Download Chapter 11 Molles Notes – Population Growth

Chapter 11 Molles Notes – Population Growth
Populations are dynamic – they increase and decrease in response to changes in the biotic and abiotic
factors in their environments.
The study of population growth includes the factors that determine the patterns and rates of growth,
in addition to the factors that limit population size.
There are two traditional methods of studying population growth. One is mathematical modeling; the
other is lab and field observations of existing populations.
11.1: Geometric and Exponential Population Grxowth
In the presence of abundant resources, populations can grow at geometric or exponential
rates
A population that grows at its maximum rate (in the presence of unlimited resources and no external
constraints on the population size) grows slowly at first, then faster and faster. The rate of growth
accelerates. Some populations at their maximum rate grow geometrically and some grow
exponentially.
Geometric Growth
Populations that expand in pulses, like annual plants, single-season insects, and others follow a
geometric growth model, in which successive generations differ in size by a constant ratio. This
applies only to populations whose generations and reproductive cycles/seasons do not
overlap.
Because the rate is constant, the algebra is simple. The rate (λ) s defined as a ratio of the population
size at some time to the size exactly one generation previously.
λ = N(t + 1)/Nt
Data used is from Table 10.1
The initial size of the population is 996. (N0)
The number of offspring produced in the first year was 2,408. (N1)
2408/996 = 2.4177 = λ = N(t + 1)/Nt
So, N2 = N1 x λ, N3 = N2 x λ, and so on.
Because N1 = N0 x λ, by substitution, N2 = N0 x λ x λ = N0 x λ2
By extension, N3 = N0 x λ x λ x λ = N0 x λ3
This means the population number (again, only for pulsed populations) can be estimated at any point
in time by
Nt = N0λt
Eternal geometric growth is unrealistic. Increased density leads to overcrowding, which leads to
depleted or unavailable resources.
Natural populations have a tremendous capacity for increase. Unlimited growth cannot be
maintained for a long period of time.
Exponential Growth
In species with overlapping generations and continuous reproduction (bacteria, humans, some trees),
the geometric model is not appropriate because the rate is not constant. It is subject to more
variation because the rate itself doubles with each generation time.
dN/dt = rN
The result is that the increase itself, the rate itself, increases at each interval, because as N gets larger,
the constant r multiples by a larger and larger number. As population size (N) increases, the rate of
increase gets larger.
The result of this compounding is an exponential growth curve. (p 243, 244, 245 Molles; p 67
Kingsolver for examples)
For any population growing at an exponential rate, the population size at any time (Nt) can be
calculated as
Nt = N0ert
SR aside: On e: In mathematical terms, the limit as N grows larger is the number that we call e. It is
also is the sum of the infinite series, and the base of the natural log. For the purposes of this course,
you only need to recognize e as a constant in the exponential growth rate formulae, and to
understand that it represents a continuously changing (increasing or decreasing) rate.
Note that this formula is essentially the same as our equation for geometric growth, except that er
takes the place of λ.
While exponential growth does in fact happen in nature, whenever density begins low and resources
are high, at some point, growth must eventually slow and population size must level off. This is
where we begin discussing logistic population growth.
See diagram on p 244 for explanations of both equations, their parts, and how the functions are
similar and different.
Concept 11.1 Review Answers
1. Bennett’s assumption was that the rate of pollen deposition is proportional to the size of the
Scots pine population around a lake.
2. Exotic species introduced from their native geographic region to a new environment that is
suitable physically may encounter few competitors in the new environment and are generally
introduced without most of the pathogens, predators, and parasites that negatively impact their
populations in the native region.
3. Guppy population growth would be best modeled as exponential growth, while African annual
killifish population growth would be best represented by geometric growth.
11.2: Logistic Population Growth
As resources are depleted, population growth rate slows and eventually stops
Exponential growth cannot continue indefinitely. At some point, resource limits exists. As the limit is
approached, the change in the growth rate minimizes, keeping the population number constant. This
pattern produces an s-shaped curve. The size at which the population can no longer grow is the
carrying capacity, K. K is a whole number, representing the number of individuals which the
environment can sustain. The rate of growth is 0 and N is constant. For most species, K is defined
by a combination of food, disease, and habitat space. The logistic model demonstrates what happens
to populations as they approach the carrying capacity.
Recall our base model for exponential growth:
We can modify this equation to account for the leveling off of N, and the decrease of r as N
approaches K, and we can do it rather simply.
The max indicates that we’ve reached a maximum rate of per capita increase. (Remember that the rate
changes with each interval in an exponential growth curve.) rmax is the intrinsic rate of increase,
only attainable under the most ideal hypothetical conditions of plentiful resources, no emigration, no
predation, etc. In reality, this rarely exists, so realized r is generally less than rmax. And of course, r can
be negative if a population is in decline.
We can rearrange this equation to a form that emphasizes the effect of N on r. This is our logistic
growth equation.
Think about this. The rate of growth (dN/dt) slows as N increases. This makes sense because 1 –
N/K becomes smaller and smaller until N = K. When N = K, the right side becomes zero. Now that
the rate of growth is zero, N is stable, and growth ceases. On the other hand, growth is greatest when
N = K/2. See figure 11.13 for a breakdown of this equation.
This ratio, N/K, is the environmental resistance to growth. As N approaches K, environmental
factors can influence population growth more than before.
As in exponential growth, in logistic growth, r changes as N increases. When N is very small, rmax can
actually be realized because there is an abundance of resources for relatively few individuals. As N
increases, and density increases, r falls from rmax to realized r.
Concept 11.2 Review Answers
1. Population growth is initially rapid, since barnacles colonize the open substrate and N is much
less than K. Population growth begins to slow near week 2 as space approaches full occupation
and N almost equals K. After 4 weeks population density declines, perhaps because N has
exceeded K, as barnacles compete with each other for exclusive use of limited space.
2. Increase the concentration of yeast available to the population and see if the Paramecium density
increases, then decrease the yeast concentration and see if the density decreases.
3. Since logistic population growth is highest at a population size of K/2, and lower at smaller
population sizes, reducing the population below K/2 would slow the recovery of the population
from the effects of harvesting.
11.3: Limits to Population Growth
The environment limits population growth by changing birth and death rates
Both density dependent (disease, predation, availability of food) and density independent (flood,
drought, temperature) factors can be responsible for limiting the possibility of population growth.
These factors can also compound, such as in the case of cold weather causing animals to retreat to
homes, but biotically, there may not be enough nesting spaces in trees to carry the population.
Read the easy examples of finches and cacti for more description. (p 249-252)
Concept 11.3 Review Answers
1. Given the reproductive potential of all populations, if they were not under some form of control
we would see many species attain population sizes of plague proportions.
2. The amount of rainfall available to stimulate plant growth appears to set the carrying capacity.
3. While the population of medium ground finches on Daphne Major Island increases during years
of higher rainfall, long-term, persistent increases in moisture could have a negative impact on
medium ground finch populations. For instance, long-term increases in rainfall could increase
populations of mosquitoes that transmit bird pathogens or could change the composition of
plant community in ways that reduce the production of suitable food.
Terms to Know
K/carrying capacity
x2/chi-square
density-dependent factors
density-independent factors
exponential growth
geometric growth
logistic growth
sigmoidal growth
intrinsic rate of increase
End of Chapter Review Questions
1. The geometric model of population growth is appropriate for organisms with nonoverlapping
generations. The exponential model of population growth is most appropriate for organisms that
have overlapping generations. Exponential growth is possible in situations where resources are
abundant. Where resource supplies are in short supply, exponential growth is not possible.
2. The much longer interval between births among north Atlantic right whale populations indicate a
very low potential rate of population growth. Consequently, these whales can be expected to
increase very slowly even with full protection from whaling.
3. To build the logistic model for population growth from the exponential model, first substitute rm,
the intrinsic rate of increase which is a constant, for r, the per capita rate of increase which is
variable. Next, take the resulting expression for exponential grow, rmaxN, times 1-N/K. This last
factor, 1-N/K, produces sigmoidal growth. See figure 11.13 for further explanation.
4. The factors that limit population size are those aspects of the environment that affect birth and
death rates in populations. These include environmental factors such as limited habitat, food
availability, predation, disease, parasites, and so forth. Make this listing more concrete by listing
the limiting factors for a specific organism that you know something about or about which you
can readily obtain information.
5. The per capita rate of increase, r, is an actual rate that varies with environmental conditions. The
intrinsic rate of increase, rmax, is a theoretical rate of increase for a population of an organism
increasing at a maximum rate under "ideal" conditions for the species. You can estimate the rm
of a species experimentally by providing ideal conditions for population growth and measuring
rates of increase.
6. Biotic factors limiting population growth include competition, predation, disease, and parasitism.
Abiotic factors that limit population growth include fire, drought, excessively high or low
temperatures, floods, toxins, and so forth.
7. Because they are capable of killing all vulnerable individuals, an abiotic factor such as a severe
freeze can influence populations independently of local population density. In contrast, the
proportion of a population attacked by a predator or infected by a disease organism usually
changes with the density of the attacked or infected population. Typically density independent
factors may be at least partially influenced by population density if, for instance, there are a
limited number of safe sites that can be filled, leaving the remainder of the population more
vulnerable. For instance, at high population densities individuals in a population may move into
floodplain habitats where they are more vulnerable to floods.
8. See figure 11.23. Human population density tends to be highest in coastal regions and lowest in
the interior of continents and in the Arctic and Antarctic. Very few regions are completely
devoid of human inhabitants. Population growth is fastest in developing countries such as
Rwanda and approximately stable in many developed countries such as Japan and many
countries of Western Europe. Many developed countries have declining populations.
9. Human carrying capacity will be determined by many of the same environmental factors that
influence other populations, including food, space, water, toxins, and disease. The projected
global population in 2050 is over 9 billion. This population size may not be sustainable over the
long term because the resource requirements of such a large population may damage the
environment (e.g., change atmospheric composition, pollute freshwaters, destroy forests, deplete
fish stocks) to such an extent its capacity to support humans will decline. Such a population
might persist over the long term if average per capita resource consumption is greatly reduced.