518 Paul Thompson
the next subsection, I show how the age structure of a population can be incor-
porated. Ecologists have added numerous other parameters producing a robust
mathematical description of the ecological dynamics of population: competition,
predator-prey interactions, foraging, resource management, frequency dependent
selection, inclusive fitness, sex ratio impacts, sexual selection, etc.^46 In addition,
certain assumptions underlying the description of growth in terms of the logistic
equation above will not be true of some populations. In particular, the assumption
that growth raterchanges linearly and the assumption that population growth (or
decline) is continuous will not be true. Assuming thatrchanges non-linearly in-
creases the mathematical complexity (and requires special techniques to solve the
equations) but will more accurately describe many populations. Also, discrete-
time models, rather than continuous-time models, will more accurately reflect
many populations.
Including Age Stratification
Birth and death rates, the basis for determiningr, may depend on age. For these
populations, age-specific fecundities and probabilities of survival are important. To
sketch how this parameter (age) can be incorporated, I assume, for simplicity, that
for each age category fecundity and probability of survival are constant (discarding
this assumption simply increases the mathematical complexity)^47.
Consider a population which breeds once a year and in which the birth-rate is
heavily dependent on the number of breeding females. A census is taken each year
prior to breeding.
Let:
nx(t) = the number of females of agexin yeart(xranging from 0 toω)
Px = probability that a female agedxin yeartsurvives until yeart+1
mx = average number of female offspring produced by a female agedx
fx = number of these offspring surviving to age 1
Then:
fx=P 0 mx,whereP 0 is the probability of a newborn (aged 0) will survive to age 1
Ifx>1, then:
nx(t+1)=Px− 1 nx− 1 (t)
Ifx= 1, then:
n 1 (t+1)=f 1 n 1 (t)+f 2 n 2 (t)+...+fωnω(t)Writen(t) for the vector{n 1 (t),n 2 (t), ... ,nω(t)}, then the two equations can
be written in matrix form as:n(t+1) =Ln(t)
(^46) For a superb development of these aspects of ecological dynamics, see [Bulmer, 1994].
(^47) This sketch follows Bulmer [1994]. Those wishing further details should consult his work.
See also, [Vandermeer and Goldberg, 2003].