Exponential growth and limitation
How regulation differs from limitation –
sigmoid growth
Regulation and density-dependence
vs. density-independence
Rates, slopes, crossovers, and stability
A population cannot
grow forever, and will reach some kind of growth-limit eventually. How does
this occur? Why do some populations grow quickly, and others only slowly? Why
do populations stop growing at different sizes? Answering questions like these
will force us to think about the factors influencing birth and death, and how
these factors interact to determine the population’s density.
Under ideal
conditions, any population can grow exponentially (that is to say, with an accelerating
rate), growing faster the bigger it gets (like compound interest in a trust
fund). Exponential growth may be extremely fast (e.g. bacteria provided
with fresh medium), or moderately fast (mice at harvest-time in the fields), or
rather slow (humans since about 1700). The rate will be set by the time it
takes an individual to become a breeder, by the number of offspring produced by
an individual, and by the nature and intensity of mortality factors acting upon
the population, among other things. Ideal conditions, however, do not persist.
As a population becomes larger, the presence and activities of its members will
alter the conditions, for example depleting food faster than it is renewed.
Fast-multiplying
organisms (such as a population of bacteria) do not differ much in their
individual ability to access resources, and have little or no capacity to use
their behaviour to beat out rival individuals. When the population becomes
unsustainably large, nearly all individuals are equally (and perhaps fatally)
inconvenienced by food shortage, and the density crashes abruptly. The few
lucky survivors may then be able to initiate a fresh phase of growth, if the
food supply recovers, but their fate will be similar. This “lid on growth”-effect
is referred to as limitation, literally suggesting that there is a
ceiling or upper limit to density imposed by an outside force and acting
indifferently upon the individuals present. The “lid” won’t necessarily always
be held in the same position over the growing population, but on average there
will be a normal ceiling against which the population will bash itself.
Many factors aside
from sudden food-shortage can impose limitation: the onset of bad weather
conditions (cold, extreme heat, drought), disturbances like fires or floods,
and in some instances predation or disease. A population whose size is set only
by limitation will never achieve a stable, near-constant density, and will
instead fluctuate through time in a series of spikes and crashes, often
referred to as a cycle of “boom and bust”. This is an example of a relatively “r-selected” life-history.
A population in which
growth is “limited-only” is in no way controlled by its own actions. The forces
which end up limiting such a population would act whether the population was
present, or not; consider limitation imposed by the weather – the onset of
winter will kill off mosquitoes, but the winter cannot know that it is
doing this, and winter will not “hold off” if mosquitoes are absent! But there are
factors whose actions are influenced by the presence and state of the
population, and their influence may be such as to stabilize the population’s
density just as a thermostat can stabilize the temperature of air in a house.
When a factor acts at
an intensity set in part by the state of the population it is affecting – i.e.
when a factor’s strength depends on the size or growth-rate of the controlled
population – the factor is referred to as a potentially regulatory factor. If
the factor actually does produce a stable, controlled density, and maintains it
over time, the process if referred to as regulation. The main difference between
regulation and limitation is that regulation imposes both upper and lower
limits on population density, not just a ceiling. Regulation can do this
because the factor in some way “notices” the density of the population and acts
accordingly.
For example: imagine
a population regulated by a constant food supply, in which the stable state is
that every individual manages to acquire just barely enough food to survive. If
a few extra individuals were to be added to this population (by extra
births, or the arrival of immigrants), the weakest would fall short of
nutritional adequacy and would die, and the remaining individuals would
stabilize at a density near the previous stable value. Now imagine the effect
of removing a few individuals (either by accidental deaths, or
emigration, or an experimenter’s action): the remaining individuals would be
able to produce more offspring, because more food would be available, but this
would increase density only to the previous stable value, and no further.
Similarly, if food supply were to be altered, density would stabilize at the
same food-per-individual value as before, either more individuals with
increased food-supply or fewer with reduced supply. In this extended example, competition
for food is clearly the mechanism of regulation. Other potentially-regulating
forces are interspecific competition, predation, and disease – biotic factors
generally.
Regulated populations,
when they are at low density, grow more or less exponentially, but as they get
larger the influence of their own members’ presence, and/or the increasing
intensity of extrinsic regulatory forces acting upon them, prevents exponential
growth from continuing. The slowing-down of growth causes the initial
exponential rise to deflect until the growth-curve levels off, that is to say until
the population density achieves a stable, unchanging value. The result on the density-over-time
graph is a shape like a flattened S, usually referred to as a pattern of S-shaped
or sigmoid growth.
This is the form of growth-curve characteristic of K-selected populations,
and the stable maximum density achieved at the end of the S is the carrying capacity, or
K-value.
Just because a factor
is biotic, this does not prove that it is regulatory. All regulatory
factors are biotic, but not all biotic factors are regulatory. [“All crows are
black, but not all black things are crows.”] Biotic factors at least have the potential
to alter their action according to the population’s condition, though they do
not necessarily do so.
Remember that all
populations, whether regulated or not, are subject to limitation. Thus some
populations are limited-only, some are mostly limited and a bit regulated, and
some are limited and strongly regulated (perhaps by several regulatory factors,
not just one).
Another way to
describe the action of regulatory factors is to describe them as density-dependent in
their action: they differ in intensity is such a way that they stabilize the
population. Competition for food acts gently at low density (lots of food, few
individuals, thus a weak force to control growth) and strongly at high density
(less food relatively speaking, to the point where bare survival becomes
difficult, as there are more hungry mouths present). Contagious disease spreads
slowly at low density (hard to transmit when individuals are sparse) and easily
at high density (perhaps aided and abetted in killing victims by starvation and
other high-density problems). A density-dependent (or “DD”) mortality factor
kills a greater percentage of a large population than it does a small one, and
this tends to stabilize density; density-dependent birth makes fewer babies per
mother in a large population than in a small one, again stabilizing but from
the opposite effect. Some regulated populations are controlled mainly by DD
mortality, some mainly by DD birthrates, and some by both; all will of course also
be influenced by limiting factors.
Limiting factors in contrast, as described above, act in
a manner which must be described as density-independent (“DI”). DI mortality
factors certainly cause an increasing number of deaths as density increases,
but always in proportion to the size of the population – they act much like
sales-tax on items of different cost! (You pay more sales-tax on a $10 item
than on a $1 item, but the increase is by a factor of 10 exactly, just like the
item-price.) Thus although DI mortality factors can slow the exponential growth
of a population, they cannot by themselves change it to sigmoid growth, so they
cannot regulate. This is true even when a DI factor like a major catastrophe
strikes a population – it may kill 95 or 99% of the individuals present, yet it
is limiting only. Very limiting, to be sure, but not likely to lead to
any regulated stable density; instead, it will simply set the stage for steady
recovery, or it will drive the population extinct – in neither
case is there regulation.
It is also possible
for factors to act in a manner that varies with density, but not in the
mode which promotes stability; consider an example. Imagine a house thermostat which
responds to warm air by turning the furnace ON, and responds to cool air by
turning the furnace OFF. Such a device, far from creating a stable temperature,
would either burn the house down, or allow it to freeze solid! It would
be responding differentially to temperature, but “the wrong way around”,
promoting instability. In ecological terms this would be
analogous to a factor killing a smaller fraction of a large than a small
population, or one promoting more births per female in a crowded population
than in a sparse one. We refer to such factors as inverse-density-dependent (“IDD”) factors.
Such factors obviously cannot regulate, and in fact are counter-regulatory,
but they can be consistent with overall regulation as long as their effects are
relatively weak.
We can best assess
the rates of birth and death of a population as a value “per individual”, or if
you prefer the fractional rate: for death, how many individuals die
per individual present per unit time? This is the same as saying what fraction
of individuals present die per unit time? If ten die of one
hundred present in a time period, you can then say “ten per 100 die”, or
equivalently “one-tenth (or ten per cent) die”, per time period. Similarly for
birth, we say (for instance) twenty offspring are produced per hundred
breeding females per unit time, or equivalently “one-fifth of females breed
per time period”, or “twenty per cent breed per unit time”, or “on average
one-fifth of an offspring is produced by each female in a time period”.
If you graph out
these “fractional rates” over density, the shape of the resulting line tells
you if the factor is DD or DI. A DI mortality, for example, would always
kill the same fraction of individuals irrespective of density – say 10 per
cent of a small population, of a medium-sized population, or of a large one.
Thus the fraction, 1/10 or 10%, would stay constant with density, and the line
would be parallel with the density-axis.
On the other hand, a
DD mortality factor would act increasingly strongly at higher densities:
it might kill 10% of a small population, 30% of a medium-sized one, and 70% of
a large one. Such a line will have a positive slope with density. Of
course, since a birth-rate factor has the opposite effect on density to
that of a death-rate factor, a DD birth-rate will show a negative slope
with density. If a population exhibits birth- and death-rate lines which cross
on the graph – and to achieve this, one rate at least must vary with density – there
will be a point of density at which the rates are equal, and at this density
the population will have zero growth.
Be
warned! – just because a crossover point exists, this does
not necessarily mean the population will be stable at that point. The
overall slopes of the birth and death rate curves must sum to a DD state,
otherwise there will be no regulation and no stability. When there are DI
factors, they will contribute nothing to regulation, and just increase “noise” (variance)
in the population-control situation. When there are IDD factors present, they
will undercut the strength of DD factors, and must be taken into consideration
when evaluating the overall mode of control. Only when all factors of significance
are analyzed for their density-relations can we confidently assert that a
population’s growth is understood.
Whether a population
is regulated or merely limited, lives at a stable consistent density or
jerks all over the map in boom-and-bust chaos, this is no indication of the
success of the organism! Many stable K-selected organisms like pandas and
whales are highly endangered species, while the majority of spike-and-crash
r-selected organisms are ineradicable pests like bacteria or mosquitoes – a population’s
mode of growth and density-control is independent of how well it may do in its
environment.