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Logistic equation (Rev #5, changes)

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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 equation growth model is

dxdt=(r/K)(Kx(1xK))x. {d x \over d t} = r (r/K) x\left(1-\frac{x}{K}\right) (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 a the simple simpler equation: model:

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

which describing describes exponential population growth:

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

When KK is finite and positive, the logistic equation model describes population growth that is approximately exponential when the population is much less thanKK, 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 \, .

It The is logistic easy model to can find be normalised by rescaling the explicit units solution of the population logistic and equation, time. since Define it is a first-order separable differential equation. However, instead of doing this, let us consider a special case.y:=x/Ky\colon = x/K and s:=rts\colon = rt. The result is

By rescaling the time and population variables (that is, by choosing appropriate units for time and population), we can reduce the general logistic equation to the case where r=K=1r = K = 1:

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

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

One The solution growing solutions are all time-translated versions of this is the logistic function (See logistic Wikipedia function : )

x y( t s)=e t s1+e t s=11+e s. x(t) y(s) = \frac{e^t}{1 {e^s\over 1 + e^t} 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 having have that the same limiting behavior and are time-translated versions of this one, one. i.e.: After rescaling back to the original variables, we have

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

There are also decreasing solutions wherex>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