In this first module of Chapter 6
on Stage Models, Module 6.1,
I will cover the basics of population models
in which stage replaces age
as the temporal state of existence,
including the situations in which stage
is important.
In many situations, an individual state
where a state may be defined as the size,
developmental stage, health, or any other
discreet state where it transitions from
state to state are possible may be better
than age as an indicator
of demographic characteristics,
such as the chance of surviving, or reproducing,
or the number of offspring produced.
For example, in trees and perennial plants,
arthropods and mollusks, and particularly
in fish, amphibians, and reptiles that exhibit
indeterminate growth, size is a more important
factor than age in determining survival
and reproductive rates.
A second group of organisms that require
stage-based models are those with multiple
modes of reproduction.
Many organisms have both sexual
and vegetative reproduction including
many species of plants, as well is invertebrates
such as rotifers and daphnia.
If same-aged offspring from each
reproductive category differ in survival
and reproduction, then age is an inadequate
demographic category.
Matrix staged-structured models are used
in these cases to describe the population size,
structure, and dynamics at any one point
in time.
The use of matrices is important because
their long-term behavior provides a connection
to stable population theory and because
matrix elements do not have to be constant
but they can also be equations.
Matrices are especially important
in the context of size and stage-structured
models because as the mathematical
component of a model, they integrate
seamlessly into the conceptual and visual
components, such as the life cycle graphic.
I will start by illustrating the concept
of the direction of transition of from-to
using a married model of five states
including single, married, divorced,
widowed, and death.
Transitions within or between each of
these states include one-time epoch,
a single individual can remain single,
can marry, or die.
A married person can remain married,
can divorce, can become widowed,
or can die.
A divorced person can remain divorced,
can remarry, or can die.
A widowed person can remained widowed,
can remarry, or can die.
And a dead person can, of course,
only remain dead.
The single state is referred to as
a transient state because all arrows lead out.
The dead state is referred to as
an absorbing state because all arrows lead in.
The transition matrix for this married model
is show here, showing the elements of
from states (j) to stage (i).
For example, A11 is the transition probability
from single to single.
A23 is a transition probability
from discover to married and A42 is
a transition probability from married to widow.
The zeroes in the matrix are referred to
as structural zeroes.
The matrix marital model is shown here
with the transition matrix multiplied by
the vector of single, married, divorced,
widowed, and dead at time (t) to compute
the number in these respective states
at time t+1.
This transition concept was used
in a Leslie Model, as seen here,
with the lifecycle graphic in transition matrix
where the top row in the matrix contains
the reproductive values,
and the subdiagonal contains the transitional
probabilities from one stage to the next.
However, here I show a much wider variety
of between state, or between stage transitions
that are possible with this stage-based
matrix model due to Lefkovitch in 1965
who generalized the Leslie Matrix Model
by using stage instead of age groupings.
The following figures parse out
the multiple transitions that may be included
in a stage-based model.
That's three stages, including newborn stage,
stage 1 and 2, reproductive stages,
stages 2 and 3 with both lifecycle diagrams
and matrices.
In the Leslie Matrix, the only non-zero entries
are survival probability from age to the next
in the subdiagonal and in the top row
where newborns are all produced
into the first age class.
However, in the Lefkovitch Model, any entry
can be non-zero.
For example, survivors can be in any stage
at the next time step,
stay the same, advance, or regress.
And newborns may occur as different sizes
or types.
The two reproductive stages contribute
to the newborn class
with stage-specific fecundities, denoted Fx.
All of the non-zero, non-fecundity
transition probabilities represent
the proportionate individuals moving from
one stage to another in one time period.
The survival probability, denoted Gx here,
denote the transition probability from stage I
to stage I+1.
The self-loop probability, denoted Px,
denote the probability that an individual
remains in stage I over one time period.
Individuals may also skip from stage I
to stage I+2 or greater which is designated
as a probability of growth, age x.
An individual may regress, denoted Rx,
from state I to state I-1 or greater.
So this ends Module 6.1, a module that
set the stage for subsequent modules
in this chapter where I describe different
variety of stage models and compare
the general concept of stage-based models
with age structured models.