10. Using Matrix Models to Focus Research and Management Efforts
161
had a large impact on the population growth rate. Heppell and associates (1994)
examined the effects of decreasing the annual mortality of fledglings alone, birds
that nest together on a territory (fledglings, helpers, and breeders), and all stages at
once. When mortality was decreased, the population growth rate increased substantially, but the habitat restrictions led to an increase in the proportion of
nonbreeders in the population. This analysis illustrates the need for critical thinking in model choice and parameterization and suggests that managers should think
about the composition of their populations over time as well as population
numbers.
Case Study 3: Southern Appalachian Brook
Trout Populations
Brook Trout (Salvelinus fontinalis) populations in southern Appalachian mountain streams have declined in response to multiple anthropogenic effects including
the introduction of an exotic salmonid species (Rainbow Trout, Oncorhynchus
mykiss), a decrease in pH (through acid deposition), an increase in siltation and a
decrease in shade and allochthonous nutrient input (from clear-cutting), and an
increase in fishing pressure (Kelly et al. 1980; Larson and Moore 1985).
Marschall and Crowder (1996) developed a population model based on a simple size-classified projection matrix to address these multiple anthropogenic
effects. They partitioned the Brook Trout population into 15 size classes based on
total fish length; finer size divisions were used in the small size classes because
changes in size in the model affect survival through the relation between size,
number, and density dependence. The annual cycle of the model Brook Trout
population consisted of a 6-month growth period (from spring to fall), fall spawning, overwintering without growth (from spawn until spring), and finally, emergence of larvae and beginning of growth again in the spring.
A matrix of monthly transition probabilities was calculated from available data
on survival and growth rates. A distribution of growth rates was incorporated by
calculating the proportion of juveniles that grew into each size class by using
cohort-specific growth estimates from field data. Density-dependent survival was
added to the model by multiplying the linear transition matrix by a second matrix
with per capita survival probabilities on the diagonal entries; survival in the age 0
classes depended on body size and number of age 0 fish. Survival in the remaining
size classes depended on size only (Marschall and Crowder 1995, 1996). Those
fish that survived the growing season were then multiplied by a vector of sizespecific fecundities and overwintering mortalities. The complexity of this model
required the authors to calculate population size through time numerically and
conduct a sensitivity analysis by using the approach described by Equation
(10.10) (above).
Marschall and Crowder (1996) assessed the sensitivity of equilibrium population size and size class structure to parameter perturbations in several ways. First,
they estimated elasticity to size-specific changes in growth and survival rates via
simulation. Next, they assessed population response to a full factorial design of
161
had a large impact on the population growth rate. Heppell and associates (1994)
examined the effects of decreasing the annual mortality of fledglings alone, birds
that nest together on a territory (fledglings, helpers, and breeders), and all stages at
once. When mortality was decreased, the population growth rate increased substantially, but the habitat restrictions led to an increase in the proportion of
nonbreeders in the population. This analysis illustrates the need for critical thinking in model choice and parameterization and suggests that managers should think
about the composition of their populations over time as well as population
numbers.
Case Study 3: Southern Appalachian Brook
Trout Populations
Brook Trout (Salvelinus fontinalis) populations in southern Appalachian mountain streams have declined in response to multiple anthropogenic effects including
the introduction of an exotic salmonid species (Rainbow Trout, Oncorhynchus
mykiss), a decrease in pH (through acid deposition), an increase in siltation and a
decrease in shade and allochthonous nutrient input (from clear-cutting), and an
increase in fishing pressure (Kelly et al. 1980; Larson and Moore 1985).
Marschall and Crowder (1996) developed a population model based on a simple size-classified projection matrix to address these multiple anthropogenic
effects. They partitioned the Brook Trout population into 15 size classes based on
total fish length; finer size divisions were used in the small size classes because
changes in size in the model affect survival through the relation between size,
number, and density dependence. The annual cycle of the model Brook Trout
population consisted of a 6-month growth period (from spring to fall), fall spawning, overwintering without growth (from spawn until spring), and finally, emergence of larvae and beginning of growth again in the spring.
A matrix of monthly transition probabilities was calculated from available data
on survival and growth rates. A distribution of growth rates was incorporated by
calculating the proportion of juveniles that grew into each size class by using
cohort-specific growth estimates from field data. Density-dependent survival was
added to the model by multiplying the linear transition matrix by a second matrix
with per capita survival probabilities on the diagonal entries; survival in the age 0
classes depended on body size and number of age 0 fish. Survival in the remaining
size classes depended on size only (Marschall and Crowder 1995, 1996). Those
fish that survived the growing season were then multiplied by a vector of sizespecific fecundities and overwintering mortalities. The complexity of this model
required the authors to calculate population size through time numerically and
conduct a sensitivity analysis by using the approach described by Equation
(10.10) (above).
Marschall and Crowder (1996) assessed the sensitivity of equilibrium population size and size class structure to parameter perturbations in several ways. First,
they estimated elasticity to size-specific changes in growth and survival rates via
simulation. Next, they assessed population response to a full factorial design of