In these next three lectures we will be delving a bit into how the biosphere operates.

What is a population and how do you describe one?


Why might an understanding of population dynamics be relevant in an environmental geology course?

Population density map of the world taken from CIESEN.

The nature of geometric (exponential growth, unrestrained growth).

Malthus (1766-1834)

A specific case of doubling every generation. Take the case where every couple has 4 children which survive to parenthood and repeat their parents history, a doubling of population every generation. this would result in a sequence like: 2 -> 4 -> 8 -> 16 -> 32 -> 64 -> 128 -> 256 -> 516 -> 1032 -> 2064 -> 4128 ...... The general formula is y=2**t , where t = the number of generations (in years generation = average age of parent). This produces a J shape curve, and the number 2 can be considered as a growth constant.

The general case: We can describe an increment of growth with the following sentence: population growth (N) = # of reproducing units (N(i)) * growth constant (k) * time increment (t). The shorthand for this is below

N = N(i) * k * t

If you integrate then this changes into:

N(t) = N(0)* (e ** (k't )) or ln Nt = ln No + k't

where e=2.71 , where ln is the natural log (base e), N(0)=initial population at time 0, N(t) = population after time t. k' is a different growth rate. You can use this equation in plug and chug mode to predict unrestrained constant growth rate population histories (yet consider how likely is such a history to occur?).

Reiterative description. For successive generations imagine a series of numbers: N(0) - N(1) - N(2) - N(3) - N(4) - N(5) - N(6) ...- N(n). If N(n) is the population at present somewhere throughout this history, then the population in the next cycle (generation), N(n+1), is given by: N(n+1)=B*N(n) , where B is the equivalent of k, a growth constant. Notice by the way that there is no reason why B can't change with time.

Growth with a restraint term and this way to chaos.

What will happen as the population begins to reach a maximum population possible (N(max))? The growth constant must decline in value. Verhulst in 1845 came up with this equation for this situation:

N(n+1)=B*N(n)*(1- N(n)/N(max)) .

B is a inherent growth factor (reproduction rate when restraining factors are not significant). How does the last term behave as you approach or move away from N(max)? Simple rearrangement of the forumula may better show this behavior.

N(n+1)=B'*N(n) , where B'=B*(1- N(n)/N(max))

In this formula the growth 'constant' actually varies with time depending on how close or far away it is from some maximum. This relatively simple equation shows some very rich behavior.

In class demonstration with computer program for a visualization of this behavior.

Can you think of some species whose populations show chaotic and/or large fluctations, and whether their inherent growth rate is high or low? Remember that this is a model, and if it has a sound theoretical basis it informs of possibilities, not eventualities.

This formula also shows a property characteristic of chaotic systems known as "initial conditions sensitivity". This is where a very small difference in input results in a significantly different outcome at some point in the future. If you change B by .001 and compare the population at 100 generations in the future of the original input with the changed input, it can be dramatically different if B is in the range that produces chaotic behavior (e.g. >3.6). A consequence of this is that we do not (and perhaps can not) know B with enough precision to make accurate predictions in the future. This is also known as the butterfly effect.

Optional: Exploring the Verhultz equation using Excel.

Exponential population growth versus consumption rate, which is the real culprit?

An argument that is sometimes cast about is that population growth should not be the real concern, but that consumption rates are more important. Some countries with low population growth have very high per capita consumption rates (including the U.S.), and maybe the focus, and, when blame is being assigned, the blame should be laid here. A mathematical model of consumption rates for two different populations shows, however, that in the long term population growth can not be ignored and dominates as a factor. It demonstrates the 'power' of exponential growth.

Consider a high birth rate versus low use group where every person produces 4 offspring and uses 1 unit of a resource in their lifetime, versus a group of low birth rate where every person produces 2 offspring but uses 10 units of that same resource. In other words, while the birth rate of one is twice as high as the other this group has a per capita resource use rate one tenth of the other. The below table produced shows that after 5 generations the high birth rate population quickly surpasses the low birth rate in cumulative resource use. Excel can be easily used to model other versions of this type of experiment.

generations 1 2 3 4 5 6 7
population for high birth rate low use group 1 1 4 16 64 256 1024 4096
cumulative consumption for group 1 1 5 21 85 341 1365 5461
population for low birth rate high use group 2 1 2 4 8 16 32 64
cumulative consumption for group 2 10 30 70 150 310 630 1270

What might one conclude from this small experiment? Perhaps, one conclusion is that attention must be paid to both factors.

To add to the mix, one can ask the question what type of correlation exists between population growth rates and resource use rates? How does this differ for different resources, especially energy vs. agricultural?

Some web resources on population dynamics:

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