Mathematics for Biologists

Ordinary differential equations

• reminders of the main results seen in the first year (Cauchy problem, Cauchy-Lipschitz theorem, maximal solutions, Gronwall's Lemma, global existence criteria)
• explicit solution methods (equations with separate variables, some linear equations) and qualitative analysis
• asymptotic behavior, stability and instability, oscillations
• detailed study of some examples, from population dynamics (e.g. Lotka-Volterra), virus diffusion models (e.g. SIR), neuronal spike models, tumor growth models.
• projects, also with the help of R software, for particular problems and examples such as differentiation and cell cycle, feedback mechanisms and enzyme-substrate dynamics.

Calculus of Probability and Statistics

• basic elements (events, their probabilities, first rules, conditional probability and independence, Bayes formula and total probabilities)
• examples of discrete and continuous random variables, mean values ​​and their properties
• linear models in statistics (multiple linear regression, principal components method, implementation on data using R software)
• Markov chains (graphs, transition probabilities, state classification, invariant measures) and applications, also to models similar to those of ordinary differential equations.

Fourier series

• Fourier series expansion of regular periodic functions in one variable
• generalizations to less regular functions
• some elements on Fourier series in two variables
• time series analysis with R

Possible advanced topics

• stochastic processes, Brownian motion
• stochastic differential equations (Langevin), Fokker-Planck equations
• graphs and networks
• Bayesian statistics.