Population Ecology -Age structure

General objectives and questions addressed

Some terms

Basics of life table analysis

Basics of Projection Matrix approach

Basics of Integral Equation Approach

Basics of Continuous Age structure PDE approach

Survey of recent papers - topic coverage

References:

Caswell, H. 1989. Matrix Population Models. Sinauer, Sunderland, MA.

Charlesworth, B. 1980. Evolution in Age Structured Populations.
Cambridge Univ. Press.

Keyfitz, N. 1968. Introduction to the Mathematics of Population.
Addison-Wesley, Reading, MA.

Papers to Read:

 Grant, P. R. and B. R. Grant. 1992. Demography and the Genetically
 Effective sizes of Two populations of Darwin's Finches. Ecology
 73:766-784.

Saltz, D. and D. I. Rubenstein. 1995. Population Dynamics of a
Reintroduced Asiatic Wild Ass (Equus Hemionus) Herd. Ecological
Applications 5:327-335.

Objectives of Demographic analyses of Populations:

1. Determine the effects of schedules of fertility and mortality on
population structure and dynamics

2. Provide methods to allow comparisons of life histories between
differing populations and species.

3. Provide the capability to estimate the impacts of alterations in
vital statistics on popultion dynamics, e.g. both forecasting and
prediction.

Terms:

Birth-flow population - births occur continuously over the time period
of interest

Birth-pulse population - reproduction in a population in concentrated
over a brief interval  per time period of interest.

Census analysis - vertical method - use census information on a
population obtained at least at two times to estimate vital statistics

Cohort analysis - horizontal method - follow a sample of individuals
born at the same time til death to determine vital statistics

Forecasting - an attempt to predict what will happen

Maternity  function - m(x) = mean number of offspring born per
individual of age x per unit time -  the age specific rate of
reproduction

Projection - an attempt to predict what would happen given certain
assumptions

Projection interval - the time step for a projection matrix

Projection matrix - a matrix having entries which estimate the
survivorships from one age class to another and the fertilities of the
age classes, which allows for projection of the numbers of individuals
in each age class at time n+1 to be estimated from the numbers in each
age class at time n.

Survivorship function - l(x) = Probability of survivorship from birth
to age x

Vital statistics - birth and death rates of a population

Life Table analysis

This is a misnomer - a life table provides a schedule of the deaths
within a population as a function of age by specifying the survivorship
function l(x).  Sometimes it is combined with the maternity function as
well. Estimating these functions can be done a number of ways:

(a) Direct methods:

(i) cohort analysis in which one follows a cohort through time,
estimating survivorship and fertility directly for this cohort

(ii) census analysis which requires some appropriate statistical
methods to estimate survivorship as a function of age from observations
of individuals in a population at two distinct times.

(b) Indirect methods - often necessary in non-laboratory cases, and all
require additional estimates of fertility and survivorship for ages
that can't be included in these studies

(i) assume the population is constant in size and has constant
age-structure, from which one can estimate that l(x)/(Sum over all ages
of l(x)) = the fraction of individuals in the population of age x

(ii) again assume the population is constant and has constant age
ditribution and use a sample of the ages at death of individuals in the
population as an estimate of l(x) - l(x+1) =  frequency of individual
who die while in age class x

(iii) again assuming constant age structure, mark a sample of
individuals at birth (or some other age) and then recover as mnany
marked dead individuals as posible to estimate distribution of age at
death.


Basics of projection matrix approach:

Define an appropriate age class structure so all inviduals aged from
say 0 to L are in age class 1, those aged from L to 2L are in age class
2, etc. For simplicity we assume L=1and that this is also the length of
the projection interval.

Then with n sub i (t) = number of individuals in age class i at time t
= 0,1,2,... and P sub i = the survival probability of members of age
class i then n sub i (t+1)_ = P sub (i-1)   n sub (i-1) (t)  for
i=2,3,4,... and t=0,1,2,...  and if F sub I = the fertilities = the
number of age class 1 individuals at time t+1 per age class I
individuals at time t, then

n sub 1 (t+1) = F sub 1 n sub 1(t) + F sub 2 n sub 2(t) + ...  and
these can all be expressed in matrix form as per the Leslie matrix

Determining estimates of the elements of the projection matrix is
non-trivial and has been subject to a great deal of misunderstanding.
There are quite different methods for birth-pulse versus birth-flow
populations, and even within these there are several potentially
different reasonable options. The key difficulty here arises because we
are attempting to discretize what is in essence a continuous process,
and the results make assumptions about the distribution of ages of
individuals within each discrete age class.

Topics from 1990's search on age structure and demography in 4 ecology
journals:

Projection matrices 			2
Statistical issues			3
Size structure 				1
Disturbance				1
Avian demography			6
Vegetative recruitment			1
Viability analysis			2
Lizard demgraphy			3
Turtle demography 			1
Coral demography 			1
Sea mammal demography			2
Orchid demography			1
Small mammal demography			9
Large terrestrial mammal demography	3
Fish demography				4
Pathogen effects			1
Total			41