152
Selina S. Heppell, Deborah T. Crouse, and Larry B. Crowder
given by the left eigenvector, which can be obtained by transposing A (reversing
its rows and columns). A number of mathematical software packages can calculate eigenvectors and eigenvalues, including MathCAD
, MATLAB
, Gauss,
and RAMAS
.
The elements of a deterministic matrix model are often based on means, calculated for several populations over one or more time periods. It is assumed that all
individuals in an age class or stage are identical and that vital rates do not change
over time (constant environment). Populations that are declining will always go
extinct, whereas populations that are increasing will grow exponentially. Clearly,
all populations violate these assumptions to some extent. Deterministic models
cannot be used to estimate population size through time and are inappropriate for
small or isolated populations that are subject to strong demographic stochasticity.
However, these models can help us identify critical life stages and vital rates
through a sensitivity analysis of the matrix. Also, the general model form of a
transition matrix is flexible. Any number of life history stage classes can be
modeled, and our case studies illustrate a number of useful modifications to the
basic Leslie (1945) or Lefkovitch (1965) matrix.
Constructing a Deterministic Matrix Model
Compared with individual-based or stochastic models, relatively few data are
necessary to construct a deterministic matrix model. This is an advantage, because
data are limited for many species of concern to conservation biologists. In most
stage-based models, mean annual survival, growth (or shrinkage), and fecundity
probabilities are needed for each stage to be modeled. These rates are generally
calculated for an annual time step, although any time step is acceptable so long as
all the vital rates are calculated for the same time step (i.e., you cannot have eggs
on a daily time step and adults on an annual time step in the same matrix).
Generally, only females are modeled; however, some two-sex and male-only
matrix models exist (Meagher 1982; Caswell 1989; Heppell et al. 1994).
One way to construct a matrix model is to start with a conceptual model, the life
graph (Fig. 10.1) (Caswell 1982). Arrows between stages represent the probability
of surviving and growing into the next stage (G) or creation of new individuals
(F), whereas arrows that circle back into a stage represent the probability of
surviving but remaining in a stage (P) (Fig. 10.1b, c). Any remaining individuals
[1 − (P + G)] are those that die or migrate out of the population; these are
eliminated from the model population each time step. The simplest matrix is a
Leslie model (Fig. 10.1a), in which surviving individuals grow into the next age
class each time step, and all individuals die after a certain number of time steps. In
a stage-based model, surviving individuals may remain in a stage for one or more
time steps, as in our model for Loggerhead Sea Turtles (Crowder et al. 1994; Fig.
10.1b). Complex life histories may have transitions between several stages (Fig.
10.1c), as in our model for Red-cockaded Woodpeckers (RCW), which breed
cooperatively and have many nonbreeding adult stages (Heppell et al. 1994).
Selina S. Heppell, Deborah T. Crouse, and Larry B. Crowder
given by the left eigenvector, which can be obtained by transposing A (reversing
its rows and columns). A number of mathematical software packages can calculate eigenvectors and eigenvalues, including MathCAD
, MATLAB
, Gauss,
and RAMAS
.
The elements of a deterministic matrix model are often based on means, calculated for several populations over one or more time periods. It is assumed that all
individuals in an age class or stage are identical and that vital rates do not change
over time (constant environment). Populations that are declining will always go
extinct, whereas populations that are increasing will grow exponentially. Clearly,
all populations violate these assumptions to some extent. Deterministic models
cannot be used to estimate population size through time and are inappropriate for
small or isolated populations that are subject to strong demographic stochasticity.
However, these models can help us identify critical life stages and vital rates
through a sensitivity analysis of the matrix. Also, the general model form of a
transition matrix is flexible. Any number of life history stage classes can be
modeled, and our case studies illustrate a number of useful modifications to the
basic Leslie (1945) or Lefkovitch (1965) matrix.
Constructing a Deterministic Matrix Model
Compared with individual-based or stochastic models, relatively few data are
necessary to construct a deterministic matrix model. This is an advantage, because
data are limited for many species of concern to conservation biologists. In most
stage-based models, mean annual survival, growth (or shrinkage), and fecundity
probabilities are needed for each stage to be modeled. These rates are generally
calculated for an annual time step, although any time step is acceptable so long as
all the vital rates are calculated for the same time step (i.e., you cannot have eggs
on a daily time step and adults on an annual time step in the same matrix).
Generally, only females are modeled; however, some two-sex and male-only
matrix models exist (Meagher 1982; Caswell 1989; Heppell et al. 1994).
One way to construct a matrix model is to start with a conceptual model, the life
graph (Fig. 10.1) (Caswell 1982). Arrows between stages represent the probability
of surviving and growing into the next stage (G) or creation of new individuals
(F), whereas arrows that circle back into a stage represent the probability of
surviving but remaining in a stage (P) (Fig. 10.1b, c). Any remaining individuals
[1 − (P + G)] are those that die or migrate out of the population; these are
eliminated from the model population each time step. The simplest matrix is a
Leslie model (Fig. 10.1a), in which surviving individuals grow into the next age
class each time step, and all individuals die after a certain number of time steps. In
a stage-based model, surviving individuals may remain in a stage for one or more
time steps, as in our model for Loggerhead Sea Turtles (Crowder et al. 1994; Fig.
10.1b). Complex life histories may have transitions between several stages (Fig.
10.1c), as in our model for Red-cockaded Woodpeckers (RCW), which breed
cooperatively and have many nonbreeding adult stages (Heppell et al. 1994).