208
WILLIAM STREIFER
The above discussion of age-specific models has neglected two
important points. First, the models as described do not apply to bisexual populations if males and females experience different death rates
or have different age distributions, or if the sex ratio changes with time.
The modifications required to incorporate these possibilities are discussed
by Goodman (1953, 1969) and Fredrickson (1971) for age-specific
models. Second, the age-specific models implicitly assume that the
environment is homogeneous. Heterogeneous environments are also
considered in the next chapter.
The most severe limitation of age-specific models is that they ignore
the fact that individuals of many species have death and birth rates
which depend not only on their age, but also on their size and other
attributes.
I V . SINGLE SPECIES AGE-SIZE SPECIFIC MODELS
A. I N T R O D U C T I O N
Perhaps the first ecologist to note that age structure alone was
inadequate to explain the population dynamics of some species was
Slobodkin (1953, 1954). He formulated an algebra (Slobodkin, 1953)
which extended the work of Lewis (1942) and Leslie (1948). In
Slobodkin’s model, cohorts are distinguished by age and the size of
individuals. Each cohort ages in each time interval and either remains at
constant size, increases by one size unit, or dies. This model was not very
realistic in its birth function or in the manner in which environmental
effects were included; it was, however, a major advance in ecological
modeling.
.Just as the Lewis and Leslie models are discrete approximations of
the Von Foerster model, so too is Slobodkin’s algebra a discrete
approximation of a partial differential equation. Such equations were
formulated independently by Oldfield (1966) and Bell and Anderson
(1967) for cell populations and by Sinko and Streifer (1967) for populations of organisms. I follow here the notation of Sinko and Streifer
(1967).
B. M O D E L
For the present we assume that age and size are the only characteristics which govern the expressed physiology, ecology and behavior
of indivuals. We therefore define an age-size density function T ( a , m, t )
where a is age, m is mass, and t is time, and choose units of a and m
appropriate for the species being modeled. It should be noted that
WILLIAM STREIFER
The above discussion of age-specific models has neglected two
important points. First, the models as described do not apply to bisexual populations if males and females experience different death rates
or have different age distributions, or if the sex ratio changes with time.
The modifications required to incorporate these possibilities are discussed
by Goodman (1953, 1969) and Fredrickson (1971) for age-specific
models. Second, the age-specific models implicitly assume that the
environment is homogeneous. Heterogeneous environments are also
considered in the next chapter.
The most severe limitation of age-specific models is that they ignore
the fact that individuals of many species have death and birth rates
which depend not only on their age, but also on their size and other
attributes.
I V . SINGLE SPECIES AGE-SIZE SPECIFIC MODELS
A. I N T R O D U C T I O N
Perhaps the first ecologist to note that age structure alone was
inadequate to explain the population dynamics of some species was
Slobodkin (1953, 1954). He formulated an algebra (Slobodkin, 1953)
which extended the work of Lewis (1942) and Leslie (1948). In
Slobodkin’s model, cohorts are distinguished by age and the size of
individuals. Each cohort ages in each time interval and either remains at
constant size, increases by one size unit, or dies. This model was not very
realistic in its birth function or in the manner in which environmental
effects were included; it was, however, a major advance in ecological
modeling.
.Just as the Lewis and Leslie models are discrete approximations of
the Von Foerster model, so too is Slobodkin’s algebra a discrete
approximation of a partial differential equation. Such equations were
formulated independently by Oldfield (1966) and Bell and Anderson
(1967) for cell populations and by Sinko and Streifer (1967) for populations of organisms. I follow here the notation of Sinko and Streifer
(1967).
B. M O D E L
For the present we assume that age and size are the only characteristics which govern the expressed physiology, ecology and behavior
of indivuals. We therefore define an age-size density function T ( a , m, t )
where a is age, m is mass, and t is time, and choose units of a and m
appropriate for the species being modeled. It should be noted that