In this lecture we will look in greater depth at both exponential
and logistic growth. This material parallels chapter 1 of the
text.
We all know the basic equation for exponential growth, but it
is important to recognize that there are different forms for the
equation to fit different life history/reproductive patterns.
The simplest pattern is for populations that have nonoverlapping generations, i.e. parents grow, reproduce, then die.
We call such species semelparous.
The equation that tells us numbers in such a population is the
integrated form of the usual differential equation for
exponential growth:
dN/dt = N0ert
Nt = N0t or Nt = N0Rt
In the differential equation:
N0 is the starting population size
r is the instantaneous rate of increase
t is time measured in some continuous unit
However, in the integrated equation, reflective of our
semelparous population:
 is the “finite rate of increase”
by definition this is the number of individuals
‘replacing’ each adult in the previous
generation, and sometimes written as R, the ratio
of numbers in two sequential generations.
t is the number of generations
This is the way to getting you to think of reproduction that
does not occur continuously. Most species produce offspring
in discontinuous bursts (e.g. once a year or once in a lifetime).
Very few living things reproduce in the essentially continuous
way assumed as characteristic of the exponential model.
Single celled organisms (and possibly other very simple ones)
basically fit that assumption. Multicellular ones rarely do.
At latitudes where seasonality is characteristic, i.e. at
temperate latitudes or more extreme areas, almost all plants
reproduce once per year.
Responding to resultant food availability, most animals
reproduce during the plant growing season. Some reproduce
throughout that period, some have two or a few reproductive
cycles related to that period, others reproduce once each year.
Thus, most species reproduce in discrete bursts. The basic
exponential model described continuous growth; that pattern
is one of the model's assumptions.
An example from your text shows how to apply discrete
growth equations:
Gypsy moths are insects that reproduce annually, following which
the adults die, i.e. they are semelparous. They feed on various
tree species, but one of their favourites and the subject of the
example is oaks (Quercus spp.). A technician counted the number
of egg masses per hectare (each mass averages containing 40
eggs). In 2003 there were 4 egg masses/hectare, or 160 eggs. In
2004 there were 5 such masses per hectare on average, or 200
eggs.
What was the finite rate of increase (R or ) for this population?
An example from your text shows how to apply discrete
growth equations:
Gypsy moths are insects that reproduce annually, following which the adults
die, i.e. they are semelparous. They feed on various tree species, but one of
their favourites and the subject of the example is oaks (Quercus spp.). A
technician counted the number of egg masses per hectare (each mass
averages containing 40 eggs). In 2003 there were 4 egg masses/hectare, or
160 eggs. In 2004 there were 5 such masses per hectare on average, or 200
eggs.
What was the finite rate of increase (R or ) for this population?
R = 200/160 = 1.25
What would the population size (number of eggs/hectare) be in 2005?
N2005 = N2004R = 250
And in 2006?
N2006 = N2004R2 = 200 x 1.5625 = 312.5
Apparently, spraying to control this pest insect is permitted only when the
density of eggs reaches 1000/hectare.
When would this density be reached as a result of population growth?
Begin with the basic equation for growth:
N t = N 0R t
Now make the natural log transform of the equation:
ln Nt = ln N0 + t ln R
Using the starting (2003) data,
ln 1000 = ln 160 + t ln 1.25
6.91 = 5.07 + .223t
1.84 = .223t
t = 8.25
Since the rules say the density must be >1000, and gypsy moths reproduce
once per year, spraying will only be permitted after 9 years of population
growth, or in 2012.
There is a second distinction that is needed:
Some species reproduce only once, then adults die
(semelparity).
Other species reproduce repeatedly. That strategy is called
iteroparity.
Different mathematical models are needed for each of the four
‘categories’:
1) continuous semelparous reproduction – this produces
the simple model of exponential reproduction you
are used to seeing.
2) discrete semelparous reproduction
3) continuous iterparous reproduction – also can fit the
simple model.
4) discrete iteroparous reproduction
So, we need to consider what is called ‘discrete time’ models,
meaning growth that occurs in bursts, and a mathematical
form called ‘difference equations’. Let’s look at the way
difference equations present ‘exponential’ growth…
If populations are semelparous (reproduce once and adults
die), then…
Nt+1 = R0Nt
This is the version you are used to seeing. This is the growth
equation for single-celled organisms. The models are different
when adults survive (i.e. seasonal reproduction in iteroparous
organisms):
Nt+1 = RNt + Nt
or
Nt+1 = (R + 1)Nt
This need not look too different from the usual…
Nt+1 = Nt
With adult survival  = (R+1). If we follow growth for a
second generation…
Nt+2 = Nt+1 = Nt = 2Nt
And after T time steps (generations)…
Nt+T = TNt
Compare this to the usual solution for continuous growth…
dN/dt = rNt
Rearranging:
dN/N = r dt
Integrating over a time interval beginning at t=0 and ending at
time = T… The left side of the equation integrates to:
ln Nt – ln N0
And the right side to:
rT – r0 = rT
‘Exponentiate’ each side of the equation, and you get the
familiar result:
Nt/N0 = erT = T
or
Nt = N0erT = N0T
The two forms, continuous and discrete, are essentially
similar, and  = er.
It is particularly important in discrete reproductive strategies
(but really in all strategies) that we consider not only births,
but also deaths between bouts of reproduction.
Graphs of population size over time when reproduction occurs
in bursts are typically ‘sawtoothed’, with rapid increase at the
time of reproduction, then gradual decrease in the intervals
between bouts of reproduction…
That is the growth of a population of pheasants introduced
onto Protection Island off the coast of the state of Washington.
There was abundant food and were no predators; the island
was far enough from mainland that there were no pheasant
dispersers onto or off of the island.
The size of the pheasant population was monitored annually
after the introduction of 8 pheasants in 1937. Growth
followed the sawtooth curve of the discrete exponential
pattern. In 1938, the population was 30. That gives a  of 30/8
or 3.75. From that we can predict the population to be
expected if growth were exponential for each succeeding year.
The predictions was pretty accurate for 1939 and 1940, but
the predicted population for 1941 was larger than that actually
observed. The population should have reached 5932, but was
actually only about 1800.
In 1942 predicted population size was 11,488, but actual size
was 1898.
These differences are considered to be indications of food
limitation, and of logistic growth. We cannot learn what the
carrying capacity was, since the American army made
Protection Island a training center, and the soldiers got target
practice, as well as meals of unexpected excellence, by
driving the pheasant population extinct.
Doubling Time
A simple comparison of population growth rates comes from
measuring the time it takes those populations to double in
size. If growth is exponential, then:
Nt/N0 = 2 = ert
ln 2 = rt
0.693/r = t
Exponential Growth in Invasive Species
When invasive species enter a new environment, they
frequently occur in the absence of controlling agents
(predators, diseases, even competitors adapted to them). As a
result, their growth, at least initially, is exponential. That was
certainly the case for Lythrum salicaria (purple loosestrife)
and the zebra mussel. Their effects on community structure
and food chains was (and may remain) severe.
The zebra mussel initially grew very rapidly. However,
in the last few years its growth has slowed.
It first was seen in the mid-1980s (though it
probably arrived a year or two before being
found), and reached extraordinary density
in Lakes Erie and St.Clair by the mid
1990s.
Adding Stochastic Fluctuations
The exponential models described are both
deterministic models. No matter how many times we 'run'
these models, they produce the same result at any point in
time. There is a single value of the y-variable, population size,
for each x, the time. We know full well that the real world
doesn't function like that.
If we run an experiment many times and want to report
the results, we would report a mean, x , and a measure of
variation around that mean, typically the variance s2 or a
standard deviation s.
In describing population growth, how can we introduce
stochastic variation into the model? There are two basic ways.
The first suggests that the environment varies stochastically,
influencing the growth rate. We report averages for population
size and r…
The equation for the variance in population size is more
complicated, and depends a bit on the method of derivation.
The most widely accepted way, shown in May (1974) results
in this equation:
Even without the messy details, there are a few things to take
away from this equation:
1) the variation increases with time, since t appears twice in
exponents, once in the 'growth term' e2rt and once in the
'variance term' est. Populations that are growing rapidly, or
that have a highly variable growth rate show high variance
in numbers; populations growing slowly or with little
variation in r have low variance in numbers. Also,
2) populations which begin at larger size have a higher
variance in numbers.
If a population shows a high variance in numbers, there is a
real danger that the population may decrease to extinction by
chance.
A very simple way to predict whether environmental
stochasticity will drive a population to extinction was
presented by May in estimating variance. If the variance in r is
greater than twice the mean, i.e. s2r > 2r , the probability of
extinction approaches 1.
The other way in which variability in growth is described is to
examine demographic stochasticity. In this view the effects
of variability are evident in the births and deaths occurring
within a population. Randomness in the sequence of events
(births and deaths) that can produce remarkable and counterintuitive results.
If birth rate and death rate are equal, then the deterministic
model says the population remains fixed in size. The
stochastic model says there is a variance in population size:
2N = 2N0bt
If b and d are not equal, then the population is either growing
or declining on average. If Nt is declining, then the population
is headed for inevitable extinction.
The more interesting question is what happens to a population
which is growing on average, but which suffers from
stochastic variation. Pielou (1969) has examined the long term
prospects for species whose birth and/or death rates vary
stochastically, and while the mathematics are complex, the
results can be sorted into 3 cases:
Case I - if birth rate is less than death rate, we evaluate
population growth as time approaches infinity, and from that
the probability of extinction over the long term - extinction is
inevitable.
Case II - b=d is the equilibrium situation, and a population
would seem immune to extinction. Stochastic fluctuations in
birth and death change this prediction. The algebraic solution
gives a probability of extinction indicating extinction is again
inevitable.
lim t P(0)t = (bt/1+bt)n(0)  1
Case III - b>d - There is now a non-zero probability that even
this population may go extinct. The probability of extinction,
which variance indicates should include both the size of the
starting population as well as birth and death rates, is, in the
limit as time approaches infinity:
If the population is growing on average (b is greater than d),
then the probability of extinction declines as the initial
population size increases. Chance events when the population
is near its initial size are unlikely to carry it down to 0. Also,
the greater the growth rate, evident in the ratio of death rate to
birth rate, the less likely extinction is. Average growth rate
increases, making recovery rapid, even from temporarily
small populations. These are the characteristics of ‘weeds’.
Briefly, let’s remember the graphical characteristics of
exponential growth…
Nt = N0ert
If:
And take the ln of both sides:
ln Nt = lnN0 + rt
Then a plot of numbers (in the form of natural logs) against
time is a straight line with slope r and y-intercept lnN0.
slope = r
ln Nt
ln N0
time
Bibliography
May, R.M. (1974) Ecosystem pattern in randomly fluctuating
environments. Progress in Theoretical Ecology 3:1-50.
Pielou, E.C. (1969) An Introduction to Mathematical Ecology. WileyInterscience, N.Y.