Modeling Biological Systems

Goals

Formulate Models Mathematical modelling Biology/Ecology Computers

Basic Programming Oral presentation Plan your work

CourseOutline

Lecture Work on project Oral presenation of

project

New chapter

Mon 16 jan 13:15-15:00 Galaxen Lect Uno Wennergren  Chapter 0-1

Tue  17 jan  13:15-14:00 Galaxen Lect Stefan Sellman Matlab and excel 

Thu 19 jan 15:15-17:00 Galaxen SE Stefan Sellman Available for questions 

Fri 20 jan 13:15-15:00 Galaxen SE Uno Wennergren Project presentations 

Mon 23 jan 13:15-15:00 Galaxen Lect Uno Wennergren Chapter 2

Wed 25 jan 13:15-15:00 Galaxen SE Peter BrommessonAvailable for questions 

Chapter 2

Fri 27jan 13:15-15:00 Galaxen SE Uno WennergrenProject presentations 

Chapter 2

Mon 30 jan 13.15-15.00 Galaxen Lect Uno Wennergren Chapter 3

Tue 31 feb 13:15-15:00 Galaxen SE Peter BrommessonAvailable for questions 

Chapter 3

wed 1 feb 13:15-15:00 Galaxen SE Uno WennergrenProject presentations 

Chapter 3

Fri 3 feb 08:15-10:00 Galaxen Lect Uno Wennergren Chapter 5.1

Mon 6 feb 13:15-15:00 Galaxen Lect Uno Wennergren Chapter 5.2

Tue 7 feb 13:15-15:00 Galaxen SE Peter BrommessonAvailable for questions

Chapter 5.1-2 

Wed 8 feb 13:15-15:00 Galaxen SE Peter BrommessonAvailable for questions

Chapter 5.1-2 

Fri 10 feb 13:15-15:00 Galaxen SE Uno WennergrenProject presentations 5.1

and 5.2

Uno Wennergren

Professor Theoretical Biology

Organic Farming Threatened Species Spread of disease Animal Welfare

6 PhD students5 senior researchers

SubjectsChapters 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 - Matlab

Mathematical Analysis

Methods/Tools

Planning PowerPoint Excel Oral presentations

Computer-OH projector

Project 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, Stefan and Peter

For whom it may concern: prepare for oral presentation. This year everytime!

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

Simplified Reality

II

Mathematical equations

Basic about Models

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)?

Dn/dt=rn(t)

Complex reality

I

Simplified Reality

II

Mathematical equations

Test t

hese q

uestio

ns in

your p

rojec

ts

Discrete Dynamical Systems

Discrete processes Events stepwise

perennialsreproduction (seeds) 1 time/year

Continous processes Events all the time

Small mammmalsreproduction 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(Markov chain)

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):

0,)()(

hh

xfhxf

dx

df

Rearrange recurrence eq:

Box diagram

Simple growth x(n)-x(n-1)=rx(n-1) x(n+1)=x(n)(1+r)

Population xrx

growth

Population xbx

fecundity

(1-s)x

deaths

i immigration

Mathematical analysis

Simplest linear recursive equationx(n+1)=Rx(n)has the solution

x(n)=Rnx(0)

growths exponentially: R>1decrease exponentially: 0<R<1Oscillates R<-1constant or oscillates if R0,1,-1

What about -1<R<0???

Spreadsheets Click and drag Relative addresses

=C1*B4 absolute adresses

=$C1*B5 =$C$1*B5

rate= 1.03

Time Population0 1001 103

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 is the equlibrium point

test in your equation for example

x(n+1)=Rx(n) +aset all for big nThen

xnx )(

Ra

xaxRx

1

x

Matematical analysis

Equlibrium points x(n+1)=Rx(n) +a

gives

Note initial value doesn’t effect whre the equlibrium is

The quilibriumpoint is stable if and only if

xnx )(

Ra

xaxRx

1

1)(' xf

Compare with xn=f(xn-1)

Cobweb Diagram

Graphic method to find the equlibrium points

y=x

y

x

y=f(x)Stable equlibrium

y=f(x) is a discrete linear modelFor examplex(n+1)=-0.5x(n)+4 can be written asy=-0.5x+4

Cobweb diagram

Initial value x* Next step is y=f(x)

y=x

y

xx*

y=f(x)

Cobweb diagram

Next step to take is x=y

y=x

y

x

y=f(x)

x*

Cobweb diagram

And then y=f(x)

y=x

y

x

y=f(x)

x*

Cobweb diagram

And then this proceeeds, next step is: x=y

y=x

y

x

y=f(x)

x*

Cobweb diagram

And y becomes y=f(x)

y=x

y

x

y=f(x)

x*

Just proceed and the curve will stepwise move towards the equilibrium if it’s a stable one

Cobweb diagram

If it steps away from the equlibrium then it’s an unstable one.

y=x

y

x

y=f(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:

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)

0)())1((.......

....)2()1()(

1

210

mnxamnxa

nxanxanxa

mm

Characteristic equation

Assume the solution:

after some calculations:

This is the charactersitisc equation, use Matlab funktion r = roots(c)

nCnx )(

0.... )1(22

110

mmn

mnnn aaaaa

Characteristic equation

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

» r=roots([1 -2 -1])r =2.4142 -0.4142

0.... )1(22

110

mmn

mnnn aaaaa

02 21 nnn CCC 012 12

Charactersitic equation

Roots to x(n)-2x(n-1)+x(n-2)=0

» r=roots([1 -2 -1])r =2.4142 -0.4142

General solution isx(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

02 21 nnn CCC 012 12

Charactersitic equation

Roots of x(n)-2x(n-1)+x(n-2)=0

x(n)=C12.4142n - C20.4142n

C1=1/2, C2=-1/2gives 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 When population is 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)

growthr(x)

population xK

R

Linear model:

)0()( xK

RRxr

)0()( xK

RRxr

Growthr(x)

population xK

R

Linear model:

)0()( xK

RRxr

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

)1))(

1()(()1( K

nxRnxnx

Logistic growth

)1))(

1()(()1( K

nxRnxnx

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

)1)1(( K

xRxx

This is a second degree equation with roots: .,0 Kxx

Determine the character of the eq. points::

12

)´( K

RxRxf

Test:

)´( i ,0 xfKxx

0211R

if stable

,1)0´(,0

R

Rfx

12

)´( K

RxRxf

2011

if stable

,1)´(,

RR

RKfKx

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

Add a term that represent how survival decease as the number of parasites, P, increase

)1))(

1()(()1( K

nNRnNnN

)()()1))(

1()(()1( nPnCNK

nNRnNnN

Host parasite model

Host population equaion

The growth of the parasite population also depend on the probability that a host and parasite meet: Assuming proportional to these occasions:

)()()1( nPnQNnP

)()()1))(

1()(()1( nPnCNK

nNRnNnN

Host parasite model

System of non linear difference equations

Look for equlibriums

Solution (N,P): (K,0) (1/Q,R/C(1-1/(QK))) (0,0)

)()()1( nPnQNnP )()()1)

)(1()(()1( nPnCN

K

nNRnNnN

PNCK

NRNN )1)1((

PNQP

Bloom’s Taxanomy

A Hierarcical Knowledge Taxonomy

Critical Thinking Activity [arranged lowest to highest]

Relevant Sample Verbs

Sample Assignments Sample Sources or Activities

1. Remembering Retrieving, recognizing, and recalling relevant knowledge from long-term memory, eg. find out, learn terms, facts, methods, procedures, concepts

Acquire, Define, Distinguish, Draw, Find, Label, List, Match, Read, Record

1. Define each of these terms: encomienda, conquistador, gaucho 2. What was the Amistad?

Written records, films, videos, models, events, media, diagrams, books.

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

Compare, Demonstrate, Differentiate, Fill in, Find, Group, Outline, Predict, Represent, Trace

1. Compare an invertebrate with a vertebrate. 2. Use a set of symbols and graphics to draw the water cycle.

Trends, consequences, tables, cartoons

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

Convert, Demonstrate, Differentiate between, Discover, Discuss, Examine, Experiment, Prepare, Produce, Record

1. Convert the following into a real-world problem: velocity = dist./time. 2. Experiment with batteries and bulbs to create circuits.

Collection of items, diary, photographs, sculpture, illustration

4. Analyzing Breaking material into constituent parts, determining how the parts relate to one another and to an overall structure or purpose through differentiating, organizing, and attributing. Take concepts apart, break them down, analyze structure, recognize assumptions and poor logic, evaluate relevancy

Classify, Determine, Discriminate, Form generalizations, Put into categories, Illustrate, Select, Survey, Take apart, Transform

1. Illustrate examples of two earthquake types. 2. Dissect a crayfish and examine the body parts.

Graph, survey, diagram, chart, questionnaire, report

5. Evaluating Making judgments based on criteria and standards through checking and critiquing. Set standards, judge using standards, evidence, rubrics, accept or reject on basis of criteria

Argue, Award, Critique, Defend, Interpret, Judge, Measure, Select, Test, Verify

1. Defend or negate the statement: "Nature takes care of itself." 2. Judge the value of requiring students to take earth science.

Letters, group with discussion panel, court trial, survey, self-evaluation, value, allusions

6. Creating Putting elements together to form a coherent or functional whole; reorganizing elements into a new pattern or structure through generating, planning, or producing. Put things togther; bring together various parts; write theme, present speech, plan experiment, put information together in a new & creative way

Synthesize, Arrange, Blend, Create, Deduce, Devise, Organize, Plan, Present, Rearrange, Rewrite

1. Create a demonstration to show various chemical properties. 2. Devise a method to teach others about magnetism.

Article, radio show, video, puppet show, inventions, poetry, short story

CourseOutline

Lecture (+ read the chapter, Monday)

Work on project (Tuesday-Thursday)

Oral presenation of project (Friday)

New chapter (Monday)

Faster……..

Projects:Choose between1.2, 1.3, 1.4, 1.6, 1.8And if you choose 1.7 you may have to adjust/add something. Discuss with teachers. (Thursday)

Uno WennergrenStefan SellmanPeter Brommesson

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.