## Birth rate of a population question

The birth rate 'Beta (t)' of a population 'P (t)' decreases exponentially with time, so that 'Beta (t)'= 'Beta0 exp^(-Alpha t)', where 'Alpha, Beta > 0'. Therefore the population dynamics are governed by the differential equation, (dP/dt)= Beta (t)P, P(0)=Po

Solve the above initial value problem to find an expression for P(t) in terms of Po, Alpha, Beta. Use this expression to deduce the behaviour of P(t) as t tends to infinity.