1. Modeling Biological Systems Goals n Formulate Models n Mathematical modelling n Biology/Ecology n Computers n Basic Programming n Oral presentation n Plan your work

2. Course Outline n Lecture n Work on project n Oral presenation of project n New chapter

3. Mon16 jan13:15-15:00GalaxenLectUno Wennergren Chapter 0-1 Tue 17 jan 13:15-14:00GalaxenLectStefan SellmanMatlab and excel Thu19 jan15:15-17:00GalaxenSEStefan SellmanAvailable for questions Fri20 jan13:15-15:00GalaxenSEUno WennergrenProject presentations Mon23 jan13:15-15:00GalaxenLectUno WennergrenChapter 2 Wed25 jan13:15-15:00GalaxenSEPeter Brommesson Available for questions Chapter 2 Fri27jan13:15-15:00GalaxenSEUno Wennergren Project presentations Chapter 2 Mon30 jan13.15-15.00GalaxenLectUno WennergrenChapter 3 Tue31 feb13:15-15:00GalaxenSEPeter Brommesson Available for questions Chapter 3 wed1 feb13:15-15:00GalaxenSEUno Wennergren Project presentations Chapter 3 Fri3 feb08:15-10:00GalaxenLectUno WennergrenChapter 5.1 Mon6 feb13:15-15:00GalaxenLectUno WennergrenChapter 5.2 Tue7 feb13:15-15:00GalaxenSEPeter Brommesson Available for questions Chapter 5.1-2 Wed8 feb13:15-15:00GalaxenSEPeter Brommesson Available for questions Chapter 5.1-2 Fri10 feb13:15-15:00GalaxenSEUno Wennergren Project presentations 5.1 and 5.2

4. n Uno Wennergren Professor Theoretical Biology u Organic Farming u Threatened Species u Spread of disease u Animal Welfare u 6 PhD students 5 senior researchers

5. Subjects Chapters in the book n Basic about models n Discrete Processes u Deterministic models u Stochastic models n Continous processes u Deterministic models u (Stochastic models – excluded)

6. Methods/Tools n Graphic methods - Cobweb n Spreadsheets - Excel n Programing - Matlab n Mathematical Analysis

7. Methods/Tools n Planning n PowerPoint n Excel n Oral presentations Computer-OH projector

8. Project n Plan your time, time schedule n Formulate the problem n Choose u Type of mathematical model u What methods and tools to use u How to present the results n Re-plan n Construct the model u If possible use critical test n Implement the model by excel or matlab n Re test the model u If possible use critical test n Make the code and a ppt presentation tidy – presentable to Uno, Stefan and Peter n For whom it may concern: prepare for oral presentation. This year everytime!

9. Basic about Models n A model is a description of reality n A mathematical model uses equations to describe reality n Two levels of modeling Dn/dt=rn(t) Complex reality I Simplified Reality II Mathematical equations

10. Basic about Models n A model usually has a purpose n The questions: u Is the reality simplified enough to be represented by equations? u 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 these questions in your projects

11. Discrete Dynamical Systems n Discrete processes u Events stepwise F perennials reproduction (seeds) 1 time/year n Continous processes u Events all the time F Small mammmals reproduction year around n Perennials survival? n insects reproduction? u in temperate climates?

12. Deterministic models n Models don’t include variation/chance probability. Parameters are constant n All process are the same (within a specific model) and simply a specific chain of events. n The result is deterministic: one value n Stochastic models include variation/chance probability n The result is a set of values u Every test generates a new chain of events with its specific result

13. Recurrence equations (Markov chain) n The equation generates a sequence of numbers n 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….. n Note: specific step lengths

14. Recurrence equations (linear) n General form x(n)=f(x(n-1),x(n-2),….) n 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. n How many initial values (numbers) do you need to start the equation to roll? n Assume simple growth: x(n+1)=Rx(n)

15. Model type: Difference equaions (number sequence) n Of first order: f(x(n-1)) =x(n)-x(n-1) n Compare with differential The derivative of f(x): Rearrange recurrence eq:

16. Box diagram n Simple growth u x(n)-x(n-1)=rx(n-1) u x(n+1)=x(n)(1+r) Population x rx growth Population x bx fecundity (1-s)x deaths i immigration

17. Mathematical analysis n Simplest linear recursive equation x(n+1)=Rx(n) has the solution n x(n)=R n x(0) growths exponentially: R>1 decrease exponentially: 0<R<1 Oscillates R<-1 constant or oscillates if R  0,1,-1  n What about -1<R<0???

18. Spreadsheets n Click and drag n Relative addresses u =C1*B4 n absolute adresses u =$C1*B5 u =$C$1*B5

19. Matematical analysis n Equlibrium points u Will the sequence stop at a point? Comes back to itself. u Is it stable or unstable? Compare with valley and hilltop. n Find and calculate the equlibrium point: n Assume is the equlibrium point test in your equation for example x(n+1)=Rx(n) +a set all for big n Then

20. Matematical analysis n Equlibrium points u x(n+1)=Rx(n) +a gives Note initial value doesn’t effect whre the equlibrium is u The quilibriumpoint is stable if and only if Compare with x n =f(x n-1 )

21. Cobweb Diagram n Graphic method to find the equlibrium points y=x y x y=f(x) Stable equlibrium 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

22. Cobweb diagram n Initial value x* n Next step is y=f(x) y=x y x x* y=f(x)

23. Cobweb diagram n Next step to take is x=y y=x y x y=f(x) x*

24. Cobweb diagram n And then y=f(x) y=x y x y=f(x) x*

25. Cobweb diagram n And then this proceeeds, next step is: x=y y=x y x y=f(x) x*

26. Cobweb diagram n 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

27. Cobweb diagram n If it steps away from the equlibrium then it’s an unstable one. y=x y x y=f(x) x*

28. Linear recurrence equation with constant coefficients n Look for a solution, compare with x(n)=R n x(0) n 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 n How to solve it? u Calculate the roots to the characteristic equation u Matlab funktion r = roots(c)

29. Characteristic equation n Assume the solution: after some calculations: n This is the charactersitisc equation, use Matlab funktion r = roots(c)

30. Characteristic equation n Use Matlab funktion r = roots(c) n Or just try it yourself without compuer….. n for x(n)-2x(n-1)+x(n-2)=0 n The charac equation becomes » r=roots([1 -2 -1]) r =2.4142 -0.4142

31. Charactersitic equation n Roots to x(n)-2x(n-1)+x(n-2)=0 » r=roots([1 -2 -1]) r =2.4142 -0.4142 n General solution is x(n)=C 1 2.4142 n - C 2 0.4142 n n Particular solutions, we know that x(0)=0 and x(1)=1 gives that n C 1 +C 2 =0 which we can use in n 1= C 1 2.4142 - C 2 0.4142 n C 1 =1/2, C 2 =-1/2

32. Charactersitic equation n Roots of x(n)-2x(n-1)+x(n-2)=0 x(n)=C 1 2.4142 n - C 2 0.4142 n C 1 =1/2, C 2 =-1/2 gives particular solution n x(n)=1/2(2.4142 n - 0.4142 n ) for big n the first tem dominates (large absolute value) hence: x(n)  1/2(2.4142 n )

33. Finite limited growth n Simple assumptions Simplified reality u When population is zero there is no reduction in individual growth, no competition, i. e. max growth R u 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)

34. Finite limited growth The curve of individual growth in relation to density shall pass the points: (0,R),(K,0) growth r(x) population x K R Linear model:

35. Growth r(x) population x K R Linear model: 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 Logistic growth

36. 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 This is a second degree equation with roots: Determine the character of the eq. points:: Test:

37. If individual maximum (unlimited) growth, R, is larger or qual to 2 there is no stable eq. and chaos and oscillations will appear.

38. Host parasite model n 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

39. Host parasite model n Host population equaion n The growth of the parasite population also depend on the probability that a host and parasite meet: Assuming proportional to these occasions:

40. Host parasite model n System of non linear difference equations Look for equlibriums Solution (N,P): n (K,0) n (1/Q,R/C(1-1/(QK))) n (0,0)

41. Bloom’s Taxanomy A Hierarcical Knowledge Taxonomy

42. Critical Thinking Activity [arranged lowest to highest] Relevant Sample Verbs Sample AssignmentsSample 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

43. Course Outline n Lecture (+ read the chapter, Monday) n Work on project (Tuesday-Thursday) n Oral presenation of project (Friday) n New chapter (Monday) n Faster…….. Projects: Choose between 1.2, 1.3, 1.4, 1.6, 1.8 And if you choose 1.7 you may have to adjust/add something. Discuss with teachers. (Thursday) Uno Wennergren Stefan Sellman Peter Brommesson

44. n Kunskapstaxonomi fritt efter Benjamin Bloom n Fakta. Ange, räkna upp fakta, definiera begrepp. n Enkel begränsad kunskap. n Beskrivning. Innebörden av begrepp och fakta. Tolka, motivera, relatera till varandra. n Tillämpning. Vad är innehållet användbart till. Observera, beräkna, kalkylera, formulera, konstruera, lösa givna problem. n Analys. Bryta ner innehållet, dela upp, gruppera om, jämföra, generalisera se nya problem. n Syntes. Dra slutsatser, formulera regler, se samband också med annan kunskap, resonera, diskutera, skapa nytt. n Värdering. Avge omdömen, kritisera, värdera olika kunskap, hypoteser och teorier mot varandra. n Komplex, vidsträckt kunskap.