How to solve the Gompertz equation
The Gompertz equation is: dy
dt = ry ln (K/y), where r and K are positive
constants and the function y is positive. This equation is separable and autonomous. Dividing by K, we can rewrite it as:
y y
d y
= −r
ln
.
dt K
K
K
Make the substitution z =
y
K
> 0. Then we obtain
dz
= −rz ln z.
dt
So (assuming z 6= 1)
Z
dz
=−
z ln z
Z
rdt,
which implies that (we can use the substitution w = ln z, where dw =
see this)
ln |ln z| = −rt + C.
Exponentiating then yields
ln z = ±eC e−rt .
Combining this with the z = 1 case (an equilibrium solution), we obtain
ln z = ce−rt ,
where c is any real constant. Exponentiating again,
z = ece
−rt
b
So (note that ecb = (ec ) )
y = Kz = K (ec )
e−rt
.
If we assume the initial condition y (0) = y0 , then we get
e0
y0 = y (0) = K (ec )
so that ec =
y0
K.
= Kec ,
We conclude that
y=K
y e−rt
0
K
.
An equivalent form is
y0
y = Keln( K )e
−rt
= exp [ln (y0 /K)] e−rt .
See Problem 17 in §2.5.
1
dz
z ,
to