? Population Growth

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?Population Growth

• Exponential and Logistic
• IB Environmental Studies
• 2004 Darrel Holnes

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Population Growth

• 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.

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Breaking It Down to Exponential

• 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.

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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

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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

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Formula Key

• 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.
• 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

• 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

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Exponential Model

• Exponential model is associated with the name of
  Thomas Robert Malthus (1766-1834) 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
  t0,1,2,... 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 lt 0)
• Population exponentially increases (r gt 0)
• Population does not change (r 0)
• Parameter r is called
• Malthusian parameter
• Intrinsic rate of increase
• Instantaneous rate of natural increase
• Population growth rate

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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 lt lt 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.

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Logistic Formula

• 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

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After Thoughts

• 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.