The Belgian mathematician Pierre Verhulst came up with the idea of limiting the population by assuming that the growth decays again then a maximal population is approached. More precisely, he proposed what is known today as the logistic equation. > [!NOTE] Definition (Logistic Equation) > $\frac{d}{dt} p(t) = kp(t)\left( 1 - \frac{p(t)}{p_{m}} \right)$where $p_{m}>0$ stands for the maximal population (the assumption is that there are not enough resources for a bigger population) and $k>0$ is the growth rate factor. # Properties Note that when $p(t)$ is very small in comparison to with $p_{m}$ then $1-p/p_{m}\approx{1}$ and we recover [[Malthus' Population Growth Model|Malthus' population growth model]]. However when $p(t)$ approaches $p_{m}$ then $1-p(t)/p_{m}$ becomes very small, so the growth $\frac{d}{dt}p(t)$ becomes small. The [[Scalar Ordinary Differential Equation|equation]] is [[Linear Scalar Ordinary Differential Equation|non-linear]] since it involves the $p(t)^{2}$ term. It is [[Autonomous Scalar Ordinary Differential Equation|autonomous]] since $t$ solely appears as an argument of $p.$ The equation is [[Homogeneous Ordinary Differential Equation|homogenous]] as there is no additive constant. See [[Validation of Population Growth Model]]. See [[Stationary Points of Population Growth Model]]. See [[Solution to Population Growth Model]].