Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 218

0.33 0.71 0.77 0.06

0.04

0.14

One-
leaved
plant

Two-
leaved
plant

Three-
leaved
plant

Four-
leaved
plant

0.01 0.21

0.33

seed seedling

0.04 0.22

0.02

21.78

28.29

FIGURE 7.2
Life stages, transition probabilities, and reproductive values for Population 3 in [ 10 ].

made by plants in the other stages; therefore they may be greater than 1 and they are
values that represent reproduction . 1 As previously mentioned, if we sum the transition
probabilities for a particular (non-seed) stage at time t 0 +

1, they do not necessarily
total to 1; that is because the probabilities only reflect the fraction of individuals that
survive from one stage to the next.
Exercise 7.1. Construct the projection matrix for the ginseng population dia-
grammed in Figure 7.2 .

7.6 PREDICTING HOW A POPULATION CHANGES
AFTER ONE YEAR
We have constructed a projection matrix to summarize how plants growing in a wild
population are changing from year to year; the matrix can now serve as a model for the
wild population, and we can manipulate the model to learn more about the population.
First, we will use this matrix to predict the number of individuals we expect in each
stage after one year, given a starting number of individuals in each stage chosen by
us. The number of individuals in each stage at time can be expressed as a vector ;
for example, if we have a population of ginseng with 800 seeds, 90 seedlings, 56
one-leaved plants, 23 two-leaved plants, 31 three-leaved plants and 11 four-leaved

1 It is somewhat easy to confuse the projection matrix with the transition matrix of Markov chains,
because some entries in the projection matrix are transition probabilities. We caution against trying to
make connections between the two.