Logistic growth

A logistic growth model looks like this

\dot y = (a-by)y=ay-by^2.

Intuitively it represents something that grows more slowly the larger it is. For instance, logistic growth can be used to represent population growth, where resource constraints limit the growth rate for large populations.

The ODE can be solved by separation of variables and partial fraction decomposition to find

y(t) = \frac{a e^{a t + a C}}{1 + b e^{a t + a C}}.

We can also extract some information without solving the ODE. By inspection the critical points at which $\dot y(y_0) = 0$ are $y_0 = \{0, \frac ab\}$

The slope field below shows the slope field for $\dot y = (1- \frac{1}{4} y)y$ . We see critical points at $y_0 = \{0,4\}$ as expected.

$y$ is stable about $4$ and unstable about $0$ .