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

$\frac{dP}{dt} = c ln(\frac{K}{P})P$

where c is a positive constant and K is the carrying capacity.

(a) Solve this differential equation (assume P(0)=P₀).

(b) As time goes on (to infinity), does the population die off, grow without bound, or settle on some finite number?

Answer 1

Solution (a) the given differential equanim is, cln (*) Р de at dP cdt oth (H P in we get, Integrating dp dp - fedt Spin (4) fig et K de Z z - In C) dz 1 => dz $ I dp p2

from 6 (i) we get 1 -J ₂ dz Jedt => s dz c-cf dr to(2) =) -ct t ln ci [c is constant] In a ln z =) -ct q -) en -ct Cl -ct -) dilu e -ct => Z cie -ct cie (ii) => >> on <^) = , Now Given that POO Po when t 0, P = P. K en ( ci Po from (1) we get, act 1 (9) - to (t)e- -ct en ce -> K Þ e

en ( 4 / ( 4 ) ect => is the solution of the given This differential equation (6) cuts see what happens when ta for this we Compute lim Plt) tto -In ( 4 / é et lim ke ttoo cenco). o lim ke [ é o ta kie K on (to infinity); m AS time goes the population settle some finity number which is, in particular, is the carrying capacity K.