to be on the y-axis and the time duration to be on the x-axis. The values on the graphs are completely arbitrary.

27-02-2023 EDIT: Added equation which depicts sigmoidal growth.