Math 1241 Description
Course overview and goals
Life never sits still. Biological systems are always in motion. Whether it's regulation of gene expression in response to environmental cues, a fight between antibodies and germs, processing of information among networks of neurons, or competition among species, the action of a biological system depends on the dynamics of the players and how they respond to each other. In such complex systems, it's no simple matter to understand or predict the system behavior, and our intuition can lead to false conclusions. Careful mathematical modeling can help us where our intuitions fail us, as it can allow us to deduce the ways in which the parameters of the system and the properties of its components influence the behavior of the system.
Although mathematics can be a valuable tool for biologists, not everyone instinctively turns to mathematics when encountering a challenge. Math 1241, Calculus and dynamical systems in biology, is designed to teach mathematics as an invaluable tool for understanding biology. By introducing key mathematical tools in the context of biological processes, the course aims to make the application of mathematics a natural response to the complexity of biology.
In contrast to a typical calculus course, Math 1241 focuses more on elucidating the concepts underlying the mathematics than intensive training on performing highly technical computations. The underlying philosophy is that a solid conceptual understanding of the mathematics is more likely to provide an enduring toolbox that can be accessed on demand. By elevating the mathematical facility of biology students, the goal is to better prepare them to address the complex nature of living systems.
Mathematical topics covered
- One-dimensional discrete dynamical systems
dynamical system basics
equilibria, stability of equilibria
cobwebbing
modeling
exponential growth/decay, logistic growth - Differentiation
tangent line, limit definition of derivative
derivative of basic functions: polynomials, exponentials, logarithms
brief overview of methods of differentiation: product, quotient, chain rules
minimization and maximization
partial derivatives - Integration
indefinite integral as solution to ODE
basic anti-derivatives: polynomials, exponentials, sinusoids
definite integral as change in solution to ODE
definite integral as signed area under curve
fundamental theorem of calculus
Euler's method as approximate solution of ODE and numerical integration
- One-dimensional continuous dynamical systems
exponential as solution to linear differential equation
equilibria and stability
phase line, direction field
estimating solutions graphically
Euler's method to approximate solutions
bifurcations of equilibria