Fixed points and equilibria

As we have seen here, population dynamics modeled with logistic growth run up to a steady state defined by the carrying capacity K. K is a so called fixed point in these dynamics. It serves as an attractor. This can be seen when starting the model with different initial values, i.e. different initial population sizes, as in the right plot below. Irrespective of the size of the population, the dynamics always develop towards the carrying capacity. If population size is less than K it grows, if it is more than K, it declines.

Log 1
Log 2

These dynamics could be pictured as the ones of a ball in a bowl. At the lowest point of the bowl the ball is in equilibrium. At each other point it will move towards this equilibrium.


Systems with more than one fixed points - the Allee-effect

The logistic equation for modeling population dynamics has been criticized for not accounting for particular problems that populations might encounter at low density. Some animals for instance have problems finding a partner when population density is low, others - like flamingos - need to congregate in order to protect against predators. Without this possibility they won't reproduce. Their reproduction rate is constrained at low densities. This effect is called the Allee-effect after the zoologist Warder Allee. It can be expressed with the following formula


which causes the dynamics beneath a so called Allee extinction threshold to decline to zero, and above this threshold to grow to K, the carrying capacity. As a consequence, population growth becomes zero not only at the carrying capacity, but also at the extinction threshold a for a = N. These dynamics thus have two non-trivial fixed points (and a trivial one which is given with r = 0). The bowl has two sinks.


Orbits and more complicated attractors

Now, consider the two Lotka-Volterra systems as they were discussed here and here.

\[\frac{dR}{dt} = a R (1 - \frac{R}{K}) - b R N\] \[\frac{dN}{dt} = c a R N - d N\]
System 1


\[\frac{dH}{dt} = a H - b H L\] \[\frac{dL}{dt} = - c L + d L H\]
Food web 2

As the phase space portrait of these systems shows, both approach some kind of steady state. The first system follows a spiral like movement to a fixed point. The other system seems to cycle in a sort of oval orbit. This orbit is called a limit cycle. Both forms, the fixed point as well as the limit cycle, are instances of attractors.

Attractors are interesting, because they can be seen as a defining feature of systems. Attractors are an expression of a system's Eigen-behavior, which is the kind of behavior a system shows independently of external influences or inputs. What this means shall be illustrated with the following simple examples.

The recursive operation 'divide x by 2 and add 1'

This recursive operation \(op_{n}=\frac{op_{n-1}}{2}+1\) has the property of asymptotically approaching 2, and it does this from every possible number that it is started from, as is shown on the right.

Divide by 2 add 1

A self-referential eigenform as attractor

Consider the following proposition as expressing something about itself:

This sentence has ... letters

Obviously, this proposition will have a correct solution, which is not externally given, but generated by the (internal) structure of the sentence. Trying to find this solution by guessing, then counting, then correcting the first guess for a second one, again counting, and so on, will iteratively approach this solution, no matter from where the first guess started. The solution is an attractor of this proposition (a hint: in English the solution varies slightly in dependence of whether the hyphen - is considered a letter or not).