Mathematical Modeling in Evolutionary and Environmental Biology
A.Y. 2026/2027
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
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: 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, extension to a fitness landscape (*).
Parts indicated by (*) might not be carried out due to lack of time.
· 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: 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, extension to a fitness landscape (*).
Parts indicated by (*) might not be carried out due to lack of time.
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
Frontal lectures are given at the blackboard and sometimes integrated with numerical simulations. A few lectures will focus on carrying out the exercises. The homeworks represent additional training and a valuable opportunity to deepen the program.
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 assessment consists of a written examination, normally lasting two hours and comprising three or four exercises concerning the main mathematical tools and models presented during the course.
The exercises require students to apply mathematical methods to the analysis of discrete- and continuous-time biological models. Some questions also assess students' ability to interpret the results from a biological perspective and to use the model to describe or predict the behaviour of the phenomenon under consideration.
Unless otherwise explicitly stated in the examination paper, all exercises carry equal weight in determining the overall grade.
The assessment is based on the following criteria:
-\item correctness of the mathematical formulation and calculations;
-\item appropriateness of the methods used;
-\item clarity, completeness and logical coherence of the solution;
-\item command of mathematical language;
-\item ability to interpret the results in their biological context;
-\item ability to use the model to formulate well-supported descriptions or predictions.
The examination is graded on a scale of 30. To be awarded honours, students must obtain a score of at least 29/30 in the exercise-based component and must provide a correct, complete and rigorously argued answer to an optional theoretical question included in the written examination.
If homework is assigned, its assessment accounts for one third of the final grade, while the written examination accounts for the remaining two thirds. The homework content, completion requirements, deadlines and assessment criteria are communicated by the end of the first month of the course.
The exercises require students to apply mathematical methods to the analysis of discrete- and continuous-time biological models. Some questions also assess students' ability to interpret the results from a biological perspective and to use the model to describe or predict the behaviour of the phenomenon under consideration.
Unless otherwise explicitly stated in the examination paper, all exercises carry equal weight in determining the overall grade.
The assessment is based on the following criteria:
-\item correctness of the mathematical formulation and calculations;
-\item appropriateness of the methods used;
-\item clarity, completeness and logical coherence of the solution;
-\item command of mathematical language;
-\item ability to interpret the results in their biological context;
-\item ability to use the model to formulate well-supported descriptions or predictions.
The examination is graded on a scale of 30. To be awarded honours, students must obtain a score of at least 29/30 in the exercise-based component and must provide a correct, complete and rigorously argued answer to an optional theoretical question included in the written examination.
If homework is assigned, its assessment accounts for one third of the final grade, while the written examination accounts for the remaining two thirds. The homework content, completion requirements, deadlines and assessment criteria are communicated by the end of the first month of the course.
MATH-04/A - Mathematical Physics - University credits: 6
Lessons: 48 hours
Professor:
Penati Tiziano
Professor(s)
Reception:
to be fixed by email
office num. 1039, first floor, Dep. Mathematics, via Saldini 50