Environmental Geology Lecture Outline - Population dynamics and environmental pressures.

What makes the human species distinct from other species (if anything)?

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

The nature of unrestrained exponential growth.

Malthus (1766-1834): credited with an early recognition that there were social implications of exponential growth.

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 a generation = average age of parent). This produces a J shape curve, and the number 2 can be considered as a growth constant. If instead each individual on average reproduced to create 1.5 offspring that then went on to do the same, we could simply replace the 2 with the number 1.5, Note that this description only considers the number born every generation. It does not take into account (add) survivors of previous generations still surviving at anyone time, and so the actual number for humans would be greater. If the parents are still alive the formula might look like y=(2**t) +(2**(t-1))

The general case: We can describe the incrementl population growth at any given time in a history with the following sentence:

population growth (N) = # of reproducing units (N(i)) * growth constant (k) * time increment (t).

The shorthand for this is as follows:

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 , 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 calculate the population at any point in time for unrestrained constant growth rate population histories (yet consider as you do so consider how likely is such a history to occur in the real world).

Reiterative description. For successive generations imagine a series of numbers that describe the population history, either by year, by generation, or by some other increment of time:

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:


where B is a different form of k, a growth constant. Notice by the way that there is no reason why B can not change throughout a history. We will see why this form of reinterative description is useful below. It is particularly fit for computer modeling.

It is hard for most people to recognize or appreciate the 'power' of geometric growth. The image below may help.

This is a satellite image of the ocean off the coast of of Cornwall, England (taken from the Visible Earth site of NASA). The white clouds above the water are evident, but what are the light patches of torquoise in the water. These are population explosions known as algal blooms. These are microscopic blue-green algae in the water, that have become dense enough they discolor the water. This is a biologic event visible from space! It turns out that these blue-green algae are also important players in the carbon cycle and play a significant role in how the earth works.

How long would it take for a microbial bloom to equal the weight of the world? This of course depends on the rate of reproduction, and the weight of a microbe, but for reasonable values it is only a matter of months (try it for yourself in Excel). Obviously this can not happen!

Maintaining present behavior (unsustainable), how long till 1 person for every square meter of land on earth?

Conclusion: present behavior can't be maintained and the growth constant is not constant, but continuously changes as a function of conditions and might be better thought of as growth factor or parameter.

What are the implications for economic systems and models?

These are not the famous white cliffs of Dover, but the white cliffs of Brighton some 100 km SW of Dover along the southern coast of England. These are chalk deposits of Cretaceous age, and they can also be found across the channel on the shores of France. These are amazing deposits that provide real insight into the role of life in geology, and into the power of exponential growth, if you know how they formed. They are largely the skeletal remains of plankton that used to live in a shallow sea here and which formed a small calcareous shell. The extensive deposits exposed in these cliffs are an immense biologic graveyard of countless microscopic shells, microscopic but made quite visible by their shear numbers. They demonstrate the power of exponential growth in a very concrete way. Similar deposits are found in Nebraska.

Growth with a restraint term and this way to chaos.

The above description of population growth is clearly deficient. In real life there are constraints, a carrying capacity for an area or a planet. 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 simple equation for this situation:

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

Note that this is nothing but the same formulat you have above with a second term added -> (1- N(n)/N(max)). Again, B is a inherent growth factor (reproduction rate when restraining factors are not significant, similar to the above). 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 the maximum (N(max)). See the computer program for a visualization of this behavior, or follow the summary in the sequence of images produced in Excel below, where we look at the effect of increase B. How well does this equation reflect real life? Can you think of some species whose populations show chaotic and/or large fluctations, and whether their inherent growth rate is high or low?

In this graph produced in Excel the x (horizontal) axis represents generations or time, and the y (vertical) axis bars represents the population. Note here that until generation15 or so there is a J shaped exponential increase, but afterwards there is a steady equilibrium.

When we increase the inherent growth constant a bit, above 3, then we see a very different behavior. The population fluctuates between two different equilibrium values.

After the initial unimpeded J shape growth, the population is now fluctuating between how many levels? If you look carefully you will see it is four. You may be able to guess where this pattern is going.

Can you detect any pattern here? It is chaotic. With successively smaller increases the number of levels jumps to 8, then 16, and then beyond a certain value of the growth constant the pattern becames chaotic.

The change here is the amplitude of the population level fluctuations increases. Remember that you can develop your own little Excel model to explore this.

This is only a model, but what would be the implications of being linked to an organism with such a history? Can you think of such organisms and how humans might be linked to them? Models like this one can help you understand the 'behavior' of equations, which if they capture natural behavior, then can provide useful insights. They can help develop your intuition.

Beyond the value 4 this equation blows up and goes to zero, which equates to extinction. It is interesting that such a simple equation can 'capture' all this rich behavior including extinction.

Summary of Verhultz equation behavior as B, the inherent growth rate, changes:

In the chaotic realm a very small change in the growth constant changes the exact history in a major way, making it impossible to make long term predictions. This is one example of what is known as "initial conditions sensitivity", or the "butterfly effect".

Non-constant growth constants and carrying capacity

What will cause changes in the growth "constant"?

Concept of carrying capacity and determinants of such capacity?

Have local carrying capacities been reached and what were/are the consequences?

What is the role between growth constant and carrying capacity at any one time and the environmental conditions?

Exponential population growth versus consumption rate, which is more important in the long term?

Population density

What is the average human population density for the planet?

What is the density distribution of humans on the earth, and what is the environmental significance of that distribution?


Human population density map taken from CIESEN's web site. The more intense colored reds and purples represent higher population densities. What are the factors controlling this distribution?

Urban versus rural living and environmental stewardship and sustainability.

Dynamics of linked populations

Populations are linked in a variety of ways, and an obvious set of examples are prey-predator links. The system diagram below models a simple relationship between rabbits and foxes.

These diagrams were made in a software package called STELLA, a very powerful and yet relatively accessible software package for modeling system behavior. I strongly encourage people to look into it. To the left is the object oriented software model that produced the graph to the right.

What are other types of linkages between populations?

There are many others.

What are other populations that human welfare is connected to?


Microbial populations as geologic agents.

There are two very good reasons to talk about population dynamics in an environmental geology course. Above we discussed one set - the human population and distribution is a crucial factor in the magnitude and type of environmental problems that exist. HABs, or Harmful Algal Blooms, are just one example we will look at a bit in this course. Another good reason is that biologic populations play an important role in how the earth operates, as recognized by considering the white cliffs of Dover (and other such deposits - see photo above) and in the Gaia hypothesis.

The above is image of where bacteria have eaten volcanic glass (from NOAA). We have just begun to learn about extensive microbial populations that live not only in the soil, but much deeper inside the earth. Some minerals and hot fluids in the earth have energy that can and is tapped by microbes, just as plants tap into solar energy through photosynthesis. This alternate way of making a living is known as chemosynthesis. One estimate is that there is more biomass inside the rocks underneath the ocean waters than in the ocean. These microbial populations act as a geologic agent in several ways, often strongly influencing chemical conditions and processes.

Some web resources on population dynamics:

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