Nonlinear Processes in Biology

Introduction

Population dynamics

• Continuous models for single species
  • Malthus model
  • The Verhulst (logistic) model
  • Generalizations and modifications of the Verhulst model
• Models of interacting populations: prey-predator
  • The Lotka-Volterra model
  • More realistic: the May model
• Ludwig et al. Model for Tumor Growth
  • Stationary states
  • Graphical solution
  • Bifurcation diagram
  • Problem 3: stability of stationary states
  • Time evolution

Notebooks:

• The Lotka-Volterra model with SAGE
• Analysis of the May model with SAGE
• Plot of stability condition in 3d

Chemical Kinetics

• Introduction to the kinetics of chemical reactions
• Autocatalytic reactions
  • Example 1
  • Example 2
• Enzyme reactions
• Theoretical analysis a’la Michaelis-Menten
  • The quasi-steady state approximation
  • Rate of the enzymatic reaction

Reaction diffusion systems

• Reaction-diffusion equations
  • Example: diffusion
• The Malthus model with migration: The Skellam equation
• Verhulst model with migration: Fisher-Kolmogorov equation
  • Analysis of the Fisher-Kolmogorov equation
  • Numerical analysis of the Fisher-Kolmogorov equation:

Modelling epidemics

• Epidemic model of Kermack-McKendrick
  • Example of a set of parameters
  • Example of epidemic
• Geographical spread of epidemic
• Realistic modeling of spatial epidemic spread
  • Model of epidemic spread in homogeneous media
  • Model of epidemic spread in random geometry
  • Laplace operator with the reflected wall

Appendices

List of Sage notebooks

• Lotka …

Analysis of stationary states