## Introduction to Systems Analysis: Mathematically Modeling Natural Systems (I)

“This book was originally developed alongside the lecture Systems Analysis at the Swiss Federal Institute of Technology (ETH) Zürich, on the basis of lecture notes developed over 12 years. The lecture, together with others on analysis, differential equations and linear algebra, belongs to the basic mathematical knowledge imparted on students of environmental sciences and other related areas at ETH Zürich. […] The book aims to be more than a mathematical treatise on the analysis and modeling of natural systems, yet a certain set of basic mathematical skills are still necessary. We will use linear differential equations, vector and matrix calculus, linear algebra, and even take a glimpse at nonlinear and partial differential equations. Most of the mathematical methods used are covered in the appendices. Their treatment there is brief however, and without proofs. Therefore it will not replace a good mathematics textbook for someone who has not encountered this level of math before. […] The book is firmly rooted in the algebraic formulation of mathematical models, their analytical solution, or — if solutions are too complex or do not exist — in a thorough discussion of the anticipated model properties.”

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I finished the book yesterday – here’s my goodreads review (note that the first link in this post was not to the goodreads profile of the book for the reason that goodreads has listed the book under the wrong title). I’ve never read a book about ‘systems analysis’ before, but as I also mention in the goodreads review it turned out that much of this stuff was stuff I’d seen before. There are 8 chapters in the book. Chapter one is a brief introductory chapter, the second chapter contains a short overview of mathematical models (static models, dynamic models, discrete and continuous time models, stochastic models…), the third chapter is a brief chapter about static models (the rest of the book is about dynamic models, but they want you to at least know the difference), the fourth chapter deals with linear (differential equation) models with one variable, chapter 5 extends the analysis to linear models with several variables, chapter 6 is about non-linear models (covers e.g. the Lotka-Volterra model (of course) and the Holling-Tanner model (both were covered in Ecological Dynamics, in much more detail)), chapter 7 deals briefly with time-discrete models and how they are different from continuous-time models (I liked Gurney and Nisbet’s coverage of this stuff a lot better, as that book had a lot more details about these things) and chapter 8 concludes with models including both a time- and a space-dimension, which leads to coverage of concepts such as mixing and transformation, advection, diffusion and exchange in a model context.

How to derive solutions to various types of differential equations, how to calculate eigenvalues and what these tell you about the model dynamics (and how to deal with them when they’re imaginary), phase diagrams/phase planes and topographical maps of system dynamics, fixed points/steady states and their properties, what’s an attractor?, what’s hysteresis and in which model contexts might this phenomenon be present?, the difference between homogeneous and non-homogeneous differential equations and between first order- and higher-order differential equations, which role do the initial conditions play in various contexts?, etc. – it’s this kind of book. Applications included in the book are varied; some of the examples are (as already mentioned) derived from the field of ecology/mathematical biology (there are also e.g. models of phosphate distribution/dynamics in lakes and models of fish population dynamics), others are from chemistry (e.g. models dealing with gas exchange – Fick’s laws of diffusion are e.g. covered in the book, and they also talk about e.g. Henry’s law), physics (e.g. the harmonic oscillator, the Lorenz model) – there are even a few examples from economics (e.g. dealing with interest rates). As they put it in the introduction, “Although most of the examples used here are drawn from the environmental sciences, this book is not an introduction to the theory of aquatic or terrestrial environmental systems. Rather, a key goal of the book is to demonstrate the virtually limitless practical potential of the methods presented.” I’m not sure if they succeeded, but it’s certainly clear from the coverage that you can use the tools they cover in a lot of different contexts.

I’m not quite sure how much mathematics you’ll need to know in order to read and understand this book on your own. In the coverage they seem to me to assume some familiarity with linear algebra, multi-variable calculus, complex analysis (/related trigonometry) (perhaps also basic combinatorics – for example factorials are included without comments about how they work). You should probably take the authors at their words when they say above that the book “will not replace a good mathematics textbook for someone who has not encountered this level of math before”. A related observation is also that regardless of whether you’ve seen this sort of stuff before or not, this is probably not the sort of book you’ll be able to read in a day or two.

I think I’ll try to cover the book in more detail (with much more specific coverage of some main points) tomorrow.

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