Mathematical Biology 

This course is meant for graduate students of the MSc program in Biomedical Engineering and Physics, offered at Cinvestav Monterrey. It's objective is to introduce the students to basic concepts and techniques in Nonlinear Dynamics and Bifurcation Theory, as well as to the fascinating world of Mathematical Modeling of Biological Systems. To this end, we study some classical examples of mathematical models of biological phenomena, at scales ranging from a single cell to ecosystems. While studying these models, new mathematical and numerical-analysis techniques are introduced. At the end, the students are expected to have a working knowledge of the subject known as Mathematical Biology.

Course outline:

  1. Introduction to ordinary differential equations.
    1. Terminal velocity.
    2. Why clouds don't fall?
    3. Why microbes swimming differs so much from that of vertebrates.
  2. Exponential growth and decay.
    1. Stochastic processes underlying these models.
    2. Growth and decay rate constants.
    3. Half time life.
    4. Life expectancy.
    5. Replication and death propensities.
    6. How long an exponentially growing bacterial culture would take to weight as much as planet Earth.
    7. Radiocarbon dating. How does it work.
  3. Logistic growth model.
    1. Inter-species competition for resources and its effect upon growth rate constant.
    2. Influence on Darwin's theory of evolution.
    3. Analytical solution vs. qualitative analysis.
    4. Steady state definition.
    5. Steady state stability.
    6. Logistic growth plus additional externally caused death rate.
    7. Population extinction as a transcritical bifurcation.
  4. Competitive Lotka-Volterra equations.
    1. Steady state stability analysis of 2-dimensional dynamic systems.
    2. Saddle node bifurcations, bistability, and competitive exclusion principle.
  5. SIR epidemiology model.
    1. Transcritical bifurcation and disease extinction.
    2. Basic reproductive number.
    3. Vaccination
  6. Biological oscillators.
    1. Lotka-Volterra predator-prey oscillatory model and Hopf bifurcation.
    2. Van der Pol oscillator.
    3. Hodgkin-Huxley model
    4. Fitzhugh-Nagumo model

References:

  1. J. Milton and T. Ohira, Mathematics as a laboratory tool; dynamics, delays and noise, 2014, Springer, ISBN: 978-1-4614-9096-8
  2. S. H. Strogatz, Nonlinear dynamics and chaos; with applications to Physics, Biology, Chemistry and Engineering, second edition, 2015, CRC Press, ISBN: ISBN-13: 978-0813349107