Wikipedia:Reference desk/Archives/Mathematics/2011 February 7

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February 7

Human Population - Two Questions

Can human population be graphed using an exponential function? Say there were about 4 million people in The United States in 1790, and now there are 310 million. Would that be consistent with the function? Also, using the same function, what year was it when there was only about 1 Human? Thanks. 129.120.195.10 (talk) 19:25, 7 February 2011 (UTC)

As to the second question, exponential models tend to fall down before going back to 10,000 BC or so; they grow too quickly compared to human growth. Thus most models I've seen do not consider the case for world population being one (bearing in mind that would be an odd thing to find, given it's widely believed more than one human evolved, or else two). Grandiose (me, talk, contribs) 20:03, 7 February 2011 (UTC)

Exponential growth means that the logarithm grows linearly. The log of 4 million is 15.2 and the log of 310 million is 19.6. So the log has grown by 4.4 during the 220 years that have passed since 1790. So log of the population grows 4.4/220=0.02 per year. An approximate formula for the population N in year y is N(y)=e^{19.6+0.02(y-2010)}. Using this formula you get that N=1 in the year 1030, long before the States of America were United. Bo Jacoby (talk) 21:51, 7 February 2011 (UTC).

I thought the OP meant world population growth, my mistake. Clearly it depends on the population as to what additional problems there might be, as below. Grandiose (me, talk, contribs) 18:07, 8 February 2011 (UTC)

One could view the matter like this: The number of years from 1790 until year x is x − 1790. It has now been 221 years since then, and the number of 221-year periods from then until the year x is (x − 1790)/221. Every time a 221-year period passes, the population multiplies by 310/4. The number of times it multiplies by 310/4 from 1790 until the year x is therefore the number of 221-year periods. So it multiplies by (310/4)^{(x − 1790)/221} between the year 1790 and the year x. It starts at 4 in the year 1790. So the population in the year x is 4 × (310/4)^{(x − 1790)/221}.

That's provided growth is exponential. But I think the rate of growth relative to the size of the whole population varies, so that equation is probably not realistic.

So year, the numbers you give are consistent with exponential growth (look at that article if you haven't already), but that doesn't make that equation a realistic model. Michael Hardy (talk) 22:47, 7 February 2011 (UTC)

A logistic function is often a more accurate tool to model population growth. In some regimes, the growth looks exponential, but in either extreme (very far in the past or future), the behavior is more realistic. Sophisticated population models are often multi-parameter models that are optimized to fit the measured data. Nimur (talk) 23:09, 7 February 2011 (UTC)

Quibble: A logistic function behaves just like an exponential in the far past. -- Meni Rosenfeld (talk) 07:26, 8 February 2011 (UTC)

OP here (different IP). So if a logistic function is more appropriate, what would the function be using that type of math? Also, I stopped liking math after I learned that y=mx+b, so if you can keep it simple, I would appreciate it. Thanks. 71.21.143.33 (talk) 02:01, 8 February 2011 (UTC)

It's not realistic to think you can describe human population with a simple curve. There are too many fluctuations due to epidemics, climate change, techological progress like the green revolution or the earlier invention of agriculture, etc. There is genetic evidence that humans had a population bottleneck a few hundred thousand years ago, in which the species almost collapsed. Some related branches like the Neanderthals actually did collapse. But you can look at the graph of logistic function: it shows basically a population stabilizing after an exponential ramp-up, as it hits the limits imposed by some resource. 71.141.88.54 (talk) 03:14, 8 February 2011 (UTC)

Yet it's completely reasonable to model population growth. All models are wrong to some degree, that is why we call them models. The logistic curve is one of the simpler models, but it can be a good fit for certain populations and time periods. Similarly, exponential growth is always a good model on short time scales. However, even stochastic processes like the ones you mention above can be incorporated into more sophisticated models. Complaining that the logistic model is too simple is to completely miss the spirit of using mathematical models to understand population growth. The logistic model is the classic 'next best' choice after exponential growth has been discarded. SemanticMantis (talk) 14:54, 8 February 2011 (UTC)

(ec) The problem is one of interpolation. Having only the two datapoints (1790,4000000) and (2010,310000000) you should not use a logistic function which has three parameters. This means that infinitely many logistic curves fit the data exactly, while only one linear function, and only one exponential function, fits the data. Linear interpolation does not take into account the fact that no population is negative. So it is a good idea to take the logarithm and interpolate between the two datapoints (1790,15.2) and (2010,19.6). The exponential growth is not realistic for sufficiently late times where the predicted population grows unrealisticly large, nor for sufficiently early times where the predicted population grows unrealisticly small. Bo Jacoby (talk) 15:45, 8 February 2011 (UTC).

I'll second SemanticMantis' contributions regarding the purpose of mathematical modeling. We have an article on this subject: mathematical model. The original questioner should be aware that no model is intended to be a 100.0% exact predictor or representation of the time-history of the population: the point of the model is to elucidate the way the population trends are behaving, and to quantify some parameters that might be affecting it. A very simple model, like an exponential growth or a logistic curve-fit, will have very limited accuracy and almost no useful capability for extrapolation far from the measured data. Nimur (talk) 19:51, 8 February 2011 (UTC)

Modelling is fine over small intervals. The problem is the OP's wish to extrapolate all the way back to when there was just one person. A more extreme example would be something like weather prediction. If it's sunny now, it will probably still be sunny 2 minutes from now. With enough measurements and fancy models you can predict fairly accurately about tomorrow or even next week. But beyond that, say if you want to know if it will rain on the 4th of July, you can only make a general guess based on climate. 71.141.88.54 (talk) 21:44, 8 February 2011 (UTC)