Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 231

Table 7.5 Steady state distribution of individuals across stages for a population
of American ginseng (data from [ 10 ]).
Stage Number of Individuals

Seeds 0.8067

Seedlings 0.1022

One-leaved plants 0.0352

Two-leaved plants 0.0168

Three-leaved plants 0.0170

Four-leaved plants 0.0221

will actually happen in the future. In addition, the quality of our estimates is directly
related to the accuracy of the initial data that were used to generate the projection
matrix. Estimating projection matrices from multiple years of data and incorporating
environmental uncertainty into the model with a stochastic approach can lead to more
biologically accurate estimates; see [ 8 ] for more details.
We can calculate the dominant eigenvalue and its associated eigenvector using
MATLAB and R, as demonstrated below. When we do so for our ginseng example,
we find that the estimated growth rate of the population is 1.1841. Because this
value is larger than one, we expect that if the projection matrix did not change this
population would grow. Through the computations below, we obtain again the steady
state distribution of individuals across stages (see Table 7.5 ). As expected, they are
the same as the approximated values we obtained for large k in the previous sections.
We predict that if the population illustrated in Figure 7.1 were to reach a stable state,
the majority of individuals in the population would be seeds; this is not surprising
from a biological perspective, given the low rate at which seeds become seedlings. It
is also interesting to note that we predict more four-leaved individuals than two- and
three-leaved individuals.

7.9.1 Calculating Eigenvalues and Eigenvectors in MATLAB
We can define the projection matrix ( 7.2 ) as A in MATLAB with the following com-
mand:

A = [0 0 0 0.27 3.90 40.00; 0.15 0 0 0 0 0;
0 0.21 0.55 0.05 0 0; 0 0 0.35 0.45 0 0;
0 0 0 0.41 0.78 0; 0 0 0 0.05 0.19 1.0]

We can generate all eigenvalues of A using the command eig (A), and we want to give
the vector of eigenvalues a name so we can use it in later commands, so we define
eig (A) as E with the command

E = eig (A)

The largest eigenvalue is the dominant eigenvalue, and the growth rate of the popula-
tion. To identify the largest eigenvalue, first we ask MATLAB to identify the index of