Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 227

7.8.3 RaisingaMatrixtoaPowerinMATLABandR
After you have entered a matrix and named it A, the MATLAB command for this
calculation is:

Aˆ5
In R, you canmultiply the matrix A by itself five times using the following commands:

B <- A%*%A
C <- B%*%A
D <- C%*%A
D%*%A
Exercise 7.3. The life stages, transitions, and reproduction for chan, a South Amer-
ican weed, are illustrated in Figure 7.5 :

113.90

Seed Adult

0.0135 0.6500

Juvenile

FIGURE 7.5
Life stages, transitions between stages, and reproduction (in italics) for chan ( Hyptis
suaveolens ), an annual savanna weed. Seeds germinate in late spring to early summer
and grow until the end of the wet season in December and January, when they produce
seeds and die. Modified from [ 15 ].

a. Build a matrix from the figure.
b. Is the matrix positive? Why or why not? If it is not positive, is it nonnegative?
c. Is the matrix irreducible? Why or why not?
d. Raise the matrix to the fifth power. Can you claim it is primitive?

7.8.4 Finding the Stable Distribution
To determine the stable state of a population, we want to know the vector that describes
the relative distribution of individuals across stages as the number of time intervals
goes to infinity.While in reality no populationwill exist an infinite amount of time, this
distribution can act as a descriptor of the population and gives us the ability to make
comparisons among life stages within the distribution and with stable distributions
from other populations.
Here, we will work through the steps to explain why the eigenvector associated
with the largest eigenvalue allows us to determine the stable distribution of individuals