# MAT5133

## Mathematical Biology - MAT4133/5133

**Catalog entry**

*Prerequisite*: MAT1214/MAT1213 Calculus I.

*Content*: The Mathematical Biology course is a broad introduction to nonlinear dynamics. Students are assumed to have an operative knowledge of single-variable calculus. Topics are introduced with interrelated biological examples of increasing difficulty. Students are exposed to discrete and continuous models. The mathematical content of the class includes: Flows on the line, linear stability analysis, matrix operations and eigenvalues, flows on the plane, bifurcations, discrete dynamical systems, and higher-dimensional systems. The biological problems studied include: Molecular processes (glycolysis, lactose operon, etc.), physiological processes (single neuron), and ecological processes (predator-prey, competing species, infectious disease modeling).

Suggested text(s) and/or readings:

- Open educational resources available in the Department of Mathematics for this course, available at https://mathresearch.utsa.edu/wiki
- Britton, Nick. Essential Mathematical Biology. Springer, 2005 (UMS Series).
- Fall, Chistopher P., Marland, Eric S., Wagner, John M., and Tyson, John J. (eds). Computational Cell Biology. Springer, 2002 (IAM Series).
- Keener, J. P., and James Sneyd. Mathematical physiology 1: Cellular physiology. New York, NY: Springer New York, 2009.
- Keener, James, and James Sneyd, eds. Mathematical physiology: II: Systems physiology. New York, NY: Springer New York, 2009.
- Strogatz, Steven H. Nonlinear Dynamics and Chaos. Westview Press, 2000.

Session | Lecture | Session | Topics | Book Section | Competency Required. The student needs to know… | Competency Gain. The student will learn to… | Examples |
---|---|---|---|---|---|---|---|

1 | Introduction to Dynamical Systems. | Historical remarks. Concept of the qualitative analysis of a dynamical system. Linear vs. nonlinear problems. Examples from biology and physics. | Strogatz Chap. 1 | Single-variable calculus. | Define a dynamical system. Understand systems of equations. Provide general examples of dynamical systems in biology and physics. | Poincaré’s three-body problem. Lorentz’s system. Mandelbrot’s fractals. | |

2 | L1 | Flows on the Line - Introduction | Simple example dx/dt = sin(x). | Strogatz 2.0, 2.1. | How to solve the separable differential equation dx/dt = sin(x) by integration. | Give a qualitative description of a simple dynamical system. | dx/dt = sin(x) |

3 | L2 | Fixed Points | Fixed points and stability. | Strogatz 2.2, 2.3. | Single-variable calculus. | Give a qualitative description of a simple dynamical system. | Population dynamics. Logistic equation of population growth |

4 | L3 | Linear Stability Analysis | Linearization about a fixed point. Existence and Uniqueness. | Strogatz 2.4, 2.5, 2.6. | Flows on the line. Taylor series. | Perform a linearization. Describe importance of fixed points. | No examples. Discuss homework |

5 | HOMEWORK # 1 | ||||||

5 | L4 | Linear Stability Analysis | Linearization about a fixed point. Hyperbolic vs. non-hyperbolic fixed points. | Strogatz 2.4, 2.5, 2.6. | Flows on the line. Taylor series. | Perform a linearization. Describe importance of fixed points. | dx/dt = r-x^2 |

6 | L5 | Bifurcations | Saddle node bifurcation. Taylor expansions and normal forms. | Strogatz 3.0, 3.1. | Single-variable calculus. Taylor series. | Determine changes in a dynamical system with parameter variation. | dx/dt = r+x^2, dx/dt = r-x^2, dx/dt = r+x+e^x |

7 | L6 | Bifurcations | Transcritical bifurcation (supercritical, subcritical), pitchfork bifurcation. | Strogatz 3.2, 3.3, 3.4. | Single-variable calculus. Taylor series. | Determine changes in a dynamical system with parameter variation. Know how to eliminate dimensions of a system. Learn stability analysis of 1-D systems | dx/dt = rx-x^2, dx/dt = rx-x^3 |

8 | COLLECT HOMEWORK # 1 | ||||||

8 | L6 | Dimensional analysis and scaling | Dimensionless groups. Buckingham’s Pi theorem. Stability of 1-D discrete systems in the context of the last problem in HW 1. | Murray 1.2, Strogatz 3.7. | Flows on the line. Bifurcations. | Make a dynamical system non-dimensional. | Population dynamics. Insect outbreak model: Spruce budworm. |

9 | L7 | Linear Systems | Introduction. Examples of two-dimensional linear systems. Lexicon. Eigenvalues and eigenvectors. | Strogatz 5.1. Class notes. | Single-variable calculus. How to solve the separable differential equation dx/dt = f(x) by integration. | Identify a linear system | Selected set of two examples that demonstrate stiff and non-stiff systems. |

10 | PRACTICE TEST # 1 | ||||||

REVIEW SESSION | |||||||

RETURN HOMEWORK # 1 GRADED | |||||||

11 | L8 | Linear Systems | Eigenvalues and eigenvectors. Reduction of second order ODEs to a system of first order ODEs | Strogatz 5.1. Class notes. | Single-variable calculus. How to solve the separable differential equation dx/dt = f(x) by integration. | Identify a linear system | Selected set of two examples that demonstrate stiff and non-stiff systems. |

12 | L9 | Linear Systems | Classification of linear systems. Types of fixed points. Classification of fixed points. | Strogatz 5.2. Class notes. | Linear systems. | Classify linear systems. Identify and classify fixed points. | Continue use of selected examples. |

13 | COLLECT HOMEWORK # 1 CORRECTIONS | ||||||

COLLECT PRACTICE TEST # 1 | |||||||

TEST # 1 | |||||||

14 | SOLUTION TO TEST # 1 | ||||||

GIVE HOMEWORK # 2 | |||||||

15 | L10 | Two Dimensional Flows | Phase portrait. Consequences of the Hartman-Grossman theorem. Nullclines. Fixed points and linearization in two and more dimensions. Jacobian matrix. | Strogatz 6.1, 6.2. | Linear systems. | Read a phase portrait. | Selected examples. |

16 | L11 | Two Dimensional Flows | How eigenvalues determine repellers, attractors, saddles, centers, and fixed points. | Strogatz 6.3 | Linear systems. | Interpret a phase portrait based on eigenvalues. | Simple linear system. |

17 | L11 | Limit Cycles | Periodic motion. Poincaré-Bendixon Theorem. Introduction to limit cycles. Hopf bifurcation | Strogatz 4.0, 4.1, 6.3, 8.2. | Two dimensional flows. | Identify limit cycles. Determine existence of limit cycles. | Simple periodic system with a limit cycle. |

18 | L12 | Two Dimensional Flows | Applications in ecology. Population dynamics. Rabbit vs. sheep. | Strogatz 6.4. | Linear systems. | Interpret a phase portrait based on eigenvalues. | Species competition. Lokta-Volterra model of competition. |

19 | COLLECT HOMEWORK # 2 | ||||||

19 | L13 | Limit Cycles | Applications in molecular biology. Glycolytic oscillations. | Strogatz 7.3, 7.5 | Two dimensional flows. | Describe qualitatively and quantitatively shape and period of closed orbits. | Cell biology: Glycolisis. Cell cycle. |

20 | L14 | Numerical Mathematics | Numerical solutions of ODEs | Kreyszig 19.1, Quarteroni, Sacco, Salieri 11.1, 11.8 | Linear algebra | Compute numerically a system of ODEs | None |

21 | RETURN HOMEWORK # 2 GRADED | ||||||

ADVISE STUDENTS TO SEARCH FOR PAPERS FOR HW # 5 | |||||||

HAVE A BRIEF DISCUSSION ABOUT SCHOLAR DATABASES | |||||||

21 | L15 | Numerical Mathematics | Runge-Kutta 2nd order method | Kreyszig 19.3, Quarteroni et al. 11.9 | Linear algebra | Compute numerically a system of ODEs | None |

22 | L16 | Numerical Mathematics | Newton-Raphson method in two dimensions | Kreyszig 17.2 | Linear algebra | Compute numerically the steady states of a system of ODEs | Numerical roots of x^2 – x = 0 |

23 | L17 | Numerical Mathematics | Newton-Raphson method in multidimensions | Nash & Sofer 10.3, Quarteroni et al. 7.1 | Linear algebra | Compute numerically the steady states of a system of ODEs | Numerical roots of x^2 – x = 0 |

24 | COLLECT HOMEWORK # 2 CORRECTIONS | ||||||

GIVE HOMEWORK # 3 | |||||||

INTRODUCTION TO COMPUTER SYSTEMS FOR NUMERICAL SOLUTIONS OF ODEs - XPP | |||||||

25 | ADVISE STUDENTS TO SEARCH FOR PAPERS FOR HW # 5 | ||||||

PRACTICE TEST # 2 & REVIEW SESSION | |||||||

26 | L18 | Applied dynamical systems | Hodgkin-Huxley model. J. Physiology, 1952 | Fall et al. 2.1, 2.2, 2.5 | Dynamical systems, linear stability, bifurcations | Apply mathematical concepts to biological problems | Hodgkin-Huxley model of neural activity. |

27 | L19 | Applied dynamical systems | Fitzhugh-Nagumo model | Fall et al. 2.6, Strogatz problem 7.5.6 | Dynamical systems, linear stability, bifurcations | Apply mathematical concepts to biological problems | Fitzhugh-Nagumo model of neural activity. |

28 | COLLECT PRACTICE TEST # 2 | ||||||

TEST # 2 | |||||||

29 | SOLUTION TO TEST # 2 | ||||||

30 | L23 | Limit Cycles | Global bifurcations | Strogatz 8.4 | Dynamical systems, linear stability, bifurcations | Identify global bifurcations | Selected examples from the book |

31 | L20 | Stochastic Systems | Introduce the concept of PDF. Ion channels | Fall et al. 11.1 | Definition of probability | Apply stochastic concepts to dynamical systems | None |

32 | L20 | Stochastic Systems | Ion channels | Fall et al. 11.1 | Definition of probability | Apply stochastic concepts to dynamical systems | None |

33 | L21 | Stochastic Systems | Dwell times | Fall et al. 11.1 | Definition of probability | Apply stochastic concepts to dynamical systems | None |

34 | COLLECT HOMEWORK # 3 | ||||||

GIVE HOMEWORK # 4 | |||||||

34 | L22 | Stochastic Systems | Ion channel ensemble | Fall et al. 11.1 | Definition of probability | Apply stochastic concepts to dynamical systems | None |

35 | L24 | Quasi-equilibrium | Michelis-Menten kinetics | Fall et al. 4.7 | Dynamical systems, linear stability, bifurcations | Identify functional responses | Bacterial glucose binding |

36 | RETURN HOMEWORK # 3 GRADED | ||||||

36 | L25 | Chaos | Intro, definitions | Strogatz 9.2, 9.4 | Two dimensional flows. | Understand the concept of chaos. | Lorenz equations. |

37 | L26 | Power-laws | Intro, definitions | PPT | Chaos | Identify when a power law is at play | None |

38 | COLLECT HOMEWORK # 3 CORRECTIONS | ||||||

38 | L27 | One-dimensional maps | Introduction. Fixed points and cobwebs. | Strogatz 10.0, 10.1, 10.2, 10.3 | Flows on the line. | Produce and analyze a one-dimensional map. | Population dynamics: Logistic map. |

39 | COLLECT HOMEWORK # 4. | ||||||

GIVE HOMEWORK # 5. | |||||||

39 | L28 | One-dimensional maps | Age-structured populations in space time. Leslie matrices. | Britton 1.9 | One-dimensional maps | Understand age-structured discrete models. | Physiology: Red blood cell count. |

40 | PRACTICE FINAL & FULL REVIEW SESSION | ||||||

41 | PRACTICE FINAL & FULL REVIEW SESSION | ||||||

RETURN HOMEWORK #4 |