life table of Red Deer on Rhum in 1970's shows that the population was increasing rapidly -- number of hinds in one area of the island increased from 50 to just over 150 between 1971 and 1979 ( = 1.12) before leveling off

as the population increased the average number of calves per female decreased because . . .

• the proportion of three-year-olds reproducing decreased steadily from about 0.6 to 0
• the proportion of hinds having calves two years in a row decreased from about 0.9 to about 0.4
• survival of calves during winter decreased from about 0.95 to about 0.80
• altogether the number of calves surviving to April decreased from about 45 to about 27 per 100 hinds

• males in their first summer produced smaller antlers (2 cm versus 15 cm)
• adult males also produced smaller antlers (650 gm versus 800 gm)
• fights became more frequent during the rut
• hinds conceived later in the season (on average about October 25 rather than October 10)

all of these changes might have resulted from lower condition of the animals as a consequence of less food to go around -- or from increased stress as a consequence of more frequent social interactions -- evidence for these effects would require experiments -- experiments on other species of deer confirm that food availability affects reproduction

our original equation for the rate of increase of a population assumed that birth and death rates were constant (and hence that instantaneous rate of increase was constant) . . .

or when solved . . .

or if we take logarithms of both sides of the equation . . .

this equation represents exponential growth of a population -- the larger N is, the faster the population increases -- N_t is an exponential function of t -- a log-linear plot of this function (ln N_t versus plain t) is a straight line with slope = r (remember that the general equation of a straight line is y = b + mx)

natural populations cannot increase exponentially forever -- but occasionally they do for a relatively short time -- for instance, when a species is introduced by humans into a suitable but previously unoccupied area -- or when a population increases after being reduced to very low numbers by hunting -- censuses have revealed that Northern Elephant Seals increased exponentially on the coast of California after near-extinction (as a result of commercial hunting for oil in the last century)

natural populations eventually reach an upper limit -- to make the above equation more realistic, we can add a term that makes the rate of increase become 0 once N has reached some value K, called the carrying capacity of the habitat -- in other words, we get zero population growth once population size reaches K . . .

so when N is very small, (K-N)/K almost equals 1.0, so the rate of population increase dN/dt is close to rN . . . we have exponential increase

but when N = K, then (K-N)/K = 0, so the rate of population increase equals 0 . . . we have ZPG !

as N approaches K, the rate of population growth drops steadily until it reaches 0

K is an equilibrium (if N is shifted away from K, it always returns to K) -- if N is less than K, r is positive -- if N is ever greater than K, r becomes negative -- when r (rate of population increase) varies inversely with population size (positive r when N is small, negative r when N is large), there is density-dependent population regulation

many vertebrate populations have zero population growth at least in the long run -- population sizes fluctuate from year to year but over longer periods remain more or less stable -- does this stability result from density-dependent population regulation?

Great Tits (related to chickadees and titmice) have been studied in Wytham (Marley) Wood near Oxford, England, for about 50 years without any long-term changes in population size -- population size (N) each spring has varied about 5-fold in an apparently irregular way -- yet high N one year usually leads to a decrease in population the next, low N usually leads to an increase

a plot of proportional changes in population size (N_t+1 - N_t)/N_t versus N shows this inverse relationship -- negative slope indicates density-dependent population regulation

a rationale for this plot ( N / N versus N) comes straight from the equation above, which we can rearrange thus . . .

or for a change in numbers over a measureable period of time (one year, for instance) we have . . .

(N_t+1 - N_t) / N_t = N / N = - (r/K) N + r

this equation is in the form of a straight line y = mx + b . . . so if we plot ( N / N) versus N, we get a straight line with slope -r/K and intercept r

question: based on the preceding graph, what is the carrying capacity of Marley Wood for Great Tits? remember that K is the number of individuals in a population at ZPG!

what is the carrying capacity of the earth for humans?   how could we know?

what stops population growth?   remember that the rate of population growth ( r) equals the difference between birth and death rates (b - d) (provided immigration and emigration can be ignored)

what makes b - d = 0? does birth rate decrease as population size increases? or does death rate increase?

we already have seen the answer for Red Deer on Rhum

for Great Tits in Wytham Wood, in years with large numbers of breeding birds,

• average size of clutches is low and
• a low proportion of eggs hatch

low clutch sizes result from stiff competition for food in early spring (when food is scarce and females are producing a lot of eggs)

low proportion of eggs hatching results from high predation on eggs -- weasels switch to searching for Great Tit eggs when there are a lot to be found

in contrast, population density does not seem to affect survival -- there is no correlation between population size and survival of nestling birds or adults

for management of natural populations (including game birds and mammals and commercial and sport fish), density-dependent survival and reproduction have crucial importance . . .

if fecundity and sources of mortality other than hunting are density-dependent, then mortality from hunting (provided it was not too much) would have little or no effect on population size

the decrease in population size as a result of hunting would instead be compensated by decreased mortality from other causes or increased fecundity -- the population would remain close to the carrying capacity despite mortality from hunting

mortality from hunting, when compensated by increased survival and reproduction, is called compensatory mortality -- harvesting a population in this way would produce a sustained yield

if hunting removes too many individuals from a population, mortality is no longer compensatory and the population then declines

notice that it is possible to manage a population to produce a sustained yield at any density -- the only essential requirement is that human-induced mortality must be compensatory -- in contrast, there is only one density that produces the maximal sustained yield -- at the inflection point (usually near K/2) on the graph of population growth -- where the rate of density-dependent increase is greatest

so understanding the demography of exploited populations is essential for prudent management