MATH0011---Numbers and
Patterns in Nature and Life
Population Dynamics
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Lecture 2
Population Dynamics I
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http://147.8.101.93/MATH0011/
Population dynamics is the study of changes
in the number and composition of
individuals in a population, and the factors
that influence those changes.
Population dynamics involves five basic
components of interest to which all changes
in populations can be related: birth, death,
sex ratio, age structure, and dispersal.
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How theoretical biologists were thinking about
animal population until the late 1960?
An understanding of population dynamics is
needed if one wants to
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(1) predict when a species or population is
endangered with extinction,
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(2) understand how environmental changes
affect populations,
The ecologists at that time knew that many
animal populations in isolated communities
generally stayed roughly constant or
fluctuate with a rather regular periodicity or
sometimes without any pattern.
(3) estimate how many animals can be
harvested,
(4) understand how one population might
affect another (e.g, competition, predation).
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How theoretical biologists were thinking about
animal population until the late 1960?
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How theoretical biologists were thinking about
animal population until the late 1960?
They do not know the underlying reason
for these population changes.
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At that time, there are two schools of
thought about how animal population
behaved.
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On one side, the Australian Charles Birch
believed that most natural populations
are driven by external effects so that they
fluctuate a lot, driven by changes in the
environment.
Birch and his colleagues tended to draw
their examples from insect populations
that do just that.
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On the other side, another Australian John
Nicholson, took the view that populations
are regulated by effects that depend
primarily not on the environment but on
the density of the population, i.e., the
number of animals living in a given space.
This latter picture indicates that the
populations tend to increase when the
densities are low and decrease when they
are high, and that they therefore tended
on average to be relatively steady.
Nicholson and his colleagues drew
examples from relatively steady
populations.
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How theoretical biologists were thinking about
animal population until the late 1960?
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Both opposing views of the problem seem
to be partially right.
The problem turned out, can be
understood much more easier in terms of a
different way of thinking---mathematical
thinking.
In fact, Robert May discovered in the early
1970s a mathematical model that would
solve this problem.
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Difference equations.
Robert May
(1936 - )
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ƒ Born and raised in Sydney, Australia.
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ƒ He began his higher education in chemical engineering
and then switched to theoretical physics and earned
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a Ph.D. in superconductivity.
ƒ He started to work in biology in the late 1960s.
ƒ He is now the president of the Royal Society
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The mathematical tool used by Robert
May to study the population dynamics is
difference equations.
In general, it is impossible to solve a
difference equation explicitly.
However, we shall develop some
methods to understand these
equations.
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We shall first demonstrate how difference
equations arise in modeling biological
phenomena and to develop the
mathematical techniques to solve the
following problem:
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Given a particular starting population levels
and a recursion relation, predict the
population level after an arbitrary number of
generations have elapsed.
To acquire a familiarity with difference
equations, we will begin with two rather
elementary examples: cell division and
insect growth.
Cell Division
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Suppose a population of cells divides at the same
speed, with each member producing r daughter
cells.
Let us suppose that initially there are P0 cells.
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Let P1 , P2 , … , Pn be respectively the number of
cells in the first, second, … , n th generations.
Applying equation (1) recursively results in
the following:
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Then the successive generations are related by a
simple equation
Pn+1=r Pn
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If we know what Pn is, then we know how
big the population will be after n generations.
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Pn+1 = r (r Pn –1) = r [r (r Pn –2) ]
(1)
We would like to solve the difference equation (1),
i.e. find Pn satisfying (1).
= " = r n +1 P 0 .
Thus, for the n th generation
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Pn = r n P 0 .
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Note that the magnitude of r will determine
whether the population grows or becomes
less with time. That is,
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An Insect Population
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r > 1,Pn increases over successive
generations,
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r < 1,Pn decreases over successive
generations,
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r = 1, Pn is constant.
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Insects generally have more than one stage in
their life cycle from offspring to maturity.
The complete cycle may take weeks, months, or
even years. However, it is customary to use a
single generation as the basic unit of time when
attempting to write a model for insect population
growth.
Several stages in the life cycle can be depicted by
writing several difference equations. Often the
system of equations condenses to a single
equation in which combinations of all the basic
parameters appear.
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Insect Growth
Aphid
ƒ Some fraction of these will emerge and survive
to adulthood.
ƒ Consider the reproduction of the poplar
gall aphid.
ƒ Adult female aphids produce galls on the
leaves of poplars.
ƒ All the offspring of a single aphid are
contained in one gall.
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Let us first ignore these effects and study a naive
model in which all parameters are constant.
pn+1 = f an .
an = number of adult female aphids in the nth
generation (initially there were a0 females),
pn = number of offspring in the nth generation,
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m = fractional mortality of the young aphids,
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f = number of offspring per female aphid,
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r = ratio of female aphids to total adult aphids.
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We would like to find an .
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Each female produces f offspring: thus
First we define the following:
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ƒ In general, the capacity for producing offspring
and the likelihood of surviving to adulthood
depends on their environmental conditions, on
the quality of their food, and on the population
sizes.
no. of
offspring in
(n + 1) st
generation
(2)
no. of females in
previous generation.
no. of offspring per
female
Of these, the fraction 1– m survives to adulthood,
yielding a final proportion of r females. Thus
an+1 = (1 – m ) r pn+1 .
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(3)
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an+1 = f r (1 – m ) an .
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What is a difference equation ?
While equation (2) and (3) describe the aphid population,
note that these can be combined into the single statement
(4)
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Equation (4) is again a first-order linear difference equation
where f r (1 – m ) is a constant.
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The expression f r (1 – m ) is the per capita number of
adult females that each mother aphid produces.
A difference equation is a formula
expressing values of some quantity P in
terms of previous values of P.
For example, if F(x) is any function, then
Pn+1=F(Pn )
(5)
is a difference equation.
For the rather theoretical case where f, r, and m are
constant, the solution of the difference equation (4) is
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an = [f r (1 – m )]n a0 ,
where a0 is the initial number of adult females.
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References
a) This lecture is mainly based on chapter one of
the textbook,
Elizabeth s. Allman,John A.Rhodes,
Mathematical models in biology
This chapter can be downloaded at
http://us.cambridge.org/titles/catalogue.asp?isbn=0521525861
b) Blue haven [videorecording] / BBC, the Open
University ; producer, Anne-Marie Gallen
(call number EX591.788 B65).
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If F(x)=rx, then we have the previous
difference equation Pn+1=r Pn .
To solve the difference equation (5)
means to find a function Pt satisfying (5).
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