8 In the year 2000 the population density, \(P\), of a village was 100 people per \(\mathrm { km } ^ { 2 }\), and was increasing at the rate of 1 person per \(\mathrm { km } ^ { 2 }\) per year. The rate of increase of the population density is thought to be inversely proportional to the size of the population density. The time in years after the year 2000 is denoted by \(t\).

1. Write down a differential equation to model this situation, and solve it to express \(P\) in terms of \(t\).
2. In 2008 the population density of the village was 108 people per \(\mathrm { km } ^ { 2 }\) and in 2013 it was 128 people per \(\mathrm { km } ^ { 2 }\). Determine how well the model fits these figures.