### Duration

Introduction to marine ecosystems modelling : 15h Th, 15h Pr

Mathematical analysis and modelling methods applied to the environment : 20h Th, 20h Pr

### Number of credits

Master in oceanography (120 ECTS) (Erasmus Mundus ECT+ : Environmental Contamination and Toxicology) | 6 crédits |

### Lecturer

Introduction to marine ecosystems modelling : Marilaure Grégoire

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

### Coordinator

### Language(s) of instruction

English language

### Organisation and examination

Teaching in the first semester, review in January

### Schedule

### Units courses prerequisite and corequisite

Prerequisite or corequisite units are presented within each program

### Learning unit contents

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?

- Conceptual model
- Mathematical model formulation
- Formulation of ecological interactions
- Chemical reactions
- Inhibition
- Coupled model equations
- Impact of physical conditions

- Taonomy of spatial models
- Spatial boundary conditions
- Example: competitive interactions in a lattice model

- In situ measurement
- Literature-Derived parameters
- Calib_ters.43

- Initial conditions
- Analytical solutions of differential equations
- Numerical solution of differential equations
- Steady-state and stability analysis

- 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

- 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

- Taxonomy of differential equations
- Solving difference equations

#### Introduction to marine ecosystems modelling

CHAPTER 1 Introduction

- What is a model?
- Why do we need models?

- Conceptual model
- Mathematical model formulation
- Formulation of ecological interactions
- Chemical reactions
- Inhibition
- Coupled model equations
- Impact of physical conditions

- Taonomy of spatial models
- Spatial boundary conditions
- Example: competitive interactions in a lattice model

- In situ measurement
- Literature-Derived parameters
- Calib_ters.43

- Initial conditions
- Analytical solutions of differential equations
- Numerical solution of differential equations
- Steady-state and stability analysis

- 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

- 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

- Taxonomy of differential equations
- Solving difference equations

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

Mathematical basic concepts and tools applied to the analysis and modeling of physical, biological and chemical aspects of the environment. in particular, a focus is put on theoritical and mathematical ecology with a lot of practical examples.

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) Dynamical modelling with one equation: the malthusian growth model, Verhulst logistic model, equilibrium and stability, linear perturbation analysis, solution of basic ordinary differential equations,

4) 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).

### Learning outcomes of the learning unit

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.

### Prerequisite knowledge and skills

Basic mathematics

#### Introduction to marine ecosystems modelling

Basic mathematics

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

Basic mathematics: calculus and matrix algebra, basic differential equations, solving linear systems, ...

Refresher exercises in mathematics are organized.

### Planned learning activities and teaching methods

### Mode of delivery (face to face, distance learning, hybrid learning)

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.

### Recommended or required readings

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

#### Introduction to marine ecosystems modelling

Lecture notes (theory, exercices) are available via eCampus. Each lecture is podcasted to allow the student to revise the lecture at his own pace.

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

Lecture notes will be made available via eCampus as well as practical exercices in R (Rmd files). Each lecture is podcasted and will be made available to allow the student to revised the lecture at his own pace.

### Assessment methods and criteria

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

**Exam(s) in session**

Any session

- In-person

written exam ( open-ended questions )

*Additional information:*

A homework is due for January 7th at midnight. Description of the homework will be given during the lecture 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.

For those who failed in January, a second session exam will be planned in August/September.

**All the exams are exclusevely in person.**

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

**Exam(s) in session**

Any session

- In-person

written exam ( open-ended questions )

*Additional information:*

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) similar to that in January.

**All the exams are exclusevely in person. **

### Work placement(s)

Not foreseen

#### Introduction to marine ecosystems modelling

Not foreseen

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

Not foreseen

### Organizational remarks

None

#### Introduction to marine ecosystems modelling

None

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

None

### Contacts

Marilaure Grégoire

MAST research group

Department of Astrophysics, Geophysics and Oceanography (AGO)

#### Introduction to marine ecosystems modelling

Prof. Marilaure Grégoire

MAST research group

Department of Astrophysics, Geophysics and Oceanography (AGO)

mgregoire@uliege.be

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

Prof. Marilaure Grégoire

MAST research group

Department of Astrophysics, Geophysics and Oceanography (AGO)

mgregoire@uliege.be