Mass extinction in a dynamical system of evolution with variable dimension.

Area of Science:

• Mathematical Biology
• Dynamical Systems Theory
• Paleontology

Background:

• Replicator equations are fundamental in evolutionary game theory and mathematical biology.
• Existing models typically assume a constant number of species (fixed dimensionality).
• The impact of extinction on ecological dynamics, particularly mass extinction events, requires robust theoretical frameworks.

Purpose of the Study:

• To develop a general mathematical model incorporating extinction into replicator equations.
• To investigate how species diversity, as a time-dependent variable, influences ecological dynamics.
• To provide a theoretical basis for paleontological theories of mass extinction.

Main Methods:

• Constructed a generalized replicator equation model where the number of species (dimension) is a time-dependent variable.
• Analyzed the system's behavior, contrasting it with the standard replicator equation.
• Simulated scenarios to observe the effects of initial species diversity on extinction dynamics.

Main Results:

• The model exhibits significantly different dynamics compared to the original replicator equation.
• High initial species diversity was found to lead to mass extinction events within the model.
• Demonstrated that variable dimensionality is an inherent characteristic of certain dynamical systems.

Conclusions:

• The developed theory offers a novel mathematical foundation for understanding mass extinction events in paleontology.
• The model serves as a prototype for dynamical systems where changes in dimensionality are inevitable.
• Highlights the critical role of species diversity in ecological stability and extinction risk.