Anyways, I have been working on a developing a set of models for predicting population growth of introduced species based on different assumptions. Now I know this has been done half a billion times by half a billion scientists, and that some of those scientists have spent a fair bit of time and effort in developing their models but I figure half a billion and one models can't hurt. So I've started playing around with some formulas and came up with this first equation to show what a population could do, when you have a high reproduction rate, low mortality and no limits to growth. You can think of Cane Toads and Australia to give you the type of scenario that I am presenting.
The exponential growth of an unchecked high reproduction rate and low mortality
As you can see from the graph, that although all of the coefficients in the equations are constant, the resultant population explosion is exponential (that is because we apply these constants to the previous breeding cycle's product - and in effect have an exponential equation). The equation that I use is very basic, where we determine the new additions to the population by: IP*SR*BSR*BR*(1-MRY) and add that to the existing population, after accounting for mortality IP*(1-MRA). There is some other stuff going on in the real equation - to account for things like rounding up to account for fractions of species which doesn't work in the real world - but this is the basic equation to tell you what a population would do if there were nothing else to consider. I will explain that:
- IP=Initial population and
- SR=sex ratio (what percent of the population is female) and
- BSR=breeding success ratio (eg. how many females in the population get bred)
Since we all know that a population can not grow exponentially forever, I adjusted the model to allow for a feedback mechanism that accounts for a limit to growth. This limit is what is called the "carrying capacity" of the ecosystem. So my new model employed a dynamic feedback structure to one of the coefficients to say that when the population grew towards the carrying capacity of the ecosystem, the birthrate would fall off as a result of increased stressed to the pregnant females.
Again, although this is a simplistic model, it does demonstrate the idea of a limit to growth and how one can represent it in a relatively simple equation.Over the next week or so I will adjust the equation to account for catastrophic collapses (as when there is a drought) and then try to find a dataset that I can test my equation against.
Now for the competition: The first person to explain what all of the variables in the following equation stand for: IP*SR*BSR*(BR-(pop/UPL*BR))*(1-MRY)+IP*(1-MRA) will win your choice of a "Rounders and Sinners" CD or "the thing that has recently quit moving around in the bottom of my backpack."
For those not interested in hurting their brains on the sharp corners of logic they can kick back and watch an outing with the Kamloops Naturalist Club.