Degree | Type | Year | Semester |
---|---|---|---|
2501915 Environmental Sciences | OT | 4 | 2 |
The prerequisite is to have passed the subjects of Mathematics and Statistics of the degree.
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.
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 |
Basic:
Complementary:
Maxima: computational algebra system.