From Wikipedia,
the free encyclopedia.
Population models are used in
population ecology to model
the dynamics of wildlife or human
populations. Matrix population
models are a specific type of
population model that uses
matrix algebra. Matrix
algebra, in turn, is simply a form
of algebraic shorthand for
summarizing a larger number of
often repetitious and tedious
algebraic computations.
All
populations can be modeled by
one simple equation:
Nt+1 = Nt + B - D + I - E,
where:
Nt+1 = abundance at time t+1 Nt =
abundance at time t B = number of
births within the population
between Nt and Nt+1 D = number of
deaths within the population
between Nt and Nt+1 I = number of
individuals immigrating into the
population between Nt and Nt+1 E =
number of individuals emigrating
from the population between Nt and
Nt+1
This equation is called a BIDE
model. Although BIDE models are
conceptually simple, reliable
estimates of the 5 variables
contained therein (Nt, B, D, I and
E) are often difficult to obtain.
Usually a researcher attempts
to estimate current abundance, Nt,
often using some form of
mark and recapture technique.
Estimates of B might be
obtained via a ratio of immatures
to adults soon after the breeding
season, Ri.
Number of deaths can be
obtained by estimating annual
survival probability, usually via
mark and recapture methods,
then multipling present abundance
and survival rate.
Many times immigration and
emigration are ignored because
they are so difficult to estimate.
For added simplicity it may
help to think of time t as the end
of the breeding season in year t
and to imagine that one is
studying a species that has only
one discrete breeding season per
year.
The BIDE model can then be
expressed as:
Nt+1 = Nt_a * Sa + Nt_a * Ri * Si
where:
Nt_a = number of adult females at
time t Nt_i = number of immature
females at time t Sa = annual
survival of adult females from
time t to time t+1 Si = annual
survival of immature females from
time t to time t+1 Ri = ratio of
surviving young females at the end
of the breeding season per
breeding female
In matrix notation this model can
be expressed as:
| Nt+1_i | | SiRi SaRi | | Nt_i |
| | = | | | |
| Nt+1_a | | Si Sa | | Nt_a |
Suppose that you are studying a
species with a maximum lifespan of
4 years. The following is an
age-based Leslie matrix for this
species. Each row in the first and
third matrices corresponds to
animals within a given age range
(0-1 years, 1-2 years and 2-3
years). In a Leslie matrix the top
row of the middle matrix consists
of age-specific fertilities: F1,
F2 and F3. Note, that F1 = SiRi in
the matrix above. Since this
species does not live to be 4
years old the matrix does not
contain an S3 term.
| Nt+1_1 | | F1 F2 F3 | | Nt_1 |
| | = | | | |
| Nt+1_2 | | S1 0 0 | | Nt_2 |
| | = | | | |
| Nt+1_3 | | 0 S2 0 | | Nt_3 |
These models can give rise to
interesting cyclical or seemingly
chaotic patterns in abundance over
time when fertility rates are
high.
The terms Fi and Si can be
constants or they can be functions
of environment, such as habitat or
population size. Randomness can
also be incorporated into the
environmental component.
Reference
- Caswell, H. 2001. Matrix
population models: Contruction,
analysis and interpretation, 2nd
Edition. Sinauer Associates,
Sunderland, Massachusetts. ISBN:
0-87893-096-5.
