for any stable population, number of individuals added to the population (births plus immigrants) in any unit of time must equal the number of indiviudals subtracted (deaths plus emigrants) -- otherwise the population is not stable and instead increases or decreases

if we let b = number of births per individual in the population per year (annual per capita birth rate),

i = number of immigrants per individual in the population,

d = number of deaths per individual in the population per year (annual per capita death rate),

e = number of emigrants per individual in the population, and

N = total number of individuals in the population,

then total change in the number of individuals per year

N = (b+i-d-e)N

for continuous change, the instantaneous rate of change is best expresssed by a derivative

define r = b + i - d - e (r is the instantaneous per capita rate of increase per year)

then

when we solve this simple differential equation, we obtain

with N_t = population size at time t, N₀ = original population size, and e = 2.71828... (a constant that is mathematically convenient for working with differential equations)

so if r = 0.017 / year (rate of population growth of humans in the U.S.), then every year there are 1.7% more individuals in the population (note that e^x with x < 0.3 equals approximately 1 + x, so e^0.017 = 1.0171... or 1.7 more additions than losses per 100 individuals) -- in 20 years we get

N₂₀ = N₀e^(0.017)(20) = N₀ (1.40) . . . a 40% increase in population size

if r = 0.035 / year (3.5% / year as in some African countries)

N₂₀ = N₀e^(0.035)(20) = N₀ (2.01) . . . a doubling of population in 20 years!

a problem in applying this simple procedure to actual populations is that not all individuals in a population have the same chance of reproducing or dying in a given year -- in particular, individuals of different ages usually differ in their fecundity and mortality -- we thus need to take account of changes in age-specifc fecundity (average number of offspring produced at a particular age) and in age-specific survival (probability of surviving from one age to next, usually one year to next)

if we know age-specific fecundity and survival, we can calculate ...

(1) the eventual rate of increase or decrease of the population (once the population has reached its equilibrial age distribution),

(2) the size of the population at some future time, and

(3) the expectation of further life, the generation time, and the eventual proportion of the population at each age (equilibrial age distribution)

field studies of marked individuals provide a way to determine survival and fecundity -- one of the first such studies was by Frank Blair -- he studied the Texas Spiny (or Rusty) Lizard Sceloporus olivaceus on 10 acres in Austin TX for four years from 1952-1956 -- S. olivaceus is a close relative of the widespread Fence Lizard (S. undulatus)

females laid eggs throughout late spring and summer, late April to late August -- thus the population consisted of distinct age-classes -- a few lived as long as five years -- but Blair could distinguish only four age-classes (hatchlings in their first summer, yearlings about one year old, two year olds, and those older than two) -- he could not find all the clutches of eggs -- nor could he catch all the hatchlings

but he could catch all the lizards one year old or older -- he marked individuals so he could identify those he recovered the following summer -- of those present one summer, he recaptured on average 21.8% the next summer regardless of their age after age 1 -- so he estimated annual survival = 0.218

for survival of eggs and hatchlings to age 1, he had to use indirect methods -- he estimated how many eggs were laid on his property each year (see below) and he knew the number of one-year-olds caught the following year -- by dividing the second number by the first he had an estimate of the proportion of eggs laid that survived to be lizards one year old -- the four years of his study gave an average of only 3.45%

how did he know how many eggs were laid? he knew the number of females in each age-class, and he estimated how many eggs females in each age-class laid on average by dissecting gravid females collected away from his property -- one-year-olds usually laid one clutch averaging 11.3 eggs -- since a few laid two clutches, the average number of eggs/year for a one-year-old female was 12.7 -- older females laid four clutches averaging 18.3 (two-year-olds) or 24.5 (older females) eggs -- older females are bigger and thus lay more eggs (the usual case for fish, amphibians, and reptiles)

these data can be summarized in a life table as follows . . .

age (years)	age-specific survivorship	age-specifc fecundity
x	lx	mx	lxmx	xlxmx
0	1.0000	0	0	0
1	0.0345	6.36	0.2194	0.2194
2	0.0075	36.60	0.2752	0.5504
3	0.0016	49.00	0.0784	0.2352
4	0.0004	49.00	0.0196	0.0784
		sums =	0.5926	1.0834

three important (and confusing) points to keep in mind: