Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

Area of Science:

• Stochastic processes
• Mathematical biology
• Computational mathematics

Background:

• Birth-death processes model population dynamics but lack efficient methods for finite-time transition probability calculation.
• Existing methods struggle with arbitrary birth and death rates, limiting applications in complex systems.

Purpose of the Study:

• To develop a robust and efficient algorithm for computing finite-time transition probabilities in general birth-death processes.
• To provide a method that handles arbitrary birth and death rates accurately.

Main Methods:

• Revisiting continued fraction theory to derive expressions for Laplace transforms of transition probabilities.
• Developing an error-controlled computational algorithm based on these continued fraction representations.

Main Results:

• Explicit derivation connecting transition probabilities and continued fractions.
• An efficient, error-controlled algorithm that agrees with known solutions.
• Demonstrated superior performance compared to previous computational approaches.

Conclusions:

• The novel continued fraction-based algorithm provides an efficient and accurate solution for birth-death process transition probabilities.
• This method enhances the study of ecological, evolutionary, and genetic systems with complex population dynamics.