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Population Growth
Exponential and Logistic
IB Environmental Studies
2004 – Darrel Holnes
Population Growth
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What is a population? Very
simply, a population is a group of
organisms of the same species
that live in a particular area. The
number of organisms in a
population changes over time
because of the following: births,
deaths, immigration, and
emigration.
The increase in the number of
organisms in a population is
referred to as population growth.
There are factors that can help
populations grow and others than
can slow down and even prevent
populations from growing. Factors
that limit population growth are
called limiting factors. However,
before we go into the limiting
factors, let's talk about the biotic
potential of a population.
Breaking It Down to Exponential
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If things were perfect for a population and all
the individuals in the population survived
and reproduced at the maximum rate, that
growth rate is called the biotic potential. The
biotic potential is used as a reference when
looking at growth rates of populations. Are
the growth rates close to the biotic potential
or far from it and how far? That type of
analysis helps population ecologists
understand if the conditions for the
population are adequate.
It is certainly not common for a population to
grow at its biotic potential for a considerable
period of time; however, there are situations
where this can happen. For example, when
fish are introduced into a lake where there is
plenty of food and space and there are no
predators, the fish can reproduce at their
biotic potential, but not for a long time.
Another example is when a scientist grows
E. coli2 on a petri dish with ideal nutrients.
The bacteria will reproduce and grow at its
biotic potential, which for E. coli means that
the population doubles in size every 20
minutes!. The graph of a population growing
at its biotic potential, which is called
exponential, can be very steep.
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For all populations, there are factors that will limit their
growth. Some of these factors depend on the population
density. The most common density-dependent factors
that limit population growth are:
Food and water supplies - A given supply of food and
water might be enough for a small population density.
However, that same supply might not be enough for a
high-density population, and competition among the
individuals of the population would develop.
Light - Light is a very common resource needed by
plants. A plant density increases those plants that don't
get enough light will not grow strong enough and might
even die.
Space - This is an obvious limiting factor, especially if
you think of fish in a lake, bacteria in a petri dish, or
plants in a forest.
Predators - Higher densities of a prey population attract
more predators and as the number of prey increases, so
does the number of predators. On the other hand, if the
number of prey decreases, so does the number of
predators.
Diseases - Diseases can certainly have an impact on
birth rate and thus affect growth rate. Since many
diseases are contagious, they are therefore dependent
on density.
Parasitism - Parasitism is a relationship where one
organism (the parasite) feeds on the tissues or body
fluids of another organism (the host). In this type of
relationship the parasite benefits and the host is
harmed, sometimes to the point of killing the host. Like
diseases, since parasites spread easier in a high-density
host, their impact depends on the density.
Breaking It Down to Logistic
• There are also limiting factors that don't depend on the population
density. These density-independent factors are abiotic factors such
as weather storms, fires, earthquakes, or floods. Any of these
factors can have a severe impact on population sizes regardless of
density.
• To conclude this section, we will describe the carrying capacity of an
ecosystem. The area occupied by a population does not have
unlimited resources such as food, water, and supplies to build and
keep a dwelling. These factors limit the population growth and many
times bring about death rates that equal the birth rates. When this
happens, the population size reaches a stable balance. So one
could say that there is a certain number of individuals of the
population that can be supported by the environmental resources in
a given ecosystem. That is called the carrying capacity of that
ecosystem. The graph of a population that grows until it reaches a
stable size based on the carrying capacity of the ecosystem is called
an S-shaped curve. - Logistic
It is divided into two groups
• The logistic curve, sometimes
referred to as an "S-shaped"
curve, initially follows a similar
pattern as the exponential
growth curve; that is,
population growth is slow
initially, then enters into a point
where growth is rising rapidly.
• The logistic model is useful in
describing populations which
exhibit exponential growth at
small populations but who live
in environments which enforce
an upper limit on population
size. The logistic population is
usually written:
• Exponential
• Formula 1
• Formula 2
Formula Key
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Formula 1
dN/dt refers to the population growth rate,
rm refers to the maximum instantaneous per capita rate of population growth,
r refers to the instantaneous per capita rate of population growth,
N refers to the population size,
K refers to the carrying capacity,
t refers to a specific time,
tau refers to the length of a time
lag, and theta can be used to create a model in which the response of per capita
population growth decreases non-linearly with population size.
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Formula 2
Parameter r in the exponential model can be interpreted as a difference between the
birth (reproduction) rate and the death rate:
where b is the birth rate and m is the death rate. Birth rate is the number of offspring
organisms produced per one existing organism in the population per unit time. Death
rate is the probability of dying per one organism. The rate of population growth (r) is
equal to birth rate (b) minus death rate (m).
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Exponential model
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Applications of the exponential model
microbiology (growth of bacteria),
conservation biology (restoration of disturbed populations),
insect rearing (prediction of yield),
plant or insect quarantine (population growth of introduced species),
fishery (prediction of fish dynamics
Assumptions of Exponential Model:
Continuous reproduction (e.g., no seasonality)
All organisms are identical (e.g., no age structure)
Environment is constant in space and time (e.g., resources are
unlimited
• Graphical unchecked or nonregulated growth is commonly represented
by the exponential growth curve
Exponential Model
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Exponential model is associated with the
name of Thomas Robert Malthus (17661834) who first realized that any species can
potentially increase in numbers according to
a geometric series. For example, if a
species has non-overlapping populations
(e.g., annual plants), and each organism
produces R offspring, then, population
numbers N in generations t=0,1,2,... is equal
to:
When t is large, then this equation can be
approximated by an exponential function:
There are 3 possible model outcomes:
Population exponentially declines (r < 0)
Population exponentially increases (r > 0)
Population does not change (r = 0)
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Parameter r is called:
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Malthusian parameter
Intrinsic rate of increase
Instantaneous rate of natural increase
Population growth rate
Logistic Model
• Logistic model was developed
by Belgian mathematician
Pierre Verhulst (1838) who
suggested that the rate of
population increase may be
limited, i.e., it may depend on
population density:
• At low densities (N < < K), the
population growth rate is
maximal and equals to ro.
Parameter ro can be
interpreted as population
growth rate in the absence of
intra-specific competition.
Logistic Formula
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Population growth rate declines
with population numbers, N, and
reaches 0 when N = K. Parameter
K is the upper limit of population
growth and it is called carrying
capacity. It is usually interpreted
as the amount of resources
expressed in the number of
organisms that can be supported
by these resources. If population
numbers exceed K, then
population growth rate becomes
negative and population numbers
decline. The dynamics of the
population is described by the
differential equation:
After Thoughts
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Eventually the population stops
increasing and reaches its
maximum level or "carrying
capacity." The maximum
population size that can be
reached is based on the
availability of light, in the case of
plants, or food, shelter, etc. Most
populations never approach the
"carrying capacity" but instead
remain at lower levels because of
the regulating effects of both
abiotic and biotic factors.
Note that populations do not
typically remain at a steady state
continually but instead tend to
fluctuate or oscillate around some
characteristic density.