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

Informações Básicas

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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



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