2021-2022 / OCEA0229-1

# Mathematical analysis and modelling methods applied to the environment / Introduction to marine ecosystems modelling

## Mathematical analysis and modelling methods applied to the environment

### Durée

Introduction to marine ecosystems modelling : 15h Th, 15h Pr
Mathematical analysis and modelling methods applied to the environment : 20h Th, 20h Pr

### Nombre de crédits

 Master en océanographie, à finalité (Erasmus Mundus ECT+ : Environmental Contamination and Toxicology) 6 crédits

### Enseignant

Introduction to marine ecosystems modelling : Marilaure Grégoire
Mathematical analysis and modelling methods applied to the environment : Marilaure Grégoire

Langue anglaise

### Organisation et évaluation

Enseignement au premier quadrimestre, examen en janvier

### Unités d'enseignement prérequises et corequises

Les unités prérequises ou corequises sont présentées au sein de chaque programme

### Contenus de l'unité d'enseignement

Introduction to the basics of environmental modelling with exercises in R.
Part Mathematical analysis:
The course will involve the following chapters:
1) Concepts and tools of mathematical analysis: revision of basic mathematics: function, limit and asymtotic behavior, derivative function (simple, composite and material, Taylor expansion), primitive and integration, basics of modelling (mass balance equation), (moving) averaging of continuous function, ..Remediation exercises will be organized.
2) Dimensional analysis: dimensions, principle of dimensional homogeneity, characteristic length and time scales. Dimensional analysis, Pi theorem, systematic determination of dimensionless products, ..
3) Interpolation: unidimensional and multi-dimensional interpolation, linear estimation, objective analysis,
4) Analysis of time series: generalities, Fourier series and transform, filtering,
5) Dynamical modelling with one equation: the Malthusian growth model, Verhulst logistic model, equilibrium and stability, linear perturbation analysis, solution of basic ordinary differential equations,
6) Dynamic modelling with interactions: modelling of biochemical transformation, composed reactions, prey-predator, species competition, search for steady state solution, space phase analysis, and analyze the stability (linear perturbation, determination of the Jacobian matrix). R exercises will be organized.
7) Modelling with partial differential equations: continuity equations, advection-diffusion equation in 3D and 1D , spectral window, ..
Part marine modeling:
CHAPTER 1 Introduction

• What is a model?
• Why do we need models?
CHAPTER 2 Model formulation

• Conceptual model
• Mathematical model formulation
• Formulation of ecological interactions
• Chemical reactions
• Inhibition
• Coupled model equations
• Impact of physical conditions
CHAPTER 3 Spatial components

• Taonomy of spatial models
• Spatial boundary conditions
• Example: competitive interactions in a lattice model
CHAPTER 4 Parameterisation

• In situ measurement
• Literature-Derived parameters
• Calib_ters.43
CHAPTER 5 Model solution

• Initial conditions
• Analytical solutions of differential equations
• Numerical solution of differential equations
CHAPTER 6 Testing and validating the model

• Dimensional homogeneity and consistency of units
• Conservation of energy and mass
• Testing the correctness of the model solution
• Testing the internal logic of the model
• Model verification
• Model validity
• Model sensitivity
• Example_74
• Example of the conservation principle: a mass budget of a marine bay
CHAPTER 7 Taxonomy of ecological models

• Strategic versus tactic models
• Continuous and discrete time models
• Deterministic and stochastic models
• Density-biomass specific models
• Physiological - individual-based - population - ecosystem models
• Example: growth of a Daphnia individual
CHAPTER 8 Appendices

• Taxonomy of differential equations
• Solving difference equations
CHAPTER 9 Books for further reading

#### Introduction to marine ecosystems modelling

Introduction to the basics of environmental modelling with exercices in R.
Part Mathematical analysis:
The course will involve the follwoing chapters:
1) Concepts and tools of mathematical analysis: revision of basic mathematics: function, limit and asymtotic behavior, derivative function (simple, composite and material, taylor expansion), primitive and integration, basics of modelling (mass balance equation), (moving) averaging of continuous function, ..Remediation exercices will be organized.
2) Dimensional analysis: dimensions, principle of dimensional homogeneity, characteristic length and time scales. Dimensionnal analysis, Pi theorem, systematic determination of dimensionless products, ..
3) Interpolation: unidimensional and multi-dimensional interpolation, linear estimation, objective analysis,
4) Analysis of time series: generalities, Fourier series and transform, filtering,
5) Dynamical modelling with one equation: the malthusian growth model, Verhulst logistic model, equilibrium and stability, linear perturbation analysis, solution of basic ordinary differential equations,
6) Dynamic modelling with interactions: modelling of biochemical transformation, composed reactions, prey-preadtor, species competition, serach for steatdy state solution, space phase analysis, and analyse the stability (linear pertrubation, determination of the Jacobian matrix). R exercices will be organized.
7) Modelling with partial differential equations: continuity equations, adevctionn-diffusion eqaution in 3D and 1D , spectral window, ..

CHAPTER 1 Introduction

• What is a model?
• Why do we need models?
CHAPTER 2 Model formulation

• Conceptual model
• Mathematical model formulation
• Formulation of ecological interactions
• Chemical reactions
• Inhibition
• Coupled model equations
• Impact of physical conditions
CHAPTER 3 Spatial components

• Taonomy of spatial models
• Spatial boundary conditions
• Example: competitive interactions in a lattice model
CHAPTER 4 Parameterisation

• In situ measurement
• Literature-Derived parameters
• Calib_ters.43
CHAPTER 5 Model solution

• Initial conditions
• Analytical solutions of differential equations
• Numerical solution of differential equations
CHAPTER 6 Testing and validating the model

• Dimensional homogeneity and consistency of units
• Conservation of energy and mass
• Testing the correctness of the model solution
• Testing the internal logic of the model
• Model verification
• Model validity
• Model sensitivity
• Example_74
• Example of the conservation principle: a mass budget of a marine bay
CHAPTER 7 Taxonomy of ecological models

• Strategic versus tactic models
• Continuous and discrete time models
• Deterministic and stochastic models
• Density-biomass specific models
• Physiological - individual-based - population - ecosystem models
• Example: growth of a Daphnia individual
CHAPTER 8 Appendices

• Taxonomy of differential equations
• Solving difference equations
CHAPTER 9 Books for further reading

#### Mathematical analysis and modelling methods applied to the environment

Introduction to the basics of environmental modelling with exercices in R.
Part Mathematical analysis:
The course will involve the follwoing chapters:
1) Concepts and tools of mathematical analysis: revision of basic mathematics: function, limit and asymtotic behavior, derivative function (simple, composite and material, taylor expansion), primitive and integration, basics of modelling (mass balance equation), (moving) averaging of continuous function, ..Remediation exercices will be organized.
2) Dimensional analysis: dimensions, principle of dimensional homogeneity, characteristic length and time scales. Dimensionnal analysis, Pi theorem, systematic determination of dimensionless products, ..
3) Interpolation: unidimensional and multi-dimensional interpolation, linear estimation, objective analysis,
4) Analysis of time series: generalities, Fourier series and transform, filtering,
5) Dynamical modelling with one equation: the malthusian growth model, Verhulst logistic model, equilibrium and stability, linear perturbation analysis, solution of basic ordinary differential equations,
6) Dynamic modelling with interactions: modelling of biochemical transformation, composed reactions, prey-preadtor, species competition, serach for steatdy state solution, space phase analysis, and analyse the stability (linear pertrubation, determination of the Jacobian matrix). R exercices will be organized.
7) Modelling with partial differential equations: continuity equations, adevctionn-diffusion eqaution in 3D and 1D , spectral window, ..

### Acquis d'apprentissage (objectifs d'apprentissage) de l'unité d'enseignement

To teach the students the basics of mathematical modeling with practical applications.

#### Introduction to marine ecosystems modelling

To teach the students the basics of mathematical modeling with practical applications.

#### Mathematical analysis and modelling methods applied to the environment

To teach the students the basics of mathematical modeling with practical applications.

### Savoirs et compétences prérequis

Basic mathematics

#### Introduction to marine ecosystems modelling

Basic mathematics

#### Mathematical analysis and modelling methods applied to the environment

Basic mathematics

### Mode d'enseignement (présentiel, à distance, hybride)

Face to face lecture and exercices. The student will have to prepare exercies at home that will be corrected during the next lecture.

#### Introduction to marine ecosystems modelling

Face to face lecture and exercices. The student will have to prepare exercies at home that will be corrected during the next lecture.

#### Mathematical analysis and modelling methods applied to the environment

Face to face lecture and exercices. The student will have to prepare exercies at home that will be corrected during the next lecture.

### Lectures recommandées ou obligatoires et notes de cours

Lecture notes will be maed available as well as practical exercices in R (Rmd files).

#### Introduction to marine ecosystems modelling

Lecture notes will be maed available as well as practical exercices in R (Rmd files).

#### Mathematical analysis and modelling methods applied to the environment

Lecture notes will be maed available as well as practical exercices in R (Rmd files).

### Modalités d'évaluation et critères

Examination:
A homework will be required and is due for January 15th.
For those who failed in January, another exam will be planned in August/September.

#### Introduction to marine ecosystems modelling

Examen(s) en session

Toutes sessions confondues

- En présentiel

évaluation écrite ( questions ouvertes )

Explications complémentaires:

A written test in january (first session). The stduent has to be able to define the theoritical concepts seen during the lecture and to use the mathematical methods for answering practical examples and to interpret the results of the analysis. Questions will be similar to those seen during the theoritical and practical sessions.
A written test in August/September (retake).

#### Mathematical analysis and modelling methods applied to the environment

Examen(s) en session

Toutes sessions confondues

- En présentiel

évaluation écrite ( questions ouvertes )

Explications complémentaires:

A written examination will be organized in presence in January.  This exam will not require the use of a computer. It will involve questions on the theory, practicals and homework.
A homework  is due for January 15th, 2022. Description of the homework will be given at the lecture of November 16th 2021 and will be made available via eCampus. Students will have to develop, implement and analyze a simple biogeochemical model to solve an environmental problem.  This homework has to be realized by group of 3-4 students. A written report of ~10 pages describing the results and answering a list of questions has to be provided as well as the Rmd file describing the model code. Questions on the homework will be part of the written exam.
For those who failed in January, a  second session exam will be planned in August/September.

Not foreseen

Not foreseen

Not foreseen

None

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### Contacts

Marilaure Grégoire
MAST research group
Department of Astrophysics, Geophysics and Oceanography (AGO)

#### Introduction to marine ecosystems modelling

Marilaure Grégoire
MAST research group
Department of Astrophysics, Geophysics and Oceanography (AGO)

#### Mathematical analysis and modelling methods applied to the environment

Marilaure Grégoire
MAST research group
Department of Astrophysics, Geophysics and Oceanography (AGO)