Oct

24

If one resource is critical for a species' survival (such as is sometimes the case in predator/prey relationships) there is a simple formula that often pertains to this closed loop situation.

If survival depends on a consistently regenerative resource (say in a small geographic area, such as a limited hunting area, and a singular source) this formula can be appropriate.

It is a recursive formula for the population size over time:

M X(t) ( 1 - X(t) ) = X(t+1)

This can be thought of as the inter-generational relationship when resource regeneration is consistently limited for each generation and there is a constant fertility rate.

M would be the multiplier, or fertility per generation.

X = Max maturity %, this would be the percentage of the current generation's use of the maximum resource.

The factor ( 1 - X(t)) or (1 - Max maturity % ) limits the next generation ability to survive due to overuse of the critical resource. In an extreme case if the prior generation ever completely used the critical resource (X= 100%), so that resource could not replenish, the next generation would be driven to extinction.

What I find intriguing about this formula is the 5 phases and the shifts between phases. And the final outcome depends on "M".

For a random X example, start with X(0)= 30%

if M is less than 1, not enough fertility to replenish, clearly the population will soon be driven to extinction.

if M greater than 1 but less than 2 there are two types asymptotic curves. One with a decline, the second with an incline.

The shift occurs at M = 1.429 = 1/ (1 - 0.30) when the result is a horizontal straight line = 30%.

The next shift is between M = 2 to 3.

Near 2 it results in a couple of high low alternations above and below the eventual asymptotic line.

As M approaches 3 the results are more alternation before the settling on the asymptote.

M a little more than 3 however, you start seeing a simple "fractal" pattern alternating up and down above and below an asymptotic line.

After about 3.5 but before 4 the up down fractal patterns become more complex with more iterations before repeating the cycle.

After 4 but before 4.76 or 1 / ( 30% X70%) rather than patterns it breaks out into chaos. Not just any chaos, but the chaos particular to chaos theory. The resulting range bouncing from near 100% to near 0% and the time of the bounces, while clearly deterministic, are impossible to determine by measurement of the initial "M". For example 4.0001 looks completely different from 4.00011 in terms of resulting peaks and valleys. (May I suggest that you set up a spreadsheet and graph the first 100 iterations with several "M" in each phase and shift.)

Like many simple formulas its application to the real world is somewhat limited. However, the formula is somewhat resilient to changes. Say the lack of the critical resource is not deadly to all the offspring, but a set percentage. that is: ( 1 - C% X(t)) the resulting patterns are similar, except X% can become greater than 100% in this case.

The formula's application does, therefore, appear to spring up in surprising diverse circumstances.

Of course, every business student knows the life cycle of a niche market can result in an asymptotic curve for sales levels.

For other phases, for example a reader wrote here a few years ago about how timber wolves and snow bunnies can turn into a consistent repeating cycle of wolves/bunnies prosperity. As the wolves prosper they drive the snow bunnies numbers down until few of the wolves can survive the winters. Then the bunnies prosper. This can result in very predictable populations of bunnies and timber wolves. Despite the myriad of other aspects that reflect survival chances for any one individual, such as disease, weather, etc. it boils down, for the wolves as a group, to one critical resource, snow bunnies.

For the bunnies however, it may require a more general formula. Where the larger the last generation was, the more wolves exist, the less likely their survival to reproduce is.

Similarly, pre 1960s when fertilizer use for farming became commercially viable a repeating pattern of up and down year to year wheat production was common in Oklahoma, with its poor soil. Here the critical resource was time for the soil to replenish. Now after many years of fertilizer use it appears that Oklahoma may again be reverting to this cycle due to harder to replenish lack of organic matter in the worked soils.

Finally, the more chaotic pattern may occur with gypsy moth caterpillars as they defoliate the trees leaves. See The Chaos of Population Growth [8 page MS Word document].

While I will leave it to the reader to ponder how this pertains to the markets in everyday use, let me suggest: with the boomers' government entitlements beginning to kick in and an unprecedented push to maintain the status quo with housing and financial companies surely this has long term application to the economy.

Phil McDonnell adds:

The recursive formula which Mr. Sears cited is also a second degree polynomial which has a long history. It can be re-written as:

M * ( - x^2 + x + 0 )

revealing its 2nd order nature.

It also was once proposed by John von Neumann as a way of generating pseudo random numbers on a computer using M between 4 and 4.76. Another name for this formula is the logistic map. Wikipedia has an interesting article on this which includes a dynamic graph of its behavior for various levels of M.

For what it is worth in a study of various simple transformations of return today to return tomorrow the logistic map showed the strongest relationship.

Dr. McDonnell is the author of Optimal Portfolio Modeling, Wiley, 2008


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