Biology Reference
In-Depth Information
area can also help conservation biologists determine the size and number of reserves
needed to protect a species from extinction [
9
].
In this chapter, we review the basics of stage-structured population growth models
and explore the underlying matrix algebra that makes them possible. We provide code
for MATLAB and R that assumes only a basic understanding of the software interface
and work through an example from a population of wild ginseng threatened by over-
harvesting. We close with a summary of additional analytical techniques involving
projection matrices.
7.2
LIFE CYCLES AND POPULATION GROWTH
When studying wild populations, and especially species of conservation interest,
biologists frequentlywant to predict whether the number of individuals in a population
is growing, stable, or shrinking. The simplest models of population growth involve
data on the number of births and deaths in a population within a given time frame.
Frequently, biologists also collect additional data on the
life cycle
of an organism,
such as how often it reproduces, how many offspring it makes, and how long it lives.
Because each of these factors is often dependent on the age or size of an organism,
biologists also include data that permits the separation of the life cycle into ages or
stages. For example, the chance that an individual plant will survive and reproduce
varies considerably with the age and size of the plant; the chance that a seed will
survive and germinate, and then that the seedling will establish and grow, is typically
very small. Many plants make thousands of seeds with only a few surviving to become
new plants. But once a plant becomes established, often its chance of surviving to the
next year is relatively high. Expanding a population growth model to include stages in
the life cycle allows us to estimate the influence of each stage on population growth,
and provides additional information for conservation management decisions.
Therefore, we need a mathematical model that describes a population in the
following ways: (1) divides the life cycle of an individual into multiple stages,
(2) keeps track of
transitions
, or how individuals move through those stages,
(3) estimates the probability that an individual will die during a stage, and (4) keeps
track of reproduction, both in terms of how many new individuals are made for each
individual in a given stage and in terms of the stage in which new individuals appear.
First, we will construct a mathematical framework that does these four things. No
model reflects the natural world with 100% accuracy, but describing population struc-
ture with a mathematical model will allow us to make observations and predictions
that can guide our understanding of the dynamics in wild populations. Because the
model is essentially a mathematical description of the population, we can use it to
explore how the population might grow under certain conditions or how a single stage
is affecting overall population growth. We will explain the mathematical theory that
allows us to conduct these analyses, and then we will work through examples that use
the models to predict the future of the population and identify the life stages that are
critical for population growth.
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