Anomalous Growth of Aging Populations

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.

Appendix: Exact Solution for a Fixed Death Age Model at the Critical Regime

We derive the exact solution for the particular case with $r = 1$, $q_k = 0$ for $k \le T$, and $q_k = 1$ for $k > T$. In other words, a new individual is added at each time step, while a uniformly chosen individual is removed if its age exceeds T.

We split the population into “young” ($k \le T$) and “old” ($k > T$) individuals according to their age. For the first T steps, the population grows linearly with time: the number of young individuals is t, while there is no old individuals. For $t > T$, the number of young individuals is fixed because at every time step, a new young individual is added while one young individual becomes old. In turn, the number $\tilde{\eta }_t$ of old individuals at time t is a random variable which follows simple dynamics:

$$\begin{aligned} \tilde{\eta }_{t+1} = {\left\{ \begin{array}{ll} \tilde{\eta }_t, &{}\,\,\, {\text {with probability }} \tilde{\eta }_t/(\tilde{\eta }_t + T), \\ \tilde{\eta }_t + 1, &{}\,\,\,{\text {with probability }} 1 - \tilde{\eta }_t/(\tilde{\eta }_t+T). \end{array}\right. } \end{aligned}$$

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In fact, at time t, the probability to choose an old individual is $\tilde{\eta }_t/(\tilde{\eta }_t + T)$, in which case this individual is removed, and the number of old individuals is not changed. In turn, with the complementary probability $1 - \tilde{\eta }_t/(\tilde{\eta }_t +T)$, the chosen individual is young and is thus not removed. As a consequence, the number of old individuals is increased by 1. This process is Markovian since the next state is fully determined by the current state.

Let $Q_n(t)$ be the probability that $\tilde{\eta }_t = n$ or, equivalently, $\eta (t) = n+T$, where $\eta (t) = \tilde{\eta }_t + T$ is the total number of individuals (i.e., the population size). In analogy with random walks, one can write the master equation on $Q_n(t)$:

$$\begin{aligned} Q_n(t) = \kappa _n Q_n(t-1) + (1 - \kappa _{n-1}) Q_{n-1}(t-1) , \end{aligned}$$

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where $\kappa _n \equiv n/(n+T)$. In fact, the state $\tilde{\eta }_t = n$ can be achieved either from the same state at the earlier time (with probability $\kappa _n$ for not moving), or from the state $\tilde{\eta }_{t-1} = n-1$. Applying this relation repeatedly, one can express $Q_n(t)$ in terms of the probabilities $Q_{n-1}(t')$ at earlier times $t'$:

$$\begin{aligned} Q_n(t) = (1 - \kappa _{n-1}) \sum \limits _{j=1}^{t-n+1} \kappa _n^{j-1} Q_{n-1}(t-j) . \end{aligned}$$

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Note that the upper limit was set to $t-n+1$ instead of t since $Q_{n-1}(j) = 0$ for $j < n-1$ (since the population can only increase by one per time step). The initial condition for the master equation is $Q_n(T) = \delta _{n,0}$, i.e., no old individual at the beginning of the diffusive phase.

The above equations can be solved exactly:

$$\begin{aligned} Q_n(T+t) = {\left\{ \begin{array}{ll} \sum \limits _{j=1}^n c_{n,j} \kappa _j^t , &{}\,\,\, (t \ge n),\\ 0, &{}\,\,\, (t < n), \end{array}\right. } \end{aligned}$$

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where the coefficients $c_{n,j}$ are

$$\begin{aligned} c_{n,j} = \frac{1}{\kappa _j} \left( \prod \limits _{i=1}^{n-1} (1-\kappa _i)\right) \left( \prod \limits _{i\ne j}^n \frac{1}{\kappa _j - \kappa _i}\right) = (-1)^{n+j} \frac{(j+T)^{n-1}(n+T)}{j! (n-j)!} . \end{aligned}$$

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The moments of the old population size are

$$\begin{aligned} \langle [\tilde{\eta }_{T+t}]^m \rangle = \sum \limits _{n=1}^t n^m Q_n(T+t) = \sum \limits _{j=1}^t b_{j,t}^{(m)} \kappa _j^t , \end{aligned}$$

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where

$$\begin{aligned} b_{j,t}^{(m)} = \sum \limits _{n=j}^t n^m c_{n,j} . \end{aligned}$$

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For instance, one finds

$$\begin{aligned} \begin{array}{ll} b_{j,t}^{(0)} &{}= \frac{(j+T)^t (-1)^{t+j}}{j! (t-j)!} , \\ b_{j,t}^{(1)} &{}= t b_{j,t}^{(0)} - \frac{(j+T)^j}{j!} \sum \limits _{n=0}^{t-j-1} (-1)^n \frac{(j+T)^n}{n!} , \\ \end{array} \end{aligned}$$

from which

$$\begin{aligned} \langle \tilde{\eta }_{T+t} \rangle = t - \sum \limits _{j=1}^{t-1} \frac{j^t}{j!} \sum \limits _{m=0}^{t-j-1} (-1)^m \frac{(j+T)^{j+m-t}}{m!} . \end{aligned}$$

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The mean population size is therefore

$$\begin{aligned} \langle \eta (t)\rangle = {\left\{ \begin{array}{ll} t, &{}\,\,\, (t \le T), \\ T + \langle \tilde{\eta }_{t} \rangle , &{}\,\,\, (t > T). \end{array}\right. } \end{aligned}$$

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We checked numerically that this solution is very close to the mean population size obtained in Sect. 4.1.