Modelling and Quantitative Methods in Fisheries

452 Pages·2001·11.592 MB·English

by  Haddon, Malcolm

Statistics/Biometrics Second
Edition
With numerous real-world examples, Modelling and Quantitative
Methods in Fisheries, Second Edition provides an introduction to
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the analytical methods used by fisheries’ scientists and ecologists.
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By following the examples using Excel, readers see the nuts and
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bolts of how the methods work and better understand the underlying e
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principles. Excel workbooks are available for download from CRC l
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Press Online. g

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In this second edition, the author has revised all chapters and n
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improved a number of the examples. This edition also includes two

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entirely new chapters:
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• Characterization of Uncertainty covers asymptotic errors and
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likelihood profiles and develops a generalized Gibbs sampler to n
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run a Markov chain Monte Carlo analysis that can be used to i
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generate Bayesian posteriors.
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• Sized-Based Models implements a fully functional size-based v
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stock assessment model using abalone as an example.
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This book continues to cover a broad range of topics related to e
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quantitative methods and modelling that have direct relevance to h
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fisheries science, biological modelling, ecology, and population
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dynamics. It offers a solid foundation in the skills required for the s

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quantitative study of marine populations. Explaining important and n

relatively complex ideas and methods in a clear manner, the author F
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presents full, step-by-step derivations of equations as much as s
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possible to enable a thorough understanding of the models and e
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methods. i
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s
Haddon
C561X
C561X_Cover.indd 1 2/9/11 11:18 AM
Modelling and
Quantitative Methods
in Fisheries
Second Edition

Modelling and
Quantitative Methods
in Fisheries
Second Edition
Malcolm Haddon
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2011 by Taylor & Francis Group, LLC
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Version Date: 20150217
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Contents
Preface to the Second Edition ............................................................................xiii
Preface to the First Edition ...................................................................................xv
1 Fisheries and Modelling ...............................................................................1
1.1 Fish Population Dynamics ...................................................................1
1.2 The Objectives of Stock Assessment ..................................................4
1.2.1 Characterizing Stock Dynamics ............................................4
1.2.2 Characterizing Uncertainty ....................................................7
1.2.3 Management Objectives ..........................................................8
1.3 Characteristics of Mathematical Models ...........................................9
1.3.1 General Properties ...................................................................9
1.3.2 Limitations Due to the Modeller ...........................................9
1.3.3 Limitations Due to Model Type ...........................................10
1.3.4 The Structure of Mathematical Models ..............................10
1.3.5 Parameters and Variables .....................................................11
1.4 Types of Model Structure...................................................................11
1.4.1 Deterministic/Stochastic ......................................................11
1.4.2 Continuous versus Discrete Models ...................................13
1.4.3 Descriptive/Explanatory ......................................................13
1.4.4 Testing Explanatory Models.................................................14
1.4.5 Realism/Generality ...............................................................16
1.4.6 When Is a Model a Theory? ..................................................17
2 Simple Population Models .........................................................................19
2.1 Introduction .........................................................................................19
2.1.1 Biological Population Dynamics .........................................19
2.1.2 The Dynamics of Mathematical Models ............................19
2.2 Assumptions—Explicit and Implicit ................................................20
2.2.1 All Assumptions Should Be Explicit ...................................20
2.3 Density-Independent Growth ...........................................................21
2.3.1 Exponential Growth ..............................................................21
2.3.2 Standard Transformations ....................................................23
2.3.3 Why Consider Equilibrium Conditions? ............................23
2.4 Density-Dependent Models ...............................................................25
2.4.1 An Upper Limit and Persistence .........................................25
2.4.2 The Logistic Model of Growth .............................................25
2.4.3 Discrete Logistic Model ........................................................29
2.4.4 Stability Properties ................................................................30
2.4.5 Dynamic Behaviour ...............................................................32
2.5 Responses to Fishing Pressure ..........................................................36
© 2011 by Taylor & Francis Group, LLC v
vi Contents
2.6 The Logistic Model in Fisheries ........................................................38
2.7 Age-Structured Models ......................................................................39
2.7.1 Age-Structured and Exponential Growth Models ...........39
2.7.2 Annual versus Instantaneous Mortality Rates ..................40
2.7.3 Selection of a Target Fishing Mortality...............................42
2.8 Simple Yield-per-Recruit ....................................................................43
2.8.1 Is There an Optimum Fishing Mortality Rate? .................43
2.8.2 What Is the Optimum Age or Size at First Capture? ........44
2.8.3 From Empirical Table to Mathematical Model ..................47
2.8.4 The Model Structure and Assumptions .............................47
2.8.5 The Model Equations ............................................................49
2.8.6 Yield-per-Recruit Management Targets ..............................52
2.8.7 Management Targets and Limits .........................................53
2.8.8 Uncertainties in Yield-per-Recruit Analyses .....................54
2.8.9 Types of Overfishing .............................................................54
3 Model Parameter Estimation ......................................................................57
3.1 Models and Data .................................................................................57
3.1.1 Fitting Data to a Model .........................................................57
3.1.2 Which Comes First, the Data or the Model? ......................58
3.1.3 Quality of Fit versus Parsimony versus Reality ................59
3.1.4 Uncertainty .............................................................................60
3.1.5 Alternative Criteria of Goodness of Fit ...............................61
3.2 Least Squared Residuals ....................................................................61
3.2.1 Introduction ............................................................................61
3.2.2 Selection of Residual Error Structure .................................65
3.3 Nonlinear Estimation .........................................................................65
3.3.1 Parameter Estimation Techniques .......................................65
3.3.2 Graphical Searches for Optimal Parameter Values ..........66
3.3.3 Parameter Correlation and Confounding Effects .............69
3.3.4 Automated Directed Searches .............................................70
3.3.5 Automated Heuristic Searches ............................................71
3.4 Likelihood ............................................................................................72
3.4.1 Maximum Likelihood Criterion of Fit ................................72
3.4.2 The Normal Distribution ......................................................72
3.4.3 Probability Density ................................................................73
3.4.4 Likelihood Definition ............................................................78
3.4.5 Maximum Likelihood Criterion ..........................................81
3.4.6 Likelihoods with the Normal Probability Distribution ...81
3.4.7 Equivalence with Least Squares ..........................................85
3.4.8 Fitting a Curve Using Normal Likelihoods .......................86
3.4.9 Likelihoods from the Lognormal Distribution .................87
3.4.10 Fitting a Curve Using Lognormal Likelihoods .................91
3.4.11 Likelihoods with the Binomial Distribution .....................93
© 2011 by Taylor & Francis Group, LLC
Contents vii
3.4.12 Multiple Observations ...........................................................98
3.4.13 Likelihoods from the Poisson Distribution .....................100
3.4.14 Likelihoods from the Gamma Distribution .....................104
3.4.15 Likelihoods from the Multinomial Distribution .............108
3.5 Bayes’ Theorem .................................................................................110
3.5.1 Introduction ..........................................................................110
3.5.2 Bayes’ Theorem ....................................................................112
3.5.3 Prior Probabilities ................................................................114
3.5.4 An Example of a Useful Informative Prior ......................115
3.5.5 Noninformative Priors ........................................................117
3.6 Concluding Remarks ........................................................................118
4 Computer-Intensive Methods ..................................................................121
4.1 Introduction .......................................................................................121
4.2 Resampling ........................................................................................122
4.3 Randomization Tests ........................................................................123
4.4 Jackknife Methods ............................................................................123
4.5 Bootstrapping Methods ....................................................................124
4.6 Monte Carlo Methods .......................................................................125
4.7 Bayesian Methods .............................................................................126
4.8 Relationships between Methods .....................................................126
4.9 Computer Programming .................................................................128
5 Randomization Tests .................................................................................129
5.1 Introduction .......................................................................................129
5.2 Hypothesis Testing ...........................................................................129
5.2.1 Introduction ..........................................................................129
5.2.2 Standard Significance Testing ............................................130
5.2.3 Significance Testing by Randomization Test ...................132
5.2.4 Mechanics of Randomization Tests ...................................133
5.2.5 Selection of a Test Statistic ..................................................136
5.2.6 Ideal Test Statistics ...............................................................140
5.3 Randomization of Structured Data ................................................141
5.3.1 Introduction ..........................................................................141
5.3.2 More Complex Examples ....................................................142
6 Statistical Bootstrap Methods ..................................................................145
6.1 The Jackknife and Pseudovalues ....................................................145
6.1.1 Introduction ..........................................................................145
6.1.2 Parameter Estimation and Bias ..........................................145
6.1.3 Jackknife Bias Estimation ...................................................150
6.2 The Bootstrap .....................................................................................151
6.2.1 The Value of Bootstrapping ................................................151
6.2.2 Empirical versus Theoretical Probability Distributions ...152
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viii Contents
6.3 Bootstrap Statistics ............................................................................154
6.3.1 Bootstrap Standard Errors ..................................................155
6.3.2 Bootstrap Replicates ............................................................156
6.3.3 Parametric Confidence Intervals .......................................158
6.3.4 Bootstrap Estimate of Bias ..................................................159
6.4 Bootstrap Confidence Intervals .......................................................160
6.4.1 Percentile Confidence Intervals .........................................160
6.4.2 Bias-Corrected Percentile Confidence Intervals ..............160
6.4.3 Other Bootstrap Confidence Intervals ..............................162
6.4.4 Balanced Bootstraps ............................................................163
6.5 Concluding Remarks ........................................................................164
7 Monte Carlo Modelling .............................................................................165
7.1 Monte Carlo Models .........................................................................165
7.1.1 The Uses of Monte Carlo Modelling .................................165
7.1.2 Types of Uncertainty ...........................................................165
7.2 Practical Requirements ....................................................................167
7.2.1 The Model Definition ..........................................................167
7.2.2 Random Numbers ...............................................................167
7.2.3 Nonuniform Random Numbers ........................................168
7.2.4 Other Practical Considerations ..........................................170
7.3 A Simple Population Model .............................................................172
7.4 A Nonequilibrium Catch Curve .....................................................173
7.4.1 Ordinary Catch Curve Analysis ........................................173
7.4.2 The Influence of Sampling Error .......................................176
7.4.3 The Influence of Recruitment Variability .........................180
7.5 Concluding Remarks ........................................................................183
8 Characterization of Uncertainty ..............................................................187
8.1 Introduction .......................................................................................187
8.2 Asymptotic Standard Errors ...........................................................189
8.3 Percentile Confidence Intervals Using Likelihoods .....................192
8.4 Likelihood Profile Confidence Intervals ........................................195
8.5 Percentile Likelihood Profiles for Model Outputs .......................200
8.6 Markov Chain Monte Carlo (MCMC) ............................................206
8.7 Concluding Remarks ........................................................................214
9 Growth of Individuals ...............................................................................215
9.1 Growth in Size ...................................................................................215
9.1.1 Uses of Growth Information ..............................................215
9.1.2 The Data ................................................................................216
9.1.3 Historical Usage ...................................................................217
9.2 Von Bertalanffy Growth Model ......................................................217
9.2.1 Growth in Length ................................................................217
9.2.2 Growth in Weight ................................................................220
© 2011 by Taylor & Francis Group, LLC
Contents ix
9.2.3 Seasonal Growth ..................................................................221
9.2.4 Fitting to Tagging Data .......................................................225
9.2.5 Extensions to Fabens Method ............................................226
9.2.6 Comparability of Growth Curves ......................................229
9.2.7 Growth from Modal Progression ......................................230
9.3 Alternatives to Von Bertalanffy ......................................................231
9.3.1 A Generalized Model ..........................................................231
9.3.2 Model Selection—AIC and BIC ..........................................232
9.3.3 Polynomial Equations .........................................................234
9.3.4 Problems with the von Bertalanffy Growth Function ...234
9.4 Comparing Growth Curves .............................................................235
9.4.1 Nonlinear Comparisons .....................................................235
9.4.2 An Overall Test of Coincident Curves ..............................236
9.4.3 Likelihood Ratio Tests .........................................................238
9.4.4 Kimura’s Likelihood Ratio Test .........................................241
9.4.5 Less than Perfect Data .........................................................243
9.4.6 A Randomization Version of the Likelihood Ratio Test ...244
9.5 Concluding Remarks ........................................................................247
Appendix 9.1: Derivation of the Fabens Version of the
von Bertalanffy Growth Equation ..................................................249
Appendix 9.2: Derivation of the Maximum Likelihood Estimator
for the von Bertalanffy Curve ..........................................................251
10 Stock Recruitment Relationships ............................................................255
10.1 Recruitment and Fisheries ...............................................................255
10.1.1 Introduction ..........................................................................255
10.1.2 Recruitment Overfishing ....................................................255
10.1.3 The Existence of a Stock Recruitment Relationship .......256
10.2 Stock Recruitment Biology ..............................................................257
10.2.1 Properties of “Good” Stock Recruitment Relationships ...257
10.2.2 Data Requirements—Spawning Stock .............................258
10.2.3 Data Requirements—Recruitment ....................................259
10.3 Beverton–Holt Recruitment Model ................................................259
10.3.1 The Equations .......................................................................259
10.3.2 Biological Assumptions/Implications ..............................261
10.4 Ricker Model ......................................................................................262
10.4.1 The Equation .........................................................................262
10.4.2 Biological Assumptions/Implications ..............................264
10.5 Deriso’s Generalized Model ............................................................264
10.5.1 The Equations .......................................................................264
10.6 Residual Error Structure ..................................................................265
10.7 The Impact of Measurement Errors ...............................................269
10.7.1 Appearance over Reality .....................................................269
10.7.2 Observation Errors Obscuring Relationships..................270
10.8 Environmental Influences ................................................................271
© 2011 by Taylor & Francis Group, LLC