154
Selina S. Heppell, Deborah T. Crouse, and Larry B. Crowder
Are stage-based models “better” than age-based Leslie models? As with all
model selection, what kind of model you choose depends on what data are available and what questions you wish to answer. Caswell (1989) discusses some
statistical evaluations (e.g., log-linear models) that have been used to test the
adequacy of age and stage as state variables. For some organisms, such as RCW,
there are obvious life history stages that are independent of age. Sea turtles, fish,
and other organisms with indeterminate growth may be more easily classified by
size than age. In these cases, the sensitivity analysis is most useful to managers
when individuals are grouped into stages that have identified mortality sources.
But the population-level effects of a change in vital rates may not be immediately
apparent in populations with long generation times. Leslie models can reveal
transient effects on population dynamics that are caused by the time lag between
birth and reproduction (Crowder et al. 1994; see Loggerhead case study below).
These “waves” in the adult population are completely deterministic but may not
be discernible in simulations based on stochastic population models. Both Leslie
and stage-based models can be useful for population dynamics analysis.
Sensitivity and Elasticity Analyses of Linear Models
A sensitivity analysis is a quantitative comparison of the relative impact of model
parameters on a population response that can be used to compare management
alternatives qualitatively. The analysis calculates the change in the outcome of a
model (e.g., the population growth rate or stage distribution) when a parameter in
the model is altered. In linear models (i.e., those without density dependence) and
models that do not include demographic stochasticity, the sensitivity of the
asymptotic population growth rate (λ) can be used as an index to make predictions
such as “increasing large juvenile [Loggerhead Sea Turtle] survival will have a
relatively greater effect on population growth than saving eggs and hatchlings”
(Crouse et al. 1987).
For simple linear models, the first step in a sensitivity analysis is to calculate the
stable stage distribution (w) and the stage-specific reproductive values (v) of the
model. These are the right and left eigenvectors associated with the dominant
eigenvalue λ, which are usually scaled such that Σ (w i ) = 1 and the reproductive
value of stage or age 1 individuals (v 1 ) = 1. Caswell (1978) used these eigenvectors to calculate the sensitivity of λ to changes in any matrix entry (A i,j ):
∂λ
∂A i,j
=
v i w j
〈v | w〉
(10.7)
where 〈v | w〉 is the inner product of the two vectors, {v 1 × w 1 + v 2 × w 2 . . . }.
Biologically speaking, the sensitivity analysis tells us how λ will change if a
model parameter is increased or decreased by an infinitismal amount. An elasticity analysis ( = proportional sensitivity; deKroon et al. 1986) calculates proportional changes in λ when matrix entries are changed by a small percentage. The
elasticity of each matrix parameter is
Selina S. Heppell, Deborah T. Crouse, and Larry B. Crowder
Are stage-based models “better” than age-based Leslie models? As with all
model selection, what kind of model you choose depends on what data are available and what questions you wish to answer. Caswell (1989) discusses some
statistical evaluations (e.g., log-linear models) that have been used to test the
adequacy of age and stage as state variables. For some organisms, such as RCW,
there are obvious life history stages that are independent of age. Sea turtles, fish,
and other organisms with indeterminate growth may be more easily classified by
size than age. In these cases, the sensitivity analysis is most useful to managers
when individuals are grouped into stages that have identified mortality sources.
But the population-level effects of a change in vital rates may not be immediately
apparent in populations with long generation times. Leslie models can reveal
transient effects on population dynamics that are caused by the time lag between
birth and reproduction (Crowder et al. 1994; see Loggerhead case study below).
These “waves” in the adult population are completely deterministic but may not
be discernible in simulations based on stochastic population models. Both Leslie
and stage-based models can be useful for population dynamics analysis.
Sensitivity and Elasticity Analyses of Linear Models
A sensitivity analysis is a quantitative comparison of the relative impact of model
parameters on a population response that can be used to compare management
alternatives qualitatively. The analysis calculates the change in the outcome of a
model (e.g., the population growth rate or stage distribution) when a parameter in
the model is altered. In linear models (i.e., those without density dependence) and
models that do not include demographic stochasticity, the sensitivity of the
asymptotic population growth rate (λ) can be used as an index to make predictions
such as “increasing large juvenile [Loggerhead Sea Turtle] survival will have a
relatively greater effect on population growth than saving eggs and hatchlings”
(Crouse et al. 1987).
For simple linear models, the first step in a sensitivity analysis is to calculate the
stable stage distribution (w) and the stage-specific reproductive values (v) of the
model. These are the right and left eigenvectors associated with the dominant
eigenvalue λ, which are usually scaled such that Σ (w i ) = 1 and the reproductive
value of stage or age 1 individuals (v 1 ) = 1. Caswell (1978) used these eigenvectors to calculate the sensitivity of λ to changes in any matrix entry (A i,j ):
∂λ
∂A i,j
=
v i w j
〈v | w〉
(10.7)
where 〈v | w〉 is the inner product of the two vectors, {v 1 × w 1 + v 2 × w 2 . . . }.
Biologically speaking, the sensitivity analysis tells us how λ will change if a
model parameter is increased or decreased by an infinitismal amount. An elasticity analysis ( = proportional sensitivity; deKroon et al. 1986) calculates proportional changes in λ when matrix entries are changed by a small percentage. The
elasticity of each matrix parameter is