# Course guide of Mathematics (2001114)

## Grado (bachelor's degree)

## Branch

## Module

## Subject

## Year of study

## Semester

## ECTS Credits

## Course type

## Teaching staff

### Theory

- Antonia María Delgado Amaro. Grupo: C
- María Clotilde Martínez Álvarez. Grupo: A
- Juan José Nieto Muñoz. Grupo: C
- Aureliano M. Robles Pérez. Grupo: D
- Rafael José Yáñez García. Grupo: B

### Practice

- Antonia María Delgado Amaro Grupos: 10 y 12
- María Clotilde Martínez Álvarez Grupos: 3 y 4
- Alberto Mayorgas Reyes Grupo: 3
- Aureliano M. Robles Pérez Grupos: 10, 11, 13, 14, 15, 16 y 9
- Rafael José Yáñez García Grupos: 1, 2, 5, 6, 7 y 8

## Timetable for tutorials

### Antonia María Delgado Amaro

Email- Tuesday de 11:00 a 13:00
- Wednesday de 09:00 a 13:00

### María Clotilde Martínez Álvarez

Email- First semester
- Tuesday de 10:00 a 13:00 (Despacho Nº50 Planta 2 Matemática Aplicada)
- Friday de 10:00 a 13:00 (Despacho Nº50 Planta 2 Matemática Aplicada)
- Second semester
- Monday de 10:00 a 14:00 (Despacho Nº50 Planta 2 Matemática Aplicada.)
- Thursday de 11:30 a 13:30 (Despacho Nº50 Planta 2 Matemática Aplicada)

### Juan José Nieto Muñoz

Email- First semester
- Tuesday de 11:00 a 14:00
- Friday de 10:00 a 13:00
- Second semester
- Tuesday de 11:00 a 14:00
- Wednesday de 10:00 a 13:00

### Aureliano M. Robles Pérez

Email- First semester
- Monday de 12:30 a 14:00
- Tuesday de 17:15 a 18:45
- Friday de 10:30 a 13:30
- Second semester
- Tuesday de 10:30 a 13:30
- Thursday de 10:30 a 13:30

### Rafael José Yáñez García

Email- First semester
- Monday de 11:00 a 12:00 (F Ciencias. Desp 0.11)
- Wednesday de 11:00 a 14:00 (F Ciencias. Desp 0.11)
- Thursday de 10:00 a 12:00 (F Ciencias. Desp 0.11)
- Second semester
- Wednesday de 08:00 a 14:00 (F Ciencias. Desp 0.11)

### Alberto Mayorgas Reyes

Email- Wednesday de 10:00 a 12:00 (Despacho 60, Planta 2 del Ala de Matemáticas)

## Prerequisites of recommendations

- It is recommended to have studied Mathematics in high school.

## Brief description of content (According to official validation report)

- Differential equations.
- Solutions of ordinary differential equations.
- Systems of differential equations: species interaction models.
- Parameter estimation.
- Discrete models in biology.
- Matrix population models in biology.
- Discrete differentation. Geometric interpretation. Biological interpretation.

## General and specific competences

### General competences

- CG01. Organisational and planning skills
- CG03. Applying knowledge to problem solving
- CG04. Capacity for analysis and synthesis
- CG06. Critical reasoning
- CG16. Creativity
- CG17. Information management skills

### Specific competences

- CE39. Aplicar los procesos y modelos matemáticos necesarios para estudiar los principios organizativos, el modo de funcionamiento y las interacciones del sistema vivo
- CE76. Knowing mathematics and statistics applied to Biology.

## Objectives (Expressed as expected learning outcomes)

Formative

- The main objective is for the student to understand mathematics as a useful tool in their training as a biologist. Emphasis will be placed on:
- obtaining information about a real biological situation from a mathematical model and
- criticism of the results obtained from the models and, where appropriate, criticism on the models themselves.

Skills

- Qualitative and quantitative knowledge of elementary functions.
- Handling of derivatives of functions.
- Interpretation of the ordinary differential equations and the systems that appear in some models of Biology.
- Identification of properties of the solutions, of an ordinary differential equation and of the systems of ordinary differential equations, from the equations.
- Recognition of the interaction between species from a mathematical model.
- Solving systems of linear algebraic equations.
- Interpretation of difference equations and systems of difference equations that appear in some models of Biology.
- Use of matrices in Gauss method and in discrete models.

## Detailed syllabus

### Theory

- Unit 0. Review of basic concepts. Equations and inequalities. Functions: derivation, handling of tables, sketch of graphs. Matrices, systems of linear equations and its resolution.
- Unit 1. Continuous models of population growth through differential equations. Qualitative study of the solutions. Malthus, Verhulst, Gompertz, and von Bertalanffy models.
- Unit 2. Continuous models of interaction between species through systems of differential equations. Equilibrium point and orbits. Phase portrait. Stability.
- Unit 3. Discrete models of population growth through difference equations. Fixed points, cycles and stability. Malthus, logistic and Ricker models.
- Unit 4. Growth models structured by age and state models through systems of linear difference equations.
- Unit 5. Parameter estimation throught least squares method. Linear and nonlinear cases. Linearization.

### Practice

Computer practices with software to be determined by the teaching staff

Practice 1. Simulation of continuous models of population dynamics.

Practice 2. Simulation of interaction models between species.

Practice 3. Simulation of discrete models of population dynamics.

Practice 4. Simulation of matrix models of population dynamics.

Practice 5. Tools for parameter estimation in discrete and continuous models of biology.

## Bibliography

### Basic reading list

- H. Anton. Introducción al álgebra lineal. Editorial Limusa, 1990.
- C. Rorres, H. Anton. Aplicaciones de álgebra lineal. Editorial Limusa, 1979.
- D.G. Zill. Ecuaciones diferenciales con aplicaciones. Editorial Iberoamérica, 1988.

### Complementary reading

- F. Brauer, C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Second Ed., Springer-Verlag, New York, 2012
- Caswell, H. (2001) Matrix Population Models: Construction, Analysis and Interpretation, 2nd edn. Sinauer Associates, Sunderland, Massachusetts, USA.
- L. Edelstein-Keshet. Mathematical Models in Biology. SIAM, Philadelphia, 2005.
- S.P. Ellner, J. Guckenheimer. Dynamic Models in Biology. Princeton University Press, 2006.
- M. Kot. Elements of Mathematical Ecology. Cambridge University Press, 2001.
- J.D. Murray. Mathematical Biology I: An Introduction (3rd Edition). Springer, 2002.
- J.D. Murray. Mathematical Biology II: Spatial Models and Biomedical Applications. (3rd Edition). Springer, 2003.
- J. Rodríguez. Ecología. Ediciones Pirámide, 2001.
- H.R. Thieme. Mathematics in Population Biology. Princeton University Press, 2003.

## Recommended links

- Prado (https://prado.ugr.es/)

## Teaching methods

- MD01. Lección magistral/expositiva
- MD02. Sesiones de discusión y debate
- MD03. Resolución de problemas y estudio de casos prácticos
- MD06. Prácticas en sala de informática
- MD07. Seminarios
- MD08. Ejercicios de simulación
- MD10. Realización de trabajos en grupo
- MD11. Realización de trabajos individuales

## Assessment methods (Instruments, criteria and percentages)

### Ordinary assessment session

In accordance with the Evaluation and Qualification Regulations for students at the University of Granada (can be consulted at https://www.ugr.es/sites/default/files/2017-09/examenes.pdf), for this subject it is proposed both a continuous evaluation and a single final one. By default, all students will follow the continuous assessment system, unless they indicate otherwise in a timely manner to the Head of the Department by virtue of the previous regulations.

A) For the ordinary call, the continuous evaluation will have the following components:

- Evaluation of theoretical knowledge and problem solving, through
- A scheduled test N12 (on topics 1 and 2), weighted at 32.5% of the grade.
- A scheduled test N3 (on topic 3), weighted at 16.25% of the grade.
- A test N4 (on topic 4), on the date assigned to the ordinary call, weighted at 16.25% of the grade.

- Resolution of problems, questionnaires and / or any other activity proposed by the teacher, (N5), weighted at 10% of the grade
- Evaluation of computer practices (N6) weighted at 25% of the grade and distributed as follows: submission of proposed exercises (10%) and completion of a group work (15%).

In all the proposed activities, the evaluation may be complemented with interviews with the teaching staff. The explanations given in the interviews will be binding when grading the activities carried out by the student.

The grade will be the obtained using the formula **N=(26N12+13N3+13N4+8N5+20*N6)/80 **(where the grades N12, N3, N4, N5, and N6 are scored out of 10). The subject will be considered passed as long as the following two conditions are verified:

- (i.) The grade
**N is equal to or higher than 5 points out of 10**. - (ii.) The grades
**N12, N3, N4 and N6 are equal to or greater than 3 points out of 10 each of them**.

In this case, the grade for continuous evaluation will be **N**.

Those students who wish to do so may examine the contents corresponding to tests N12 and / or N3 on the date scheduled for the ordinary call by the Teaching Committee, in which case, the grade will replace the one previously obtained.

In the case of not passing the subject for:

- not comply (i.) then the final grade will be
**N**, - not comply (ii.) although (i.) is verified, then the final grade will be 4.5.

It is also recalled that, according to the evaluation regulations of the UGR referred to above (chapter VI, Article 22, point 4):

"When the student has carried out activities and tests of the Continuous Evaluation process contemplated in the Teaching Guide of the subject that constitute more than 50% of the total weight of the final grade of the subject, they will appear in the record with the corresponding grade."

regardless of the performance of the ordinary call exam.

### Extraordinary assessment session

For the extraordinary call, the qualification will be obtained through the following components:

- Evaluation of knowledge by resolution of problems and theoretical-practical questions, through a written test with a weight of 75% of the grade.
- Evaluation of practices, by carrying out a practical test in a computer room, with a weight of 25% of the qualification. In case the student agrees, the N6 grade obtained by continuous evaluation will be considered.

The course will be considered passed if the sum of both parts reaches 50% of the total.

### Single final assessment

The student who takes advantage of the single final evaluation system will be evaluated on the date scheduled for the ordinary call by the Teaching Commission as follows:

- Evaluation of knowloedge by solving problems and theoretical-practical questions, through a written test with a weight of 75% of the grade.
- Evaluation of practices, by carrying out a practical test in a computer room, with a weight of 25% of the qualification.

The course will be considered passed if the sum of both parts reaches 50% of the total.