Download Dynamics of dinosaurs Dynamics of dinosaurs Infectious disease

Introductory Lecture
MATH0011
MATH0011
Numbers and Patterns in Nature and Life
Numbers and Patterns in Nature and Life
Teachers
Dr. K.H. Chan
Course website:
Dr. TSING Nam Kiu
http://147.8.101.93/MATH0011/
Teaching Assistant
or visit homepage of HKU Mathematics Department,
> Teaching
> Undergraduate Courses
> MATH0011 (Course homepage)
Miss LAU Lai Ngor
Important
Please switch off your mobile phones during classes
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Dynamics of dinosaurs
MATH0011
Introductory lecture
„
Overview of topics:
Dynamics of dinosaurs
Infectious disease modeling
„ Population dynamics
„ Fractals in biology and geology
„ Animal size and problems of scaling
„ Fibonacci patterns in plants
„ Bioinformatics: molecular evolution and
phylogenetic trees
„
„
„
How to weigh a
dinosaur?
Difficulty: No live
specimens available,
nor even dead ones
with their whole body
(including blood and
body fluids, etc.)
intact.
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Dynamics of dinosaurs
„
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Infectious disease modeling
Would sand support a
big dinosaur just as
well as a small one?
How about clay?
Epidemics:
„
Black Death
rat flea
„
How fast could
dinosaurs move?
Europe lost 1/3 of population in 1347 - 1350.
Great Plague of London, 1664–66. Killed
more than 75,000 of total population of
460,000.
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Introductory Lecture
Infectious disease modeling
Infectious disease modeling
Epidemics:
Epidemics:
„
Influenza
„
AIDS
„
SARS
AIDS virus
influenza virus
killed 25 million in 1918-19 in Europe
Coronavirus
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Infectious disease modeling
Infectious disease modeling
Epidemics:
„
„
„
The members of the population progress through the
three classes in the following order.
Chicken flu
bovine spongiform
encephalopathy
(mad cow disease)
Susceptible
St
Infectious
It
Immune/
Removed
Rt
SIR model
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Infectious disease modeling
Infectious disease modeling
Mathematical models can:
„ predict rate of spread, peak, etc., of epidemics
„ predict effects of different disease control
strategies
.
The WHO's eradication project
reduced smallpox (variola)
deaths from two million in
1967 to zero in 1977–80.
Smallpox was officially declared
smallpox virus
eradicated in 1979.
A basic epidemic model
„ S (susceptible class)
t
„ I (infective class)
t
„ R (removed class)
t
St + It + Rt = N
Mathematical tools: Difference equations,
differential equations
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Introductory Lecture
Population dynamics
Population dynamics
Mathematical models of population growth (or
decline).
Examples
„ formulas relating population of a species in a
certain year to that of a subsequent year.
„ birth rates and death rates of a population, as a
function of the population size.
Applications
„ Cell division, insect growth, species population, …
Population models:
„ Fibonacci: reproduction of rabbits --Fibonacci sequence: 1,2,3,5,8,13,21,34,…
„ Malthusian population model
„
Pt = (1+b－d)t P0 .
Leslie model of structured populations: using
matrix theory
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Population dynamics
Population dynamics
More applications:
Population models:
„ Discrete logistic model, which is a non-linear
difference equation:
„
Forest management
logging
⎛
⎛ P ⎞⎞
Pt +1 = Pt ⎜1 + r ⎜1 − t ⎟ ⎟
⎝ K ⎠⎠
⎝
„
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„
This model, unlike the previous mentioned
models, can produce very complicated
behaviours
Sustainable harvesting of animals
bisons
fishery
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Fractals in biology and geology
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Fractals in biology and geology
Question: how to model shapes of objects?
The Earth
Paul Cezane
Salt crystal
French Painter, 1839-1906
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Introductory Lecture
Fractals in biology and geology
Fractals in biology and geology
Mathematical description of shapes of:
„
And how to describe …
clouds
„
coastlines
„
meanders and rivers
„
branching of trees
„
airways in lungs
„
surface of proteins
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Fractals in biology and geology
Fractals in biology and geology
Characteristics of fractals
„ scaling
„ self-similarity
fern
construction of Koch curve
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Fractals in biology and geology
Fractals in biology and geology
Dimension
Fractal objects
„
d=1
d =2
cauliflower
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d=3
„
What is the “dimension” of
clouds ?
„
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B.B. Mandelbrot, the pioneer
in fractals research, coined the
word “fractals” to indicate:
(i) these objects have
fractional dimensions and
(ii) they are fragmented into
ever finer pieces.
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Introductory Lecture
Animal size and problems of scaling
R∝M
Animal size and problems of scaling
z
This 1958 movie is
about a woman of
50 ft tall. Will the
scenario in the
movie be possible?
Power law
Y = cXz
How much food will each of these
animals consume ?
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Animal size and problems of scaling
Animal size and problems of scaling
Are ants “super strong”?
Antelopes can jump 2m.
Fleas can jump 4cm.
But
2m
4cm
height of antelope
height of flea
„
An ants is able to carry an object
5 times its own weight
2m
>>
„
Does it mean fleas are
stronger than antelopes?
„
4cm
An average man can carry only
about his own weight and walks
around not too uncomfortably
Question: Does it imply that ants are much stronger
than human?
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Fibonacci patterns in plants
„
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Observation: number of spirals in each direction
Phyllotaxis: patterns of arrangements of
seeds on a sunflower head, a pine cone, etc.
sunflower head
pinecone
sunflower head
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Introductory Lecture
Observation: number of spirals in each direction
Observation: number of spirals in each direction
34 counter-clockwise spirals
55 clockwise spirals
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Observation: number of spirals in each direction
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„
Other types of sunflowers may have numbers
(13,21), or (21,34), or (55,89), or (89,144), …
Pine cones : (2,3), (3,5), (5,8), (8,13), …
Dragon fruits : (3,5)
Pineapples : (8,13)
Broccoli and cauliflower : (5,8)
„
„
„
„
„
pine cone
Spiral numbers =
(34,55).
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„
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Fibonacci patterns in plants
A third set of spirals: 21 spirals
„
„
Fibonacci sequence:
1, 2, 3, 5, 8, 13, 21, 34, 55, …
fn+ fn+1 = fn+2
Golden ratio:
fn+1: fn approaches
φ=
1+ 5
= 1.61803398 ...
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Leonardo Pisano
(Fibonacci) 1170-1250
triplet of spiral numbers: (21,34,55)
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Introductory Lecture
Molecular evolution
Fibonacci patterns in plants
„
„
A “mystery”: how can the plants know
about the golden ratio and make use of it
in their growth pattern ?
In 1992, two experimental physicists
S.Douady and Y.Couder successfully
devised a physical experiment using
magnetic fluid and magnetic field to verify
that phenomenon of patterns in plants
may be results of simple physical and
mathematical laws
DNA (deoxyribonucleic acid)
is molecule of heredity:
parents transmit copied
portions of their own DNA
to offspring during
reproduction.
Though rarely happened, errors in formation of
new double strands may occur --- DNA mutation.
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Molecular evolution
Molecular evolution
DNA evolution -- base substitution:
Ancestor S0: ACCTGCGCTA
Descendant S1: ACGTGCACTA
S0
S1
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Markov models of molecular evolution (uses
probability theory and matrix algebra)
„ Jukes-Cantor model
„ Kimura model
S0
S1
Formulate phylogenetic distances among
species (uses graph theory and trees for
modeling ancestor-descendant relations)
S2
Question: What is the “distance” between
two species ?
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Molecular evolution
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Synergy between Biology and Mathematics
Mathematics Is Biology's Next Microscope, Only Better; Biology
Is Mathematics' Next Physics, Only Better by J.E. Cohen, PLoS
Biology Vol. 2, No. 12, December 14, 2004:
Phylogenetic
trees:
ancestordescendant
relationship
among
species.
“The discovery of the microscope in the late 17th century caused
a revolution in biology by revealing otherwise invisible and
previously unsuspected worlds. …
“Mathematics broadly interpreted is a more general microscope.
It can reveal otherwise invisible worlds in all kinds of data, not
only optical. …
“Today's biologists increasingly recognize that appropriate
mathematics can help interpret any kind of data. In this sense,
mathematics is biology's next microscope, only better. …“
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Introductory Lecture
Synergy between Biology and Mathematics
Nature and Mathematics
Mathematics Is Biology's Next Microscope, Only Better; Biology
Is Mathematics' Next Physics, Only Better by J.E. Cohen:
Nature’s Numbers by Ian Stewart:
“… We live in a universe of patterns. … there is a formal system
of thought for recognizing, classifying and exploiting patterns. …
It is called mathematics. Mathematics helps us to organize and
systemize our ideas about patterns; in so doing, not only we
admire and enjoy these patterns, but also we can use them to
infer some of the underlying principles that govern the world of
nature. … There is much beauty in nature’s clues, and we can all
reorganize it without any mathematical training. There is beauty
too in the mathematical stories that … deduce the underlying
rules and regularities, but it is a different kind of beauty, applying
to ideas rather than things. Mathematics is to nature as Sherlock
Holmes is to evidence.”
“… Conversely, mathematics will benefit increasingly from its
involvement with biology, just as mathematics has already
benefited and will continue to benefit from its historic involvement
with physical problems. In classical times, physics, as first an
applied then a basic science, stimulated enormous advances in
mathematics. …
“In the coming century, biology will stimulate the creation of
entirely new realms of mathematics. In this sense, biology is
mathematics' next physics, only better.”
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Nature and Mathematics
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Nature and Mathematics
Patterns in Nature by Peter S Stevens:
By Nature’s Design by Pat Murphy:
“ Why does nature appear to use only a few fundamental forms in
so many different contexts? Why does the branching of trees
resemble that of arteries and rivers? Why do crystal grains look
like soap bubbles and the plates of a tortoise shell? Why do some
fronds and fern tips look like spiral galaxies and hurricanes? Why
do some meandering rivers and meandering snakes look like the
loop pattern in cables? Why do cracks in mud and markings on a
giraffe arrange themselves like films in a froth of bubbles?”
“ Nature, in its elegance and economy, often repeats certain forms and
patterns … like the similarity between the spiral pattern in the heart of a
daisy and the spiral of a seashell, or the resemblance between the
branching pattern of a river and the branching pattern of a tree … ripples
that flowing water leaves in the mud … the tracing of veins in an autumn
leaf … the intricate cracking of tree bark … the colorful splashings of
lichen on a boulder. … The first step to understanding --- and one of the
most difficult --- is to see clearly. Nature modifies and adapts these basic
patterns as needed, shaping them to the demands of a dynamic
environment. But underlying all modifications and adaptations is a
hidden unity. Nature invariably seeks to accomplish the most with the
least --- the tightest fit, the shortest path, the least energy expanded.
Once you begin to see the basic patterns, don’t be surprised if your view
of the natural world undergoes a subtle shift.”
“… among nature’s darlings are spirals, meanders, branchings,
hexagons, and 137.5 degree angles. … Nature’s productions are
shoestring operations, encumbered by the constraints of 3dimensional space, and the necessary relations among the size
of things, and the eccentric sense of frugality.”
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Assignment
Read the article:
Mathematics Is Biology's Next Microscope, Only Better;
Biology Is Mathematics' Next Physics, Only Better by
J.E. Cohen, PLoS Biology Vol. 2, No. 12, December 14,
2004 (can be accessed through the “Links” page of
course website)
Reference
Mathematics in Nature – Modeling Patterns in the
Natural World, John A. Adam, Princeton University
Press, 2003.
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