2009 MJC P1 Q4

ToTo bear is an endangered species on Too Island. The number of births per unit time is proportional to the population of ToTo bears at any time and the number of deaths per unit time is proportional to the square of the population of ToTo bears.

If the population at time \(t\) is \(x\), show that \(\frac{\mathrm{d}x}{\mathrm{d}t}=ax-b{{x}^{2}}\), where \(a\) and \(b\) are positive constants.

It is also known that the population of ToTo bears remain constant when \(x=200\).

Hence, show that \(x=\frac{200}{B{{e}^{-kt}}+1}\), where \(k\) is a positive constant.