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One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics
Eugenia Franco1, Mats Gyllenberg1, Odo Diekmann2
1Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland.
Acta Applicandae Mathematicae
|November 1, 2021
Summary
This study characterizes renewal equations with specific kernels, enabling their measure-valued solutions to be linked to scalar renewal equations. This allows for the derivation of large-time behavior using established renewal theorems.
Area of Science:
- Mathematical Biology
- Probability Theory
- Analysis
Background:
- Renewal equations are crucial in mathematical biology.
- Limited general results exist on the asymptotic behavior of measure-valued solutions for renewal equations based on kernel properties.
Purpose of the Study:
- To characterize a class of renewal equations whose measure-valued solutions can be related to scalar renewal equations.
- To derive the large-time behavior of these solutions.
Main Methods:
- Kernel characterization of renewal equations.
- Expressing measure-valued solutions via scalar renewal equation solutions.
- Application of Feller's classical renewal theorem.
Main Results:
- A specific class of renewal equations is identified.
- The asymptotic behavior of measure-valued solutions is linked to scalar renewal equation solutions.
- Large-time behavior is derived through established renewal theorems.
Conclusions:
- The study provides a method to analyze the asymptotic behavior of a class of renewal equations.
- This approach simplifies analysis by relating complex equations to simpler scalar forms.
- The findings contribute to a better understanding of renewal processes in mathematical biology.
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