# Anomalous Growth of Aging Populations

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## Abstract

We consider a discrete-time population dynamics with age-dependent structure. At every time step, one of the alive individuals from the population is chosen randomly and removed with probability \(q_k\) depending on its age, whereas a new individual of age 1 is born with probability *r*. The model can also describe a single queue in which the service order is random while the service efficiency depends on a customer’s “age” in the queue. We propose a mean field approximation to investigate the long-time asymptotic behavior of the mean population size. The age dependence is shown to lead to anomalous power-law growth of the population at the critical regime. The scaling exponent is determined by the asymptotic behavior of the probabilities \(q_k\) at large *k*. The mean field approximation is validated by Monte Carlo simulations.

## Keywords

Queue Population Birth-death process Markov model Aging## Notes

### Acknowledgments

The author acknowledges partial support under Grant No. ANR-13-JSV5-0006-01 of the French National Research Agency.

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