There are some analytical solutions of the Penna model of biological aging; here, we discuss the approach by Coe et al. (Phys. Rev. Lett. 89, 288103, 2002), based on the concept of self-consistent solution of a master equation representing the Penna model. The equation describes transition of the population distribution at time t to next time step (t + 1). For the steady state, the population n(a, l, t) at age a and for given genome length l becomes time-independent. In this paper we discuss the stability of the analytical solution at various ranges of the model parameters--the birth rate b or mutation rate m. The map for the transition from n(a, l, t) to the next time step population distribution n(a + 1, l, t + 1) is constructed. Then the fix point (the steady state solution) brings recovery of Coe et al. results. From the analysis of the stability matrix, the Lyapunov coefficients, indicative of the stability of the solutions, are extracted. The results lead to phase diagram of the stable solutions in the space of model parameters (b, m, h), where h is the hunt rate. With increasing birth rate b, we observe critical b (0) below which population is extinct, followed by non-zero stable single solution. Further increase in b leads to typical series of bifurcations with the cycle doubling until the chaos is reached at some b (c). Limiting cases such as those leading to the logistic model are also discussed.

中文翻译：

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Penna生物衰老模型的解析解的稳定性。

佩纳生物衰老模型有一些解析解。在这里，我们讨论Coe等人的方法。（Phys。Rev. Lett。89，288103，2002），基于代表Penna模型的主方程的自洽解的概念。该方程式描述了人口分布在时间t到下一个时间步长（t +1）的过渡。对于稳态，年龄为a且给定基因组长度为l的种群n（a，l，t）变得与时间无关。在本文中，我们讨论了解析参数在模型参数的各种范围（出生率b或突变率m）下的稳定性。构造了从n（a，l，t）到下一时间步人口分布n（a + 1，l，t +1）的转换图。然后固定点（稳态解）带来了Coe等人的恢复。结果。从稳定性矩阵的分析中，提取了Lyapunov系数，这些系数表示溶液的稳定性。结果导致在模型参数（b，m，h）空间中稳定解的相图，其中h是寻线率。随着出生率b的增加，我们观察到临界值b（0），低于该值时，种群灭绝，接着是非零的稳定单一解决方案。b的进一步增加导致典型的分叉系列，其中周期加倍，直到在b（c）处达到混乱为止。还讨论了导致逻辑模型的极限情况。我们观察到临界值b（0），低于该值时，种群消失了，接着是非零的稳定单个解。b的进一步增加导致典型的分叉系列，其中周期加倍，直到在b（c）处达到混乱为止。还讨论了导致逻辑模型的极限情况。我们观察到临界值b（0），低于该值时，种群消失了，接着是非零的稳定单个解。b的进一步增加导致典型的分叉系列，其中周期加倍，直到在b（c）处达到混乱为止。还讨论了导致逻辑模型的极限情况。