# Decline and half life

If $$\gamma$$, in $$\frac{dN}{dt}=\gamma*N$$, should be negative, indicating decrease instead of increase, this would be a case of exponential decline.

An important concept in exponential decline is half life which is defined as the time it takes to loose half of the stock’s (e.g. the population’s) size, or in other words, to reach $$\frac{𝑁_0}2$$. From $$\frac{𝑁_0}2=N_0*𝑒^{\gamma 𝑡}$$ one obtains $$\frac 1 2=−\gamma$$ or $$t=ln \frac 2 \gamma$$. With $$ln2\sim 0.69$$ this means that half life in the case of exponential decline is about $$\frac {0.69}{\gamma}$$. Hence with $$\gamma = 0.03$$, a population of 100 would take about $$\frac {0.69}{0.03}=23$$ time units to decrease to 50.