4 The number of insects in a population \(t\) weeks after the start of observations is denoted by \(N\). The population is decreasing at a rate proportional to \(N \mathrm { e } ^ { - 0.02 t }\). The variables \(N\) and \(t\) are treated as continuous, and it is given that when \(t = 0 , N = 1000\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = - 10\).

1. Show that \(N\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = - 0.01 \mathrm { e } ^ { - 0.02 t } N .$$
2. Solve the differential equation and find the value of \(t\) when \(N = 800\).
3. State what happens to the value of \(N\) as \(t\) becomes large.