Stability of the replicator equation for a single species with a multi-dimensional continuous trait space

Area of Science:

• Evolutionary Biology
• Mathematical Modeling
• Population Genetics

Background:

• Traditional adaptive dynamics models behavior evolution using simplified trait spaces.
• A more comprehensive model is needed for multi-dimensional continuous trait evolution.

Purpose of the Study:

• To develop a replicator equation model for species with multi-dimensional continuous traits.
• To compare the stability of this measure-theoretic model with traditional adaptive dynamics.
• To identify conditions under which established stability concepts predict model outcomes.

Main Methods:

• Developed a replicator equation model on the set of probability measures.
• Analyzed stability of monomorphisms using weak topology.
• Compared results with continuously stable strategy (CSS) and neighborhood invader strategy (NIS) from adaptive dynamics.

Main Results:

• For quadratic fitness and normal distributions, CSS often predicts stability but can be too stringent.
• For general fitness/distributions, CSS relates to dominance solvability for distributions without support gaps.
• The neighborhood invader strategy (NIS) is required for arbitrary trait distributions.

Conclusions:

• The measure-theoretic replicator model offers a robust framework for studying continuous trait evolution.
• Existing adaptive dynamics concepts like CSS and NIS provide valuable insights but require careful application.
• The study highlights the importance of trait distribution support in determining evolutionary stability predictions.