12) Scientists are interested in the population of a particular species. They attempt to model the population $P$ at time $t$ days using a differential equation. Initially the population is observed to be 50 and after 10 days the population is 100 .

The first model the scientists use assumes that the rate of change of the population is proportional to the population.
(a) Write down a differential equation for this model and solve it for $P$ in terms of $t$.


To allow for constraints on population growth, the model is refined to
\frac{\mathrm{d} P}{\mathrm{~d} t}=\lambda P(500-P)
where $\lambda$ is a constant.
(b) Solve this differential equation to find $P$ in terms of $t$.


(c) Using the refined model, state the population of this species in the long term. Comment on how this value suggests the refined model is an improvement on the first model.