On a Fractional Variant of Linear Birth-Death Process
Manisha Dhillon, Pradeep Vishwakarma, Kuldeep Kumar Kataria
TL;DR
This work introduces the generalized fractional linear birth-death process (GFLBDP) by applying the regularized Hilfer-Prabhakar derivative ${}^{hp}\mathcal{D}_{\alpha,-\beta}^{\gamma,\rho}$ to the state-probability equations, unifying time-fractional BD models with more flexible memory via additional parameters. A time-changed representation ${\mathcal N}_{hp}(t)\overset{d}{=}\mathcal N(\mathscr{Q}(t))$ connects the GFLBDP to a standard BD process through a fractional Cauchy-problem-driven time-change, and the limiting behavior of the time-change relates to inverse stable subordinators with index $\rho-\alpha\gamma$ under suitable conditions. The authors derive explicit state probabilities for three rate regimes, along with formulas for the mean and second moments, and they analyze extinction probabilities and asymptotics, recovering known results in special cases when $\gamma=0$ or $\rho=1$. They also study Prabhakar-type integrals of the process, joint distributions with its path integral, and provide a genetic-model application featuring an upper bound on population size, illustrating the model's utility for systems with rapid dynamics and carrying capacity.
Abstract
We introduce and study a fractional variant of the linear birth-death process, namely, the generalized fractional linear birth-death process (GFLBDP) which is defined by taking the regularized Hilfer-Prabhakar derivative in the system of differential equations that governs the state probabilities of linear birth-death process. For a particular choice of parameters, the GFLBDP reduces to the fractional linear birth-death process that involves the Caputo derivative. Its time-changed representation is obtained and utilized to derive the explicit expressions of its state probabilities. The explicit expressions for its mean and variance are derived. In a particular case, it is observed that the limiting distribution of the time changing process coincides to that of an inverse stable subordinator. A relation between the extinction probability of GFLBDP and the density of inter arrival times of a generalized fractional Poisson process is obtained. Later, we study some integrals of the GFLBDP and discuss the asymptotic distributional characteristics for a particular integral process. Also, an application of the path integral at random time to a genetic population with an upper bound is discussed.