Current solution has been obtained under the assumption that the birth-rate and the death-rate are variable. However, there are another matters which have to be taken into account. One of these is harvesting of fishes. As we know harvesting has strongly been influenced on the popultation of many species of fishes.
Exercise
For the continuous logistic model obtain the solution for M = 50, c = 0.5, P0 = 10. Plot the solution and investigate the behaviour of the solution for a range of values P0.