Reaction-diffusion and age-structured equations with applications to biological populations

Informações Básicas

Carga horária: 

45h

Ementa: 

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

Bibliografia

Obrigatória: 

  • 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.

Complementar: 

  • 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.