Biology Reference
In-Depth Information
k
matrix
A
that has an entry for each possible transition between the
k
life stages. The entry
in the
i
th row and
j
th column of
A
is given by
To generate a
projection matrix
for a population, we construct a square
k
×
(
A
)
i
,
j
=
the average number of members in stage
s
i
at time
t
0
+
1 that are
produced by a single member in stage
s
j
at time
t
0
.
Estimates for the probabilities
p
ji
and the average number of offspring
f
ji
are based
on the relative number of individuals and offspring production by those individuals
in each stage in natural populations, typically over multiple years (see [
8
]formore
details on how this is done).
To illustrate the creation of a matrix from transition probabilities, we will work
through an example from the biological literature on American ginseng (
Panax quin-
quefolius
). Ginseng is a plant native to the eastern United States and grows in forest
understories. It is a perennial, and new leaves come up from an underground stem
every year. This stem is enlarged and stores nutrients and starch. It also produces a
variety of compounds that give ginseng a distinctive odor and taste, and give it value
as an herbal medicine. The plants are often harvested from wild populations and the
underground stems sold, sometimes for a considerable sum. Ginseng is becoming
increasingly rare in many places, and its decline is likely a result of habitat loss and
overharvesting. Therefore, it has been the subject of studies that use projection matri-
ces to model the future viability of populations. The data we use for the examples
below are from Charron and Gagnon [
10
].
The life cycle of ginseng lends itself easily to classification into life stages. Seeds
spend a year and a half in the soil before germinating, and when they do germinate,
they produce a seedling that becomes established in its first year and stores resources
in its underground stem for the next year. At the end of the summer, it dies back above
ground but the stem stays alive below ground. The next spring, the underground stem
sends up a shoot with one compound leaf on it (compound leaves typically have five
leaflets). The following year, the plant may make one leaf again, or it may make
two leaves. In subsequent years, plants may make three or four leaves; rarely, plants
produce five or six leaves. Plants sometimes flower when they have two leaves but
more commonly wait until they have three or four leaves. The flowers will develop
into fruits with seeds in them, and then these seeds will disperse away from the parent
plant and become a new plant. The interval between time
t
0
and time
t
0
+
1 is a year,
and because the data we are using were collected from the wild populations in the
fall when the seeds were ripe, we can imagine the year starting at this time in the fall.
In our model, we have six life stages: seed, seedling, one-leaved plant, two-leaved
plant, three-leaved plant, and four-leaved plant.
An individual in each life stage has a certain probability of dying, staying in its
stage, or moving into another stage. For example, we expect that some seedlings will
become one-leaved plants in the next year, and some will die (the probability that
a seedling will stay a seedling is 0). All nonzero transitions among life stages are
illustrated in Figure
7.1
.
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