Klausur zum Modul

Komplexe Systeme in der BiologieProf. Dirk Brockmann

Sommersemester 2015

Institut für theoretische Biologie - Institut für Biophysik

Institut für Biologie

Fakultät für Lebenswissenschaften

Humboldt Universität zu Berlin

22. Juli 2015 - 8:30 - 10:00 Uhr - Bearbeitungszeit 90 min.

• On every problem you can earn 10 points, the maximum score for the test is 100 points.

• Write your name on every sheet of paper (lower left corner).

• If you need additional sheets of paper ask for them.

• Calculators are neither required nor allowed.

• You can earn up to 20 points drawn from your score on the midterm exam.

Name, Matr.Nr.: 1

1. [10 pts.] Below are three images. What process generated these structures? Whatquantity would you employ to quantify differences in their structure?

Name, Matr.Nr.: 2

2. [10 pts.] Below are snapshots of four two-dimensional patterns generated by four differentpattern forming mechanisms (listed below, also). Identity which pattern is generated bywhich process.

a) a system that is driven by local excitation and global inhibition.

b) a lattice system in which, at each time step, a lattice site randomly picks a neighborand changes its own state to the neighbors state.

c) a lattice system of phase-coupled oscillators in which each lattice point is a phaseoscillator that interacts with its neighbors according to the Kuramoto equation.

d) a spreading process near the critial point.

Name, Matr.Nr.: 3

3. [10 pts.] Give one example of critical phenomena and explain how you would design asimulation to implement them on lattice numerically?

Name, Matr.Nr.: 4

4. [10 pts.] Analyse the following dynamical systems graphically, find their fixpoints anddetermine their stability.

a) x = �x

3 + x

b) x = x

2 + x� 2

Name, Matr.Nr.: 5

5. [10 pts.] The Viszek Model desribes the basic mechanism that could underly collectivemotion and swarming. Explain the Viskzek model, the basic ingredients, its propertiesand behavior.

Name, Matr.Nr.: 6

6. [10 pts.] Consider the dynamical system x = f(x) that corresponds to the graph shownbelow. Now consider the dynamical system

x = f(x) + �

where � is a parameter. Sketch the bifurcation diagram as you increase the parameter �qualitatively.

Name, Matr.Nr.: 7

7. [10 pts.] Consider a two dimensional dynamical system x = f(x, y), y = g(x, y). Assumethat it has three fixpoints (x1, y1), (x2, y2) and (x3, y3). Assume that for each fixpoint youhave computed the Jakobian, its trace s and its determinant � and you’ve found that

• for (x1, y1) : s = �1, � = 5

• for (x2, y2) : s = 3, � = 1

• for (x3, y3) : s = 1, � = �2

a) Classify the fixpoints and their stability

b) Draw the phase portrait of a hypthetical dynamical system with these fixpoints, in-clude a few trajectories.

Name, Matr.Nr.: 8

8. [10 pts.] Consider the two-dimensional dynamical system

x = 2� xy

y = x� 2y

a) Find the fixpoint of the system

b) Find its stability and classify the fixpoint

c) Draw the nullclines of the system

d) Sketch the phase portrait

Name, Matr.Nr.: 9

9. [10 pts.] What is the Kuramoto model for coupled oscillators? Discuss the properties andsynchronisation conditions for two oscillators. List biological examples where synchroni-sation is observed.

Name, Matr.Nr.: 10

10. [10 pts.] Assume you want to model a contagion phenomenon by the following interac-tions

A+B

↵�! 2A

A

��! B

a) Derive a system of differential equations of the concentration a(t) and b(t) of A andB “particles” in the system.

b) Show that you can simplify it to a one-dimensional dynamical system by using theconservation of particles (a(t) + b(t) = 1)

c) What are the conditions that a(t) > 0 as t ! 1.

d) How would you implement such a dynamical system on a lattice for example innetlogo. Just sketch your implementation idea, no explicit code necessary.

Name, Matr.Nr.: 11