# Exponential Growth and Decay Models

## Key Questions

Population [P]= Ce^[kt

#### Explanation:

If the rate of growth $P$ is proportional to itself, then with respect to time $t$,

$\frac{\mathrm{dP}}{\mathrm{dt}} = k P$, ....inverting both sides, .....dt/[dP]=[1]/[kP and so integrating both sides

intdt=int[dP]/[kP, thus,..... $t = \frac{1}{k} \ln P +$ a constant............$\left[1\right]$

Suppose $P$ is some value $C$ when$t = 0$, substituting

$0 = \frac{1}{k} \ln C +$ constant, therefore the constant $= - \frac{1}{k} \ln C$ and so substituting this value for the constant in ...$\left[1\right]$ we have ,

$t = \frac{1}{k} \left[\ln P - \ln C\right]$ = $\frac{1}{k} \ln \left[\frac{P}{C}\right]$, therefore , $k t = \ln \left[\frac{p}{C}\right]$[ theory of logs] and so

${e}^{k t} = \frac{P}{C}$......giving  P=Ce^[kt. The constant $k$ will represent the excess of births over deaths or vice versa for a decreasing rate.