Growth Rate: The Speed of Getting Bigger
Imagine you are watching a bamboo shoot in a time-lapse video. One day it is 10 cm tall; the next day it is 15 cm. That increase of 5 cm in one day is the growth — the raw change in size. But the growth rate is the answer to a more interesting question: how fast is that change happening?
Growth rate is simply the change in size per unit of time. If the bamboo grows 5 cm in one day, its growth rate is 5 cm per day. That is the arithmetic view: a fixed amount added each time interval.
But living things do not grow like a bank account earning a fixed amount of interest. A seedling adds a tiny amount of new tissue each day; a mature tree adds a huge amount. The same plant, at different ages, grows at very different speeds. This is where the geometric view comes in.
Arithmetic vs. Geometric Growth
Arithmetic growth means a constant amount is added per unit time. If a plant adds 2 cm every day, its height after t days follows a simple straight-line rule: height at time t equals the starting height plus (2 × t). The graph is a straight line. This is rare in biology — it happens in some elongating roots or stems for short periods, but not for long.
Geometric growth means the increase is proportional to the current size. A population of bacteria that doubles every hour is growing geometrically. If you start with 100 bacteria, after 1 hour you have 200, after 2 hours 400, after 3 hours 800. The amount added each hour keeps getting larger (100, then 200, then 400), but the rate per individual is constant (each bacterium divides once per hour).
For a plant, geometric growth means the new tissue produced depends on how much tissue is already there to photosynthesise and divide. In words: final size equals the starting size multiplied by an exponential factor that grows with time and with the intrinsic growth rate — written compactly as W(t) = W₀ × e^(rt), where W₀ is the initial size, r is the intrinsic growth rate, and t is time. The graph curves upward steeply.
In biology, growth rate usually means the relative growth rate — the increase per unit of existing size per unit time. This is the intrinsic rate r in the exponential relationship, not the raw increase in cm.
The Sigmoid Curve: Why Growth Cannot Stay Exponential Forever
If growth were always geometric, a plant would become infinitely large. That does not happen. Resources (light, water, nutrients) are limited, and the plant's own structure imposes constraints.
A typical growth curve for a plant or population looks like an S-shape — the sigmoid curve. It has three phases:
- Lag phase: The organism is small and establishing itself. Growth is slow.
- Log (exponential) phase: Resources are abundant, and growth is geometric — the steepest part of the curve.
- Stationary phase: The organism approaches its maximum size (carrying capacity). Growth slows and eventually stops.
The growth rate is not constant. It starts low, rises to a maximum during the log phase, then falls to zero. …