Logistic Equation

The logistic equation is a first-order nonlinear ordinary differential equation used to model growth that is initial exponential but slows down due to limited resources. It is an example of a Bernoulli differential equation.

It is given by

$$\frac{dp}{dt}=ry(1-\frac{r}{K})$$

where:

• $p(t)$ is the quantity of interest at time $t$ (e.g., population size).
• $r\in \mathbb{R}>0$ is the intrinsic growth rate (a positive constant).
• $k\in \mathbb{R}>0$ is the carrying capacity of the environment (a positive constant).

Common Use Cases

1. Population dynamics
2. Spread and saturation processes

Link to Bernoulli Differential Equation

Beginning with the logistic model:

$$\frac{dp}{dt}=rp-\frac{r}{k}{p}^2$$

Rearrange into Bernoulli form:

$$\frac{dp}{dt}-rp=-\frac{r}{k}{p}^2$$

Thus, the logistic equation is a Bernoulli equation with exponent $\alpha =2$.

Solution

The logistic equation is separable. The general solution is

$$p(t)=\frac{k}{1+A{e}^{-rt}}$$

where $A$ is a constant determined by the initial condition.

If $p(0)=p_0$, then

$$A=\frac{k-p_0}{p_0}$$

Key Properties

• Equilibrium solutions: $p=0$ and $p=k$.
• For $0<p_0<k$, solutions increase monotonically toward $k$.
• The solution curve is sigmoidal