# The Azimuth Project 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

 {d x \over d t} = r (r/K) x\left(1-\frac{x}{K}\right) (K-x) x \, .

Here $x$ is the population, which is a function of time $t$. $K$ is the equilibrium population, and $r$ is the growth rate.

Note that in the limit $K \to \infty$ , we get a the simple simpler equation: model:

${d x \over d t} = r x$

which describing describes exponential population growth:

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

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

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

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.$y\colon = x/K$ and $s\colon = rt$. 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 $r = K = 1$:

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

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(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 $0$ to $1$ as $t$ 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) = \frac{e^{t-t_0}}{1 {K\over 1 + e^{t-t_0}} e^{-r(t-t_0)}} \,. \, .

There are also decreasing solutions where$x \gt 1$ and solutions (irrelevant to population biology) where $x \lt 0$ 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