Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 216

Table 7.2 The second column of the projection matrix for a population of
American ginseng. The values in this column are the probability of transitioning
from the seedling stage to another stage (data from [ 10 ]).
Stage at time t 0
Seed Seedling One- Two- Three- Four-

leaved leaved leaved leaved

plant plant plant plant

Stage Seed ∗ 0 ∗ ∗ ∗ ∗

at Seedling ∗ 0 ∗ ∗ ∗ ∗

time One-leaved plant ∗ 0.21 ∗ ∗ ∗ ∗

t 0 + 1 Two-leaved plant ∗ 0 ∗ ∗ ∗ ∗

Three-leaved plant ∗ 0 ∗ ∗ ∗ ∗

Four-leaved plant ∗ 0 ∗ ∗ ∗ ∗

year of 0.55, and a 0.35 probability of becoming a two-leaved plant in the next year.
They don't flower, so they can't make seeds, and they can't revert back to being
seedlings. Therefore:
The probability that a one-leaved plant at time t 0 :
makes seeds at time t 0 + 1 is 0, therefore A 1 , 3 = 0,

becomes a seedling at time t 0 + 1 is 0, therefore A 2 , 3 = 0,

stays a one-leaved plant at time t 0 + 1 is 0.55, therefore A 3 , 3 = 0 . 55,

makes a two-leaved plant at time t 0 + 1 is 0.35, therefore A 4 , 3 = 0 . 35,

makes a three-leaved plant at time t 0 + 1 is 0, therefore A 5 , 3 = 0,

makes a four-leaved plant at time t 0 + 1 is 0, therefore A 6 , 3 = 0.

The fourth column of the matrix represents transitions from the two-leaved stage to
other stages. There are several possible fates for a two-leaved plant; they occasionally
transition back into one-leaved plants between one year and the next, they often stay
two-leaved plants in the next year, they sometimes become three-leaved plants in the
next year, and they rarely become four-leaved plants in the next year. Occasionally,
a two-leaved plant flowers, which means it is generating seeds that are present in the
following year. Therefore, based on the data from Figure 7.1 :

The average number of seeds per plant generated by two-leaved plants between
time t 0 and time t 0 +

27.
The probability that a two-leaved plant is a seedling at time t 0 +

1 is 0.27, therefore A 1 , 4 = 0 .

1 is 0, therefore

A 2 , 4 = 0.

The probability that a two-leaved plant is a one-leaved plant at time t 0 + 1is

0.05, therefore A 3 , 4 = 0 . 05.

The probability that a two-leaved plant is a two-leaved plant at time t 0 + 1is

0.45, therefore A 4 , 4 = 0 . 45.