Discrete-time population dynamics on the state space of measures

Area of Science:

• Mathematical Biology
• Population Dynamics
• Measure Theory

Background:

• Population models use measures on metric spaces to represent individuals with specific characteristics.
• Population turnover maps model changes in population structure over time.
• Approximations of these maps yield basic population turnover operators.

Purpose of the Study:

• Investigate the spectral radius of the basic population turnover operator as a threshold for population extinction versus persistence.
• Address limitations of the variation norm in ensuring spectral radius properties.

Main Methods:

• Utilizing measures on the Borel subsets of a metric space to model population structure.
• Defining a discrete-time population model via a population turnover map.
• Analyzing the first-order approximation of the turnover map as a basic population turnover operator.
• Employing the flat norm (dual bounded Lipschitz norm) as an alternative to the variation norm.

Main Results:

• The spectral radius of the basic population turnover operator can indicate population persistence.
• The variation norm is too strong, preventing the spectral radius from being an eigenvalue with a positive eigenmeasure.
• The flat norm provides necessary compactness, though it restricts operator continuity.

Conclusions:

• The flat norm is a suitable alternative for analyzing population persistence thresholds using spectral radius.
• This approach offers a refined understanding of population dynamics in structured populations.