# Cellular Dynamics and Complex Systems

Master Maths En Action - Université Lyon 1 - 2022-2023

Cette UE permet d’acquérir des bases solides sur les modèles usuels en dynamique des populations cellulaires. Les formalismes de systèmes dynamiques les plus courants seront introduits : Processus stochastiques, équations différentielles ordinaires et stochastiques, systèmes discrets. L’accent sera mis sur l’étude qualitative des systèmes dynamiques et des méthodes de résolution et d’analyse numérique.

In this class you will acquire solid background on modelling cell population dynamics. The most widely used dynamical system formalisms will be intriduced: Stochastic processes, ordinary and stochastic differential equation, and discrete systems. Emphasis will be put on qualitative study of dynamical systems and on the numerical methods and analysis.

### Synopsis

• Chapter 1: Stochastic Processes and the birth-death process
Équation maîtresse, Équation de Fokker-Planck, Algorithme de simulation stochastique. Lien avec les systèmes déterministes. Exemples de modèles de prolifération cellulaire
• Chapter 2: Nonlinear Systems of ODEs
Existence/unicité des solutions, Théorème de Hartman-Grobman, Linéarisation et stabilité linéaire, Classification des points fixes, Bifurcations de co-dimension 1 et 2 pitchfork, col-nœud, transcritique, Hopf, systèmes bistables. Étude numérique avec logiciels d’analyse de stabilité et de continuation de bifurcation. Exemples de la dynamique des populations cellulaires (dynamique du HIV, croissance tumorale, cycle cellulaire).
• Chapter 3: Discrete Systems
Existence/unicité des solutions, Linéarisation et stabilité linéaire, comparaison avec les EDO, Application de Poincaré, Bifurcations de doublement de périodes, Chaos. Applications : Équation logistique, Matrices de Leslie.
• Chapter 4: Large Soupled Systems and Collective Dynamics
Oscillateurs (oscillateur de phase, modèle de Goodwin), Réseaux, Synchronisation d’oscillateurs. Étude du Modèle de Kuramoto, Entrainement de systèmes périodiques. Exemples et étude numérique de modèles pour la synchronisation d’oscillateurs circadiens, synchronisation du cycle cellulaire par l’horloge circadienne.
• Chapter 5: Selected Topics

### Evaluations

• 1 Project (~2/3)

### Reference books

• L Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics Volume 7, 1991, Springer
• S Strogatz, Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering, Studies in Nonlinearity, Westview Press
• J Guckenheimer, P Holmes Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Science, Springer

### Numerical softwares

Codes used in class on Github (github/samubernard/popdyn). Mostly Matlab, but also XPPAUT, C Python.

For the reading assigments, you can use the template fiche de lecture (French), or fiche de lecture (English).

### Contact/Questions

Samuel Bernard to bernard@math.univ-lyon1.fr ## Project

The term project will deal with a recent modelling article. For the project, each student will hand in a report along with the codes used to generate the results presented in the report.

The project is in three parts: 1/ Understanding the model, 2/ its analysis (stability, bifurcation, ...), 3/ an extension of (stochastic or discrete version)

### List of articles for the projet

• A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth (pdf)
• A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences (pdf)
• Environmental Brownian noise suppresses explosions in population dynamics (pdf)
• Oscillatory Regulation of Hes1: Discrete Stochastic Delay Modelling and Simulation (pdf)
• Nonlinear Dynamics of Immunogenc Tumors: Parameter Estimation and Global Bifurcation Analysis (pdf)
• Mathematical modelling of stem cell differentiation: the PU.1–GATA-1 interaction (pdf)
• Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect (pdf)

Write the report for yourself, not for your teacher. If a question is not relevant for the article, do not answer it. For instance, biological experiments do not apply for mathematics papers. You can hand in your assignment as a .doc, pdf or scanned file, or in paper format in class.
Here are templates for the reading assignments in French or in English.

1. Reading assignment on variation in cancer risk. Date due: Friday, Jan 20th

1. Reading assignment on variation in cancer risk. Date due: Friday, February 3rd

## Chapter 1 - Stochastic and Deterministic Processes

Solutions for the birth and death process: Kendall (1948) On the Generalized "Birth and Death" Process

Stochastic equations in Matlab: An algorithmic introduction to numerical simulation of stochastic differential equations

## Chapter 2 - Modelling with ODEs

Numerical bifurcation continuation methods

• XPPAUT oldish but still widely used
• MATCONT Matlab package for ODEs
• DDE-BIFTOOL - Matlab package for delay differential equations
• PyDSTool - Dynamical systems in Python (ODEs and algebraic differential equations, hybrid systems). Based on numpy et scipy

## Chapter 3 - Finite Difference Equations

Li and Yorke (1975) Period Three Implies Chaos

## Chapter 5 - Selected topics

A few examples in mathematical biology from the past 60 years