Stochastic population models, spring 2011
News

Summary
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 massaction; 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 delaydifferential equations; impulse response; frequency response; transfer function; filter characteristics of the population model.
The population as the source of noise: singletype and multitype birthdeath processes; demographic noise; stochastic processes and ergodicity; the FokkerPlanck equation; stochastic differential equations; autocorrelation function and spectral density.
Lecturer
Scope
10 cu.
Type
Advanced studies / Applied mathematics / Biomathematics
Prerequisites
Some acquaintance with differential equations would be handy.
Lectures
Weeks 39 and 1118, Tuesday 1012 in room B321, Thursday 1416 C124.
Easter holiday
Lecture notes
PART I: "The population as a filter of external noise"
1. Introduction
2. Fluctuating parameters in singleODE models
3. Delay differential equations (DDE)
4. Fluctuating parameters in singleDDE models
5. Stochastic differential equations (SDE)
7. Models with randomly fluctuating parameters
PART II: "The population as a generator of internally produced noise"
8. Introduction to birthdeath processes
9. Diffusion approximation for semilarge systems
10. Multitype branching processes
APPENDICES
A1. Massaction and the bimolecular reaction between identical particles
A2. Local stability analysis of systems of differential equations
A3. Elements of the theory of Poincare and Bendixon
A4. The theorem of Perron and Frobenius
EXERCISES
Part I:
Part II:
Exams
Bibliography
Registration
Did you forget to register? What to do.
Exercise groups
Group 
Day 
Time 
Place 
Instructor 

1. 
Friday 
1416 
B321 
Ilmari Karonen 