MATH 1106: Modeling with Calculus for the Life Sciences

Over the summer of 2020, I worked on designing, refining, and typesetting materials for MATH 1106. A major portion of this work involved the creation of 25 worksheets used during the biweekly course recitations. The course has been designed around active learning, so the recitations comprise many problems that are completed in small discussion groups.

The course utilizes the textbook Modeling Life by Garfinkel, Shevtsov, and Guo. It covers the material in chapters 1-3, as well as the start of chapter 4 (through the Hopf bifurcation). In our analysis of previous iterations of the course, we felt that a more formal treatment of the introduction to calculus (chapter 2) was warrented, including a discussion of limits and the limit definitions of derivation and integration. I wrote a companion text that does this, which is also included below.

Worksheets
Recitation 1: The Shark-Tuna Model (Sec. 1.1)
MATH 1106 ∙ July 2020
Covers: Shark-tuna model, introduction to vector fields, feedback and feedback loops
Recitation 2: Vectors (Sec. 1.3)
MATH 1106 ∙ July 2020
Covers: Vectors, addition, scalar multiplication, norms, normalization
Recitation 3: Writing Change Equations (Sec. 1.3-1.4)
MATH 1106 ∙ July 2020
Covers: State variables, state space, word and symbolic equations
Recitation 4: More about Change Equations (Sec. 1.4)
MATH 1106 ∙ July 2020
Covers: Modeling Disease Transmission activity (intro to SIR models), the mass-spring system
Modeling Disease Transmission Activity:

TA Instructions   Version 1   Version 2   Version 3   Version 4
Recitation 5: Compartmental Models (Sec. 1.4)
MATH 1106 ∙ July 2020
Covers: Change equations and compartmental models, model parameters, modifying compartmental models
Recitation 6: Vector fields (Sec. 1.5)
MATH 1106 ∙ July 2020
Covers: Matching vector fields to change equations, drawing vector fields
Recitation 7: Time Series (Sec. 1.6)
MATH 1106 ∙ July 2020
Covers: Drawing time series from trajectories
Recitation 8: Trajectories (Sec. 1.6)
MATH 1106 ∙ July 2020
Covers: One dimensional time series, matching time series and trajectories, drawing trajectories from time series
Recitation 9: Approximating Trajectories (Sec. 1.7)
MATH 1106 ∙ July 2020
Covers: Euler’s Method in one and two dimensions, approximation vs. computation trade-off
Recitation 10: Limits (Supp. Notes)
MATH 1106 ∙ July 2020
Covers: Computing limits, limits at infinity, sigmoid functions, continuity
Recitation 11: Basic Derivatives (Sec. 2.2)
MATH 1106 ∙ July 2020
Covers: Limit definition of the derivative, the power, sum and constant multiple rules
Recitation 12: Derivative Rules (Sec. 2.3,2.5)
MATH 1106 ∙ July 2020
Covers: Derivative rules, geometric interpretation of the derivative
Recitation 13: Applications of the Derivative (Sec. 2.4)
MATH 1106 ∙ July 2020
Covers: Linear approximation, approximation error
Recitation 14: Integrals (Sec. 2.6)
MATH 1106 ∙ July 2020
Covers: Riemann sums, area under a curve, integral bound rules
Recitation 15: The Fundamental Theorem of Calculus (Supp. Notes)
MATH 1106 ∙ July 2020
Covers: The fundamental theorem, indefinite and definite integrals, computing area between curves
Recitation 16: Differential Equations (Sec. 2.7)
MATH 1106 ∙ July 2020
Covers: Solving separable differential equations, general and particular solutions, checking solutions to differential equations
Recitation 17: Equilibria in One Dimension (Sec. 3.2)
MATH 1106 ∙ July 2020
Covers: Finding equilibria, classifying equilibria using test points or linear stability analysis, the Allee effect
Recitation 18: Classifying Equilibria in Two Dimensions (Sec. 3.3)
MATH 1106 ∙ July 2020
Covers: Classifying equilibria of Shark-tuna and Mass-spring systems, review of solving linear systems
Recitation 19: The Method of Nullclines (Sec. 3.4)
MATH 1106 ∙ July 2020
Covers: Sectors and nullclines, finding and classifying equilibria in two dimensional systems
Recitation 20: The Spruce Budworm System (Sec. 3.6)
MATH 1106 ∙ July 2020
Covers: The “over-under” method, visualizing a bifurcation
Recitation 21: Bifurcations (Sec. 3.6)
MATH 1106 ∙ July 2020
Covers: Pitchfork bifurcations, constructing and interpreting bifurcation diagrams
Recitation 22: Attractors (Sec. 4.1)
MATH 1106 ∙ July 2020
Covers: Definition of attractors, periodic attractors, time series in systems with attractors
Recitation 23: Change Equations with Time Delays (Sec. 4.2)
MATH 1106 ∙ July 2020
Covers: Writing change equations in systems with time delays
Recitation 24: The Holling-Tanner Model (Sec. 4.3)
MATH 1106 ∙ July 2020
Covers: Deriving the equations of the Holling-Tanner model
Recitation 25: Hopf Bifurcations (Sec. 4.1,4.3)
MATH 1106 ∙ July 2020
Covers: Visualizing a Hopf bifurcation, review of bifurcations
Supplementary Calculus Notes
Chapter 2: An Introduction to Calculus
MATH 1106 ∙ July 2020
A more formal treatment of introductory calculus, these notes are meant to be a companion to chapter 2 of Modeling Life. They motivate calculus, and present the definition of limits, derivatives, and integrals. In addition, applications such as linear approximation and using the Fundamental Theorem of Calculus to solve definite integrals are discussed. Finally, a discussion of solving separable differential equations uses calculus to provide a derivation of the Lotka-Volterra predation equations.