Friday, May 12, 2006
Cost Theory in Population Scenarios
Walter ReMine recently published a clarification of cost models in a recent issue of TJ.
His goal was to clarify issues surrounding the cost of substitution in populations, and to modify it so that it is tied to more physical entities and useful in a wider variety of scenarios.
Haldane said that his cost model would require revision, and ReMine has said he revised the concept in the following ways:
- Rebuilt the cost model on concepts tied more to physical reality than "genetic death".
- Generalized the concept to remove assumptions that were previously necessary
- Eliminates matters of confusion, specifically why there is no such thing as a change which has "no cost" or "pays for itself"
- Show that the cost of substitution is set by the growth rate, and cannot be reduced by other means
ReMine's cost model works essentially like this:
The "cost" of a given scenario is the minimum amount of reproduction required to achieve that scenario.
For instance, the basic cost is the cost of replacement. In order for the next generation to have the same population as the current one, the minimum cost is for each member of the previous generation to have exactly one offspring. This is the Cost of Continuity. If we also add in various mortalities, you have slightly additional reproduction required to keep the population steady.
So, if you have a trait in a single animal in a population, and you want to know what the cost is for that trait to be in 27 animals in three generations, his cost model will give you the minimum reproduction required. In this simple case, it is 3 offspring per generation, plus a few to account for miscellaneous deaths.
In this model, substitution to fixity can occur in a single generatiton, provided all of the original-type members die off in one she-bang. But that leaves a new problem -- the population size is now very small, so the chances of a beneficial mutation occurring are much, much less.
For example, let's say that you have a population of a million. One of them comes in with a novel mutation. Let's consider a scenario. Let's say that all of the original population dies off, and only a few organisms remain, one of which is the one carrying the novel trait. That trait can reach fixity very quickly. However, it is now a million times less likely for a given new novel trait to emerge (beneficial or otherwise). Therefore, while this particular trait was able to come to fixity quickly, it slows down the ability for another novel trait to enter the population. If on the other hand you keep most of your original-type, you have a better chance of getting new traits, but it requires a much larger cost to achieve fixity.
Darwin noted that his theory required reproductive excess. The cost model is meant to determine how much excess is required.
Using this cost model, while other factors can come into play, they can only _increase_ the costs associated. There is no way to decrease costs, since they are a physical reproduction requirement of the scenario (actually, there is one way -- and that is for multiple organisms to arrive at the same novel type, but that requires pre-planned mutations).
[NOTE -- I have not finished the paper, and am unlikely to be able to have time to do so any time soon, so this is a very rough sketch of the paper's contents. Any clarifications or corrections please post below.]
Well, at least I understand how populations actually work in the real world...
Find any more dinosaur drawings proving that humans and dinos coexisted?