# Spectral theory of metastability and extinction in birth-death systems.

@article{Assaf2006SpectralTO, title={Spectral theory of metastability and extinction in birth-death systems.}, author={Michael Assaf and Baruch Meerson}, journal={Physical review letters}, year={2006}, volume={97 20}, pages={ 200602 } }

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…

## 77 Citations

### Spectral theory of metastability and extinction in a branching-annihilation reaction.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2007

The spectral method is applied to calculate the statistics of a reaction-limited multistep birth-death process that includes as elementary steps branching A-->2A and annihilation 2A-->0 and settles the issue of the "lacking" boundary condition in the spectral formulation.

### Extinction Rates for Fluctuation-Induced Metastabilities: A Real-Space WKB Approach

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A real space WKB method based on the master equation is presented, and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state.

### Extinction of an infectious disease: a large fluctuation in a nonequilibrium system.

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A theory of first passage processes in stochastic nonequilibrium systems of birth-death type using two closely related epidemiological models as examples is developed and a nonmonotone, spiral path to extinction of a disease is uncovered.

### Phase Diagram for Logistic Systems under Bounded Stochasticity.

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To study the power-law phase of logistic and logisticlike systems under the combined effect of demographic and bounded environmental stochasticity, a new WKB scheme is presented, applicable both in the diffusive and in the nondiffusive regime.

### Stochastic fluctuations as a driving force to dissipative non-equilibrium states

- PhysicsJournal of Physics A: Mathematical and Theoretical
- 2020

Most natural complex systems exhibit fluctuations-driven processes, which work at far from equilibrium states, and are generally dissipative processes, for instance living cells. We studied this…

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- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2008

This work studies the relation between the phase portrait of the Hamiltonian system representing a set of chemical reactions with constant rates and the corresponding system when these rates vary in time, and shows that the topology of the phase space is robust for small time-dependent perturbations.

### Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

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- 2022

The large deviations at level 2.5 are applied to Markov processes with absorbing states in order to obtain the explicit extinction rate of metastable quasi-stationary states in terms of their…

### Stochastic foundations in nonlinear density-regulation growth.

- MathematicsPhysical review. E
- 2017

In this work we construct individual-based models that give rise to the generalized logistic model at the mean-field deterministic level and that allow us to interpret the parameters of these models…

### Population extinction in a time-modulated environment.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2008

This work investigates the extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment using, as an example, a stochastic branching-annihilation process with a time-dependent branching rate.

### Extinction of metastable stochastic populations.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large, and yields both entropic barriers to extinction and pre-exponential factors.

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