Mathematical methods in biology 4, Lectures, Lectures

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A practical introduction to mathematical concepts and methods applied in the life sciences. Part 4 focuses on models of wide interest, such as interacting populations and biochemical reaction networks.

We learn mathematics through solving problems of biological interest, with emphasis on applicable skills and hands-on experience. The course is specifically tailored for biology students and assumes no background in mathematics.

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 e.g. in population cycles and 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.

Parts 1 and 3 (or equivalent knowledge) are necessary for part 4.

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