Logistic Equation

What?

When dealing with exponentials and growth of populations, the logistic equation better represents the the population.

$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$
Where:

• $P(t)$ is population with respect to time
• $t$ is time
• $\frac{dP}{dt}$ is the instantaneous rate of change of population
• $K$ is the carrying capacity (the max population the environment is capable of sustaining)
• $r$ is the growth rate

The solution is:
$P(t)=\frac{P_{0}K{e}^{rt}}{(K-P_0)+P_0{e}^{rt}}$
Where:

• $P_0$ is the initial population