Stochastic population models, spring 2013
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Lecturer
Scope
10 cu.
Type
Advanced studies
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.
Prerequisites
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.
Lectures
Weeks 39 and 1118, Tuesday 1416 in room B321, Thursday 1416 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 birthdeath processes
9. FokkerPlanck approximation for semilarge systems
10. Multitype birthdeath processes (update is forthcoming)
....
APPENDICES
A1. Massaction 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
....
EXERCISES
Part I:
Exercises 13 ; Exercises 46 ; Exercises 79 ; Exercises 1012 ;
Part II:
Exercises 13; ....
Exams
Bibliography
Registration
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Exercise groups
Group  Day  Time  Place  Instructor 

1.  Friday  1416  C129 