10. Using Matrix Models to Focus Research and Management Efforts
153
Once the form of the model has been chosen, the transition probabilities and
annual fecundities are calculated from available data. Because the parameters in a
deterministic model are fixed, it is generally unwise to combine data from several
populations that may experience different vital rates. Also, the modeler should
consider when the “census” of the model takes place (e.g., just before or after
breeding [birth-pulse models; Caswell 1989] or at an arbitrary “date” for populations with continuous reproduction [birth-flow populations; cf. Caswell 1989;
Levin and Huggett 1990]). In birth-pulse models, transition matrices calculated
for a prebreeding census do not have a row/column for newborns; instead, the
fertilities are equal to fecundity × first-year survival. In models with a postbreeding census, adults that are counted at the beginning of the year must survive most
of that year before reproducing, and any subadults that grow into adulthood that
year must also reproduce. The fecundities in a postbreeding census are multiplied
by the adult survival rate, and an additional fertility value appears in any subadult
stages that have some probability of surviving and growing into adults (e.g., Fig.
10.1b, c).
Annual growth is straightforward in a Leslie model, in which individuals age
each year and transition probability equals survival probability. But in a stagebased model, individuals may remain in a stage for one or more time steps, and the
annual probability that an individual will survive and grow must be measured or
calculated. In well-studied populations, average annual growth (γ) may be measured directly from transitions observed in marked individuals (e.g., Teasel;
Werner and Caswell 1977). The matrix parameters then become (Caswell 1989)
P i = σ i (1 − γ i )
(10.2)
G i = σ i γ i
(10.3)
F i = fec i σ 1
(10.4)
(prebreeding census), or
F i = fec i P i + fec i+1 G i
(10.5)
(postbreeding census), where fec i is stage-specific fecundity and σ is newborn
survival. In some organisms, such as sea turtles, a growth curve may provide the
approximate number of years spent in each stage (Frazer 1983a; Crouse et al.
1987). A size-based model dependent on stage length is really an age-based model
with blocks of ages grouped together. Caswell (1989) gives an equation for the
annual growth probability (γ) of individuals in stage i:
γ i =
[σ i /λ init ]
T i − [σ i /λ init ]
T i−1
[σ i /λ init ]
T i − 1
(10.6)
where σ ι is annual survival probability, T i is stage duration, and λ init is an initial
estimate of the population growth rate (Caswell 1989). The final λ of the matrix
can be calculated through an iterative procedure in which λ init is replaced by the
calculated λ until λ = λ init .
153
Once the form of the model has been chosen, the transition probabilities and
annual fecundities are calculated from available data. Because the parameters in a
deterministic model are fixed, it is generally unwise to combine data from several
populations that may experience different vital rates. Also, the modeler should
consider when the “census” of the model takes place (e.g., just before or after
breeding [birth-pulse models; Caswell 1989] or at an arbitrary “date” for populations with continuous reproduction [birth-flow populations; cf. Caswell 1989;
Levin and Huggett 1990]). In birth-pulse models, transition matrices calculated
for a prebreeding census do not have a row/column for newborns; instead, the
fertilities are equal to fecundity × first-year survival. In models with a postbreeding census, adults that are counted at the beginning of the year must survive most
of that year before reproducing, and any subadults that grow into adulthood that
year must also reproduce. The fecundities in a postbreeding census are multiplied
by the adult survival rate, and an additional fertility value appears in any subadult
stages that have some probability of surviving and growing into adults (e.g., Fig.
10.1b, c).
Annual growth is straightforward in a Leslie model, in which individuals age
each year and transition probability equals survival probability. But in a stagebased model, individuals may remain in a stage for one or more time steps, and the
annual probability that an individual will survive and grow must be measured or
calculated. In well-studied populations, average annual growth (γ) may be measured directly from transitions observed in marked individuals (e.g., Teasel;
Werner and Caswell 1977). The matrix parameters then become (Caswell 1989)
P i = σ i (1 − γ i )
(10.2)
G i = σ i γ i
(10.3)
F i = fec i σ 1
(10.4)
(prebreeding census), or
F i = fec i P i + fec i+1 G i
(10.5)
(postbreeding census), where fec i is stage-specific fecundity and σ is newborn
survival. In some organisms, such as sea turtles, a growth curve may provide the
approximate number of years spent in each stage (Frazer 1983a; Crouse et al.
1987). A size-based model dependent on stage length is really an age-based model
with blocks of ages grouped together. Caswell (1989) gives an equation for the
annual growth probability (γ) of individuals in stage i:
γ i =
[σ i /λ init ]
T i − [σ i /λ init ]
T i−1
[σ i /λ init ]
T i − 1
(10.6)
where σ ι is annual survival probability, T i is stage duration, and λ init is an initial
estimate of the population growth rate (Caswell 1989). The final λ of the matrix
can be calculated through an iterative procedure in which λ init is replaced by the
calculated λ until λ = λ init .