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Spatial models in ecology and evolution, spring 2009


Eva Kisdi


8 op. 2 x 2h lectures every week and 2h exercise classes every other week. Course code: 57392

Type and prerequisites

Advanced studies.

I assume familiarity with basic analysis, linear algebra and probability theory; some experience with mathematical modelling is useful but not indispensable. Some exercises require basic computer programming (Maple, Mathematica, C++, Pascal or any other language). The course starts with an introduction to mathematical ecology, and can be taken as a first course in biomathematics.


Weeks 3-9 and 11-18, Tuesday 12-14 and Thursday 12-14 in room B120.

Easter holiday 9.-15.4.

The lecture of 3 February is cancelled.To make up for this lecture, try to figure out how to establish the asymptotic stability of a fixed point in discrete time systems (also in higher dimensions): pdf. The solution will be discussed in the exercise class of 13 February, but questions are welcome any time.


This course will explore how to model the dynamics and evolution of populations with spatial movement, spatial constraints and spatial interactions between organisms. After a brief introduction to classic mathematical ecology, we study diffusion, travelling waves, pattern formation and Turing instability, stochastic patch occupancy models, structured metapopulation models, probabilistic cellular automata and coupled map lattices. Next, we investigate three topical issues of evolutionary biology where spatial structure plays a crucial role: the evolution of mobility (dispersal); the origin of new species via specialisation to different environments; and the evolution of altruistic behaviour.

This is a course in applied mathematics. Instead of choosing the problem to suit a method, we emphasise the use of versatile techniques, opening the appropriate toolbox to study a problem of interest, and approaching the same problem using different methods. On the way, we shall introduce/review methods to study ordinary differential equations and difference equations, partial differential equations, Fourier analysis, stochastic processes, pair approximation methods, game theory and adaptive dynamics. When necessary, we turn to numerical analysis.

Requirements for course completion 

A written exam or an individual research project with a written report. The latter can be chosen only by those who followed the exercise classes satisfactorily (min 75%).

Exercise groups

Exercise classes are in weeks 5, 7, 9 (period III) and weeks 12, 14, 16, 17 (period IV).








10 - 12


Ilmari Karonen

Homework exercises

Exercises 1-5  (30 January)

stability of fixed points in discrete-time systems (13 February)

Exercises 6-10 (13 February)

Exercises 11-15 (27 February)

Exercises 16-20  (20 March)

Exercises 21-25 (17 April)

Exercises 26-30 (24 April)

Exercises 31-32 (30 April)


Proposed course projects in pdf

Quadratic map:
     Cobweb diagrams: Java applet by Andy Burbanks, Loughborough University
     Plots of iterated maps: pdf
     Bifurcation diagram: Java applet by Richard Dallaway

Coupled logistic map: pdf

Local adaptation in heterogeneuous metapopulations: pdf 

Visual PDE solver by Matti Määttä

Introduction to metapopulation models:  PPTPDF


Lecture notes on the introductory part of the course, by Margarete Utz, are available in pdf.

Cellular automata (MSc thesis) by Ilmari Karonen (gives a good summary of approximation methods): English (full), Finnish (short)

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