Matematiikan ja tilastotieteen laitoksen kurssisivualue

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Stochastic population models, spring 2013




Stefan Geritz


10 cu.


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. This noise may be exogenous, i.e., due to autonomous processes external to the population itself but nevertheless affecting it by causing population parameters to fluctuate in time. The noise may also be endogenous, i.e., due to stochastic demographic fluctuations in the number of births and deaths within any given time interval.

The course addresses the following issues:

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.


Some acquaintance with ordinary differential equations, complex analysis and probability theory would come in handy.

All the other concepts mentioned in the Summary above will be introduced during the course, and no prior acquaintance is required here.


Weeks 3-9 and 11-18, Tuesday 14-16 in room B321, Thursday 14-16 in room B322.

Easter holiday 28.3.-3.4.

Lecture notes

(We'll be largely using the lecture notes of the SPM 2011 course which can be found here.)

PART I: "The population as a filter of external noise"

1. Introduction

2. Fluctuating parameters in single ODE population models 

2½. Fourier integral transform and systems of ODEs (Sorry, the lecture notes "2½" of Tuesday are not ready yet, but most of the info you find in the appendix A2 (local stability analysis) and in the lecture notes "4" (Fourier transform))

3. Delay differential equations (DDE)

4. Fluctuating parameters in DDEs

5. Stochastic differential equations (SDEs) 

6. Fluctuation statistics of stationary processes

7. Populations with stochastic parameters

PART II: "The population as a generator of internally produced noise"

8. Introduction to birth-death processes

9. Fokker-Planck approximation for semi-large systems

10. Multi-type birth-death processes (update is forthcoming)



A1. Mass-action and the bimolecular reaction between identical particles

A2. Local stability analysis in systems of ODEs

A3. Numerical integration of stochastic differential equations

A4. Numerical analysis of the stochastic delayed logistic equation

A5. Table of Fourier transform pairs



Part I:

Exercises 1-3 ; Exercises 4-6 ; Exercises 7-9 ; Exercises 10-12

Part II:

Exercises 13; ....




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Thomas Vallier

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