Course guide of Mathematics (2001114)

Curso 2022/2023
Approval date: 13/06/2022

Grado (bachelor's degree)

Bachelor'S Degree in Biology

Branch

Sciences

Module

Materias Básicas Instrumentales para la Biología

Subject

Matemáticas

Year of study

1

Semester

1

ECTS Credits

6

Course type

Core course

Teaching staff

Theory

  • Juan Campos Rodríguez. Grupo: D
  • Antonia María Delgado Amaro. Grupo: C
  • María Clotilde Martínez Álvarez. Grupo: A
  • Juan José Nieto Muñoz. Grupo: C
  • Rafael José Yáñez García. Grupo: B

Practice

  • José Alfredo Cañizo Rincón Grupos: 12 y 4
  • Antonia María Delgado Amaro Grupo: 9
  • Lazaro Rene Izquierdo Fabregas Grupos: 11 y 3
  • María Clotilde Martínez Álvarez Grupos: 1 y 2
  • Aureliano M. Robles Pérez Grupos: 13, 14, 15 y 16
  • Rafael José Yáñez García Grupos: 10, 5, 6, 7 y 8

Timetable for tutorials

Juan Campos Rodríguez

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  • Wednesday de 10:00 a 13:00 (Despacho 56)
  • Thursday de 10:00 a 13:00 (Despacho 56)

Antonia María Delgado Amaro

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  • Tuesday de 11:00 a 13:00
  • Wednesday de 09:00 a 13:00

María Clotilde Martínez Álvarez

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  • First semester
    • Tuesday
      • 09:00 a 12:00 (Facultad de Ciencias, Dpto. Matemática Aplicada, Despacho Nº 50)
      • 09:30 a 13:00 (Facultad de Ciencias, 2ª Planta, Dpto. Matemática Aplicada, Desp 50)
    • Wednesday de 10:00 a 11:00 (Facultad de Ciencias, Dpto. Matemática Aplicada, Despacho Nº 50)
    • Friday
      • 09:30 a 12:00 (Facultad de Ciencias, 2ª Planta, Dpto. Matemática Aplicada, Desp 50)
      • 10:00 a 12:00 (Facultad de Ciencias, Dpto. Matemática Aplicada, Despacho Nº 50)
  • Second semester
    • Tuesday
      • 09:30 a 13:30 (Facultad de Ciencias, 2ª Planta, Dpto. Matemática Aplicada, Desp 50)
      • 17:30 a 19:30 (Facultad de Ciencias, Dpto. Matemática Aplicada, Despacho Nº 50)
    • Friday de 09:30 a 13:30 (Facultad de Ciencias, Dpto. Matemática Aplicada, Despacho Nº 50)

Juan José Nieto Muñoz

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  • First semester
    • Monday
      • 09:30 a 12:00
      • 13:00 a 14:00
    • Tuesday
      • 09:30 a 11:00
      • 12:00 a 13:00
  • Second semester
    • Monday de 09:30 a 12:30
    • Tuesday de 09:30 a 12:30

Rafael José Yáñez García

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  • First semester
    • Monday de 10:00 a 12:00 (F Ciencias. Desp 0.11)
    • Wednesday de 10:00 a 12:00 (F Ciencias. Desp 0.11)
    • Thursday
      • 10:00 a 11:00 (F Ciencias. Desp 0.11)
      • 12:00 a 13:00 (F Ciencias. Desp 0.11)
  • Second semester
    • Wednesday de 08:00 a 14:00 (F Ciencias. Desp 0.11)

José Alfredo Cañizo Rincón

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No hay tutorías asignadas para el curso académico.

Lazaro Rene Izquierdo Fabregas

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No hay tutorías asignadas para el curso académico.

Aureliano M. Robles Pérez

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  • First semester
    • Tuesday de 10:00 a 13:00
    • Thursday de 10:00 a 13:00
  • Second semester
    • Tuesday de 17:30 a 20:30
    • Thursday de 10:30 a 13:30

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 and linear systems: reduced form of a matrix and system resolution. 
  • Unit 1. Continuous models of population growth. Differential equations. Qualitative study of the solutions. Malthus, Verhulst, Gompertz and von Bertalanffy models. 
  • Unit 2. Continuous models of interaction between species. Systems of differential equations. Equilibrium point and orbits. Phase portrait. Stability. 
  • Unit 3. Discrete models of population growth. Difference equations. Fixed points, cycles and stability. Malthus, logistic and Ricker models. 
  • Unit 4. Growth models structured by age. State models. Systems of equations in linear differences. Powers of a matrix. Positive matrices.
  • Unit 5. Parameter estimation. 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
    • two scheduled tests (N1 and N2),  with weights of 30% and 20%, respectively, of the grade.
    • A test (N3), on the date assigned to the ordinary call, with a weight of 15% of the grade.
  • Resolution of problems, questionnaires and / or any other activity that the teacher proposes, (N4), with a weight of 10% of the grade
  • Evaluation of computer practices (N5) with a weight of 25% of the grade, distributed as follows: delivery of proposed exercises (10%) and a group work (15%).

In all the proposed evaluable 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 rating will be the result of the sum N = 0.3 N1+0.2 N2+0.15 N3+0.1 N4+0.25 N5 (where grades N1, N2, N3, N4 and N5 are scored out of 10 points). The course will be considered passed as long as the following two conditions are verified:

      i. The N sum is equal to or greater than 5 out of 10.

      ii. The grades N1, N2, N3 and N5 are equal to or greater than 3 points out of 10 in 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 N1 and / or N2 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 equal to the sum 0.3 N1+0.2 N2+0.15 N3+0.1 N4+0.25 N5,
  • 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 Assessment process contemplated in the Teaching Guide of the subject that constitute more than 50% of the total weighting of the final grade for the subject, they will appear in the minutes 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

  • Knowledge assessment 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. In case the student agrees, the N4 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:

  • Knowledge assessment 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.