Regarding Carrying Capacity

April 25th, 2011

Supplement to Class Notes on Logistic Models

My first meaningful work with the logistic model came about in a class called, “Psychometrics” which, as you might guess, is the study of assigning values to psychological phenomena such as, learning and being emotional.   Most who hear of such a class revolt at the thought of numbers being used to measure something as sacred as the mind, but I assure you the work in this field is progressing at a pace more rapid now than most probably at any other time in our history.  The reason for the increased level both of academic activity and of attention given both to psycho-metrics and to logistic analysis, stems from the many parties concerned with the generation of an artificial intelligence.  Before we move too far astray into these dark, murky and intriguing waters, let us very neatly define the mathematical essence of the logistic model.

We have practiced problems involving population increases and decreases, whether the population under question was that of a peopled territory or that of a block of cheese being overrun by bacterium; regardless, the form of the mathematical model remained the same.

As a brief recap of these exercises, let us remind ourselves that, generally speaking, you were given some initial population of whatever:  For today, let us assume that the population in question is the population of interest in learning math on a Monday.  If the pattern of growth resembles a “J” then it might be exponential growth and of course the reverse “J” might mean exponential decay. 

J-Curve Review

As a good introduction to logistic modeling and to some of the theoretical support for the parameters of the logistic model, recall the J-curve model and understand that it is an essential component of our general logistic model.   We will focus on an algebraic modification of the equation for the J-curve that allows for “levels of saturation”, or carrying capacities.

To Self:            (Already, I have reservations regarding the correctness of “voice” here; could it be that this “level of saturation” to which I refer has already been reached by some of my students? How presumptuous am I to even consider that there was ever an upward trend?  I shall sketch an upward trend in the hopes that this will suggest optimism.)

Please note the upward trend below!  How glorious is this trend that it would extend to infinity!

If our domain is time (as it so often is) and the range is “interest in class”, then this paints an unbelievably optimistic picture of "interest in Precalculus material". 

To Self:
          Could it be that there is really no saturation; no plateau?!

Am I really suggesting that there could be no point at which the material being presented is simply too boring? 

Must the material be not too something?

Did I not lose 7 to 8 students when I mentioned the word, “exponential”?  If not, then surely when I mentioned, “logarithmic”? 

(I did mention logarithmic, didn’t I?)

          If it is too something, then what... is it, too... what of...? 

          Too….too….

         ....too much NOT the weekend?

           Too much NOT a holiday.  

           Too much NOT \

           This picture is entirely too optimistic indeed. 

           In any event, the shorthand we use for such optimism, as a matter of review         is as follows:

            (Your Interest, End of Class) =  (Your Interest, Now)  x  e  ^{(your time)}

But this is not shorthanded enough, so we reduce our syntax even further to:

                                                P_F = P_O  x   e^t

This then is something familiar and we have worked many, many problems involving this model and we have seen quite enough of it, you must be saying.

                       

(Note to self:    Be sure to check if anyone is in fact saying this.)

But we are to alter this today.  Today, we are realists: Today, we acknowledge our boredom and attempt to model just this, and if we are bolder still, we enlarge our world and model a standard week at school, generally speaking.  We begin our weekly studies with some enthusiasm but perhaps not as much as we will look forward to during the thick of it, the Wednesdays, the Tuesdays, of our school weeks.    Slow are the fits and starts of our Monday mornings and quick is the pace of our mid-week exams and papers.  By Friday we are indeed saturated and can stand little more than to end the day with a last gasp before heading home to truly comfortable surroundings. 

Let us begin sketching pictures of our interest in today’s class bearing in mind the following simple truths:   (1)  It is Monday.   (1.a.)  It is a post-holiday-Monday. (2)  This is a math class.  and  (3) Studies show that 50 minutes is simply too long to ask anyone to pay attention to anything.

Just for fun, let’s assume that your interest, right now, as we speak, is at a Level “1” and also that your interest could rise up to a Level “10”:  If we use the natural base, e, as our base of increase, is it possible for you to reach an interest level of 10 within this now 45 minute interval where t is measured in minutes and if it is possible what is the rate of increase, or the "k" value?   

The S-Curve

However much we adjust our “attention index” from the above exercise and however much I would like to believe that your interest will only continue to rise as the lesson moves forward, let us be honest and introduce the appropriate modifications.

The following picture might be a bit more realistic, though we will have to move it around a bit to have it fit our current conditions.   In any event, saturation, or more formally “carrying capacity” has been introduced in the form of the limiting value as defined by the curve approaching a maximum value.  This is the famous “S-curve”.

This “S-curve” is most commonly associated with a socio-economic dictum known as the, “Law of Diminishing Returns” which is self-explanatory and represents for us today a kind of drifting away of our attentions; our interests floating over to the end of the school day, to friends, to family, and to all things unrelated to math.

And yet this law is greatly underestimated in both its reach and its depth.  We will first look at how the function works, its technical matters as it were, and then we will attempt to provide this model with a proper context.

So, our modified model, taking saturation into account, is represented by the following code:

                                                G(x) = a /  (1 +  be ^-x)

Where a and b can both be “1” initially in the simplest form of the model.

There are two inverses in the form of this function:  The part that includes the exponentiation is in the denominator of the expression on the right and the variable power term is being negated.  This is generally how the function form is introduced and there is really very little context given to help develop our intuition for its form.  So, let’s test the function under a variety of conditions to get a feel for what it does:  i.e., are there any domain or range restrictions, any asymptotes, vertical or horizontal, and that sort of thing.

First, then we will ask how the behavior of the denominator changes as the power changes through various sets of values.  For now, we are assuming that the numerator of our function is “1” which stands for 100% of whatever it is that we are investigating.

Please refer to your own notes here for details of the construction of this model and refer to the questions from our previous blog discussion for answers to some of the questions posed there.

Regarding Points of Inflection

As read in class:

“Thus inflection is the pure Event of the line or of the point, the Virtual, ideality par excellence.  It will take place following the axes of the axes of the coordinates, but for now it is not yet in the world: it is the World itself, or rather its beginning…”

From “The Folds in the Soul, Chapter 2, p. 15:  The Fold:  Leibniz and the Baroque Deleuze, Gilles

The point of inflection, as it occurs within the S-curve, is formally ambiguous, even according to the co-founder of the Standard Calculus, Leibniz, in that there exists a region over which there is a steady rate of increase and a region over which there is a steady rate of decrease, and the point of inflection is to be found precisely between the two states, belonging to neither and both simultaneously.  Of course we may calculate its exact location:  Later, when studying calculus you will learn, if you have not already, that this point is well-defined when the derivative of this particular function equals, 0.  How much more exact can we be?  But alas, 0 has not the properties of any other number or set of numbers, and so, yes, there is still much left to its interpretation and to its use.

Let us consider that this logistic function defines for us a so-called law and inquire as to the nature of this law: Is it a natural law?  Is it a law as the law of gravity is a law?  In other words, are we bound to this law under any and all circumstances, or are there exceptions?  Can we break this law?

I proffer that contemporary American culture is defined by the bending of this very law and I have students who are presently disgusted, outraged by our gross abuse of free speech, as if there is no limit, no point of saturation to language.  If it can be said, then it must be said, regardless of diminishing returns.  Moreover and far worse, if it can be done,then it must be done, regardless of how swiftly the fall of our return on effort, shock and awe rule the day as if to say, this law does not apply in a Democracy.

Are there any exceptions to the Law and how often do these points of inflection occur in our daily lives?


Experiment

I submit the following experiment:  As you are moving through your daily activities, try to reflect on exactly when a point of inflection occurs for you.  This Easter weekend past may provide a good case study for us in that some of us received chocolate and will consider eating this chocolate at a very rapid pace (speaking for myself, anyway!)   And so, I will try to be mindful of that point at which I am no longer really enjoying the chocolate but may want to continue eating it nonetheless.  At this point, I will reflect and submit a ("POI") tweet.  Perhaps there will be a spike of such tweets at certain times and on certain days of the week?

We will add some modifications to this code so that we can differentiate our tweets as we move through the experiment.  So, an, "eating POI", could be simply, FOODPOI.