On a global representation of dynamical characteristics in ESS-models

Area of Science:

• Evolutionary Game Theory
• Dynamical Systems Theory
• Behavioral Ecology

Background:

• Evolutionary Stable Strategy (ESS) models are crucial for understanding strategy dynamics.
• Existing ESS models often lack comprehensive methods for global characteristic representation.
• Dynamical systems provide a framework for analyzing complex evolutionary processes.

Purpose of the Study:

• To develop a novel method for global representation of dynamical characteristics in ESS models.
• To introduce and analyze ESS models with changing matrices.
• To identify and clarify the significance of structurally stable and catastrophic cases within ESS models.

Main Methods:

• Framing ESS models within adequate dynamical systems.
• Developing a global representation method for dynamical characteristics.
• Introducing and analyzing ESS models with changing matrices.
• Applying catastrophe theory concepts, including cusp catastrophe analogues.

Main Results:

• A comprehensive view of relations among ESS matrices is achieved.
• Structurally stable and catastrophic cases in ESS models are identified and clarified.
• A surface representation analogous to the elementary cusp catastrophe is obtained for two-strategy cases.
• The concepts are applied to a model of directed behavioral evolution.

Conclusions:

• The dynamical systems approach offers a powerful tool for analyzing ESS models.
• The concept of ESS models with changing matrices enhances applicability.
• Catastrophe theory provides insights into evolutionary stability and transitions.
• The developed methods are applicable to understanding evolutionary processes in behavioral components.