Models induced from critical birth-death process with random initial conditions.

Area of Science:

• Stochastic processes
• Probability theory
• Mathematical biology

Background:

• Linear birth-death processes are fundamental models in population dynamics and branching phenomena.
• Understanding the impact of initial conditions is crucial for accurate process modeling.

Purpose of the Study:

• To analyze the behavior of linear birth-death processes with various random initial conditions.
• To investigate how different probability distributions for initial particle numbers affect process dynamics.
• To extend the analysis to more complex initial conditions, such as those from a Pólya urn scheme.

Main Methods:

• Analytical derivations for probabilistic models.
• Numerical simulations to validate theoretical findings.
• Extension of numerical models to incorporate advanced initial condition schemes.

Main Results:

• Particle number distributions at any positive time follow the same law as the initial condition, with time-dependent parameter changes.
• Random initial conditions do not affect the critical parameter of the branching mechanism.
• Extinction probability is significantly impacted by the choice of initial conditions.

Conclusions:

• The probability law of particle numbers in a linear birth-death process is preserved over time, irrespective of the initial distribution.
• Initial conditions play a critical role in determining the extinction probability of the process.
• The framework can be extended to analyze complex initial conditions, showing potential links to distributions like Pólya-Aeppli.