10. Using Matrix Models to Focus Research and Management Efforts
149
sensitive stages; (3) which of an array of management alternatives is most (or
least) likely to produce the desired result when the relative effects of each alternative can be estimated; and (4) where to focus limited research efforts to identify
critical mortality sources and to refine parameters for future analyses (Schemske
et al. 1994). In models that incorporate density dependence or stochasticity, the
sensitivity of other response variables to changes in vital rates can also be examined. Matrix models can help managers make prudent management decisions
currently, while simultaneously focusing limited research dollars on estimating
those vital rates that most need refining to increase the accuracy of predictions.
Sensitivity analysis of simple deterministic models can give insight on which
parameters are likely to be most critical in more complex models that incorporate
stochasticity. Additionally, these analyses have been used to compare population
responses with perturbation across a range of life history strategies (Silvertown et
al. 1993; Saether et al. 1996).
In this chapter, we emphasize the development and analysis of deterministic
linear matrix models and briefly outline three case studies in which matrix models
have provided useful insights for evaluating management and research alternatives in the field.
Matrix Modeling Approach
Matrix population models are surveyed exhaustively in the work of Caswell
(1989). Since then, more specialized treatments have appeared, focusing on stochastic environments (Tuljapurkar 1990), optimal harvesting (Getz and Haight
1989), and density dependence (Ginzburg et al. 1990; Marschall and Crowder
1996; Grant 1998). Presentations of the basic theory and equations can be found in
the work of Caswell (1986, 1989), van Groenendael and co-workers (1988),
MacDonald and Caswell (1993), Noon and Sauer (1992), and Ferriere and associates (1996) and in the case studies reviewed in this chapter.
Deterministic linear models are relatively simple to produce and easy to interpret and provide analytical rather than simulation results. With a matrix model, we
can calculate the proportion of individuals in an age or stage class each year,
dependent on the survival and growth rates of individuals within each class and
the fecundity of individuals that contribute to each class. A transition matrix (A)
contains one row and column for each age or stage in the model, with each entry
representing the probability of survival and transition to another stage or fecundity
(Fig. 10.1). The asymptotic growth rate of a population (λ) is given by the
dominant eigenvalue of the transition matrix. Because the models do not include
variability in the matrix entries, populations converge to a constant proportion of
individuals in each age or stage class (w, the right eigenvector of the matrix), so
that
A(w) = λ(w)
(10.1)
The value of future reproduction by individuals in each age or stage class (v) is
149
sensitive stages; (3) which of an array of management alternatives is most (or
least) likely to produce the desired result when the relative effects of each alternative can be estimated; and (4) where to focus limited research efforts to identify
critical mortality sources and to refine parameters for future analyses (Schemske
et al. 1994). In models that incorporate density dependence or stochasticity, the
sensitivity of other response variables to changes in vital rates can also be examined. Matrix models can help managers make prudent management decisions
currently, while simultaneously focusing limited research dollars on estimating
those vital rates that most need refining to increase the accuracy of predictions.
Sensitivity analysis of simple deterministic models can give insight on which
parameters are likely to be most critical in more complex models that incorporate
stochasticity. Additionally, these analyses have been used to compare population
responses with perturbation across a range of life history strategies (Silvertown et
al. 1993; Saether et al. 1996).
In this chapter, we emphasize the development and analysis of deterministic
linear matrix models and briefly outline three case studies in which matrix models
have provided useful insights for evaluating management and research alternatives in the field.
Matrix Modeling Approach
Matrix population models are surveyed exhaustively in the work of Caswell
(1989). Since then, more specialized treatments have appeared, focusing on stochastic environments (Tuljapurkar 1990), optimal harvesting (Getz and Haight
1989), and density dependence (Ginzburg et al. 1990; Marschall and Crowder
1996; Grant 1998). Presentations of the basic theory and equations can be found in
the work of Caswell (1986, 1989), van Groenendael and co-workers (1988),
MacDonald and Caswell (1993), Noon and Sauer (1992), and Ferriere and associates (1996) and in the case studies reviewed in this chapter.
Deterministic linear models are relatively simple to produce and easy to interpret and provide analytical rather than simulation results. With a matrix model, we
can calculate the proportion of individuals in an age or stage class each year,
dependent on the survival and growth rates of individuals within each class and
the fecundity of individuals that contribute to each class. A transition matrix (A)
contains one row and column for each age or stage in the model, with each entry
representing the probability of survival and transition to another stage or fecundity
(Fig. 10.1). The asymptotic growth rate of a population (λ) is given by the
dominant eigenvalue of the transition matrix. Because the models do not include
variability in the matrix entries, populations converge to a constant proportion of
individuals in each age or stage class (w, the right eigenvector of the matrix), so
that
A(w) = λ(w)
(10.1)
The value of future reproduction by individuals in each age or stage class (v) is