Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation dP/dt=cln(K/P)P

Question

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation dP/dt=cln(K/P)P where c is a constant and K is the carrying capacity.

a)Solve this differential equation for c=0.25, K=1000, and initial population P0=100. P(t)=???

b)Compute the limiting value of the size of the population. limt→∞P(t)= ??

c) At what value of P does P grow fastest? P= ??

Answer 1

a)We separate variables: dP/Pln(K/P) = cdt; ∫dP/P(lnK - lnP) = ∫cdt; - ln(lnK - lnP) = ct + C;

ln(ln1000 - lnP) = 0.25t + C; - ln(ln1000 - ln 100) = C; C = - ln(ln10);

ln(ln1000 - ln P) = - 0.25t + ln(ln10);

ln1000 - lnP= e^{- 0.25t} ln10;

lnP = 3ln10 - e^-0.25t·ln10= (3 - e^-0.25t)ln10

P(t) = 10^(3 - e^{- 0.25t});

b) limP(t) = 10³ = 1000

t→∞

c)You have to find maximum of P'(t). So you have to find C''(t) =0 to get t