Consider a rabbit population which satisfies the logistic mo

Consider a rabbit population which satisfies the logistic model as given in the previous exercise. If the initial population is 120 rabbits and there are 8 births per month and 6 deaths per month occurring at time t = 0, how many months does it take for P to reach 95% of the limiting population M?

Solution

Solution :

You have a population that is evolving according to the logistic model :

dP/dt = a*P - b*P²

a*P is the unfettered (exponential) growth rate and - b*P² is the death rate. We are told that initially, the population is P_o and that there are Bo births and Do deaths occurring at that time. That means that at P_o :

a*P_o = B_o

a = B_o/P_o

and

b*P_o² = D_o

b = D_o/P_o²

so we can write the logistic equation in terms of Po, Bo, and Do instead of the constants a and b :

dP/dt = (B_o/P_o)*P - (D_o/P_o²)*P²

The limiting (i.e., long-term, steady state) value(s) of the population, is (are) the values P such that dP/dt = 0 (i.e., there the population is not changing with time).

In this case :
0 = (B_o/P_o)*(P_lim) - (D_o/P_o²)*(P_lim)²

0 = P_lim * ((B_o/P_o) - (D_o/P_o²)*(P_lim))

Obviously, one value of P_lim that satisfies this equation is P_lim = 0, but this is not a very interesting case. The other value of P_lim i satisfies :

0 = ((B_o/P_o) - (D_o/P_o²)*(P_lim))

P_lim = B_o*P_o/D_o

To answer the last part of this problem, we need to solve the logistic model:

dP/dt = (B_o/P_o)*P - (D_o/P_o²)*P²

dP/dt = (D_o/P_o²)*P*( B_o*P_o/D_o - P)

dP/(P*(P - B_o*P_o/D_o)) = (-D_o/P_o²) dt

Expanding the left hand side in terms of partial fractions:

(D_o/(B_o*P_o))*[1/(P - B_o*P_o/D_o) - 1/P] dP = (- D_o/P_o²) dt
[1/P - 1/(P - B_o*P_o/D_o)] dP = (B_o/P_o) dt

ln(P) - ln(P - B_o*P_o/D) = (B_o/P_o)*t + c

ln(P/(P - B_o*P_o/D_o)) = (B_o/P_o)*t + c

For an initial population of P_o, we have :

ln(P_o/(P_o - B_o*P_o/D_o)) = c

ln(1/(1 - B_o/D_o)) = - ln(1 - B_o/D_o) = c

so the solution is :

ln(P/(P - B_o*P_o/D_o)) = (B_o/P_o)*t - ln(1 - B_o/D_o)

ln[(P - P*B_o/D_o)/(P - P_o*B_o/D_o)] = (B_o/P_o)*t

In general, the time it takes for the population to reach 95% of the limiting value (P(T_95) = 0.95*P_lim = 0.95*B_o*P_o/D_o) is :

ln[0.95*(1 - B_o/D_o)/(0.95 - 1)] = (B_o/P_o)*T_95

T_95 = (P_o/B_o)*ln[0.95*(1 - B_o/D_o)/(0.95 - 1)]

In this case, P_o = 120 rabbits, B_o = 8 rabbits/month D_o = 6 rabbits/month, so :

T_95 = (120/8)*month * ln[0.95*(1 - 8/6)/(-0.05)]

T_95 = 27.69 months