Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 211

7.3 DETERMINING STAGES IN THE LIFE CYCLE
In many models of population growth, life stages are defined based on morphological
changes during growth, or changes in size. In some organisms, development leads
to natural categories; seeds, seedlings, and reproductive plants, for example, or egg,
larva, pupa, and adult in butterflies. In other organisms, sometimes it makesmore sense
to categorize individuals on the basis of age. Here we will only discuss models that use
discrete categories based on life stage, but see [ 8 ] for an introduction to age-structured
models (also called Leslie matrices). In the mathematical model we are building, the
number of life stages must be finite (fixed). To simplify the mathematics, we impose
a discrete time structure, where the time t can only take the values t If

t 0 is any non-negative integer value, any changes that occur in the interval between
time t 0 and time t 0 +

= 0 , 1 , 2 ,...

1 are treated as occurring simultaneously at time t 0 +

1.

1, an individual could die, reproduce, move to
another life stage, or stay unchanged. In the most general version of the model, all
four of these events are possible for every life stage. In addition, the new individuals
that appear could start life in any life stage, and individuals could switch between any
two life stages in one time interval. Biologically, this model seems unlikely; an adult
chicken can't go back to being a chick, and a mature tree can't develop in a year. But
for the purposes of building a general model, we are going to allow for all possible
transitions and the possibility that new individuals appear in every life stage. When
we develop models specific to a particular organism, this general framework can be
restricted by setting biologically impossible transitions to zero.

In between time t 0 and time t 0 +

7.4 DETERMINING THE NUMBER OF INDIVIDUALS
IN A STAGE AT TIME t 0 + 1

If we know how many members there are in each stage at time t 0 and we know some-
thing about birth rates, death rates, and transitions to other stages, we can predict the
number of members in each stage at time t 0 +

1. This is the first step in understanding
whether the population is growing or shrinking and how each life stage contributes
to that change. Mathematically, we divide the life cycle of the organism into k stages,
which we designate s 1 , s k . These stages could be based on changes in devel-

opment or size; models often begin with a juvenile stage. The number of stages k is
dependent on the biology of the system and the nature of the data. We designate the
number of individuals in stage s j at time t 0 as n j (

s 2 ,...,

. Once we have defined the stages
and number of individuals in each, we want to find n i (

t 0 )

t 0 + 1 ) , the expected number

of individuals in stage s i at time t 0 + 1. Note that we are considering n j ( t 0 ) to be a

known number and n i ( to be a prediction. Finding this predicted number of

individuals in a stage at time t 0 +

t 0 + 1 )

1 depends on knowing the number of individuals
in all stages at time t 0 because each stage could potentially contribute to n i ( t 0 + 1 ) .

Knowing n j ( t 0 ) for all stages s j , j = 1 , 2 ,..., k , we want to find the expected value