Integrability of stochastic birth-death processes via differential Galois theory

Primitivo B. Acosta-Humánez^1, 2 ; José A. Capitán³ ; Juan J. Morales-Ruiz⁴

¹ School of Basic and Biomedical Sciences, Universidad Simón Bolívar, Barranquilla, Colombia.
² Instituto Superior de Formación Docente Salomé Ureña, Dominican Republic.
³ Depto. de Matemática Aplicada & Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, Madrid, Spain.
⁴ Depto. de Matemática Aplicada & Grupo Modelos Matemáticos no Lineales, Universidad Politécnica de Madrid, Madrid, Spain.

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 70.

Résumé

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals.

Détail

DOI : 10.1051/mmnp/2020005

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