In this module of chapter 5
on population basic models, module 5.5
I cover population projection
including the Leslie metrix, example iterations of this matrix,
and population projection
of drosophila fruit flies.
The Leslie Matrix
Leslie in 1945
reframed the continuous time, Lotka model,
as a discrete time model
using matrix algebra.
This model, known as Leslie matrix,
provides a numerical tool
for determining growth rate
in age-structured populations
that can be used to study
the transient properties of populations
as they converge to the stable state.
Before I introduce this matrix model,
I will take one minute
to present a quick refresher
on matrices and matrix multiplication
Matrices are used in population analysis
and projection to efficiently
express and manipulate a set of equations
that look like the following:
a series of formulae with variables and coefficients.
The coefficients can be arranged in matrix form
as rows and columns like the following
and the variables arranged in vector form like this.
The multiplication then involves
the elements of each of the rows
times the elements of the vectors.
For example A11 times X + A12 times Y + A13×Z
equals A in the right most vector
Likewise A21 times X + A22 Times Y +A23 times Z
equal B. And so forth.
Matrices in population projection
will look like this where the coefficients
are denoted F and P,
zeros are considered structural elements,
and the elements in the two vectors
denote population number by age
at two different times.
This will become clear in the next slide.
The Leslie matrix is of this form
where the top row of the matrix
contains the birth elements, Fx,
and sub-diagonals, Px, are the survival elements
for the different age classes.
The number of individuals age x
at time t+1, denoted Nx, t+1,
is the product of the matrix and the age vector
containing the numbers at age x and time t, Nx, t.
The birth elements are computed as follows,
where mx and mx+1
denote the number of offspring produced
by females age x and x+1, respectively.
Here's an Example Iteration
A population is projected through time
by first entering an initial number of individuals
into one or more age classes
and multiplying the Leslie matrix
by the age vector through a series
of one step iteration.
For example, consider a population
with three age classes starting with this matrix
and this vector.
Multiplying yields this new vector
based on the matrix multiplication
and seem to the right for each of the elements
The numerical results of the first iteration
are substituted in the left vector
which, when multiplied with the matrix elements again
yields new population numbers in each age class.
Then this outcome, in turn,
is substituted for the left vector
and again multiplied with the matrix elements
and yields the population numbers
for the third iteration.
This table shows the results
of this hypothetical population projected
over 20 time periods.
The first column is time from 0 to 20.
The next three columns contain the number
in each of three age classes,
this next column is total number
in the population over all age classes in each time.
And the next column is growth rate
which is the ratio of
the number at time T +1
divided by the number of time T.
And the last next three columns
contain the percentage in each age class
at each time.
This rectangle contains the
results of the first three iterations
shown in the last slide.
Note that initially there were equal numbers
of individuals in each age class.
But after a few time periods
the largest percentage of individuals
shifts to age classes 0 and 1.
That is the younger ones.
There is a numerical pattern of convergence
to a stable (fixed) rate of increase
and fixed age structure.
Note that this relatively simple projection model,
there are several fundamental properties of populations
included can be identified.
The first is oscillations,
the model projections show oscillations
where both the growth rate
and age structure proportions wax and wane
due to the lag in reproductive output.
These oscillations are caused by
the time required for newborn individuals
to begin reproducing
such that when the proportions
of the population in the reproductive age classes
are maximal, births will surge
and growth rate will be at its highest,
and when individuals are primarily
in the non-reproducing age classes
the growth rate will be at its lowest.
The second is convergence.
Convergence of both age structure
and growth rates occurs
when births are spread out
over at least two age classes.
This means that the peaks and troughs
in birth sequences are thrown together
and thus smoothed out in the replacement process.
The third fundamental property is ergodicity
That is the property that the present state
of a population is independent
of its make-up in the remote past,
and is determined only by the birth and death rates
recently experienced.
The fourth property is stability
which is, the state when the age structure
is unchanging because the fixed fraction
of reproducing adults
will generate a fixed number of offspring
relative to the whole population,
In this example, as the population approaches
its stable growth rate, lambda = 2.1,
its stable age distribution (the SAD)
is approximately 67%, 27% and 6%
in age classes 0, 1 and 2, respectively.
Next I present the results of a population projection
for the fruit fly drosophila melanogaster.
This projection is purely theoretical
because in the wild many constraints
limit growth rate
including physical (temperature, humidity),
nutrition (food shortages, quality)
and natural enemies (predators, parasites, pathogens).
However, the numerical projection
captures characteristics of growing populations
that can be generalized.
Moreover, at certain times even natural populations
of D. melanogaster may experience
optimal conditions of resources
for short periods and thus experience
near exponential growth rates.
In the laboratory, colonies of flies
that are maintained under optimal rearing conditions
(that is: diet, mating, climate control)
may also experience rates of population increase
that approximate the rates described below.
I use matrices with the elements shown here
for a 90 x 90 Leslie matrix
in which we have age in these two columns
from 0 to 90
we have stage corresponding to age
the sub diagonal elements shown here in the top row
of f(x) based on the fertility mx.
This table contains the result of the Leslie matrix projection
with age in this column
number of individuals at age x at times
0, 10, 20, 30 and 40 days
and then the stable fraction in each age class
at time T.
This figure gives the stage structure
of the population from 0 to 100 days
on the X axis
and percentage in each of these different stages
on the y-axis.
The inset shows the convergence
of the growth rate to lambda equal 1.3
Population at time t=10 days.
The Drosophila population increased
from an initial total number of N(0)=4
to a total at 10 days of N(10)=332,
which is an increase of 83-fold/10 days
or an average of 1.56-fold per day.
Of this total number of 332 individuals
there were 52 eggs, 165 larvae,
111 pupae and 4 adults.
In other words, slightly over 1%
of the population was in the adult stage
and 99% were in the pre-adult stages.
Since the pre-adult stages are
found either in fruit (that is the eggs or larvae)
or in the soil (that is pupae),
only a tiny fraction of the population (that is the adults)
would be visible flying around.
Population at time t= 20 days.
In only 20 days the population expanded
by over 3,250-fold
from 4 individuals to over 13,000.
The 10-day growth rate at this time
was an increase of 40-fold,
or a 1.45-fold daily increase.
Across life stages there were 5,304 eggs,
7,526 larvae, 180 pupae and 213 adults
in the population, which translates to
40% eggs, 57% larvae
and less than 2% for both the pupal and adult stages.
Population at time t=40 days.
In around 6 weeks, the hypothetical fly population
exploded to over 3.5 million individuals
or around 40-fold over the last 10 days
and nearly a million-fold
over the initial population size.
This is equivalent to 20 population doublings
in a 40-day period.
At 40 days the population is approaching
its stable state, with a daily finite growth rate
of 1.3-fold and a stage age (stage) distribution
of 31.8, 61.9, 4.8 and 1.5%
of the total population in the egg, larval,
pupal and adult stages, respectively.
These values are all quite close to the stable state
described next.
Population in the long term, that is the stable state.
The characteristics of this population
when it has converged to the stable state
include a daily finite growth rate of 1.3-fold
and a stable stage distribution of
32, 62, 5 and 1.5%
for eggs, larvae, pupae and adults, respectively.
Two aspects of these projection
merit comment.
First, the fraction of adults in the population
is extremely small.
In as much as this stage is typically the most easily sampled
in the wild (using for eaxample, banana baits),
observations of changes in the numbers of adults
are not unlike the iceberg metaphor.
In other words, 1.5% of population as adults
means that most of the population is out of sight.
Second, with a doubling time of only 2 to 3 days,
a population doubling from 10 to 20
or from 50 to 100 may go unnoticed.
However, several doublings from a few thousand flies
can become millions of flies very quickly.
Thus, what appears to be a
sudden population explosion of flies
may have nothing to do with changes
in population growth rates
but is rather an issue of detectability.
This ends Module 5.5 on population projection.