Ecological Modelling and Ecophysics: Agricultural and environmental applications

PRELIMS.pdf
	Preface
		References
	Acknowledgements
	Author biography
		Hugo Fort
CH000.pdf
	Chapter 0 Introduction
		0.1 The goal of ecology: understanding the distribution and abundance of organisms from their interactions
		0.2 Mathematical models
			0.2.1 What is modelling?
			0.2.2 Why mathematical modelling?
			0.2.3 What kind of mathematical modelling?
			0.2.4 Principles and some rules of mathematical modelling
		0.3 Community and population ecology modelling
			0.3.1 Parallelism with physics and the debate of the ‘biology-as-physics approach’
			0.3.2 Trade-offs and modelling strategies
		References
CH001.pdf
	Chapter 1 From growth equations for a single species to Lotka–Volterra equations for two interacting species
		Summary
		1.1 From the Malthus to the logistic equation of growth for a single species
			1.1.1 Exponential growth
			1.1.2 Resource limitation, density dependent per-capita growth rate and logistic growth
		1.2 General models for single species populations and analysis of local equilibrium stability
			1.2.1 General model and Taylor expansion
			1.2.2 Algebraic and geometric analysis of local equilibrium stability
		1.3 The Lotka–Volterra predator–prey equations
			1.3.1 A general dynamical system for predator–prey
			1.3.2 A first model for predator–prey: the original Lotka–Volterra predator–prey model
			1.3.3 Realistic predator–prey models: logistic growth of prey and Holling predator functional responses
		1.4 The Lotka–Volterra competition equations for a pair of species
			1.4.1 A descriptive or phenomenological model
			1.4.2 Stable equilibrium: competitive exclusion or species coexistence?
			1.4.3 Transforming the competition model into a mechanistic model
		1.5 The Lotka–Volterra equations for two mutualist species
		Exercises
			Exercise 1.1
			Exercise 1.2
			Exercise 1.3
			Exercise 1.4
			Exercise 1.5
			Exercise 1.6
			Exercise 1.7
			Exercise 1.8
			Exercise 1.9
			Exercise 1.10
			Exercise 1.11
			Exercise 1.12
			Exercise 1.13
			Exercise 1.14
			Exercise 1.15
		References
CH00A1.pdf
	Chapter A1 Extensive livestock farming: a quantitative management model in terms of a predator–prey dynamical system
		A1.1 Background information: the growing demand for quantitative livestock models
		A1.2 A predator–prey model for grassland livestock or PPGL
			A1.2.1 What is our goal?
			A1.2.2 What do we know? and what do we assume?: identifying measurable relevant variables for grass and animals
			A1.2.3 How? Adapting a predator–prey model
			A1.2.4 What will our model predict?
		A1.3 Model validation
			A1.3.1 Are predictions valid?
			A1.3.2 Sensitivity analysis
			A1.3.3 Verdict: model validated
		A1.4 Uses of PPGL by farmers: estimating gross margins in different productive scenarios
		A1.5 How can we improve our model?
		MATLAB codes
			Main code: LVPPGL_Ap1%
			Function ‘Digest’
		References
CH002.pdf
	Chapter 2 Lotka–Volterra models for multispecies communities and their usefulness as quantitative predicting tools
		Summary
		2.1 Many interacting species: the Lotka–Volterra generalized linear model
		2.2 The Lotka–Volterra linear model for single trophic communities
			2.2.1 Purely competitive communities
			2.2.2 Single trophic communities with interspecific interactions of different signs
			2.2.3 Obtaining the parameters of the linear Lotka–Volterra generalized model from monoculture and biculture experiments
		2.3 Food webs and trophic chains
		2.4 Quantifying the accuracy of the linear model for predicting species yields in single trophic communities11This section is based on Fort (2018a).
			2.4.1 Obtaining the theoretical yields: linear algebra solutions and simulations
			2.4.2 Accuracy metrics to quantitatively evaluate the performance of the LLVGE
			2.4.3 The linear Lotka–Volterra generalized equations can accurately predict species yields in many cases
			2.4.4 Often a correction of measured parameters, within their experimental error bars, can greatly improve accuracy
		2.5 Working with imperfect information
			2.5.1 The ‘Mean Field Matrix’ (MFM) approximation for predicting global or aggregate quantities
			2.5.2 The ‘focal species’ approximation for predicting the performance of a given species when our knowledge on the set of parameters is incomplete
		2.6 Conclusion
		Exercises
			Exercise 2.1. Redoing calculations for the experiment involving four species of winter annuals plants of Rees et al (1996)
			Exercise 2.2.
			Exercise 2.3 (from Goh 1977)
			Exercise 2.4. A Lyapunov function for the Lotka–Volterra competition equations with symmetric interaction coefficients
			Exercise 2.5. The reference point for the modified index of agreement d1
			Exercise 2.6. Working with relative yields rather than yields
			Exercise 2.7. Using the mean field approximation for predicting the RYT of a BIODEPTH 32 plant species experiment
			Exercise 2.8. An example of application of the focal approximation
		References
CH00A2.pdf
	Chapter A2 Predicting optimal mixtures of perennial crops by combining modelling and experiments
		A2.1 Background information
		A2.2 Overview
		A2.3 Experimental design and data
		A2.4 Modelling
			A2.4.1 Model equations
			A2.4.2 Data curation
			A2.4.3 Initial parameter estimation from experimental data
			A2.4.4 Adjustment of the initial estimated parameters to meet stability conditions
			A2.4.5 On the types of interspecific interactions
		A2.5 Metrics for overyielding and equitability
		A2.6 Model validation: theoretical versus experimental quantities
			A2.6.1 Qualitative check: species ranking
			A2.6.2 Quantitative check I: individual species yields
			A2.6.3 Quantitative check II: overyielding, total biomasses and equitability
			A2.6.4 Verdict: model validated
		A2.7 Predictions: results from simulation of not sown treatments
			A2.7.1 Similarities and differences between theoretical results for sown and not sown polycultures
			A2.7.2 Using the model for predicting optimal mixtures
		A2.8 Using the model attempting to elucidate the relationship between yield and diversity
			A2.8.1 Positive correlation between productivity and species richness.
			A2.8.2 No significant correlation between productivity and SE
		A2.9 Possible extensions and some caveats
		A2.10 Bottom line
		MATLAB code
		References
CH003.pdf
	Chapter 3 The maximum entropy method and the statistical mechanics of populations
		Summary
		3.1 Basics of statistical physics
			3.1.1 The program of statistical physics
			3.1.2 Boltzmann–Gibbs maximum entropy approach to statistical mechanics
		3.2 MaxEnt in terms of Shannon’s information theory as a general inference approach
			3.2.1 Shannon’s information entropy
			3.2.2 MaxEnt as a method of making predictions from limited data by assuming maximal ignorance
			3.2.3 Inference of model parameters from the statistical moments via MaxEnt
		3.3 The statistical mechanics of populations
			3.3.1 Rationale and first attempts
			3.3.2 Harte’s MaxEnt theory of ecology (METE)33This subsection devoted to METE is based on chapter 7 of Harte’s Maximum entropy and ecology (2011).
		3.4 Neutral theories of ecology
		3.5 Conclusion
		Exercises
			Exercise 3.1. An alternative way to obtain that the Lagrange multiplier of the Boltzmann distribution is λ1=1/kBT.
			Exercise 3.2.
			Exercise 3.3.
			Exercise 3.4.
			Exercise 3.5.
			Exercise 3.6. A toy community
			Exercise 3.7.
			Exercise 3.8.
			Exercise 3.9.
			Exercise 3.10.
			Exercise 3.11.
			Exercise 3.12.
		References
CH00A3.pdf
	Chapter A3 Combining the generalized Lotka–Volterra model and MaxEnt method to predict changes of tree species composition in tropical forests
		A3.1 Background information
		A3.2 Overview
		A3.3 Data for Barro Colorado Island (BCI) 50 ha tropical Forest Dynamics Plot
			A3.3.1 Some facts about BCI
			A3.3.2 Covariance matrices and species interactions
		A3.4 Modelling
			A3.4.1 Inference of the effective interaction matrix from the covariance matrix via MaxEnt
			A3.4.2 Model equations
		A3.5 Model validation using time series forecasting analysis
			A3.5.1 Estimation of intrinsic growth rates and carrying capacities using a training set of data
			A3.5.2 Generating predictions to be contrasted against a validation set of data
			A3.5.3 Verdict: model validated
		A3.6 Predictions
		A3.7 Extensions, improvements and caveats
		A3.8 Conclusion
		MATLAB code
		References
CH004.pdf
	Chapter 4 Catastrophic shifts in ecology, early warnings and the phenomenology of phase transitions
		Summary
		4.1 Catastrophes
			4.1.1 Catastrophic shifts and bifurcations
			4.1.2 A simple population (mean field) model with a catastrophe
		4.2 When does a catastrophic shift take place? Maxwell versus delay conventions
		4.3 Early warnings of catastrophic shifts22This section is mostly a summary of the material presented in chapter 9 of Gilmore (1981).
		4.4 Beyond the mean field approximation
			4.4.1 Spatial model: cellular automaton
			4.4.2 Early warning signals
		4.5 A comparison with the phenomenology of the liquid–vapor phase transition
			4.5.1 Beyond the ideal gas: the van der Waals equation of state for a fluid and its formal correspondence with the grazing model
			4.5.2 Similarities and differences between desertification and the liquid–vapor transition
		4.6 Final comments
		Exercises
			Exercise 4.1. Spruce budworm model
			Exercise 4.2. The ‘fold curve’ of the Spruce budworm model in the control parameter space c–K
			Exercise 4.3. A simple model with a fold bifurcation
			Exercise 4.4.
			Exercise 4.5. Critical slowing down
			Exercise 4.6. Anomalous variance
			Exercise 4.7. Simulating an environmental shift
			Exercise 4.8.
		References
CH00A4.pdf
	Chapter A4 Modelling eutrophication, early warnings and remedial actions in a lake
		A4.1 Background information
		A4.2 Overview
		A4.3 Data for Lake Mendota
		A4.4 Modelling1
			A4.4.1 The Mendota Lake cellular automaton
			A4.4.2 Catastrophic shifts in lakes and their spatial early warnings
		A4.5 Model validation
			A4.5.1 Simulations and results
			A4.5.2 Verdict: model validated, but…
		A4.6 Usefulness of the early warnings
		A4.7 Extensions, improvements and caveats
		FORTRAN 77 code
		References
APP1.pdf
	Chapter
		A.1 Local and global stability
		A.2 Stability for two-dimensional systems
		A.3 Some general theorems1
		References
APP2.pdf
	Chapter
		Additional Fermi problems
		References