Contents of the thesis

Chapter 1. Theory of the scale transition

1.1 Introduction

1.1.1 The scale transition

1.1.2 Definition of scale

1.1.3 Chesson's general theory of the scale transition: an overview

1.1.4 An example: caddisflies living in riffles

1.2 Nonlinear averaging

1.2.1 Migration among local populations

1.2.2 Graphical illustration: density dependent caddisfly dynamics

1.2.3 Nonlinear averaging: quantitative approximation

1.2.4 Caddisfly dynamics: quantifying the scale transition

1.3 Nonlinear averaging with joint variation

1.3.1 Caddisfly dynamics with variation in disturbance among riffles

1.4 Multiple averaging mechanisms

1.5 Multiple scales of variation

1.5.1 The ecological neighbourhood

1.5.2 The nested variance equations

1.5.3 The concepts of grain and extent

1.6 The variance-mean relationship

1.7 Limitations and generalisations

1.8 Overview and approach of the thesis

1.9 The main equations

Chapter 2. Models

2.1 Introduction

2.2 Main components of the biofilm-grazer system

2.3 A general model for biofilm dynamics at the local scale

2.4 Models of biofilm growth in relation to light intensity

2.4.1 Shading: a mechanistic basis for density dependence

2.4.2 The photosynthesis-irradiance function

2.4.2.1 Empirical functions

2.4.2.2 Mechanistic formulations

2.4.3 Density-dependent models: mechanistic

2.4.3.1 Shading-Poisson model

2.4.3.2 Shading-Michaelis-Menten model

2.4.3.3 Shading-linear PI model

2.4.4 Density-dependent models: phenomenological

2.4.5 Nonlinearities in the production function

2.4.6 Model predictions

2.5 A model of biofilm growth in relation to nutrients

2.5.1 Nutrient diffusion: a mechanistic basis for density dependence

2.5.2 Production function and model predictions

2.6 The GRL model: a general model of resource-limited growth

2.6.1 Relationship of GRL model to mechanistic models

2.7 Effect of temperature on biofilm growth

2.8 Models for the effect of flow velocity on biofilm growth

2.8.1 Shear force: a mechanistic basis for density dependence

2.8.2 Continuous biomass loss

2.8.3 Threshold biomass loss

2.8.4 Predictions: effect of velocity on K

2.9 Linear models of biofilm growth

2.10 Grazing models

2.10.1 Foraging-related, non-consumptive losses

2.10.2 Alternative grazing models

2.11 The scale transition

2.11.1 A regional model with spatial variation in biofilm biomass

2.11.2 Migration and the scale of closure

2.11.3 The scale of the ecological neighbourhood

2.11.4 A regional model with spatial variation in biofilm biomass, photosynthetic rate, and grazer biomass

Chapter 3. Experimental design

3.1 Introduction

3.2 Study site

3.3 Nested sampling and variance components estimation

3.3.1 Variance as a function of scale

3.3.2 Variance components estimation

3.3.2.1 Independent variance estimates

3.3.2.2 Flexibility in choice of scales

3.3.2.3 Variance at intermediate scales is captured

3.4 Experimental design

3.4.1 The basic design

3.4.1.1 Modifications in 1998

3.4.2 Sampling and measurement in 1997

3.4.2.1 Sampling schedule

3.4.2.2 Biological samples

3.4.2.3 Abiotic measurements

3.4.3 Sampling and measurement in 1998

3.4.3.1 Sampling schedule

3.4.3.2 Biofilm samples

3.4.3.3 Grazer samples

3.4.3.4 Abiotic measurements

Chapter 4. Methods

4.1 Introduction

4.2 Biofilm biomass

4.2.1 Sampler

4.2.2 Biofilm biomass (AFDW)

4.2.2.1 Pre-treatment of filters

4.2.2.2 Filtering, drying and weighing

4.2.2.3 Ashing

4.2.2.4 Control of correlated errors and measurement-error variance

4.2.3 AFDW errors

4.2.3.1 Blanks

4.2.3.2 Effect of humidity

4.2.3.3 Conclusions about AFDW errors

4.3 Grazer biomass

4.3.1 Sampling technique and sample processing

4.3.2 Biomass of animals

4.3.3 Surface area of rocks

4.3.3.1 Volume and weight methods

4.3.3.2 Length method

4.3.3.3 Length-circumference method

4.3.3.4 Evaluation of surface-area methods

4.3.3.5 Choice of method

4.4 Grazer exclusion

4.4.1 Exclusion of grazers by electricity

4.4.1.1 Exclusion from rocks

4.4.1.2 Exclusion from paver bricks

4.4.1.3 Control (grazed) rocks and pavers

4.4.2 Effectiveness of grazer exclusion treatment

4.4.2.1 Laboratory observations

4.4.2.2 Field trial: grazer abundance on electrified and control rocks

4.4.2.3 Observations of behaviour in the stream

4.4.2.4 Whole rock grazer samples, 1997 experiment

4.4.2.5 Conclusions and consequences

4.5 Solar radiation

4.5.1 PAR measurement

4.5.1.1 Construction and operation of sensors

4.5.1.2 Spectral characteristics

4.5.1.3 Cosine response

4.5.1.4 Calibration

4.5.1.5 Installation of sensors in the stream bed

4.5.2 Modelling PAR from horizon profiles

4.5.2.1 CLOUDY

4.5.2.2 Calibration of CLOUDY to measured PAR

4.6 Other abiotic variables

4.6.1 Long-term stage and temperature monitoring

4.6.2 Short-term temperature monitoring

4.6.3 Flow velocity

Chapter 5. Local dynamics and nonlinearity

5.1 Introduction

5.1.1 Hypotheses

5.1.1.1Hypothesis 1. Biofilm growth is limited by grazers

5.1.1.2Hypothesis 2. Biofilm growth is nonlinear and depends on biofilm biomass

5.1.1.3Hypothesis 3: Biofilm growth rate and carrying capacity depend on solar radiation

5.1.1.4Hypothesis 4: Biofilm carrying capacity is limited by flow velocity

5.1.1.5Hypothesis 5: Biofilm growth rate depends on temperature

5.1.1.6Hypothesis 6. Biofilm removal rate by grazers is nonlinear and depends on biofilm biomass

5.1.2 Environmental conditions during the experiment in 1997 and 1998

5.2 Methods

5.2.1 Linear mixed-model regression

5.2.2 Fitting models of biofilm dynamics

5.2.2.1 Models for biofilm growth rate

5.2.2.2 Models for biofilm removal rate by grazers

5.2.3 Effects of environmental factors on biofilm growth

5.2.3.1 Regression (REML) modelling

5.2.3.2 Relationship of temperature to growth rate at the biofilm surface

5.2.3.3 Time-adjusted temperature and solar radiation data

5.3 Results

5.3.1 Effect of grazers on biofilm dynamics

5.3.2 Nonlinearity: dynamics of biofilm growth in the absence of grazers

5.3.3 Environmental effects on biofilm dynamics

5.3.3.1 Solar radiation

5.3.3.2 Flow velocity

5.3.3.3 Temperature

5.3.4 Nonlinearity: biofilm removal rate by grazers

5.4 Discussion

5.4.1 Hypotheses

5.4.1.1Hypothesis 1. Biofilm growth is limited by grazers

5.4.1.2Hypothesis 2. Biofilm growth is nonlinear and depends on biofilm biomass

5.4.1.3Potential biases in estimating local nonlinearity and density dependence

5.4.1.4Hypothesis 3: Biofilm growth rate and carrying capacity depend on solar radiation

5.4.1.5Hypothesis 4: Biofilm carrying capacity is limited by flow velocity

5.4.1.6Hypothesis 5: Biofilm growth rate depends on temperature

5.4.1.7Hypothesis 6. Biofilm removal rate by grazers is nonlinear and depends on biofilm biomass

5.4.2 Summary

Chapter 6. Scales of variation

6.1 Introduction

6.1.1 Variance components analysis

6.1.2 Data transformation

6.1.3 Components of dispersion

6.2 Methods

6.2.1 The variance model

6.2.1.1 Independent variance estimates at each scale

6.2.1.2 Checking the suitability of *b* across scales

6.2.2 Variance components

6.2.2.1 REML components of variation

6.2.2.2 GLM dispersion components

6.2.3 Significance of variance components

6.2.4 Negative variance components

6.2.5 Covariance components

6.2.6 Analysis details for different variables

6.2.6.1 Biofilm biomass

6.2.6.2 Grazer biomass

6.2.6.3 Flow and depth

6.2.6.4 Temperature

6.2.6.5 Solar radiation

6.3 Results

6.3.1 Biofilm biomass

6.3.2 Grazers

6.3.3 Environmental factors

6.3.4 Covariance components

6.4 Discussion

6.4.1 The variance model

6.4.2 Scales of variation

6.4.2.1 Biotic variation at the smallest scales

6.4.2.2 Domains of variation for biofilm compared to grazers

6.4.2.3 Spatio-temporal variation

6.4.2.4 Problems with some nested analyses

6.4.2.5 Extensions to tests for spatial structure

6.4.3 Summary

Chapter 7. Synthesis

7.1 Introduction

7.2 The nature of the biofilm-grazer system

7.3 Analysis of the scale transition

7.3.1 Quantifying the scale transition: the regional model

7.3.2 Analysis of the rate of change at a point in time

7.3.2.1 Relative contribution of four terms to the scale transition

7.3.2.2 The effect of scale: the scale of the ecological neighbourhood

7.3.2.3 The effect of scale: the scale of closure

7.3.2.4 The effect of grazer mobility on the scale transition

7.3.3 Analysis of dynamics over short time scales

7.3.3.1 Functions in the regional model compared to the mean-field model

7.3.3.2 The scale transition for the equilibrium biomass

7.3.4 Summary of analyses

7.4 Limitations and future directions

7.4.1 The dynamics of variation

7.4.2 Temporal variation in local nonlinearity

7.4.3 Accuracy of the quadratic approximation

7.4.4 Multiple grazer types

7.4.5 The scale transition in open systems

7.4.6 Statistical methods

References

**Appendix 1.** Grazers of the Bimberamala and Yadboro catchments

**Appendix 2.** Derivation of a regional model of biofilm dynamics

**Appendix 3.** Sampling schedule for 1998

**Appendix 4.** Derivation of the length-circumference method

**Appendix 5.** Evaluation of the accuracy of the quadratic approximation