1. Discrete Difference Equation Models
Potential Entry Point Models

2. Possible Entry Point Models Single Equation Models
Linear Models
Nonlinear Models
Exponential Growth/Decay 𝑥 𝑡+1 =𝑎 𝑥 𝑡
Cell culture growth
Drug decay
Cohort mean annual survival (life tables)
Linear Difference Equation 𝑥 𝑡+1 =𝑎 𝑥 𝑡 +𝑏
Drug dosing
Harvesting/fisheries maximum sustainable yield
Continent-island model (one-way gene flow)
Population Growth with Resource Constraints
Logistic Difference Equation 𝑥 𝑡+1 = 𝑥 𝑡 +𝑟 𝑥 𝑡 1+ 𝑥 𝑡 𝐾
Beverton-Holt Difference Equation 𝑥 𝑡+1 = 𝑟 𝐾𝑥 𝑡 𝐾+(𝑟−1) 𝑥 𝑡
Natural Selection Model 𝑞 𝑡+1 = 𝑤 2 1− 𝑞 𝑡 𝑞 𝑡 + 𝑤 3 𝑞 𝑡 2 𝑤 1 1− 𝑞 𝑡 𝑤 2 1− 𝑞 𝑡 𝑞 𝑡 + 𝑤 3 𝑞 𝑡 2
Comparing different modes of selection

3. Scanning electron micrograph of E. coli
Optical Density of E. coli cell culture
Table gives the optical density of an Escherichia coli cell culture measured every 30 minutes for 3 hours. The culture was grown in a nutrient solution at 37°C.
Scanning electron micrograph of E. coli
Arithmetic mean of ratios
Q: What assumptions did we make in constructing this model?
Q: Why are the ratios not exactly the same between time steps? Measurement error? Stochasticity?
Geometric mean of ratios
Median of ratios