Hello all,
Mr. Fedirko had sent out a homework assignment asking us to calculate and create a model for the carrying capacity of the United States and its population. This furthers our study of Malthus, and I've spent a few days working on this so far. The following is what I've gathered through a few different approaches.
Malthus uses arithmetic increase in available resources, and geometric increase (exponential) for population increase. The relationship between these two progressions is that the the geometric increase of population depends upon the arithmetic increase of the resources. Thus, there is a surplus of resource that allots for the growth of the population when there is such a large scale space in between the line and curve. However, these are similar to regression analysis models that bets fit the data, as various circumstances like socioeconomic (among other) factors that can change the model. A general model that could fit a very basic understanding of population growth is the following. f(x)=2^x-(mortality rate as a decimal-x)
However, an exponential model (unless piecewise) does not hover around the resources that it needs to be sustained, as this is not in the nature of exponential growth. In this, a logistic curve is the most practical course of action in order to create this model. I tried to build my own logistic model off of the basic parent function in the textbook, or y=a/(1+be^(-rx)) Here, a is the current population (2011 census), b is the change increase in population from the 1900 census to the 2011 census, and r is the percent change between the two. With taking the 111 year population difference into account, I thought it might provide a more founded model rather than one this solely uses the 2011 population in the US. Thus, the model would be y=311,915,000/(1+309,301,152.3e^(-.757x)) However, I don't understand this r value, and certainly was having difficulty adjusting my window to show the s curve of the logistic. But of all things, I didn't know how to include in the model any reference to the resources that dictate a carrying capacity. Thus, I looked up carrying capacity formulas on the internet, but haven't gone as far yet as to break down what each variable and constant in the equation mean.
Any input?
Melanie
Melanie,
ReplyDeleteConsider that your "r" value is negative, an inverse: Does this play any role in modeling the curbing of resources? Also, in the standard models, we find the natural growth function situated in the multiplicative inverse of the overall function: Might this play some role? Lastly, does the "competition" between the multiplication and exponentiation of the natural growth function itself, somehow reflect a kind of "internal adjustment" to growth? I am looking forward to some discussion of these modeling issues and to your peer feedback!
I dont quite understand how "r" can be negative isn't it modeling the natural growth, not a decrease?
ReplyDeleteThis is a good question Teddy: I'll wait a bit before responding so that your peers can give this a try!
ReplyDeleteI guess I agree with Teddy. If "r" is negative and is multiplied with "x" the two would yied a negative value. Furthermore, "r" is a constant just like "k" if that is of any use.
ReplyDeleteI feel like "r" could definitely be negative. If r is the rate of increase, then when in fact it is decreasing it would have to be negative. This makes sense because if you were measuring the rate of decrease, you would not need to specify that it was negative, simply because you already defined it as such by stating that it was a measurement of decrease.
ReplyDeleteOn the other hand, I could be completely wrong and could have misinterpreted it entirely.... so feel free to correct me.
I think I agree with Ted, because if "r" is the percent change between the different populations, then how could we have negitive population? But then I can see where Erica is coming from. But I guess if people stopped reproducing then you might have a negitive. But may be I am way off.
ReplyDeleteErica, I think you made a good point. So off of what Alex said, "r" is a constant then it couldn't really be positive or negative, it has to be one since it is a constant. ("r" is resources, right?)
ReplyDeleteAfter doing some research, i have found numerous findings indicating our carrying capacity has already been surpassed. The question is very difficult because how can one successfully measure what constitutes a "full" use of resources? in reality, one must account for socio-economic conditions, the rate of technological improvement... It is a very soft-ruler that is used at best. Therefore, our most practical approach to the question is one of simplicity: using regression analysis from a large range of our nation's past. Using data from 1860-2010, I have found the following formula: -0.0017x^4+0.078x^3-0.4282x^2+12.006x+17.971 = a broad expectation of our nation's population would near 750 million people around 2210.? Any comments? or magical predictions...Snape?
ReplyDeleteI don't expect many of you to appreciate the subtle science and exact art that is potion making, however I do predict that there will be a snow day tomorrow.
ReplyDeleteTo paraphrase Truman, "...Bring me a one-armed Economist (Meteorologist), so that he cannot say; But on the other hand..."
ReplyDeleteI find it comical that snape is able to reply to Daniel's comment in such a timely fashion. As for ted's question- can we have a negative 'r' assuming that we are working our way towards something which will leave us with 0 room for growth?
ReplyDeleteI responded a full 12 hours after Daniel. I don't like what you are insinuating Alex. 10 points from Brown. Also, if 'r' is resources as Teddy suggests, then how can you have negative resources?
ReplyDeletei think we need to define "r" more directly because some people are referring to it as resources and some people are referring to it as a measure of population...
ReplyDelete