Logistic functions are good models of biological population growth in species which have grown so large that they are near to saturating their ecosystems, or of the spread of information within societies. They are also common in marketing, where they chart the sales of new products over time; in a different context, they can also describe demand curves: the decline of demand for a product as a function of increasing price can be modelled by a logistic function, as in the figure below.
The formula for the logistic function,
Y=CAeBx
involves three parameters A, B, C. (Compare with the case of a quadratic function y = ax2+bx+c which also has three parameters.) We will now investigate the meaning of these parameters. First we will assume that the parameters represent positive constants. As the input x grows in size, the term –Bx that appears in the exponent in the denominator of the formula becomes a larger and larger negative value. As a result, the term e–Bx becomes smaller and smaller.
To identify the exact meaning of the parameter A, set x = 0 in the formula; we find that
Y=C1+AeBx0 =C1+A
Clearing the denominator gives the equation (1 + A) y(0) = C. One way to interpret this last equation is to say that the limiting value C is 1 + A times larger than the initial output y(0) An equivalent interpretation is that A is the number of times that the initial population must grow to reach C .The parameter B is much harder to interpret exactly. We will be content to simply mention that if B is positive, the logistic function will always increase, while if B is negative, the function will always decrease.
A look at a scatterplot of these data makes clear that a logistic function provides a good model. It also appears that data seems to be quickly approaching its limiting value. It makes sense for us to estimate this limiting value as C = 1250.
Next, since our data give us that f (0) = 554.8, we have the following equation for
A: (1 + A) f (0) = C, or (1 + A) 554.8 = 1250. Therefore, 1 + A =2.25,
Or A = 1.25. Finally, the data point in the middle of the plot, (5, 830.7), is a good candidate for our inflection point since its ycoordinate is roughly onehalf the limiting value. Consequently, we can set 5 = ln A / B= ln 1.25 / B to get an equation we can solve for B: B = ln 1.25 / 5 = 0.046. We have thus created a set of values for our parameters, giving the logistic function,
p=ft=12501+1.25e0.046x
On observing the graph we notice that it’s a growth curve year on year. So
it’s appropriate to first check whether exponential function fits it or not.
Two common types of mathematical models are
Exponential Growth: y = a e bx, b > 0.
Exponential Decay: y = a e bx, b > 0.
Suppose we know that a variable y can be expressed in the form aebx, but we don't know the values a and b. If we are given any two points on the graph of y, then it is possible to find the numbers a and b.
The simplest case, and one that is often encountered in applications, is where we know the value of y when x = 0 and one other point on the graph of y.
A graph of the function over the scatterplot shows the nice fit. On the other hand, your calculator will also provide a logistic regression function with different values for the parameters but it, too, provides a nice fit.
Here are additional data on population trends in China from the 2008 World Economic Outlook, published by the International Monetary Fund (IMF).
Now we scatter plot the above data
As we spot the trend line on the plotted graph we observe that exponential function fits this data. It’s a growth function as per observation.
This data is approximated well by the exponential growth model
P = 1030e0.0101x
where t is the number of years since 1980. In other words, the year 1980 corresponds to t = 0, 1981 corresponds to t = 1, etc. The data points and model are graphed below.
Population data points and model P =1030e0.0101x
where t is number of years since 1980.
Use the model to predict the population of the China in 2030.
2030 corresponds to t = 50, so our model predicts that the population will be
P = 1030 e0.0101*50 = 170669.5 million.
Therefore, the population is expected to reach 1 trillion by the year 2030.
It is important to recognize the limitations of this model. While it is obvious from the graph that for t between 0 and 20, the model values are very close to the actual population values, we should not assume that our model will give an accurate prediction of population for values of t much larger than 20. For instance, the model predicts that in the year 2030 (t = 50), the population of the city will be almost 1 trillion! That is not likely.
CHAOS IN THE LOGISTIC MODEL
Discretetime version
When the logistic equation is discretised it displays chaos. It is, in fact, the canonical ground for the study of the perioddoubling cascade.
Assume that the Euler formula is used to discretise the logistic growth model, that is,
ΔxΔt=rKKxx.
Assume tn+1−tn=Δt and xn=x(tn) for all n. Then,
xn+1=1+r∆txnrΔtKxn2
This can be normalised by letting λ=1+rΔt:
xn+1= λxn111λK xn
and measuring x in units of K1−1/λ, that is, x=Ky1−1/λ and one obtains the socalled Logistic Map.
yn+1=λyn1yn
It is known that, depending on the value of λ, this equation has a stable fixed point, multi periodic stationary states, or displays chaotic behaviour. Note that λ depends only on the growth rate and the time step, and not on the carrying capacity.
When the logistic map is interpreted as a numerical approximation to the continuous logistic growth model, the time step Δt is largely arbitrary and can be chosen to be less than 1/r, so that 1<λ<2ensuring numerical stability.
However, it is possible to interpret the continuous logistic model as an approximation to a fundamental discrete model. In that case, Δt is not arbitrary (representing, for instance, the periodicity with which a species reproduces  usually annually) and neither is r, in which case λ=1+rΔt cannot be tuned.
If the population reproduces too fast (high values of lambda), then the sizes of successive generations may be chaotic. If λ>3 the stationary state fluctuates among at least two states, and if λ>3.57(approximately) the dynamics is chaotic.
BEHAVIOUR OF THE SYSTEM
The behaviour of the system is determined by following the orbit of the initial seed value. All initial conditions eventually settle into one of three different types of behaviour.

Fixed: The population approaches a stable value. It can do so by approaching asymptotically from one side in a manner something like an over damped harmonic oscillator or asymptotically from both sides like an under damped oscillator. Starting on a seed that is a fixed point is something like starting an SHO at equilibrium with a velocity of zero. The logistic equation differs from the SHO in the existence of eventually fixed points. It's impossible for an SHO to arrive at its equilibrium position in a finite amount of time (although it will get arbitrarily close to it).

Periodic: The population alternates between two or more fixed values. Likewise, it can do so by approaching asymptotically in one direction or from opposite sides in an alternating manner. The nature of periodicity is richer in the logistic equation than the SHO. For one thing, periodic orbits can be either stable or unstable. An SHO would never settle in to a periodic state unless driven there. In the case of the damped oscillator, the system was leaving the periodic state for the comfort of equilibrium. Second, a periodic state with multiple maxima and/or minima can arise only from systems of coupled SHOs (connected or compound pendulums, for example, or vibrations in continuous media). Lastly, the periodicity is discrete; that is, there are no intermediate values.

Chaotic: The population will eventually visit every neighbourhood in a subinterval of (0, 1). Nested among the points it does visit, there is a countable infinite set of fixed points and periodic points of every period. The points are equivalent to Cantor middle thirds set and are wildly unstable. It is highly likely that any real population would ever begin with one of these values. In addition, chaotic orbits exhibit sensitive dependence on initial conditions such that any two nearby points will eventually diverge in their orbits to any arbitrary separation one chooses.
The behaviour of the logistic equation is more complex than that of the simple harmonic oscillator. The type of orbit depends on the growth rate parameter, but in a manner that does not lend itself to "less than", "greater than", "equal to" statements. The best way to visualize the behaviour of the orbits as a function of the growth rate is with a bifurcation diagram. Pick a convenient seed value, generate a large number of iterations, discard the first few and plot the rest as a function of the growth factor.