Population Growth

The differential equation describing exponential growth is

$${dN\over dt} = {N\over\tau}.$$ (1)

This can be integrated directly

$$\int_{N_0}^N {dN\over N} = \int_0^t {dt\over\tau}$$ (2)

$$\ln\left({N\over N_0}\right)={t\over\tau}.$$ (3)

Exponentiating,

$$N(t) = N_0e^{t/\tau}.$$ (4)

Defining $N(t=1)=N_0e^\alpha$ gives $\tau=1/\alpha$ in (4), so

$$N(t) = N_0 e^{\alpha t}.$$ (5)

The quantity $\alpha$ in this equation is sometimes known as the Malthusian Parameter.

Consider a more complicated growth law

$${dN\over dt} = \left({\alpha t-1\over t}\right)N,$$ (6)

where $\alpha>1$ is a constant. This can also be integrated directly

$${dN\over N} = \left({\alpha-{1\over t}}\right)\,dt$$ (7)

$$\ln N = \alpha t-\ln t+C$$ (8)

$$N(t)={Ce^{\alpha t}\over t}.$$ (9)

Note that this expression blows up at $t=0$. We are given the Initial Condition that $N(t=1)=N_0e^\alpha$, so $C=N_0$.

$$N(t)=N_0 {e^{\alpha t}\over t}.$$ (10)

The $t$ in the Denominator of (10) greatly suppresses the growth in the long run compared to the simple growth law.

The Logistic Growth Curve, defined by

$${dN\over dt}={r(K-N)\over N}$$ (11)

is another growth law which frequently arises in biology. It has a rather complicated solution for $N(t)$.

See also Gompertz Curve, Life Expectancy, Logistic Growth Curve, Lotka-Volterra Equations, Makeham Curve, Malthusian Parameter, Survivorship Curve