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Environmental Modelling

Code: 102809 ECTS Credits: 6
Degree Type Year Semester
2501915 Environmental Sciences OT 4 0


Ana Maria Cima Mollet

Teaching groups languages

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Miquel Barcelona Poza


The prerequisite is to have passed the subjects of Mathematics and Statistics of the degree.

Objectives and Contextualisation

The objective of the subject is to develop and study mathematical models of interest in environmental sciences. The mathematical techniques necessary to make predictions of the behavior of the solutions of these models will be introduced.

We intend that the student learn to:

  •      Recognize variables, hypotheses and important parameters in real mi problems.
  •      Formulate mathematical models for different problems related to environmental processes.
  •      Know how to identify different types of models.
  •      Obtain the solutions in an exact or approximate way using analytical or numerical tools.
  •      Know how to interpret and visualize the obtained solutions.
  •      Know how to contrast the mathematical results with the properties observed in the real problem.


  • Adequately convey information verbally, written and graphic, including the use of new communication and information technologies.
  • Analyze and use information critically.
  • Collect, analyze and represent data and observations, both qualitative and quantitative, using secure adequate classroom, field and laboratory techniques
  • Demonstrate adequate knowledge and use the tools and concepts of mathematics, computer science and statistics to analyze and manage environmental issues.
  • Demonstrate concern for quality and praxis.
  • Demonstrate initiative and adapt to new situations and problems.
  • Learn and apply in practice the knowledge acquired and to solve problems.
  • Teaming developing personal values regarding social skills and teamwork.
  • Work autonomously

Learning Outcomes

  1. Adequately convey information verbally, written and graphic, including the use of new communication and information technologies.
  2. Analyze and use information critically.
  3. Apply mathematical models, both deterministic and random,
  4. Demonstrate concern for quality and praxis.
  5. Demonstrate initiative and adapt to new situations and problems.
  6. Learn and apply in practice the knowledge acquired and to solve problems.
  7. Observe, recognize, analyze, measure and adequately represent mathematical concepts applied to environmental sciences.
  8. Teaming developing personal values regarding social skills and teamwork.
  9. Use computer packages numerical and symbolic computation.
  10. Using mathematical tools to describe and solve environmental sciences.
  11. Work autonomously


1. Discrete time models in dimension 1.

    The law of Malthus
    Nonlinear models The discrete logistic model. Fixed points and stability. Graphic iteration.
    Periodic behaviors and chaotic behaviors.

2. Linear models at discrete time in dimension greater than 1.

    Systems of linear equations in differences. General solution
    Populations with age structure. The Leslie model. Asymptotic behavior: the fundamental theorem of demography.
    Markov chains.

3. Continuous time models in dimension 1: Differential equations.

    Examples: Exponential growth. Migrations Radioactive decay Solutions
    Differential equations of first order separable and linear.
    The logistic differential equation. The Allee effect.
    The hysteresis effect. A model of ecology. A model about the global energy balance.

4. Continuous time models in dimension greater than 1: Systems of differential equations.

    Introduction: trajectories, equilibrium points, periodic orbits.
    The linear systems. General solution Balances and stability: centers, spotlights, chairs and nodes.
    The model of Lotka and Volterra.
    Non-linear systems Linealització. Models of Ecology and kinetics chemistry.


In the process of learning the subject is fundamental the homework of the student who at all times will have the help of the teacher.

The contact hours are distributed in:

- Lectures: The teacher introduces the corresponding basic concepts in the subject of the subject by showing several examples of its application. The student will have to supplement the teacher's explanations with the personal study.

- Problem session: The understanding and application of the concepts and tools introduced in the theory class, with the realization of exercises. The student will have lists of problems, a part of which will be solved in the problem classes. The rest will have to be solved by the student as part of his autonomous work.

- Lab session: The student will use packages of symbolic and numerical calculation programs. The practice classes will be held in the same classroom where the theory is done; students must bring their laptop to both problem classes and hands-on classes. In these classes the application of mathematical tools will be applied to models that require the use of computer software.


Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.


Title Hours ECTS Learning Outcomes
Type: Directed      
Lab session 9 0.36 2, 3, 6, 5, 4, 7, 1, 11, 8, 10, 9
Lectures 32 1.28 3, 7, 10
Problem session 9 0.36 2, 3, 6, 5, 4, 7, 1, 11, 8, 10
Type: Autonomous      
Solving problems and studying theoretical concepts 32 1.28 2, 3, 6, 5, 4, 7, 11, 8, 10, 9


Students will be asked for 4 problem submissions, one for each topic; 30% of the grade will be assessed and counted.

There will be two partial exams with a grade value of 20% each. An average of at least 4 of the two partials must be taken to be able to average with the other assessment activities.

A final project will be requested which will count for 20% of the grade.

10% of the grade will be for class attendance. Your participation in classes is considered very convenient.

Only exams can be retaken. And they can be recovered if the student has previously presented in 2/3 of the evaluable activities.


Unique assessment.


 Students who have accepted the unique assessment modality will have to take a final test which will consist of a written exam that will consist of problem solving and some theoretical question. When finished, the student will hand in all the exercise assignments and the final project.


The final grade is obtained as follows: the exam counts 40%, the delivered problems 40% and the final project 20%.


 To pass the course, the exam grade must be greater than 4 (on a scale of 10), and the final average (exams and other assessment tests) must be greater than 5


If the exam grade does not reach 4 or the final grade does not reach 5, there is another opportunity to pass the subject through the make-up exam. The recovery system is as follows: it will be possible to recover the part of the grade corresponding to the exam and the final project, which in this case must be done by the student individually (a total of 60%). The part of exercise deliveries is not recoverable.



Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Class attendance 10% 0 0 6, 7, 10
Delivery of problems 30% 32 1.28 2, 3, 6, 5, 4, 7, 1, 11, 8, 10, 9
Final project 20% 24 0.96 2, 3, 6, 5, 4, 7, 1, 11, 10
Mid-term tests 40% 12 0.48 2, 3, 6, 5, 4, 7, 1, 11, 8, 10



  • F.R. Giordano, W.P. Fox, S.B. Horton, M.D. Weir, A First Course in Mathematical Modeling. Fourth Edition. Brooks/Cote, Cengage Learning, 2009.
  • D. G. Zill, M. R. Cullen,  Ecuaciones diferenciales con problemas de valores en la frontera (sexta edición). International Thompson editores, México 2006.


  • M. Braun, Ecuaciones Diferenciales y sus aplicaciones. Grupo Editorial Iberoamericano, México, 1990.
  • J.D. Murray, Mathematical Biology, Springer-Verlag, 1993.
  • S. H. Strogatz,  Non linear dynamics and chaos with applications to  Physics, Biology, Chemistry and EngineeringWestview Press, 2011


Maxima: computational algebra system.