Course guide of Mathematics (2001114)

• It is recommended to have studied Mathematics in high school.
• When using AI tools in this course, students are expected to do so ethically and responsibly. It is mandatory to follow the "Recommendations for the Use of Artificial Intelligence at the UGR", published at the location https://ceprud.ugr.es/formacion-tic/inteligencia-artificial/recomendaciones-ia#contenido0.

• Continuous models in biology: differential equations
• Identification of the solutions of an ordinary differential equation.
• Models of relationships between species: systems of differential equations
• Parameter estimation.
• Discrete models in biology: difference equations.
• Discrete matrix models in biology.
• Derivation using tables. Geometric interpretation. Interpretation in biology.

Formative

• The main objective is for students to understand mathematics as a useful tool in their training as a biologist. Emphasis will be placed on:
  • extracting information about real biological systems from mathematical models; and
  • critically assessing the results obtained from these models and, where appropriate, evaluating the models themselves.

Skills

• Qualitative and quantitative knowledge of elementary functions.
• Manipulation of derivatives of functions.
• Interpretation of the ordinary differential equations and the systems that appear in some models of biology.
• Identification of the properties of the solutions of continuous biological models.
• Recognition of interactions between species from mathematical models.
• Solving systems of linear algebraic equations.
• Interpretation of equations and their solutions in some discrete models of biology.
• Use of matrices in discrete models.

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

Computer practices with software to be determined by the teaching staff

• Practice 0. Introduction.
• Practice 1. Simulation of continuous population models.
• Practice 2. Simulation of species interaction models.
• Practice 3. Simulation of discrete population models.
• Practice 4. Simulation of matrix population models.
• Practice 5. Tools for parameter estimation in continuous and discrete models in biology.

• H. Anton, C. Rorres. Introducción al álgebra lineal (con aplicaciones en negocios, economía, ingeniería, física, ciencias de la computación, teoría de aproximación, ecología, sociología, demografía y genética), 5ª edición. Editorial Limusa Wiley, 2013.
• D.G. Zill. Ecuaciones diferenciales con aplicaciones de modelado (11ª edición). Cengage Learning Editores, 2019

• Brauer, C. Castillo-Chávez. Mathematical Models in Population Biology and Epidemiology, Second Edition. Springer-Verlag, New York, 2012.
• H. Caswell. Matrix Population Models: Construction, Analysis and Interpretation (2nd edition). Sinauer Associates, Inc., 2006.
• 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, 4ª edición. Ediciones Pirámide, 2016.
• H.R. Thieme. Mathematics in Population Biology. Princeton University Press, 2003.

• Prado (https://prado.ugr.es/)

• 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 

According to the "Regulations Governing the Assessment and Grading of Students at the University of Granada" (which is available at https://www.ugr.es/sites/default/files/2017-09/examenes.pdf), both continuous assessment and a single final assessment are proposed for this course. By default, all students will follow the continuous assessment system, unless they request otherwise in due time and form to the Department Director in accordance with the previous regulations.

For the ordinary call, continuous assessment will have the following components:

• Evaluation of theoretical knowledge and problem-solving through:
  • A scheduled test N12 (about units 1 and 2), with a weight of 32.5% of the grade.
  • A scheduled test N34 (about units 3 and 4), with a weight of 32.5% of the grade.
• Problem solving, questionnaires, and/or any other activity proposed by the professor (N5), with a weight of 10% of the grade.
• Evaluation of computer practices (N6), with a weight of 25% of the grade and distributed as follows: submission of proposed exercises (15%) and completion of a group project (10%).

In all the proposed activities, grading may be supplemented with eventual interviews with the teaching staff. The explanations given in such interviews will be binding in order to grade the activities carried out by the students.

The grade will be obtained using the expression N = (13*N12 + 13*N34 + 4*N5 + 10*N6) / 40 (where the grades N12, N34, N5, and N6 are scored out of 10). The course will be considered passed provided that the following two conditions are met:

• (i) The grade N is equal to or greater than 5 points out of 10.
• (ii) The grades N12, N34, and N6 are equal to or greater than 3 points out of 10 each.

In this case, the grade by continuous assessment will be N.

Those students who wish to do so may take exams on the contents corresponding to tests N12 and/or N34 on the date scheduled for the ordinary call by the Teaching Commission, in which case the grade obtained will replace the previously obtained one.

In the event of not passing the course due to:

• Not meeting (i), then the final grade on the record will be N.
• Not meeting (ii) even if (i) is met, then the final grade on the record will be 4.5.

It is also reminded that, according to the evaluation regulations of the UGR mentioned 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 course that comprehend more than 50% of the total weight of the final grade of the course, then it will appear on the record with the corresponding grade"

regardless of taking the ordinary call exam.

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

• Evaluation of knowledge through problem-solving and theoretical-practical questions, via a written test with a weight of 75% of the grade. Alternatively, the grade (13N12 + 13N34 + 4N5) / 30 obtained in the continuous assessment will be considered.
• Evaluation of computer practices, via a practical test in the computer lab, with a weight of 25% of the grade. Alternatively, the grade N6 obtained in the continuous assessment will be considered.

In all the proposed activities, grading may be supplemented with eventual interviews with the teaching staff. The explanations given in such interviews will be binding in order to grade the activities carried out by the students.

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

Students who are assessed through the single final assessment system will be evaluated on the date scheduled for the ordinary call by the Teaching Commission in the following manner:

• Evaluation of knowledge: 75% of the grade. A written test will be conducted, covering the theory topics, including problem-solving and theoretical-practical questions.
• Evaluation of computer practices: 25% of the grade. A test will be conducted, using a computer, covering the practice topics.

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

Yes. LibreOffice and OpenOffice.