Mathematical models in biology

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
O
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Fundamental principles of mathematical modeling, biological motivation.
Discrete models of population dynamics. Stability in linear and nonlinear systems, Leslie matricial model, models for a single population, discrete models for interacting populations (models of parasitism and mutualism, competition, and epidemiological models).
Stochastic models in biology. Application of probability theory in ecology (Mendelian heritage, lineage extinction), Fundamental genetic models (Hardy-Weinberg and Fisher-Haldane-Wright law), evolutionary models.
Continuous models in biology. Application of dynamical systems in population dynamics, stability in linear and nonlinear systems (Lyapunov theory), various growth models, fundamentals of Poincare-Bendixson theory, predator-prey models (Lotka-Volterra), models of symbiosis, of competition, and their generalizations, concrete ecological and epidemiological models (biodiversity, systems of type SIR), molecular kinetics (Menten-Michaelis) and basic neurological models (Hodgkin-Huxley, Fitzhugh-Nagumo).

Readings

L.J.S. Allen, An Introduction to Mathematical Biology, Prentice Hall, New York 2007.
J.D. Murray: Mathematical Biology, Springer, 1993.
L. Edelstein-Keshet: Mathematical Models in Biology, McGraw-Hill, 2005.
N.F. Britton: Essential Mathematical Biology, Springer 2003.
J. Hofbauer, K. Sigmund: Evolutionary Game Dynamics, Cambridge University Press, 1998.
A.W.F. Edwards: Foundation of Mathematical Genetics, Cambridge University Press, 2000.

Objectives and competences

The main goal is the application of already obtained mathematical knowledge to the description of biological processes. The student will be prepared to the interdisciplinary work and to the collaboration with experts from other disciplines.

Intended learning outcomes

Knowledge and understanding:
To achieve understanding of principles of mathematical modeling in science. To be
acquainted with basic biological models.
Application:
Formulating and solving simple problems in biology (modeling, forecasting o!!f phenomena).
Reflection:
Through various examples one begins to appreciate the applicability of mathematics in science.
Transferable skills:
One can learn how to describe biological (and other) processes using mathematical language and achieve a general feeling for mathematical applications. The goal is also to develop the skills for using existent literature and various computer programs.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Presentation of home exercises
2 midterm exams instead of written exam, written exam
Oral exam
Estimating working knowledge on two midterm tests, home exercises and possible on the written test as well as estimating theoretical knowledge on the final oral exam.
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

PREZELJ-PERMAN, Jasna. Interpolation of embeddings of Stein manifolds on discrete sets. Mathematische Annalen, ISSN 0025-5831, 2003, band 326, heft 2, str. 275-296. [COBISS-SI-ID 12518489]
PREZELJ-PERMAN, Jasna. Weakly regular embeddings of Stein spaces with isolated singularities. Pacific journal of mathematics, ISSN 0030-8730, 2005, vol. 220, no. 1, str. 141-152. [COBISS-SI-ID 13910873]
FORSTNERIČ, Franc, IVARSSON, Björn, KUTZSCHEBAUCH, Frank, PREZELJ-PERMAN, Jasna. An interpolation theorem for proper holomorphic embeddings. Mathematische Annalen, ISSN 0025-5831, 2007, bd. 338, hft. 3, str. 545-554. [COBISS-SI-ID 14335065]
PREZELJ-PERMAN, Jasna. A relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces. Transactions of the American Mathematical Society, ISSN 0002-9947, 2010, vol. 362, no. 8, str. 4213-4228. [COBISS-SI-ID 15641433]