Stochastic population models, spring 2011

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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 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.

Lecturer

Stefan Geritz

Scope

10 cu.

Type

Advanced studies / Applied mathematics / Biomathematics

Prerequisites

Some acquaintance with differential equations would be handy.

Lectures

Weeks 3-9 and 11-18, Tuesday 10-12 in room B321, Thursday 14-16 C124.

Easter holiday

Lecture notes

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

1. Introduction

2. Fluctuating parameters in single-ODE models

3. Delay differential equations (DDE)

4. Fluctuating parameters in single-DDE models

5. Stochastic differential equations (SDE)

6. Fluctuation statistics

7. Models with randomly fluctuating parameters

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

8. Introduction to birth-death processes

9. Diffusion approximation for semi-large systems

10. Multi-type branching processes

APPENDICES

A1. Mass-action 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:

Exercises 1-3 

Exercises 4-5 

Exercises 6

Exercise 7-9

Exercise 10-15

Part II:

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