Biologian matemaattiset menetelmät 4

A practical introduction to mathematical concepts and methods applied in the life sciences. Part 4: Advanced models

9.2.2026 - 31.7.2026

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Syventävät opinnot

We learn mathematics through solving problems of biological interest, with emphasis on applicable skills and hands-on experience. The full course consists of four parts (each can be taken separately):

Mathematical methods in biology, Part 1: Fundamentals (construction of simple models and basic calculus)
Mathematical methods in biology, Part 2: Probability (handling stochastic phenomena, groundwork for statistics)
Mathematical methods in biology, Part 3: Vectors and matrices (applied to population dynamics, genetics and statistics)
Mathematical methods in biology, Part 4: Dynamic models (techniques to analyse models of ecological interactions, biochemical reactions, etc.)

In Part 4, we learn advanced methods that enable the analysis of the most common type of models, differential equations with several variables. We start with a graphical analysis (called phase plane analysis) that helps to visualize the solutions and can in some cases lead to definitive conclusions about the long-term dynamics. A core subject of Part 4 is to find equilibria and establish their stability with multiple variables and to understand how the equilibria and their stability can change via bifurcations. We study also limit cycles, outcomes possible only in models of several variables, which explain the sustained oscillations seen in population cycles and also elsewhere (e.g. in the cell cycle). Models of predator-prey systems, competitive systems, epidemics and molecular biology serve as the basis for introducing and practicing the methods of analysis.

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Koodi

EEB-014

Students will be able to use advanced methods for analysing differential equation models with several variables, including

visualization using phase plane analysis;
stability analysis of equilibria;
finding limit cycles;
understanding how the qualitative behaviour of the model may change when changing parameters (transcritical, saddle-node, and Hopf bifurcations).

We learn and practice these methods via examples mostly from population dynamics (predator-prey and competitive systems) and the dynamics of epidemics, with some additional examples from molecular biology (enzyme kinetics, cell cycle).

Completion methods

Participation in teaching (lectures, exercise classes, exam) or exam. The course can be completed entirely as distance learning.

Assessment practices and criteria

Written exam with problems similar to homework exercises, graded 0-5.

Activities and methods in support of learning

2h lectures and 2h exercise classes per week for one study period

Target groups

Open to all (bachelor's / master's students of biology, students of other degree programmes, non-degree students, exchange students).

The course is specifically tailored for biology students and assumes no background in mathematics other than parts 1 and 3 of this same course series. Both undergraduates and graduate students are welcome.

Teaching period when the course will be offered

Period 4, even years (spring 2024 and 2026)

Recommended time or stage of studies for completion

Recommended for completion in the 1st or 2nd year of MSc studies

Study module

The course is optional in Ecology and Evolutionary Biology.

Expiry of studies

The course expires in 10 years.

Language of instruction

English

EQF level

Second-cycle degree, EQF level 7

Järjestäjä

Helsingin yliopisto

Yhteystiedot

avoin-student@helsinki.fi