Arithmetic and geometric growth

Growth rate is the increase in growth per unit time, so it can be expressed mathematically. The increase may be arithmetic or geometric.

Arithmetic growth: after mitosis only one daughter cell keeps dividing, while the other differentiates and matures. A root elongating at a constant rate is the simplest example.

Plotting length against time for arithmetic growth gives a straight line: Lt = L0 + rt, where Lt is length at time t, L0 is length at time zero and r is the growth rate, the elongation per unit time.

Geometric growth: both daughter cells keep the ability to divide and go on dividing. Growth starts slowly (lag phase), then rises rapidly at an exponential rate (log or exponential phase).

With limited nutrients, growth then slows and levels off (stationary phase). Growth plotted against time gives a sigmoid or S-shaped curve, typical of living organisms growing in a natural environment and of all cells, tissues and organs of a plant.

Exponential growth is written W1 = W0 e^rt, where W1 is final size (weight, height, number and so on), W0 is initial size, r is the growth rate, t is time and e is the base of natural logarithms.

Here r is the relative growth rate, a measure of the plant's ability to produce new plant material, called the efficiency index. The final size W1 therefore depends on the starting size W0.