Kaisa_2012_3_photo by Veikko Somerpuro

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 time interval. During the course we learn how to formulate stochastic population models and how to analyze their stochastic behavior in time.

Topics:

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; Ito-calculus; autocorrelation function and spectral density.

Prerequisites:

Ordinary differential equations; elements of real and complex analysis.

Lecture notes and exercises:

Lecture notes and exercises can be found under the heading MATERIAL below. During the course the notes or the exercises may be updated.

Enrol
10.12.2018 at 09:00 - 30.4.2019 at 23:59

Messages

Stefan Geritz

Published, 17.4.2019 at 14:47

About the projects:
The deadline for returning your project is 8 May.
You can hand it in on a later date as well, but then you may get the grade only after the summer break.

Stefan Geritz

Published, 16.4.2019 at 11:09

About supervising the projects:
I realised that it is not very efficient use of the lecture room if I sit there and wait for someone who needs help with the projects. Instead, I propose you send me an email if you need help, and then we agree on a time when we meet.

Stefan Geritz

Published, 15.4.2019 at 15:09

The lecture notes on multi-type birth-death processes (SPM 2019-04-15.pdf) contained some mistakes (in particular about the calculation of the probability of eventual extinction. So, I removed the old file and replaced it by the file SPM 2019-04-15 (NEW).pdf

Stefan Geritz

Published, 15.4.2019 at 9:47

I uploaded the new lecture notes on multi-type birth-death processes (SPM 2019-04-15.pdf). I also uploaded last week's lecture turn notes on timescale separation in a birth-death process (SPM 2019-04-09.pdf).

Stefan Geritz

Published, 9.4.2019 at 7:44

I uploaded the corrected notes "SPM 08-04-2019 (NEW).pdf"
I also uploaded the appendixes "Appendix A.pdf" (linear stability analysis)
and "Appendix PF.pdf" (POerron-Frobenius theorem)

Stefan Geritz

Published, 2.4.2019 at 9:46

I uploaded some projects of the SPM course in the file "SPM Projects.pdf".
We will briefly discuss the projects during the lecture.
The projects are instead of the exam. There will be no exam.
The projects also take the place of the homework exercises from 09-04-2019
onwards.

Stefan Geritz

Published, 26.3.2019 at 16:57

I have just uploaded the lecture notes "SPM 26-03-2019" and the exercises ""Ex 02-04-2019"

Anna Suomenrinne-Nordvik's picture

Anna Suomenrinne-Nordvik

Published, 20.3.2019 at 15:28

Hi all, next weeks exercises can be found in the lecture notes of 19.3.2019. In addition we will do exercise 15 from exercise sheet 15-17 in the Materials section below.

Stefan Geritz

Published, 18.3.2019 at 13:07

I wrote a new intro to section 08 about birth-death processes.
The intro is in file "SPM 18-03-2019.pdf"

Stefan Geritz

Published, 12.3.2019 at 12:57

New lecture notes of 11/03 and 12/03 have been uploaded as the file "SPM 08-03-2019.pdf".
The typos that were discovered during the lectures have been corrected.
In the notes you find the exercises for next week Tuesday.

The lectures of next week will be about demographic stochasticity in small populations, i.e., populations as a generator of stochastic noise.

Pages

Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Mon 14.1.2019
10:15 - 12:00
Tue 15.1.2019
10:15 - 12:00
Tue 15.1.2019
14:15 - 16:00
Mon 21.1.2019
10:15 - 12:00
Tue 22.1.2019
10:15 - 12:00
Tue 22.1.2019
14:15 - 16:00
Mon 28.1.2019
10:15 - 12:00
Tue 29.1.2019
10:15 - 12:00
Tue 29.1.2019
14:15 - 16:00
Mon 4.2.2019
10:15 - 12:00
Tue 5.2.2019
10:15 - 12:00
Tue 5.2.2019
14:15 - 16:00
Mon 11.2.2019
10:15 - 12:00
Tue 12.2.2019
10:15 - 12:00
Tue 12.2.2019
14:15 - 16:00
Mon 18.2.2019
10:15 - 12:00
Tue 19.2.2019
10:15 - 12:00
Tue 19.2.2019
14:15 - 16:00
Mon 25.2.2019
10:15 - 12:00
Tue 26.2.2019
10:15 - 12:00
Tue 26.2.2019
14:15 - 16:00
Mon 11.3.2019
10:15 - 12:00
Tue 12.3.2019
10:15 - 12:00
Tue 12.3.2019
14:15 - 16:00
Mon 18.3.2019
10:15 - 12:00
Tue 19.3.2019
10:15 - 12:00
Tue 19.3.2019
14:15 - 16:00
Mon 25.3.2019
10:15 - 12:00
Tue 26.3.2019
10:15 - 12:00
Tue 26.3.2019
14:15 - 16:00
Mon 1.4.2019
10:15 - 12:00
Tue 2.4.2019
10:15 - 12:00
Tue 2.4.2019
14:15 - 16:00
Mon 8.4.2019
10:15 - 12:00
Tue 9.4.2019
10:15 - 12:00
Tue 9.4.2019
14:15 - 16:00
Mon 15.4.2019
10:15 - 12:00
Tue 16.4.2019
10:15 - 12:00
Tue 16.4.2019
14:15 - 16:00

Material

Description

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Differential equations, probability theory

Elements of complex analysis

Construction and analysis of models in form of non-autonomous differential equations, delay-differential equations, stochastic differential equations; single-type and multi-type birth-death processes; semi-large systems

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

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. Especially in small populations, 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.

Lectures and exercise classes

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later