Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 226

Post-
reproductive
adults

Reproductive
adults

yearlings juveniles

FIGURE 7.3
Life stages, transitions between stages, and reproduction for a pod of killer whales. Adults
are divided into two stages; reproductive adults, which contribute to the “yearling” stage
by having offspring, and “post-reproductive adults,” which do not reproduce or revert to
other stages, making the matrix reducible. Modified from [ 13 ].

First
instar

Second
instar

Third
instar

Fourth
instar

egg Pupa Adult

FIGURE 7.4
Life stages, transitions between stages, and reproduction for Chironomus riparius ,amidge
species. Adults have wings, and lay their eggs in water. The eggs hatch into larvae, which
pass through four larval stages before pupating. Only adults reproduce, and adults repro-
duce only once, resulting in a transition matrix that is not primitive. Modified from [ 14 ].

this test; at least one element will always remain 0 because it is not connected back
into to the rest of the matrix through any path.
If we raise matrix ( 7.2 ) to the fifth power, we get:

⎛
⎝

⎞
⎠

0 . 0427 0 . 7475 6 . 2298 15 . 873 23 . 214 40 . 119

0 . 0004 0 . 0427 0 . 5339 1 . 8309 3 . 0004 6 . 0000

0 . 0061 0 . 0258 0 . 1197 0 . 3387 0 . 7246 2 . 3562

A 5 = .

. . . . . .

0 0085 0 0397 0 1281 0 0831 0 1603 0 8820

. . . . . .

0 0080 0 0650 0 3191 0 3597 0 3063 0 1808

0 . 0020 0 . 0261 0 . 1926 0 . 4391 0 . 6164 1 . 0220

Thus, the matrix in our ginseng example is primitive. It is also nonnegative and irre-
ducible; therefore we can continue using the Perron-Frobenius theorem to determine
the stable distribution of individuals across stages.