Mathematical Modeling in Evolutionary and Environmental Biology
A.Y. 2020/2021
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: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
An initial attempt to provide "live lectures" on Zoom will be done;
lectures will take place according to the scheduled program of the
week (two lectures of two hours each). Each lecture will be recorded
and uploaded on the official web space of the Ariel platform.
In case of problems of either the teacher or part of the students,
videos with lessons and exercises collections will be made available
at the official web space of this course and they will cover the topic
of each week. Live meetings during the classes hours may be scheduled
by using Zoom.us, in order to answer to students' questions on the
topics covered by the videos: also these meetings will be recorded and
uploaded.
The details of these meetings will be available in the course web page
of the Ariel platform, as well as the videos and every kind of
material needed for the course.
The written exams will be done by following the indications suggested
on the web page of the University. The exams will have the same
structure as the ones done in presence, possibly reduced in time and
parts of exercises.
lectures will take place according to the scheduled program of the
week (two lectures of two hours each). Each lecture will be recorded
and uploaded on the official web space of the Ariel platform.
In case of problems of either the teacher or part of the students,
videos with lessons and exercises collections will be made available
at the official web space of this course and they will cover the topic
of each week. Live meetings during the classes hours may be scheduled
by using Zoom.us, in order to answer to students' questions on the
topics covered by the videos: also these meetings will be recorded and
uploaded.
The details of these meetings will be available in the course web page
of the Ariel platform, as well as the videos and every kind of
material needed for the course.
The written exams will be done by following the indications suggested
on the web page of the University. The exams will have the same
structure as the ones done in presence, possibly reduced in time and
parts of exercises.
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: introduction (fitness frequency in the model at constant fitnesses, Fisher
Theorem, games theory and cooperation/competition, stability under invasion, mutation in a 2-states
model).
· 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: introduction (fitness frequency in the model at constant fitnesses, Fisher
Theorem, games theory and cooperation/competition, stability under invasion, mutation in a 2-states
model).
Prerequisites for admission
a first course in calculus: theory of single real variable functions,
derivatives and integrals, introduction to probability and linear algebra (eigenvalues, eigenvectors,
determinants,...).
derivatives and integrals, introduction to probability and 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. Students will be asked to actively interact with the teacher, discussing the models proposed and the strategy of solution of the assigned problems.
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 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.
might include questions on the biological interpretation of the result. The written text typically lasts 2 hours and a half.
Professor(s)
Reception:
to be fixed by email
office num. 1039, first floor, Dep. Mathematics, via Saldini 50