Verhultz equation, an exploration
of chaos, and a fractal connection.
Population dynamics is of interest to geoscientists
for several reasons. Human population dynamics figure heavily
into natural hazard assessment and resource management. For many
sedimentary systems microbial action can be important. Think of
planktonic blooms and the like. A simple equation that can be
nicely modeled in Excel (or in a large assortment of other software)
shows an incredibly rich array of behavior, that may also be insightful
into one aspect of population behavior.
The equation is X(n+1) = B Xn (1-Xn/Xmax) . X
refers to population values during increments of time. This is
a reiterative formula, so you feed the result of the last increment
into the formula for the next increment. The subscript refers
to a particular increment within the history. X max is the maximum
population that can be sustained, a measure of system carrying
capacity for that population. B is an inherent growth rate, i.e.
the growth rate if or when there is no restraint on population
The first part of the equation, X(n+1)
= B Xn, is basically the unrestrained population
growth modeled by exponential growth. The second part of the equation
provides the restraint. As the population approaches the carrying
capacity, this term can not only reduce growth, it can actually
cause a population reduction. What type of behavior do we see
for different values of B?
Exploring the model in Excel (in class demonstration).
- It turns out the results are not extremely
sensitive to Xmax,
but are exquisitely sensitive to B. You can enter a reasonable value for Xmax in a cell for later reference,
and then change that value to see how it changes your results.
- Create an array of different B values you
wish to explore. values between 2.9 and 4.0 are particularly
interesting (see below).
- Put an initial population value in the top
cell of a column. Insert the formula above into the cell below
with Xn = to the
cell above, and B
is defined by the appropriate cell array containing the B values
you wish to explore, and Xmax by its appropriate cell. Remember to use $ to
fix the location of reference cells you don't want to change
as you paste this formula into other cells.
- Copy the formula down however many reiterations
you want. 100 works well.
- Do a column graph of that column and you
will get a history of the population with time (with successive
- Repeat the above using different values for
One can map out what happens to the population
history as B varies by mapping B as X and the cloud of points
of population values as Y.
- B is < 2.5: after a exponential growth
period the population gradually settles out at an equilibrium
value that is 66% of X
- B is 2.5 to 2.999 ...: after exponential
growth the population oscillates a bit before settling down to
the above equilibrium value.
- B is 3.0 to 3.4495: after exponential growth
the population oscillates in equilibrium around two values.
- B is 3.4495 to 3.56: after exponential growth
the population oscillates in equilibrium around four values.
- B is 3.56 to 3.596: after exponential growth
the population oscillates in equilibrium around 8 values.
- B reaches 3.569999.. the population oscillates
chaotically at an infinite number of positions.
- In a narrow range of values around B=3.8
stability and doubling is found again, but quickly resorts back
On a map the resulting pattern has a fractal
branching behavior, providing a link between chaos and fractals.
Above is an Excel plot of population
values for 125 generations of a 'mature' population for different
inherent growth values by increments of .0. Note that as the inherent
growth constant gets larger one proceeds from an oscillation between
two values to an oscillation between four values, to a oscillation
between 8 and then in a short x distance, 16 until one quickly
reaches chaos. The pattern here is partly a fractal one, although
there are not enough points in this plot to see the fine details.
One would want to use smaller increments of change in x and many
more reiterations to see the fine details. Note the return to
order at an x value of 3.63.
Think of what the implications might be
in understanding associated natural systems.
- One thing to recognize is initial conditions
sensitivity. Changing one variable just a little bit, in
this case, can alter subsequent long term behavior dramatically.
- This behavior challenges deterministic precise
predictions of where a system will be in the future, but may
allow us to predict general patterns of behavior. A key word
to learn more about here is strange attractors.
- There is very interesting and rich behavior
in this simple equation.
- This type of chaotic perspective is involved
in the debate as to whether earthquakes be predicted?
- There is much more to think on here!
Copyright by Harmon D. Maher Jr.. This material
may be used for non-profit educational purposes if proper attribution
is given. Otherwise please contact Harmon D. Maher Jr.