Kolmogorov's differential equations and positive semigroups on first moment sequence spaces

Area of Science:

• Mathematical Biology
• Dynamical Systems Theory
• Stochastic Processes

Background:

• Infinite systems of ordinary differential equations arise in complex population models, including metapopulation dynamics and host-parasite interactions.
• Existing theories often lack the generality to encompass these specific, structured models.

Purpose of the Study:

• To lay the linear foundations for a general theory applicable to infinite systems of ordinary differential equations.
• To derive conditions under which solutions to Kolmogorov's differential equations induce a C0-semigroup on a sequence space.
• To analyze the growth bounds and asymptotic behavior of these systems.

Main Methods:

• Focus on linear foundations of infinite systems of ordinary differential equations.
• Derivation of conditions for C0-semigroup generation on sequence spaces.
• Estimation of growth and essential growth bounds.
• Analysis of asymptotic behavior.

Main Results:

• Established conditions for solutions of infinite linear systems (Kolmogorov's equations) to induce a C0-semigroup.
• Derived estimates for growth and essential growth bounds.
• Investigated the asymptotic behavior of the system's solutions.

Conclusions:

• The developed linear theory provides a framework for analyzing infinite-dimensional population dynamics models.
• The results are applicable to continuous-time population growth processes, including birth-death processes with immigration and catastrophes.