Goals

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
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


Formulate Models
Mathematical
modelling
Biology/Ecology
Computers
Basic Programming
Oral presentation
Planning
Course
Outline




Lecture
Work on project
Oral presenation of
project
New chapter
Mon 18 jan 10:15-12:00 LECT ECO1 Uno Wennergren BL34 Chapter 0-1
Tue 19 jan 13:15-14:00 LECT ECO1 Uno Wennergren E330 Matlab and excel
Thu 21 jan 11:15-12:00 SE ECO1 Uno Wennergren Galaxen Available for questions
Fri 22 jan 13:15-15:00 SE ECO1 Uno Wennergren BL33 Project presentations
Mon 25 jan 15:15-16:30 LECT ECO1 Uno Wennergren E328 Chapter 2
Wed 27 jan 15.15-16:00 SE ECO1 Uno Wennergren Galaxen Available for questions
Fri 29 jan 13:15-15:00 SE ECO1 Uno Wennergren BL33 Project presentations
Fri 29 jan 15:15-16:30 LECT ECO1 Uno Wennergren BL33 Chapter 3
Thu 4 feb 11.15-12:15 SE ECO1 Uno Wennergren Galaxen Available for questions
Fri 5 feb 13:15-15:00 SE ECO1 Uno Wennergren BL33 Project presentations
Fri 5 feb 15:15-16:30 LECT ECO1 Uno Wennergren BL33 Chapter 5.1-2
Tue 9 feb 13:15-14:00 SE ECO1 Uno Wennergren Galaxen Available for questions
Wed 10 feb 13:15-15:00 SE ECO1 Uno Wennergren BL34 Project presentations 5.1
Wed 10 feb 15:15-16:30 LECT ECO1 Uno Wennergren BL34 Available for questions
Fri 12 feb 13:15-15:00 SE ECO1 Uno Wennergren BL31 Project presentations 5.2

Uno Wennergren
Theoretical and
Computational Biology
 Organic
Farming
 Threatened Species
 Spread of disease
 Animal Welfare
5
PhD students
2 senior researchers
Subjects
Chapters in the book

Basic about models

Discrete Processes
 Deterministic
models
 Stochastic models

Continous processes
 Deterministic
models
 (Stochastic models –
excluded)
Methods/Tools

Graphic methods - Cobweb
Spreadsheets
- Excel

Programing

Mathematical Analysis

- Matlab
Methods/Tools


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
Planning
PowerPoint
Excel
Oral presentations
Computer-OH projector
Project

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Plan your time, time schedule
Formulate the problem
Choose
 Type of mathematical model
 What methods and tools to use
 How to present the results
Re-plan
Construct the model
 If possible use critical test
Implement the model by excel or matlab
Re test the model
 If possible use critical test
Make the code and a ppt presentation
tidy – presentable to uno
For whom it may concern: prepare for
oral presentation
Basic about
Models



A model is a description
of reality
A mathematical model
uses equations to
describe reality
Two levels of modeling
Dn/dt=rn(t)
Complex
reality
I
II
Simplified
Reality
Mathematical
equations
Basic about
Models
Dn/dt=rn(t)
Complex
reality


I
II
Simplified
Reality
Mathematical
equations
A model usually has a purpose
The questions:
 Is
the reality simplified enough to be
represented by equations?
 Is the reality simplified too much and
hence the model is no longer a
description of reality (not useful)?
Discrete Dynamical
Systems

Discrete processes
 Events
stepwise
 perennials
reproduction (seeds) 1 time/year

Continous processes
 Events
all the time
 Small
mammmals
reproduction year around


Perennials survival?
insects reproduction?
 in
temperate climates?
Deterministic models





Models don’t include variation/chance
probability. Parameters are constant
All process are the same (within a
specific model) and simply a specific
chain of events.
The result is deterministic: one value
Stochastic models include
variation/chance probability
The result is a set of values

Every test generates a new chain of
events with its specific result
Recurrence
equations


The equation generates a
sequence of numbers
The equation calculates a number
by using some of the previous
number.
Example: How many were
infected previously determines
how many will be infected right
now. Which in its turn…..

Note: specific step lengths
Recurrence
equations
(linear)




General form
x(n)=f(x(n-1),x(n-2),….)
The order of the equation is set by
the number of steps backwards
used in the equation
x(n)=7x(n-5)
is of order 5.
How many initial values (numbers)
do you need to start the equation
to roll?
Assume simple growth:
x(n+1)=Rx(n)
Model type:
Difference equaions
(number sequence)

Of first order:
f(x(n-1)) =x(n)-x(n-1)

Compare with differential
The derivative of f(x):
df
f ( x  h)  f ( x )

,h  0
dx
h
Box diagram
Simple growth

 x(n)-x(n-1)=rx(n-1)
 x(n+1)=x(n)(1+r)
rx
Population x
growth
bx
fecundity
Population x
i
(1-s)x
deaths
immigration
Mathematical
analysis

Simplest linear recursive equation
x(n+1)=Rx(n)
has the solution
x(n)=Rnx(0)

growths exponentially: R>1
decrease exponentially: 0<R<1
Oscillates R<-1
constant or oscillates if R0,1,-1
What about -1<R<0???

Spreadsheets


Click and drag
Relative addresses
 =C1*B4

absolute adresses
 =$C1*B5
 =$C$1*B5
rate=
Time Population
0
100
1
103
1.03
Matematical
analysis

Equlibrium points
 Will
the sequence stop at a point?
Comes back to itself.
 Is it stable or unstable?
Compare with valley and hilltop.


Find and calculate the equlibrium
point:
Assume x is the equlibrium point
test in your equation for example
x(n+1)=Rx(n) +a
set all x( n)  x for big n
Then
a
x  Rx  a  x 
1 R
Matematical
analysis

Equlibrium points
 x(n+1)=Rx(n)
gives
+a
x ( n)  x
a
x  Rx  a  x 
1 R
Note initial value doesn’t effect
whre the equlibrium is
 The quilibriumpoint is stable if and
only if
f '(x)  1
Compare with xn=f(xn-1)
Cobweb Diagram

Graphic method to find
the equlibrium points
y
y=x
Stable equlibrium
y=f(x)
x
y=f(x) is a discrete linear model
For example
x(n+1)=-0.5x(n)+4 can be written as
y=-0.5x+4
Cobweb diagram


Initial value x*
Next step is y=f(x)
y
y=x
y=f(x)
x*
x
Cobweb diagram

Next step to take is x=y
y
y=x
y=f(x)
x*
x
Cobweb diagram

And then y=f(x)
y
y=x
y=f(x)
x*
x
Cobweb diagram

And then this proceeeds,
next step is: x=y
y
y=x
y=f(x)
x*
x
Cobweb diagram

And y becomes y=f(x)
y
y=x
y=f(x)
x*
Just proceed and the curve
will stepwise move towards
the equilibrium if it’s a
stable one
x
Cobweb diagram

If it steps away from the
equlibrium then it’s an
unstable one.
y
y=f(x)
y=x
x*
x
Linear recurrence
equation with
constant coefficients


Look for a solution, compare with
x(n)=Rnx(0)
A linear combination of x(i) terms,
for example m number of terms:
a0 x(n)  a1 x(n  1)  a2 x(n  2)  ....
.......am1 x(n  (m  1))  am x(n  m)  0
This is a homogeneous equation
since the right hand side is 0. The
simplest linear homogenous
equation is: ax=0

How to solve it?
Calculate the roots to the
characteristic equation
 Matlab funktion r = roots(c)

Characteristic
equation

Assume the solution:
x(n)  C
n
after some calculations:
a0n  a1n 1  a2n  2  ....amn ( m1)  am  0

This is the charactersitisc
equation, use Matlab funktion
r = roots(c)
Characteristic
equation
a0n  a1n 1  a2n  2  ....amn ( m1)  am  0




Use Matlab funktion r =
roots(c)
Or just try it yourself without
compuer…..
for x(n)-2x(n-1)+x(n-2)=0
The charac equation
becomes
n 1
n2
C  2C  C
2
1
  2  1  0
n
» r=roots([1 -2 -1])
r =2.4142 -0.4142
0
Charactersitic
equation

Roots to
x(n)-2x(n-1)+x(n-2)=0
n 1
C  2C
n
n2
 C
0
  2  1  0
2





1
» r=roots([1 -2 -1])
r =2.4142 -0.4142
General solution is
x(n)=C12.4142n - C20.4142n
Particular solutions, we know
that x(0)=0 and x(1)=1 gives
that
C1+C2=0 which we can use
in
1= C12.4142 - C20.4142
C1=1/2, C2=-1/2
Charactersitic
equation

Roots of
x(n)-2x(n-1)+x(n-2)=0
x(n)=C12.4142n - C20.4142n
C1=1/2, C2=-1/2
gives particular solution

x(n)=1/2(2.4142n - 0.4142n)
for big n the first tem
dominates (large absolute
value)
hence:
x(n)1/2(2.4142n)
Finite limited
growth
Simple assumptions
Simplified reality

 Whe
poulationis zero there is no
reduction in individual growth, no
competition, i. e. max growth R
 When
population is at a
equlibrium it has reached its limits
and use the resources, K, such
that mean individual growth is
zero.
Hence: The curve of individual
growth in relation to density shall
pass the points:
(0,R),(K,0)
Finite limited
growth
The curve of individual
growth in relation to
density shall pass the
points: (0,R),(K,0)
growth
r(x)
R
r ( x)  R  
R
( x  0)
K
K
population x
Linear model:
R
r ( x)  R   ( x  0)
K
Logistic growth
R
Growth
r(x)
K
population x
Linear model:
R
r ( x)  R   ( x  0)
K
Since x(n)-x(n-1)=r(x(n-1))x(n-1)
Or even better x(n+1)=x(n)(r(x(n))+1)
with r(x) as above we then have
x ( n)
x( n  1)  x( n)( R (1 
)  1)
K
x ( n)
x( n  1)  x ( n)( R (1 
)  1)
K
At right handside there is a quadratic
term, x(n),, this is a nonlinear equation!
To calculate the equilibrium: Once
again assume that there is a
equilibrium: Then this have to be true
x
x  x ( R (1  )  1)
K
This is a second degree
equation with roots: x  0, x  K .
Deterine the the character of the eq. points::
2 Rx
f ´(x )  R 
1
K
Test:
x  0, x  K i f ´(x)
2 Rx
f ´(x )  R 
1
K
x  0, f ´(0)  R  1,
st able if
R  1  1  2  R  0
x  K , f ´(K )  1  R,
st able if
1 R  1  0  R  2
If individual maximum (unlimited)
growth, R, is larger or qual to 2
there is no stable eq. and chaos and
oscillations will appear.
Host parasite
model

Assumptions, simplified reality:
The host population N growths
according to limited logistic
growth
N ( n)
N (n  1)  N (n)( R(1 
)  1)
K
Add a term that represent how
survival decease as the number
of parasites, P, increase
N ( n)
N (n  1)  N (n)( R(1 
)  1)  CN (n) P(n)
K
Host parasite
model

Host population equaion
N ( n)
N (n  1)  N (n)( R(1 
)  1)  CN (n) P(n)
K

The growth of the parasite
population also depend on
the probability that a host and
parasite meet: Assuming
proportional to such
meetings:
P(n  1)  QN (n) P(n)
Host parasite
model
System of non linear
difference equations
N ( n)
N (n  1)  N (n)( R(1 
)  1)  CN (n) P(n)
K
P(n  1)  QN (n) P(n)

Look for equlibriums
N
N  N ( R(1  )  1)  CN P
K
P  QNP



Solution (N,P):
(K,0)
(1/Q,R/C(1-1/(QK)))
(0,0)









Kunskapstaxonomi fritt efter Benjamin
Bloom
Fakta. Ange, räkna upp fakta, definiera
begrepp.
Enkel begränsad kunskap.
Beskrivning. Innebörden av begrepp och
fakta. Tolka, motivera, relatera till
varandra.
Tillämpning. Vad är innehållet
användbart till. Observera, beräkna,
kalkylera, formulera, konstruera, lösa
givna problem.
Analys. Bryta ner innehållet, dela upp,
gruppera om, jämföra, generalisera se
nya problem.
Syntes. Dra slutsatser, formulera regler,
se samband också med annan kunskap,
resonera, diskutera, skapa nytt.
Värdering. Avge omdömen, kritisera,
värdera olika kunskap, hypoteser och
teorier mot varandra.
Komplex, vidsträckt kunskap.
Bloom’s Taxanomy
A Hierarcical
Knowledge Taxonomy
Critical Thinking Activity
[arranged lowest to highest]
1. Remembering Retrieving,
recognizing, and recalling relevant
knowledge from long-term memory,
eg. find out, learn terms, facts,
methods, procedures, concepts
2. Understanding Constructing
meaning from oral, written, and
graphic messages through
interpreting, exemplifying,
classifying, summarizing, inferring,
comparing, and explaining.
Understand uses and implications of
terms, facts, methods, procedures,
concepts
3. Applying Carrying out or using a
procedure through executing, or
implementing. Make use of, apply
practice theory, solve problems, use
information in new situations
Relevant Sample Sample Assignments Sample Sources
Verbs
or Activities
Acquire, Define,
1. Define each of these Written records,
Distinguish, Draw, terms: encomienda,
films, videos,
Find, Label, List,
conquistador, gaucho models, events,
Match, Read,
2. What was the
media, diagrams,
Record
Amistad?
books.
Compare,
1. Compare an
Trends,
Demonstrate,
invertebrate with a
consequences,
Differentiate, Fill in, vertebrate. 2. Use a
tables, cartoons
Find, Group, Outline, set of symbols and
Predict, Represent, graphics to draw the
Trace
water cycle.
Convert,
1. Convert the
Collection of
Demonstrate,
following into a real- items, diary,
Differentiate
world problem: velocity photographs,
between, Discover, = dist./time. 2.
sculpture,
Discuss, Examine, Experiment with
illustration
Experiment,
batteries and bulbs to
Prepare, Produce, create circuits.
Record
4. Analyzing Breaking material into Classify, Determine, 1. Illustrate examples Graph, survey,
constituent parts, determining how Discriminate, Form of two earthquake
diagram, chart,
the parts relate to one another and generalizations, Put types. 2. Dissect a
questionnaire,
to an overall structure or purpose
into categories,
crayfish and examine report
through differentiating, organizing, Illustrate, Select,
the body parts.
and attributing. Take concepts apart, Survey, Take apart,
break them down, analyze structure, Transform
recognize assumptions and poor
logic, evaluate relevancy
5. Evaluating Making judgments
Argue, Award,
1. Defend or negate Letters, group with
based on criteria and standards
Critique, Defend,
the statement: "Nature discussion panel,
through checking and critiquing. Set Interpret, Judge,
takes care of itself." 2. court trial, survey,
standards, judge using standards, Measure, Select,
Judge the value of
self-evaluation,
evidence, rubrics, accept or reject on Test, Verify
requiring students to value, allusions
basis of criteria
take earth science.
6. Creating Putting elements
Synthesize, Arrange, 1. Create a
Article, radio show,
together to form a coherent or
Blend, Create,
demonstration to show video, puppet
functional whole; reorganizing
Deduce, Devise,
various chemical
show, inventions,
elements into a new pattern or
Organize, Plan,
properties. 2. Devise a poetry, short story
structure through generating,
Present, Rearrange, method to teach others
planning, or producing. Put things
Rewrite
about magnetism.
togther; bring together various parts;
write theme, present speech, plan
experiment, put information together
in a new & creative way