Site Index |
 |
|
| Introduction
Age-Structured Population Growth |
Models with Age Structure Demographic parameters for many populations vary with the age of individuals in the population. It is useful under these conditions to model the population as an aggregation of age classes, with distinct survival and/or reproduction rates for each class. For discrete time models of populations with age structure, one must include transition equations as above, for each age cohort in the population. Thus, surviving individuals in any age cohort except the last automatically transfer into the next age cohort. The oldest cohort represents all individuals in the population of age k or older, and surviving members of the cohort remain there. Recruitment for such an age-structured model is given as an aggregate of age-specific reproductive efforts, based on cohort sizes. A conventional model for this situation includes age-specific survival and reproduction rates, which are assumed for now to be constant over time. The transition equation for each age cohort except the first and last is
with
Reproduction in each time period is based on the reproduction rates of individuals that survived from the previous time period:
Example.
A simple extension of the discrete model (eq.
3) allows for two age classes: a "birth-year" or juvenile
class that survives at rate
and
respectively.
The finite rate of increase for each age class is
and
and the population rate of increase is
given in terms of these cohort rates:
can vary over time, the rates
and
can as well, and thus the
population rate λ changes as
the population grows. A stable rate
of growth for the population requires a stable age distribution, that is,
a constant proportion of animals in each age class.
If the population is not at stable age distribution, growth rates will
change every year until a stable age distribution is achieved, even with
constant survival and reproduction rates. Of
course, age stability is reached very quickly for a simple two-cohort
population.
Applet Exercises
Age Structure - This applet allows you to manipulate the starting population, age-class survival rates, and age-class fecundity rates over 10 generations for up to 6 age classes. The default gives you the same population size for each age class as well as the same fecundity rate and survival rates. Move the sliders for each age class to manipulate each of these factors. You will see the relative proportions of each age class will change over time, but will eventually reach a stable age distribution. r over time - Go to the second graph and you will see how r varies over time, but eventually stabilizes at a particular rate. This then obviously determines the rate growth of the population. Euler's equation - Euler developed an equation in 1760 to estimate r. This was later refined by Lotka and so is often call Lotka's equation. Basically, this uses a fixed schedule of births and deaths and an estimate of r to refine our estimate. This is because Euler showed that the sum of products of the birth rates, death rates, and exponential of rm of each age class should be 1.0. You can then modify rm until the theoretical value becomes 1.0.
|
