**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.

## Abstract

We show stochastic effects due to finite population sizes can pose a significant impediment to evolutionary branching. We investigate these effects by exploring the waiting time until evolutionary branching occurs using both individual-based simulationand analytic approximations. The accuracy of our approximation demonstrates that adaptive branching can be thought of as occurring in four phases: (1) (2) (3) (4). Different ranges of parameters will make different phases become rate-limiting. We find that the delicate balance of coexistence early in evolutionary branching is most often rate-limiting, and provide a convenient approximation to the waiting time based on this limit.

Evolution demographic stochasticity branching adaptive dynamics

# Introduction

Sympatric speciation and the adaptive dynamics of evolutionary branching

Previous work on stochasticity in branching

Summarize results and outline paper

# Theory and methods

## Model Formulation

Paragraph reviewing basics of evolutionary branching, , .

### Rosenzweig model of competition for a limiting resource

Paragraph reviewing the competition model

### Individual-based simulation

Paragraph reviewing the individual based model implementation

## Four phases of evolutionary branching

Convergence the branching point

Invasion of the coexistence region

Coexistence until next invasion

Divergence from the branching point.

Figure 1: with two panels: (a) shows eachs of these phases on the Pairwise Invasibility Plot. (b) Histograms for each showing the absolute population abundance at each trait value during each of the phases.

# Results

## Full Approximation

Figure 2: Distribution of waiting times from simulation, with full approximation fit, with rate-limiting approximation from phase 3.

## Rate-limiting coexistence until next invasion

Figure 3: Escape from potential energy well approach used to calculate the coexistence time

## Other rate-limiting steps

Figure 4: Conceptual figure showing biological scenario corresponding to each limit, the resultant approximation, and simulation from that limiting case.

# Discussion

What understanding waiting times tells us about the evolutionary branching process

How quantifying rate limiting steps helps identify the most relevant parameters to measure in determining rates of evolutionary branching.

Extensions of the approach, such as the inclusion of environmental variation

# Acknowledgements

IIASA

NAS, DoE funding

other?