Logistic equation (Rev #5, changes)

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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

$$\frac{dx}{dt}=(r/K)(K-x(1-\frac{x}{K}))x\phantom{\rule{thinmathspace}{0ex}}.$$ {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~~~~ :~~ )

$$\mathrm{xy}(\mathrm{ts})=\frac{{e}^{\mathrm{ts}}}{1+{e}^{\mathrm{ts}}}=\frac{1}{1+{e}^{-s}}\phantom{\rule{thinmathspace}{0ex}}.$$ ~~ 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

$$x(t)=\frac{{e}^{t-{t}_{0}}K}{1+{e}^{-r(t-{t}_{0})}}\phantom{\rule{thinmathspace}{0ex}}.$$ 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 $-\infty$.

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

Here chaos arises as a consequence of delayed feedback.

category: ecology