Mathematical Modeling in Evolutionary and Environmental Biology
A.Y. 2023/2024
Learning objectives
The goal of the course is to help students of Evolutionary Biology to understand how mathematical models in population dynamics are built and then studied. Special attention will be paid to biological assumptions and to the corresponding mathematical translation. Coherently with classical undergraduate courses given in foreign Universities, the student will be introduced to discrete and continuous dynamical systems, with focus on equilibria and their (linear) stability: applications to prey-predators, parassitoidism, competition and cooperation will be presented. Moreover, a small part of the course will present the mathematical treatment of fitness. In general, the overall idea is to educate students to parts of Mathematics which are internationally used for modeling biology.
Expected learning outcomes
At the end of the course the student will have:
* knowledge of simple mathematical models, in order to understand both at a qualitative and quantitative level biological phenomena.
* ability to interpret classical mathematical models in population dynamics (Ecology and Epidemiology)
* basic knowledge of a quantitative formulation of the Theory of Evolution.
* increased background in mathematical tools widespread in any field of Science, mainly Dynamical Systems, both in discrete and continuous time (ODE).
* knowledge of simple mathematical models, in order to understand both at a qualitative and quantitative level biological phenomena.
* ability to interpret classical mathematical models in population dynamics (Ecology and Epidemiology)
* basic knowledge of a quantitative formulation of the Theory of Evolution.
* increased background in mathematical tools widespread in any field of Science, mainly Dynamical Systems, both in discrete and continuous time (ODE).
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Population dynamics:
· Linear discrete time dynamics: Fibonacci, models with delay, higher dimensional Malthus models,
matrices, eigenvectors and eigenvalues. Stability of extinction state.
· Nonlinear discrete time dynamics: equilibria, stability and instability, caos in the logistic model.
Overcompensation ed undercompensation. The works of May and Hassel.
· Continuous time dynamics: equilibria. Stability and instability. Linearization.
· Modeling population growth. Exponential growth, logistic growth and other size dependent models.
· Other biological applications of exponential and logistic growth. Holiing's type functional response.
An example of Biological irreversibility (spruce-budworm and catastrophe theory).
· Interacting populations in discrete time: parassitoidism.
· Interacting populations: predation and cooperation. Lotka-Volterra model and D'Ancona paradox.
· Infection and epidemiological models: SIR model and vaccination.
· Interacting polulations: competition. The principle of competitive exclusion. Cooperation.
Mathematical theory of evolution end dynamics of fitness: fitness frequency in the model at constant fitnesses, Fisher
Theorem, mutation in a 2-fitness model and extension to many-fitness model. Numerical simulations.
· Linear discrete time dynamics: Fibonacci, models with delay, higher dimensional Malthus models,
matrices, eigenvectors and eigenvalues. Stability of extinction state.
· Nonlinear discrete time dynamics: equilibria, stability and instability, caos in the logistic model.
Overcompensation ed undercompensation. The works of May and Hassel.
· Continuous time dynamics: equilibria. Stability and instability. Linearization.
· Modeling population growth. Exponential growth, logistic growth and other size dependent models.
· Other biological applications of exponential and logistic growth. Holiing's type functional response.
An example of Biological irreversibility (spruce-budworm and catastrophe theory).
· Interacting populations in discrete time: parassitoidism.
· Interacting populations: predation and cooperation. Lotka-Volterra model and D'Ancona paradox.
· Infection and epidemiological models: SIR model and vaccination.
· Interacting polulations: competition. The principle of competitive exclusion. Cooperation.
Mathematical theory of evolution end dynamics of fitness: fitness frequency in the model at constant fitnesses, Fisher
Theorem, mutation in a 2-fitness model and extension to many-fitness model. Numerical simulations.
Prerequisites for admission
a first course in calculus: theory of single real variable functions,
derivatives and integrals, introduction to probability and to linear algebra (eigenvalues, eigenvectors,
determinants,...).
derivatives and integrals, introduction to probability and to linear algebra (eigenvalues, eigenvectors,
determinants,...).
Teaching methods
It is warmly recommended to attend most of the frontal lectures given at the blackboard and sometimes integrated with numerical simulations. A few lectures will focus on carrying out the exercises.
Teaching Resources
G. Gaeta, Modelli Matematici in Biologia; Springer 2007
Mathematical Epidemiology - Lecture Notes in Mathematics
Mathematical Models in Biology - (Leah Edelstein-Keshet)
Mathematical Models in Population Biology and Epidemiology - Texts in Applied Mathematics
In addition, some tutorial exercises will be available on the web page of the course, with some examples of written tests.
Mathematical Epidemiology - Lecture Notes in Mathematics
Mathematical Models in Biology - (Leah Edelstein-Keshet)
Mathematical Models in Population Biology and Epidemiology - Texts in Applied Mathematics
In addition, some tutorial exercises will be available on the web page of the course, with some examples of written tests.
Assessment methods and Criteria
The exam consists in a written test, with usually three/four exercises which require different mathematical knwoledges and abilities; so doing, most of the methods and models seen during the program is covered by the exam. Some exercises
might include questions on the biological interpretation of the result. The written text typically lasts 2 hours and a half. To get 30/30 cum Laude it is necessary to answer a theoretical question; the given answer will be corrected provided the exercise part has been passed with at least 29/30.
might include questions on the biological interpretation of the result. The written text typically lasts 2 hours and a half. To get 30/30 cum Laude it is necessary to answer a theoretical question; the given answer will be corrected provided the exercise part has been passed with at least 29/30.
Educational website(s)
Professor(s)
Reception:
to be fixed by email
office num. 1039, first floor, Dep. Mathematics, via Saldini 50