Description:

A Practical Guide to Ecological Modelling Karline Soetaert and Peter M.J. Herman Netherlands Institute of Ecology, Yerseke, The Netherlands Dr. Karline Soetaert Netherlands Institute of Ecology Centre for Estuarine & Marine Ecology (NIOO-CEME) 4400 AC Yerseke, PO Box 140, The Netherlands k.soetaert@nioo.knaw.nl Dr. Peter M.J. Herman Netherlands Institute of Ecology Centre for Estuarine & Marine Ecology (NIOO-CEME) PO Box 140, 4400 AC Yerseke, The Netherlands p.herman@nioo.knaw.nl Additional material, the R-examples and the R-code of all figures, is available as an R-package (ecolMod), which can be found on the official R-website (http://cran.r-project.org). The R-example files are also available on the website of this book at www.springer.com. ISBN: 978-1-4020-8623-6 e-ISBN: 978-1-4020-8624-3 Library of Congress Control Number: 2008933286 c Springer Science+Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover illustration: View at the Oosterschelde from the Netherlands Institute of Ecology, Yerseke, The Netherlands. Photograph taken by Thomas Haverkamp. Printed on acid-free paper 987654321 springer.com Preface Why Another Ecological Modelling Book? For several years we have taught courses on ecological modelling at the level of graduates or starting PhD students. The audience typically consists of students from biology, geology, bio-engineering, and less frequently from sciences such as physics and chemistry. For most of these students our course was a first acquaintance with the field of ecological mathematical modelling. Although often difficult, it was an intellectual adventure both for them and for us. The course was set up as an initiation to the subject, starting from the most basic principles but nevertheless leading to quite advanced applications. This book is based on lecture notes that were written in 2001 and accompany this modelling course. We prepared the original notes because we felt there was a shortage of a book covering all the material that we wanted to address in our lectures. The notes also served as a readable text that the students can consult while preparing for the exam. This allowed us to teach the theory in rather short lessons, leaving more time to cover practical exercises during which we directly interacted with the students individually. This book is written for young researchers who want to get more out of their data than just description. Even when they see the possibilities of modelling to help them gain insight into the processes they study, two factors might frighten them away from this path: one is mathematical formalism, theorems and detailed proofs, the bread and butter of the applied mathematician; the other is complicated, semantic and near-philosophical ecological theory. We have steered away from these two extremes. We have tried to write a book that is readable for an audience with a basic formation in ecology and a basic knowledge of mathematics. Although enough material is presented that may also interest the more experienced ecologic modeller, it is not a book only readable for either full-blown mathematicians or ecological theoreticians. Throughout the book we have tried to be practical, emphasizing the diversity that exists in mathematical models and techniques. We discuss only the essential aspects of mathematical methods without pretension to mathematical rigour: often one does not need to understand the fine details of a technique to correctly apply it, but it helps greatly to have an intuitive understanding of its foundations. This is where our emphasis has been placed. Despite our preference for practical and simple approaches, fact is that the basic methods of solution are often adequate only for the most simple models. As application of more efficient and complex mathematics may considerably speed up solutions and thus avoid frustration, we do not neglect to mention some more advanced techniques that can make the life of a modeller so much more pleasant. Ecological modelling has multiple roots. Many theoretical ecological models go back in some way to the pioneering work of Lotka and Volterra. A new approach, aiming at environmental modelling, was based on engineering principles from about the 1960’s onwards. We have taken this approach as a starting point for the book because it is much more directly based on conservation laws and therefore an easier vehicle to explain principles underlying modelling. However, we have tried to bridge, from there, to the more classical ecological approach in later sections of the book. Scope and Content A brief mention of the book’s contents reveals its scope. We start by giving arguments as to why models are useful, from the scientific point of view as well as with respect to management. This sets the scene and explains why we are doing what we are doing. Here we introduce some model applications, both simple and complex, that will be expanded on further in the book. After having introduced the semantics of models, we then proceed with the basic principles of transferring ecology into equations. This is where our book differs most from other books, which generally assume that such knowledge is already available or can be deduced from the rather complex examples these books contain. During our lectures we became aware that the mental switch from descriptive to process-based thinking is the largest leap for most of our students; it is not the maths. Therefore we spend much effort explaining and detailing the formulation of ecological interactions. Next we deal with how space can be incorporated into the mass equations. We concentrate on one-dimensional problems with various geometries, with a short excursion to three dimensions. We then continue with the mathematical solution of the models, mentioning where applicable possible sources of difficulty and error. This section deals with differential and numerical calculus; basic mathematical concepts are introduced as they are needed and compiled in an appendix. This is definitely the most demanding part of the book but necessary to put theory into practice. In a next chapter, the derivation of the steady-state solution and subsequent analysis of its properties introduces concepts such as stability, domains of attraction, multiple stable states and bifurcation. Thus far, the models that were discussed fall into the category of deterministic differential equations. In the remaining part of the book, some other types of models are dealt with: they include difference (discrete time) equations, dynamic matrix models, and sequential decision models, also known as dynamic programming models. As it is essential for making robust modelling applications, we generally spend a lot of time during our practical courses on designing, testing, validating and improving ecological equations. This is the topic of the final book chapter, which discusses various techniques for analysing model behaviour. Each chapter is organised as follows: an introduction sets the scene of what is to follow; if relevant, it puts the chapter in perspective with respect to previous chapters. The first sections give the basics and theory, if appropriate illustrated by (simple) examples. Certain sections (starred) probe beyond the elementary level and may be skipped at first reading. We also find it important to actually show how to implement models so that the reader can acquire hands-on experience. Thus, each chapter includes case studies that illustrate (nearly) all methods discussed in the main text and put the theory into practice. Where possible, we chose published models that are simple enough and were amongst the first in their kind, to illustrate concepts. The code to run these examples, implemented in the R computer language, is included and discussed in the book and can be found on the accompanying website or on the official R-website (see below). R, the Modelling Platform Used in this Book For those who are being initiated into the field, the learning of a new (programming) language, on top of the new sets of principles that surround mathematical modelling may be very demanding. Therefore, during our practical courses, problems have been kept simple such that students can implement them in a spreadsheet, a software package that most of them know or should know. These exercises and their solutions can be found on the accompanying website. For this book, we have taken a different approach and use R for our examples mainly because it is free software; it is rapidly gaining popularity; R-code is highly readable; and we simply like it. Although R was not originally developed to be used as a modelling tool, it is very well suited for this task. In our day-to-day work, we use R mainly to develop simple models or to visualise model output. We also use R to interface with compiled models written in Fortran. R is then used for post-processing the model output (making graphs, creating summaries, performing tests...). As the use of the R-language is growing rapidly, students are now becoming acquainted with R during their statistical courses. We expect (or hope) that it is only a matter of time before the use of spreadsheets in introductory modelling courses can be avoided. In an appendix, we give a very short introduction to R. A more extensive introduction can be found on the book’s website. ix Ключевые слова: springer media, implicit euler, vector, density, dynamic, simple, loaded rst, set, cylindrical isosurfaces, c t, cell, vice versa, conceptual diagram, data, vect, ecology, food, spherical isosurfaces, parameter, based, internal logic, tend, simulated annealing, problem, zero-concentration gradient, rate, db, ?ux, physical conditions, ci, ecological interactions, monod kinetics, individual, stable, squared residual, statistical distributions, steady, surface, taylor expansion, equilibrium point, dt, initial condition, equilibrium, inowing water, instance, models, labile doc, code, dr, scientic computing, numerical integration, physiological apparatus, bw, rate change, day, spatial coordinate, -cm, stability, reaction, par, numerical approximation, package, length, prey, multispecies assemblage, sinh, ?rst, age, differential, substance, interaction, type, modelling, matrix, start, note, xi, cosh, rst coincides, predatorprey couple, reasonable approximation, time step, shaded area, linear, estuaries rivers, photosynthetic apparatus, equal, phase plane, solution, water, ecological, cylindrical coordinate, dt dy, organic, suboxic condition, process, black dot, ?tness, method, ri, ni, -ricker, normal distribution, mass, boundary conditions, numerical, population, plot, organic matter, carbon, complex, appendix, ht pt, analytical methods, box, differential equation, subsequent analysis, cost, animal, differential equations, negative, mathematical routine, ni dt, low-level languages, mathematical, organism, total, constant, depth, column, et, change, term, result, biomass, dipsacus sylvestris, equation, ha, solve, function, lorenz equations, spatial, ebx, predator, eax, pt, concentration, unit, layer, robins condition, step, chapter, eal, integration, patch, light, functional dependency, r-package desolve, growth, position, ri-values, carrying capacity, number, variable, nt, dt dn, analysis, order, point, formulation, final fitness, soetaert, dispersion distance, exp, boundary, increase, van cappellen, work, space, phytoplankton, zero-gradient boundary, steady-state, model, case, oxygen, small, transport, vertical prole, basal respiration, c dt, solar radiation, functional responses, initial, oo oo, schematic representation, dx dt, -runif, user-dened functions, numerical errors, grid search, time, interface, sediment, error, cylindrical, journal, xi l, output, overlying water, condition, everyday life, mathematical formulations, zooplankton