Consider a rabbit population P(t) satisfying the logistic equation StartFraction dP Over dt EndFraction equals aP minus bP squared , where Upper B equals aP is the time rate at which births occur and Upper D equals bP squared is the rate at which deaths occur. If the initial population is 220 rabbits and there are 9 births per month and 15 deaths per month occurring at time tequals0, how many months does it take for P(t) to reach 110% of the limiting population M?

Solution:

Given :

        .............(1)

where, B = aP = birth rate

            D =  =  death rate

Now initial population at t = 0, we have

= 220 ,   = 9 ,   = 15

Now equation (1) can be written as :

   .................(2)

Now this equation is similar to the logistic differential equation which is ,

where M = limiting population / carrying capacity

This gives us M = a/b

Now we can find the value of a and b at t=0 and substitute for M

   and    

So,

          =

          = 132

Now from equation (2), we get the constants

k = b =

        =

The population P(t) from logistic equation is calculated by :

As per question, P(t) = 110% of M

Now taking natural logs on both the sides we get

t = 36.216

Number of months = 36.216