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

1. Convergence the branching point

2. Invasion of the coexistence region

3. Coexistence until next invasion

4. 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?