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