11. A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation
$$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 15 } P ( 5 - P ) , \quad t \geqslant 0$$
where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began.
Given that when \(t = 0 , P = 1\),
- solve the differential equation, giving your answer in the form
$$P = \frac { a } { b + c \mathrm { e } ^ { - \frac { 1 } { 3 } t } }$$
where \(a\), \(b\) and \(c\) are integers.
- Hence show that the population cannot exceed 5000