Abstract
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.
Highlights
Stochastic birth-death processes [10, 15, 23] are widely used in the mathematical modeling of interacting populations
In contrast to deterministic models, these kinds of processes make the assumption that population changes take place in discrete numbers, and this fact introduces variability and noise when compared to deterministic dynamics [7, 8]
In the limit of infinite system size, these models are the counterpart of deterministic dynamics that usually appear in demography and population dynamics [18, 26]
Summary
Stochastic birth-death processes [10, 15, 23] are widely used in the mathematical modeling of interacting populations. Death rates are taken as a quadratic function (d = 2) since it is commonly assumed that two individuals compete with each other in death events, whereas birth processes (asexual reproduction) are described as linear functions of N (b = 1, i.e., the probability of a birth event is proportional to the number of individuals in the population) In this contribution we will consider two combinations of exponents: (b, d) ∈ {(1, 1), (1, 2)}. The generating function has to satisfy the conditions (1.8) This case of quadratic death rates, which is the more relevant one in biological terms, remains as non-integrable, as we will show in Section 4 using results from Differential Galois Theory. Proposition 1.2 is a non-integrability result and tell us that any search for a closed-form, analytical solution for equation (1.9) is doomed to failure
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