Learning Tools for Mathematical Modeling
Welcome! The following are tools for students learning mathematical modeling in the life sciences.
Simulations
Simulations of phenomena in the natural world. Specific to particular examples.
Caffeine Simulator
Track the caffeine in your body, with three adjustable doses each day.
Shark and Tuna Interaction-Counter
Relate number of interactions to populations of sharks and tuna.
Shark and Tuna Trajectories
Trajectories in state space and time-series. Interventions.
Insulin Glucose Response
Set parameters for insulin sensitivity, beta cell mass, etc. Look at steady state or a glucose challenge test. Uses a minimal model.
Gentamicin pharmacokinetics
Adjust protocols to explore effect on drug in bloodstream. Exhibits 1st order kinetics.
Higgins-Sel'kov glycolysis model
Adjust parameters to see limit cycle and Hopf bifurcation. Identify source of oscillations in F6P and ADP.
Holling-Tanner predator-prey model
Adjust parameters to see limit cycle and Hopf bifurcation. Sharks with limited appetite.
FitzHugh-Nagumo model
Generalization of van der Pol oscillator. Simplification of Hodgkin-Huxley model for neural spikes.
Hutchinson population growth model
Logistic growth where crowding effect has a time delay. Exhibits Hopf bifurcation and oscillations
Simple muscle simulator
Muscle length with equilibrium point and time delay. Exhibits oscillations from time delay and steep feedback
Tree ring simulator
Simulates radial tree growth (tree rings) with uniformly random growth each year.
Growth and collapse simulator
Simulates things that often grow linearly but occasionally collapse back down to zero. Runs simulation 500 times, displaying histogram of results.
Yule process simulator
Runs simulation of Yule type birth/death process. Tracks populations with dozens of simulations for comparison of outcomes.
Genetic drift simulator
Runs simulation of genetic drift, essentially a Wright-Fisher model with fixed total population. Change parameters, observe fixation of traits, etc.
Atmospheric molecule simulator
Runs a random walk to model up/down motion of molecules in the atmosphere, with adjustable temperature and gravitational bias.
Brownian motion simulator
Simulation of Brownian motion, with adjustable particle size, viscosity, temperature, to replicate Perrin's experiment.
Diffusion simulator
Simulation of diffusion, with 1000 particles undergoing a random walk. Histogram of x-coordinates resembles the bell curve.
Ion channel simulator
Simulation of current from a single ion channel, imitating patch clamp recording. User can choose transition probabilities between open and closed states.
Explorers
Tools for exploring different kinds of equations.
Self-interaction Simulator
Relate the number of self-interactions to populations of moving organisms in a dish.
Dynamic Vector Field Explorer
Play with your own vector fields in the plane, and watch particles blow in the wind.
Data Plotter with Log-scaling
Enter 10 data points, experiment with log-scaling and line of best fit.
Logistic Growth Explorer
Explore parameter changes, and try to fit logistic model to your data by minimizing the residual sum of squares.
Exploration of Equilibrium in 1-D
Explore 1-dimensional dynamical systems. Visualize phase portrait, equilibria, trajectories, and time series.
Generalized Lotka-Volterra Explorer v2
(New and improved!) Control parameters in a 2-dimensional generalized Lotka-Volterra system. Allows exploration of predator-prey, competition, cooperation, coupled with logistic growth.
Simple Harmonic Oscillator (SHO) Explorer
Control parameters in the simple harmonic oscillator. Allows exploration of trajectory shape and period of oscillation.
Discrete Logistic Explorerr
Set birth rate and initial population, and see what happens. Explore bifurcation and the onset of chaos in this classic example.
Rapid Coin Flipper
Choose fair or biased coin, and flip up to 200 times. Provides statistics and random walk to accompany outcomes.
Two-compartment Markov Chain Explorer
Choose the matrix of transition probabilities, and see the populations change while individuals behave stochastically.
Matrix and vector calculator
Quickly enter a matrix, vector, scalar, and see sums and products. Works in 2, 3, 4 dimensions.
Three compartment matrix modeler
Set the transition probabilities, and run the matrix model. Displays time series plots.
2-Dimensional matrix visualizer
Input a 2x2 transition matrix. Displays eigenvalues/eigenvectors. Start with grid of points, and apply the transition matrix to see the effect.
2-Dimensional matrix smiler
Input a 2x2 transition matrix. See the effect of the matrix on a smiley-face. Animate the transition.
Study Aids
Tools to practice fundamental skills, prepare for tests, etc.
Spreadsheet Practice
Learn how to use spreadsheets! For the complete beginner. Self-contained and leads into modeling change equations.
Fundamental Modeling Practice
Practice with randomly generated situations, figure out change equations, graphs, and more. Click buttons to see correct answers.
Trajectory and Time-series Practice
Practice going back and forth between trajectories in state space and time-series plots. Click buttons to see correct answers.
Logarithm Definition Practice
Practice reframing logarithm questions as exponent questions, and answering them. Click buttons to see correct answers. Natural logarithms and unusual exponents included.
Derivative Practice
Practice drawing derivatives from a graph. Practice symbolic derivatives for key functions (linear, quadratic, cubic, exponential, power-law). Click buttons to see correct answers.
Data Sets
Tuna and Sharks in Trieste
Real fisheries data from the port in Trieste. Kilograms of sharks and tuna caught each year.