The Azimuth Project
Logistic equation (Rev #5)

Idea

The logistic equation is a simple model of population growth in conditions where there are limited resources. When the population is low it grows in an approximately exponential way. Then, as the effects of limited resources become important, the growth slows, and approaches a limiting value, the equilibrium population or carrying capacity.

The logistic growth model is

dxdt=(r/K)(Kx)x.{d x \over d t} = (r/K) (K-x) x \, .

Here xx is the population, which is a function of time tt. KK is the equilibrium population, and rr is the growth rate.

Note that in the limit KK \to \infty, we get the simpler model:

dxdt=rx {d x \over d t} = r x

describing exponential population growth:

x(t)=x 0e rt. x(t) = x_0 e^{r t} \, .

When KK is finite and positive, the logistic model describes population growth that is approximately exponential when the population is much less than KK, but levels off as the population approaches KK. If the population is larger than KK, it will decrease. Every positive solution has

lim t+x(t)=K. lim_{t \to +\infty} x(t) = K \, .

The logistic model can be normalised by rescaling the units of population and time. Define y:=x/Ky\colon = x/K and s:=rts\colon = rt. The result is

dyds=y(1y).{d y \over d s} = y(1-y) \, .

It is easy to find the explicit solution of the logistic equation, since it is a first-order separable differential equation.

The growing solutions are all time-translated versions of the logistic function (See Wikipedia)

y(s)=e s1+e s=11+e s. y(s) = {e^s\over 1 + e^s} = {1\over 1 + e^{-s}}\, .

It looks like this:

This solution goes from 00 to 11 as tt goes from -\infty to ++\infty. All other growing solutions have the same limiting behavior and are time-translated versions of this one. After rescaling back to the original variables, we have

x(t)=K1+e r(tt 0). x(t) = {K\over 1 + e^{-r(t-t_0)}} \, .

There are also decreasing solutions where x>1x \gt 1 and solutions (irrelevant to population biology) where x<0x \lt 0 decreases explosively to -\infty.

Chaos in the Logistic model

Discrete version

When the logistic equation is discretised it displays chaos. It is, in fact, the canonical ground for the studying of the period-doubling cascade.

Continuous version

Here chaos arises as a consequence of delayed feedback.

References

category: ecology