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 growth.

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 increments.
• Repeat the above using different values for B.

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 max.
• 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 to chaos!

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.