Reaction-Diffusion and Age-Structured Equations with Applications to Biological Populations

**I. Introduction to population dynamics - model building (2,5 S)**

- Describing a population and its environment; Population balance equation; Characterizing the population; Parameters and state variables
- Exponential growth; Logistic growth; Harvesting effect; Allee effect
- Predator-prey; Competition for resources; Symbiosis
- Lotka-Volterra equation
- Proliferative and quiescent cells (cell cycle); Competition between healthy and tumor cells
- Chemical reactions: Irreversible reactions; Reversible reactions; Michaelis-Menten; Hill kinetics; Law of mass action
- Ordinary Differential Equations (ODE); Delay Differential Equations (DDE); ReactionDiffusion Equations; Age-Structured Equations

**II. Ordinary differential equations with applications to biological populations (3,5 S)**

- Existence/uniqueness; Computation of steady states; Linearization of dynamics; Characteristic equation; Eigenvalues and eigenvectors; Phase portraits of dynamics in planar systems; Monotone dynamical systems; Lyapunov functions; Bifurcation theory; Poincaré- Bendixson theorem; Monotone dynamical systems

**III. Age-structured and delay differential equations (3 S)**

- Introduction of the basic concepts; Analysis of the Lotka-McKendrick equation; Renewal equation (the method of characteristics); General nonlinear model (Gurtin-MacCamy equation); Existence/uniqueness; Computation of steady states; Allee-logistic model; Stability of steady states
- Reduction to delay differential difference equations (juvenile-adult populations, proliferativequiescent cells)

**IV. Reaction-diffusion models (3 S)**

- Reaction-diffusion equations: Models of invasion; Analysis of Fisher-KPP model; Introduction to traveling waves; Model of spread of infectious diseases; Model of invasion of malignant cells
- Reaction-diffusion models with spatial bounded domain; Separation of variables; Comparison principles; Maximum principle; Fundamental solution (Heat kernel); Variation of constants formula; Principle of linearized stability; Monotone dynamical systems; Principal eigenvalues

#### Basic Information

Workload

45 hours

**Mandatory:**

- J. Murray. Mathematical Biology, I: An Introduction. Springer, 2002.
- J. Murray. Mathematical Biology, II: Spatial Models and Biomedical Applications. Springer, 2003.
- B. Perthame. Transport equations in biology. Birkhäuser Verlag, 2007.
- E.D. Sontag. Mathematical Systems Biology. Lecture Notes. Rutgers University, 2005/2015.

**Complementary: **

- J. Müller. Mathematical Models in Biology. Lecture Notes, University of Munich, 2003/2004.
- J.R. Chasnov. Mathematical Biology. Lecture Notes, University of Hong Kong, 2009.
- R.E. Baker. Mathematical Biology and Ecology. Lecture Notes, University of Oxford, 2011.
- N. Britton. Reaction-Diffusion Equations and Their Applications to Biology. Academic Press, 1986.
- D. Jones, M. Plank, and B. Sleeman. Differential Equations and Mathematical Biology. CRC Press, 2010.
- M. Kot. Elements of Mathematical Ecology. Cambridge University Press, 2001.