Matematiikan ja tilastotieteen laitoksen kurssisivualue

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Teacher: Stefan Geritz 

Scope: 10 cr

Type: Advanced studies


This is a course about population models that cannot be properly described or analysed in a purely deterministic way because of the presence of noise. We consider two kinds of noise depending on its origin: The noise may be exogenous, i.e., due to autonomous processes external to the population itself and affecting it by causing population parameters to fluctuate in time. The noise may also be endogenous, i.e., due to stochastic demographic in the number of births and deaths within any given interval of time.


Basic notions in model formulation and analysis: the principle of mass-action; growth and development; equilibria and local stability; elements of the theory of Poincare and Bendixon.

The population as a filter of externally generated noise: ordinary differential equations and delay-differential equations; impulse response; frequency response; transfer function; filter characteristics of the population model.

The population as the source of noise: single-type and multi-type birth-death processes; demographic noise; stochastic processes and ergodicity; the Fokker-Planck equation; stochastic differential equations; autocorrelation function and spectral density.


Ordinary differential equations; elements of real and complex analysis


  • Exam on Wednesday 3 May in room B321 from 14:15 to 16:00;

    Use of lecture notes is allowed. 

Teaching schedule

Weeks 3-9 and 11-19, Tuesday and Thursday 14-16 in room B322. Two hours of exercise classes per week.

Easter holiday 13.-19.4.


Exam on Wednesday 3 May in room B321 from 14:15 to 16:00;

Use of lecture notes is allowed. 

Course material

We largely follow the lecture notes of 2015:

1. SPM 2015 INTRO.pdf

2. SPM 2015 ODE.pdf                                    Demo: SI model.pdf

3. SPM 2015 DDE-1.pdf

4. SPM 2015 DDE-2.pdf                                 Demo: Delayed logistic.pdf

5. SPM 2015 SDE-1.pdf                                 Supp: SDE 1 (supplement).pdf

6. SPM 2011 SDE-2.pdf

7. SPM 2011 STOCH PARAM.pdf                  Demo: Pop+env.noise.nb.pdf

8. SPM 2011 BDP.pdf

9. SPM 2011 SEMI-LARGE POP.pdf

10. SPM 2013 EXT PROB.pdf

11. SPM 2011 MULTI TYPE BDP-1.pdf

12. SPM 2015 MULTI TYPE BDP-2.pdf         Demo: SISdemo.pdf

13. SPM 2011 INVASION.pdf


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Exercise classes

First exercise class on Wednesday 25 January.

1.Wednesday 14-16 B321 Stefan Geritz 

Course feedback

Course feedback can be given at any point during the course. Click here.

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