2023 VJC P1 Q11

Statistics show that the population of rodents in a small area follows a density-dependent population growth model. In this model, as the population approaches its carrying capacity, the growth rate slows down due to increased competition for limited resources, such as food, space, and mates.

In a particular small farm, a virus spread has a significant impact on the population of rodents. The population \(t\) days after the start of the virus spread is denoted by \(N\)(in hundreds). The density-dependent population growth model states that the population of rodents increases at a rate inversely proportional to the square of the population. At the same time rodents die due to the virus spread at a rate proportional to the population. When \(N=3\), the number of rodents remains constant.

1. Show that \(\frac{\mathrm{d}N}{\mathrm{d}t}=\frac{k\left( 27-{{N}^{3}} \right)}{{{N}^{2}}}\), where \(k\) is a positive constant.[3]
2. It is further given that when the virus spread started, there were \(500\) rodents. Show that \({{N}^{3}}=27+C{{\mathrm{e}}^{-3kt}}\), where \(C\) is a positive constant to be determined.[6]
3. Sketch the graph of \(N\) against \(t\) and explain what will happen to the population of rodents in the farm in the long run.[3]