Mathematical models in natural sciences

obligatory courses

The course concerns the application of discrete and continuous dynamical systems to describe natural phenomena. Basic concepts concerning the analysis of models described by discrete equations and ordinary differential equations, as well as the simplest partial models are discussed.

Course content:

Presentation of the basic methods of analysis of dynamical systems with continuous time (differential equations) and with discrete time (differential equations): solving systems of linear equations, methods of analysing nonlinear systems.

Discrete dynamical systems: an overview of the possible types of trajectory behaviour.

Ordinary differential equations: the simplest methods of integration, integral and phase curves, stability, phase portraits.

Applications of dynamical systems to describe various phenomena – presentation and analysis of selected mathematical models: dynamics of a single population, interactions between populations, protein production, epidemic course.

Indication of similarities and differences between continuous and discrete description on the example of selected models.

Learn about the most important linear partial differential equations of two variables. Reaction-diffusion equations.

As part of the laboratory: getting to know Matlab and Mathematica packages for numerical solving and graphical presentation of solutions to difference and differential equations.

J.D. Murray: Mathematical biology I: An introduction

J.D. Murray: Mathematical biology II: Spatial models and biomedical applications

Student finishing the course:

1) has knowledge of the basic methods of analysis od dynamical systems with continuous and discrete time,

2) knows selected mathematical models describing various natural phenomena,

3) is able to use selected mathematical packages (Maple, Matlab) to numerically solve differential equations and graphical representation of their solutions.

4) is able to apply mathematical methods to describe natural phenomena, is able to draw conclusions from specific models and is aware of the limitations of the methods used.

FINAL SCORE WILL BE GIVEN ON THE BASIS OF:

- points from classes – 70 points: short tests 40 points, activity during classes 30 points,

- points from labs (short problems to solve) – 30 points,

- written exam on difference and ordinanry differential equations (solving linear equations, solving differential equations of selected types, analysis of difference and differential equations) – 100 points.

For a positive grade, it is necessary to obtain more than 50% of the points.

Zero exam will be available to students who obtain a minimum of 80 points for classes (exercises + laboratory).

Re-take exam: the grade will be given only on the basis of the exam.