Consider a population whose size changes stepwise by its members reproducing or dying (disappearing), but is otherwise quite general. Denote the initial (non-random) size by \(Z_0\) and the size of the nth change by \(C_n\), \(n= 1, 2, \ldots \). Population sizes hence develop successively as \(Z_1=Z_0+C_1,\ Z_2=Z_1+C_2\) and so on, indefinitely or until there are no further size changes, due to extinction. Extinction is thus assumed final, so that \(Z_n=0\) implies that \(Z_{n+1}=0\), without there being any other finite absorbing class of population sizes. We make no assumptions about the time durations between the successive changes. In the real world, or more specific models, those may be of varying length, depending upon individual life span distributions and their interdependencies, the age-distribution at hand and intervening circumstances. We could consider toy models of Galton–Watson type generation counting or of the birth-and-death type, with one individual acting per change, until extinction, or the most general multitype CMJ branching processes with, say, population size dependence of reproduction. Changes may have quite varying distributions. The basic assumption is that there is a carrying capacity, i.e. a non-negative number K such that the conditional expectation of the change, given the complete past history, is non-positive whenever the population exceeds the carrying capacity. Further, to avoid unnecessary technicalities, we assume that the change \(C_n\) equals -1 (one individual dying) with a conditional (given the past) probability uniformly bounded away from 0. It is a simple and not very restrictive way to avoid parity phenomena, it is related to irreducibility in Markov settings. The straightforward, but in contents and implications far-reaching, consequence is that all such populations must die out. Mathematically, it follows by a supermartingale convergence property and positive probability of reaching the absorbing extinction state.

考虑一个人口，其大小随着其成员的繁殖或死亡（消失）而逐步变化，但在其他方面却相当普遍。用\（Z_0 \）表示初始（非随机）大小，用\（C_n \），\（n = 1，2，\ ldots \）表示第n次更改的大小。因此，种群数量会无限期地发展为\（Z_1 = Z_0 + C_1，\ Z_2 = Z_1 + C_2 \），依此类推，直到灭绝为止。因此，假设灭绝是最终的，所以\（Z_n = 0 \）意味着\（Z_ {n + 1} = 0 \），而没有其他任何有限的人口规模吸收类别。我们不对连续更改之间的持续时间做任何假设。在现实世界中，或更具体的模型中，这些模型的长度可能会有所不同，具体取决于各个人的寿命分布及其相互依存关系，当前的年龄分布以及所处环境。我们可以考虑用高尔顿-沃森类型生成计数或生与死类型的玩具模型，每个变化一个人行动直到灭绝，或者说是最普遍的多类型CMJ分支过程，例如繁殖种群的依赖性。变化可能具有非常不同的分布。基本假设是有一个承载能力，即非负数K这样，如果有完整的过去历史记录，那么只要人口超过承载能力，对变化的有条件预期就不会是积极的。此外，为避免不必要的技术性，我们假设变化\（C_n \）等于-1（一个个体死亡），且条件（给定的过去）概率均匀地远离0。这是一种简单但并非限制性的方法避免平价现象，这与马尔可夫环境中的不可约性有关。结果很直接，但在内容和意义上却具有深远的影响，那就是所有这些人口都必须消灭。从数学上讲，它具有超级mart的收敛性和达到吸收消光状态的正概率。