Phys 102: Natural Systems

1
Phys 102 Natural Systems
Vincent Conrad
2
Exponential Population Growth
Lecture 14
Phys 102 Natural Systems
Exponential Population Growth
In habitats with unlimited resources population
growth is best represented by an exponential
model The equation dN/dt rN (by
integrating) can express the population at a
given time

• In this model, the species, enjoying unlimited
  resources, reproduces at the maximum rate allowed
  by the value of r for its species.
• very rapid, exponential increase in numbers of
  individuals (N).
• positive feedback is developed in which the
  increases in N lead to faster and faster
  increases in N.

Vincent Conrad
3
Exponential Population Growth
Lecture 14
Phys 102 Natural Systems
Exponential Population Growth
Example of exponential growth Bacteria can
reproduce every 20 mins. Starting with a single
bacteria, there will be 2 in 20 mins, 4 in 40
mins etc... If the rate of this population
growth continues for 36 hours (1 1/2 days!) there
would be enough bacteria to cover the whole
planet in a layer 1 foot thick!
Vincent Conrad
4
Exponential Population Growth
Lecture 14
Phys 102 Natural Systems
Exponential Growth
Weeds and insects, for example, typically
colonize recently disturbed habitats and exhibit
exponential population growth. But as resources
become limited G then reduces, The model used
to describe this is logistic population growth.
A species can grow exponentially only with
unlimited resources and in the absence of
predators. No species can have unlimited
resources indefinitely but exponential population
growth does occur frequently in nature. It is
characteristic of pioneer populations, or those
that have recently colonized an new habitat in
which their species was not previously present.
Vincent Conrad
5
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logisitic Population Growth
In habitats with limited resources (ie real
habitats), the logistic model, prevails. In
undisturbed, stable habitats, the population
growth is limited. Competition for increasingly
scarce resources,predators, disease, and habitat
degradation and accumulation of toxic metabolites
prevent the population from realizing, or even
coming close, to the exponential model. The sum
of these negative influences on population growth
is known as environmental resistance. In stable
environments environmental resistance acts on
population growth so each individual replaces
itself, no more and no less so that B D and the
population size is stationary.
Vincent Conrad
6
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logistic Population Growth
The mathematical model describing this type of
growth is based on the G rN equation. Another
term is added to the equation that models the
effects of environmental resistance, resulting in
the logistic growth equation. As N increases
more and more resources are consumed and it
becomes more and more difficult to add
individuals to the population and increase N
further. r becomes smaller and smaller as it
becomes harder and harder for individuals to
reproduce. Eventually a maximum population size
is reached. This size is known as K, or the
carrying capacity, and it is the maximum
sustainable number of individuals of a species
that the habitat can support indefinitely without
damage to the habitat. The carrying capacity is
a function of the species and the habitat.
Vincent Conrad
7
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logistic Population Growth

• Consider a species first colonising a new habitat
  where its species is not present.
• At first (t 0), there are abundant resources
  since N is low
• the population experiences exponential growth.
  (pioneer population)
• As N increases, so does environmental resistance.
  Food, nesting sites, water, space, become scarce
  as predators and pathogens become more and more
  abundant. Also the species alters its own habitat
  in ways that are detrimental to itself (metabolic
  wastes accumulate as resources are converted to
  wastes and poison the environment).
• The increasing environmental resistance causes
  the growth curve to deviate from the exponential
  model. The growth rate decreases as environmental
  resistance increases. The growth curve develops a
  sigmoid, or "S" shape, as growth rate eventually
  reaches zero.

Vincent Conrad
8
Carrying Capacity
Lecture 14
Phys 102 Natural Systems
Carrying Capacity of the Environment
Vincent Conrad
9
Exponential V Logistic
Lecture 14
Phys 102 Natural Systems
Exponential V Logistic Growth
Vincent Conrad
10
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logistic Population Growth
The equation for logistic growth is given by
Although this looks more complicated, notice
only one new term, K, has been added. When
population first starts N is small (N lt K)
Thus G r N and growth is exponential.
When N gets near the carrying capacity K (N K)
then (K - N) is small .
It becomes a smaller and smaller fraction as N
approaches K. Thus rN is multiplied by a smaller
and smaller fraction till the population growth G
goes to zero. The population is now at its
carrying capacity and no longer growing. Notice
that this process, unlike exponential growth, is
controlled by a negative feedback mechanism.
Vincent Conrad
11
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logistic Population Growth
The speed at which populations approach K is
related to the growth rate r. The three lines on
the graph represent populations all starting with
2 individuals, growing at rates of 0.1, 0.4 and
0.8 per time interval. All pops have the same
carrying capacity, K 100. Note that the
population with a growth rate of 0.1 had not come
close to K within 30 time units, whereas the
populations with higher growth rates had reached
K.
Vincent Conrad
12
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logistic Population Growth
Populations with different carrying capacities
and the same growth rate all approach different
equilibrium population sizes. The three lines on
the graph represent populations all starting with
2 individuals, all with a growth rate of 0.4 per
time interval. The carrying capacities of 50,
100, and 150 individuals. Note that all
populations had reached their carrying capacities
within the same time 25 time units.
Vincent Conrad
13
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logistic Population Growth
Populations starting above the carrying capacity
decline to K, and populations starting below
carrying capacity rise to K. If Ngt K then (K
- N) is negative and so is (K - N)/K. Thus rN
is multiplied by a negative number and G is
negative, the populations is declining. The two
lines on the graph represent populations starting
with N0 2 individuals and N0 80 individuals.
In both cases, the growth rate is 0.4, and the
carrying capacity is 50.
Vincent Conrad
14
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logistic Population Growth

• Both the logistic and exponential population
  growth models are mathematical ideals.
• Non natural populations fit either perfectly,
  though some come very close.
• The logistic applies more generally than the
  exponential model. But it makes a number of
  assumptions that may not be valid in all
  populations
• The relation between density and the rate of
  increase is linear
• The effect of density on rate of increase is
  instantaneous
• The environment (and thus K) is constant
• All individuals reproduce equally
• There is no immigration or emigration.
• In nature, K may vary seasonally or with climate
  and often a few individual command many matings.
• They are useful starting points for ecologists to
  build more complex mathematical models to predict
  how populations will grow in certain environments.

Vincent Conrad
15
Logistic Population Growth
Lecture 14
Phys 102 Natural Systems
Logistic Population Growth
Vincent Conrad
16
Human Population Growth
Lecture 14
Phys 102 Natural Systems
Human Population Growth
Vincent Conrad
17
Human Population Growth
Lecture 14
Phys 102 Natural Systems
Human Population Growth

• In October 1998 the human population density was
  5.9 billion.
• We were adding about 1 billion every 20 years but
  that rate is slowing a little.
• The UN now estimates a world population of 9
  billion by 2050.
• Almost all growth is expected to occur in poorer
  countries, where it is already difficult or
  impossible to feed everyone.
• This estimate is lower than earlier estimates
  that did not take the effects of the AIDS
  epidemic into account.

• Global Distribution of Population Growth
• The locus of global population is shifting toward
  the Third World.
• Forty percent of world population is in three
  countries, China (20), India (15) and Former
  USSR (6)

Vincent Conrad
18
Human Population Growth
Lecture 14
Phys 102 Natural Systems
Human Population Growth

• The obvious interpretation of the global human
  population curve is that we are a pioneer species
  that has not yet filled its habitat (which is
  most of the planet).
• We have been present in measurable numbers only a
  few 10s of thousands of years.
• During the first many thousands of years our
  population grew very slowly but eventually
  numbers increased sufficiently so that rN became
  the very large number it is now.
• The logical conclusion is that our species, like
  all others, will eventually succumb to the
  effects of environmental resistance.
• This will lead to a natural decline in growth
  rate.
• This may sound like a good thing what does the
  word natural mean in this context?
• Consider what forces limit the growth rates of
  other species competition, pathogens,
  metabolites, and predation, or in human terms,
  starvation, war, and pestilence.
• If we do not voluntarily limit our population
  density to some level substantially below K,
  environmental resistance will do it for us and
  that the experience will be very unpleasant.

Vincent Conrad
19
Human Population Growth
Lecture 14
Phys 102 Natural Systems
Human Population Growth
No species can continue to grow exponentially and
we, like all others will be subjected to natural
controls when we begin to approach K. The
quality of life at K is low. There is barely
enough space, food, medical care, to sustain the
population. The quality of life below K can be
much higher. We differ from other species in
only one respect we could deliberately choose to
limit the size of our population to some level
below k where the quality of life is high. The
social, religious, and political obstacles to us
ever agreeing to make such a decision are
possibly/(probably??) insurmountable.
Vincent Conrad
20
Human Population Growth
Lecture 14
Phys 102 Natural Systems
Human Population Growth
Disease is a density dependent population
control mechanism. There is undeniable evidence
that this mechanism is depressing human
population growth rates. At present in nine
African countries 10 of adults are infected with
HIV and in Botswana the number is 25. Life
expectancy in Botswana has dropped from 61 to 40
years because of AIDS. Many African nations are
preparing for population losses of as much as
25 The big question how much time do we have
before we reach K? Are we getting close? No one
knows the answers to these questions. It seems we
are receiving unmistakable signals that we are
pushing the capacity of our environment to
sustain the current numbers of individuals, let
alone twice as many.
Vincent Conrad

\beginmyslide1Human
Population Growth \beginitemize \item
Disease is a density dependent population control
mechanism. \item There is undeniable evidence
that this mechanism is depressing human
population growth rates. \item At present in
nine African countries 10\ of adults are
infected with HIV and in Botswana the number is
25\. Life expectancy in Botswana has dropped
from 61 to 40 years because of AIDS. Many
African nations are preparing for population
losses of as much as 25\! \item The big
question how much time do we have before we
reach K? \item Are we getting close? No one knows
the answers to these questions. It seems we are
receiving unmistakable signals that we are
pushing the capacity of our environment to
sustain the current numbers of individuals, let
alone twice as many. \enditemize
\endmyslide
\beginmyslid
e1Human Population Growth evidence we are
getting near K \beginitemize \item
species extinctions, decline in biodiversity
\item acid rain, greenhouse effect, global
warming, climate deterioration, ozone
depletion, subjection of biosphere to UV
radiation \item land fills are full, no land
for new ones \item shortages of resources, oil
, copper, energy \item polluted oceans, lakes,
rivers, widespread frog developmental
abnormalities \item contaminated soil topsoil
loss through erosion \item
desertification of former agricultural areas due
to overgrazing, increasing salinity in
agricultural areas, loss of wetlands,
exhaustion of groundwater, surface water
supplies \item no safe disposal mechanism for
accumulating radioactive wastes
\enditemize \endmyslide

\beginmyslide1Human Population Growth
evidence we are getting near K (the list goes
on...) \beginitemize \item famine
widespread in many areas of the world, epidemic
diseases (AIDS, TB, hepatitis), new diseases
from formerly remote regions (HIV, ebola,
hanta) \item bleaching and death of coral
reefs due to increasing water temperature
\item red tide, algal blooms due to enriched
land runoff \item depletion/exhaustion of
marine fisheries \item decline in size of
Antarctic ice pack \item water shortage seen as
the major limiting factor for 3rd world
development \item Unfortunately you could
certainly add to this list.... \enditemize
\endmyslide
\beginmyslide
1Human Population Growth \beginitemize
\item We are probably close to K. \item In
fact some areas of the world without the benefit
of modern agriculture, industry, and medicine
are undoubtedly beyond K. \item Remember that
K is a function of the species and the
habitat. \item Some areas of the word, such as
Sudan, Ethiopia, North Korea, and Bangladesh,
have exceeded the carrying capacity and famine
is the result. \item If we are close to K,
why don't we see the predicted leveling of the
growth curve instead of the continued increase in
its steepness? \item Could this be due to the
extreme, and unique, adaptability of our
species? \enditemize \endmyslide

\beginmyslide1Human Population
Growth \beginitemize \item So far we have
been able to move K higher and higher through
our inventions and discoveries. Without modern
agriculture, industry, medicine, and an
enormous energy subsidy from fossil fuels, K
would be far lower than it is now and we would
surely have reached it long ago. \item If any
of these should be lost, especially the energy
subsidy which makes modern agriculture
possible, K would plummet and there would be
widespread famine and a decline in N to much
lower levels. \item So far our inventiveness has
kept K moving up but it can't continue that
indefinitely. \item Eventually there will be a
resource we can't replace or substitute for and
the curve will level off or decline
\enditemize \endmyslide

\beginmyslide1Human Population
Growth \beginitemize \item
\enditemize \endmyslide

\beginmyslide1
\beginitemize \item see http//members.cox.n
et/ecology/Logistic.html \enditemize
\begincenter \includegraphicswidth0.7
\columnwidth,height!ExpoLogFluct.eps
\endcenter \endmyslide

\beginmyslide1
\sidebyside0.40.45
\beginflushleft \includegraphicswidth
1.2 \columnwidth,height!
\endflushleft \beginitemize
\item \enditemize \endmyslide
\enddocument

\beginmyslide1
\beginitemize \item \enditemize
\begincenter \includegraphicswidth
0.7 \columnwidth,height!
\endcenter \endmyslide

\enddocument Local Variables
mode latex TeX-master t End
21
Human Population Growth
Lecture 14
Phys 102 Natural Systems
Human Population Growth

• Evidence we are getting near K
• species extinctions
• decline in biodiversity
• acid rain
• greenhouse effect,
• global warming
• climate
• deterioration
• ozone depletion
• subjection of biosphere to UV radiation
• land fills are full, no land for new ones
• shortages of resources, oil , copper, energy
• polluted oceans, lakes, rivers
• widespread frog developmental abnormalities
• contaminated soil topsoil loss through erosion
• desertification of former agricultural areas due
  to overgrazing
• increasing salinity in agricultural areas
• loss of wetlands
• exhaustion of groundwater and surface water
  supplies

Vincent Conrad
22
Human Population Growth
Lecture 14
Phys 102 Natural Systems
Human Population Growth
We are probably close to K. In fact some areas
of the world without the benefit of modern
agriculture, industry, and medicine are
undoubtedly beyond K. Remember that K is a
function of the species and the habitat. Some
areas of the word, such as Sudan, Ethiopia, North
Korea, and Bangladesh, have exceeded the
carrying capacity and famine is the result. If
we are close to K, why don't we see the predicted
leveling of the growth curve instead of the
continued increase in its steepness? Could
this be due to the extreme, and unique,
adaptability of our species?
Vincent Conrad
23
Human Population Growth
Lecture 14
Phys 102 Natural Systems
Human Population Growth
So far we have been able to move K higher and
higher through our inventions and discoveries.
Without modern agriculture, industry, medicine,
and an enormous energy subsidy from fossil fuels,
K would be far lower than it is now and we would
surely have reached it long ago. If any of
these should be lost, especially the energy
subsidy which makes modern agriculture possible,
K would plummet and there would be widespread
famine and a decline in N to much lower
levels. So far our inventiveness has kept K
moving up but it can't continue that
indefinitely. Eventually there will be a
resource we can't replace or substitute for and
the curve will level off or decline
Vincent Conrad
24
Phys 102 Natural Systems
Vincent Conrad