REALISTIC MODELS IN POPULATION ECOLOGY
207
The birth rate ~(0, t ) and the initial population age distribution r)(a, 0)
are required to complete the population description, and indeed must be
known before Eqn (13) can be solved. The initial age distribution,
~ ( a ,
0 ) , is simply specified and various models can be formulated for the
birth rate (see section IV). Here I state only the simplest model
(see Trucco, 1965b) where B(a, t ) is the rate at which neonates are
produced by adult individuals of age a at t .
The partial differential equation model and Bailey’s model (which are
equivalent; see Sinko and Streifer, 1967) are, in effect, a continuous
version of the Lewis and Leslie matrix models. Their formulation
implicitly employs a density function, but in discrete form as shown in
Fig. 2. The illustration is for a population divided into five cohorts; in
0
FIQ. 2. A sample age distribution function separated into five cohorts aa in the
Lewis and Leslie models.
the limit as the number of cohorts tends to infinity and the age spread
in each cohort simultaneously tends to zero, the matrix equation
approaches the partial differential equation. In numerically solving the
partial differential equation, we approximate it by a matrix equation
identical to that of Lewis and Leslie.
Clearly, age-specific models represent a conceptual advance in mathematical ecology; they are far more realistic than total population
models. Furthermore, when the only important characteristic of individuals in a population is their age and when environmental effects,
food supply, crowding, time delays etc. are incorporated in the death
and birth functions, the models can be quite useful as is illustrated in
section VIII on applications. Age-specific models are more complex and
more general than total population models. The greater generality is
demonstrated by showing that the latter follow from the former by
integrating Eqn (13) over age (Sinko and Streifer, 1967).
207
The birth rate ~(0, t ) and the initial population age distribution r)(a, 0)
are required to complete the population description, and indeed must be
known before Eqn (13) can be solved. The initial age distribution,
~ ( a ,
0 ) , is simply specified and various models can be formulated for the
birth rate (see section IV). Here I state only the simplest model
(see Trucco, 1965b) where B(a, t ) is the rate at which neonates are
produced by adult individuals of age a at t .
The partial differential equation model and Bailey’s model (which are
equivalent; see Sinko and Streifer, 1967) are, in effect, a continuous
version of the Lewis and Leslie matrix models. Their formulation
implicitly employs a density function, but in discrete form as shown in
Fig. 2. The illustration is for a population divided into five cohorts; in
0
FIQ. 2. A sample age distribution function separated into five cohorts aa in the
Lewis and Leslie models.
the limit as the number of cohorts tends to infinity and the age spread
in each cohort simultaneously tends to zero, the matrix equation
approaches the partial differential equation. In numerically solving the
partial differential equation, we approximate it by a matrix equation
identical to that of Lewis and Leslie.
Clearly, age-specific models represent a conceptual advance in mathematical ecology; they are far more realistic than total population
models. Furthermore, when the only important characteristic of individuals in a population is their age and when environmental effects,
food supply, crowding, time delays etc. are incorporated in the death
and birth functions, the models can be quite useful as is illustrated in
section VIII on applications. Age-specific models are more complex and
more general than total population models. The greater generality is
demonstrated by showing that the latter follow from the former by
integrating Eqn (13) over age (Sinko and Streifer, 1967).