editorial note: These notes pre-date the formal start of my online laboratory notebook, Feb 2 2010: The Lab Notebook Goes Open and were adapted from a LaTeX document in which I kept notes on this topic during my summer at IIASA. Lacking a proper notebook then, documents like this one were updated periodically and occassionally branched into new ones. The post date represents the last time the LaTeX document was edited in the course of that research.
The process of adaptive branching can be divided into four phases. Depending on the parameter regime, different phases would be limiting.
Approaching branching point
Invading Coexistence regime (\(P_2\))
Coexistence until next successful invader
Subsequent divergence until stable
A forward and backwards transition rate can be calculated between each phase, based on my analytical work this summer. The shape of the waiting-time distribution between each phase can be determined, and when this rate is limiting this dominates the overall shape. This allows four analytic approximations, one for each parameter regime where one of the above steps is limiting, even though such an analytic solution is not possible over the entire process. This is biologically interesting, as it identifies not only the waiting-time distribution but the mechanism responsible, and highlights the role of the 4 different mechanisms whereby stochasticity frustrates branching.
Write down the four analytic distribution predictions
create numerical simulations to compare to each regime
compare to intermediate parameter regimes (these are actually rare, for my simulations with O(1000) individuals step three is usually limiting.
write the paper describing four mechanisms whereby stochasticity frustrates evolutionary branching