Spectral Theory of Metastability and Extinction in Birth-Death Systems

Abstract

We suggest a general spectral method for calculating the statistics of multistep birth-death processes and chemical reactions of the type mA-->nA (m and n are positive integers) which possess an absorbing state. The method employs the generating function formalism in conjunction with the Sturm-Liouville theory of linear differential operators. It yields accurate results for the extinction statistics and for the quasistationary probability distribution, including large deviations, of the metastable state. The power of the method is demonstrated on the example of binary annihilation and triple branching 2A--> ø, A-->3A, representative of the rather general class of dissociation-recombination reactions.

Stochastic dynamics and logistic population growth

“…To leading order, ϕ(u) ≃ u 2 and As an example, for a = 2 we find 46) which coincides with the result in [36]. We have verified the result (3.45) by numerical simulations.…”

“…Since G(p, t) must be analytic at p = −1 for all times, we require that G(p = −1, t) = (−1) n0 . This boundary condition stems from the fact that G(p = −1, t) is the sum of all even probabilities minus the sum of all odd probabilities [36]. The steady state has to be solved by integrating the equation 20) with the boundary conditions G s (1) = 1 and G s (−1) = (−1) n0 .…”

“…Therefore, G s (p) = 1 which describes an empty population state, i.e., extinction, as t → ∞. To calculate the MET, we employ the "momentum-space" spectral method developed recently [36,37,45,47]. After a short relaxation time t r , which corresponds to the deterministic relaxation time of the system to the stable stationary state, the population typically settles into a long-lived metastable state, which is encoded by the lowest excited eigenmode ψ(p) of the probability generating function G(p, t) [47].…”

“…Our analytical results are compared with numerical simulations, performed using the first reaction method [35]. We consider the birth-and-death and birth-competition-death cases separately, making use of the "momentum space" spectral theory [36,37] and the "real-space" WKB theory [26,[38][39][40], respectively.…”

Efficient stochastic simulations of complex reaction networks on surfaces

“…While MC simulations require the accumulation of statistical data over long times, the master equation provides the probability distribution from which the reaction rates can be obtained directly. In certain cases, the master equation can be solved using a generating function [7,8,9,10,11]. The set of coupled ordinary differential equations is then transformed into a single partial differential equation for the generating function.…”