## Admission requirements

This course is open to all students that have been admitted to the minor ‘Quantitative Biology’.

Students have to select either this course or the course ‘Elementary practical biology and methods for non-biologists’, depending on their background. Please consult the course or minor coordinator when there is uncertainty about which course is the most appropriate.

## Contact information

Course coordinator: Dr. S.C. Hille

Email: shille@math.leidenuniv.nl

## Description

In the course the students will become acquainted with mathematical concepts and methods involved in the modeling of dynamical processes in terms of ordinary differential equations (ODEs). These type of equations have been used successfully in various branches of Science, after their invention by Isaac Newton. The modeling approach will be explained by means of application motivated examples: e.g. from chemical reaction networks, electrophysiology, membrane transport. It is discussed what is the meaning of solving such equations. The concept of ‘state’ of the modeled system is introduced. Typically, a model in terms of ODEs cannot be solved explicitly, i.e. there does not exist an easy functional expression that describes the state of the system at any point of time. Instead, the students will be introduced to the technique of ‘phase plane analysis’: a graphical method that allows one to gain insight into the dynamics represented by the model. It is particularly handy in analyzing models when no explicit solution is available. Basic concepts like steady state, stability of steady states and periodic solutions are introduced. The freely available software tool for phase plane analysis, XPPAUT by Bart Ermentrout, will be demonstrated by which one can examine moderately complicated mathematical models.

## Learning goals

**Course objectives:**

Students with a Life Science background will learn fundamental concepts and methods from mathematics needed in other courses in the minor to enable effective communication with mathematicians. They will get acquainted with the type of research questions considered in applied mathematics, in particular within the context of analysis of dynamic mathematical models using ordinary differential equations. Attention is given to interpreting the mathematical expressions in a model as representing particular physical, chemical or biological processes. Moreover, in exercise sessions the students will increase their practical skills in manipulating with mathematical expressions and work with various concepts. The use of particular software tools in the analysis of models will be demonstrated and examined in exercise sessions.

**Final qualifications:**

• Ability to effectively communicate about mathematical models.

• Ability to interpret mathematical models in terms of underlying processes.

• Skilled in manipulating with a selection of types of mathematical expressions and concepts.

• Ability to analyze basic mathematical models, combining explicit elementary computations with the use of software tools for more complicated problems.

• Understanding what type of information can be obtained from mathematical analysis and simulation of the discussed types of mathematical models.

## Time table

The time table is provisional and provided times are indicative. A detailed final schedule will become available before the start of the minor. It is foreseen that he course starts intensively in the first two weeks of September, then less intensively in the next two weeks. In week 5 the course is closed by a written exam.

## Mode of instruction

Lectures and exercise sessions. Instruction in the use of computational software tools is provided.

## Assessment method

a) Assignments, including the use of a software tool for analysis (XPAUT) (20%)

b) Written exam (80%)

## Blackboard

Blackboard will be used for communication and provision of course material.

## Literature

Will be announced.

## Registration

Via Usis. Enroll also for the course in Blackboard.

Exchange and Study Abroad students: please see the Prospective students website for information on the application procedure.