Differential Equations and Logistics

Euler's Method

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$, where $h$ is the step size

Logistic Growth Rate

$\frac{dP}{dt} = kP\left(1 - \frac{P}{M}\right)$, where $M$ is carrying capacity

Logistic max growth rate

occurs at $P = \frac{M}{2}$; rate $= \frac{kM^2}{4}$

Logistic curve shape

S-shaped; concave up when $P < \frac{M}{2}$, concave down when $P > \frac{M}{2}$, inflection at $P = \frac{M}{2}$

Separable Differential Equation

rewrite as $g(y)\,dy = f(x)\,dx$, then integrate both sides independently