- Suppose I am studying the evolution of a system of unicellular organims and I want to have a continuous model of its population $\,\mathrm{P}\left(t\right)$ with respect to a real time variable $t$ ( units: days ).
- We make the assumption that there is infinite place and resources for the organims to grow freely.
- The birth rate $b$ is defined to be the ratio of birth over the population, at a time $t$ and for a time interval $\Delta t$.
- Hence, $b\,\mathrm{P}\left(t\right)\Delta t$ is the population growth during a fraction of day $\Delta t$. It goes the same for the death rate $d$.
- We can then write for a small step $\Delta t$ : $$ \mathrm{P}\left(t + \Delta t\right) - \,\mathrm{P}\left(t\right) \approx b\,\mathrm{P}\left(t\right)\Delta t - d\,\mathrm{P}\left(t\right)\Delta t $$ We can then take the limit as $\Delta t$ goes to 0 to find $P(t)$.
My question : I don't understand why the equation of the change in population is an approximation $\left(~\approx~\right)$. Can anyone explain ?.