Dynamical Systems - Intro

Part 1: Simple models and calculus
Part 2: Probability
Part 3: Vectors andmatrices

Write down anything about math

Derivative of ln(x) = 1/x

Exponentials, parabola, population genetics

$f (x) = x^{2}$
Even parabola centered on 0,0

f (x) = 2 x (1 - x) = 2 x - 2 x^{2}

H = 2 p q = 2 p (1 - p)

(Hardy-Weinberg formula for heterozygote frq in population genetics)

This is an inverted(downward opening) parabola centered on $(x, y) = (0.5, 0.5)$

Hyperbola, saturating functions

φ (x) = β x / (1 + α x)

This function will converge at a point called by mathematicians an asymptote and by biologists a saturation point equal approximately to $β / α$ (limit to infinity). We call this a concave function since its rate of increase gets smaller over time.

Positive exponential functions and parabolas are convex functions since they increase more and more over time.

Exponential curve for population growth

N (t) = N_{0} e^{r t}

If the coefficient r is greater, the amount of time required to reach a given population is reduced.

Radioactive decay

$e^{- x}$ with a negative exponent can be used to show these negative values

L (t) = a - b e^{- μ t}

Where $a$ = 3, $b = 1.5$ and $μ = 0.7$

left = -0.5
bottom = -0.5
---
a = 3
b = 1.5
\mu = 0.7
y= a - be^{-\mu t}|red