Topological and random spread models with frozen symbols

Area of Science:

• Mathematical Biology
• Population Dynamics
• Matrix Theory

Background:

• The "frozen" state in population systems, where an individual produces only one offspring of the same type, alters spread patterns and system behavior.
• Standard methods like the Perron-Frobenius theorem are inapplicable to frozen systems due to non-primitive ξ-matrix and offspring mean matrix.

Purpose of the Study:

• To characterize key matrices (ξ-matrix, offspring mean matrix) in frozen systems.
• To analyze population spread rates under more general conditions in topological and random spread models with frozen symbols.
• To develop an algorithm for computing spread rates and relate them to matrix eigenvectors.

Main Methods:

• Characterization of key matrices in frozen systems.
• Development of an explicit algorithm for computing population spread rates.
• Analysis of spread rates using eigenvectors of the ξ-matrix and offspring mean matrix.

Main Results:

• The study provides a method to analyze spread rates in frozen systems where standard theorems do not apply.
• Population growth is demonstrated to be exponential.
• The composition of the population exhibits asymptotic periodicity.

Conclusions:

• The proposed methods allow for the prediction of spread rates in frozen population systems.
• The findings confirm exponential population growth and asymptotically periodic population structure.
• Numerical experiments validate the theoretical framework for analyzing frozen population dynamics.