NJC Differential Equations Tutorial Q1

A particular species reproduce at the rate of \(100k\%\), where \(k\) is a positive constant. Due to natural causes, members of the population die at a rate that is proportional to the square of its population size with constant of proportionality \(\frac{k}{n}\), where \(n\) is a positive constant. It may be assumed that there are no external factors that affect the growth of the population, whose size is represented by \(x\).

1. Write, in terms of \(k\) and \(n\), a differential equation for the population growth rate.
2. Based on your differential equation in part (i), find, in terms of \(k\) and/or \(n\), the size of the population when its growth is the fastest, and the growth rate at this instant.
3. Solve the differential equation in part (i), given that the population is growing the fastest initially. Express \(x\) in terms of \(t\).
4. Sketch the graph of \(x\) against \(t\), stating the equation of the asymptote in terms of \(n\).
5. With reference to your graph, explain the significance of \(n\) in context.