在承载力较弱的环境中的种群最终会灭绝
Populations in environments with a soft carrying capacity are eventually extinct
摘要：
考虑一个种群，其规模随着其成员的繁殖或死亡（消失）而逐步变化，但在其他方面是相当一般的。用$Z_0$表示初始（非随机）大小，用$C_n$，$n= 1, 2, \ldots$表示$n$ th变化的大小。因此，种群大小依次发展为$Z_1=Z_0+C_1,\ Z_2=Z_1+C_2$等，无限期地或直到由于灭绝而没有进一步的大小变化。灭绝因此被假定为最终的，所以$Z_n=0$意味着$Z_{n+1}=0$，没有任何其他有限的吸收类的人口规模。我们对连续变化之间的持续时间没有任何假设。在现实世界中，或者在更具体的模型中，它们的长度可能会有所不同，这取决于个人的寿命分布及其相互依赖性、手头的年龄分布和干预环境。变化可能具有相当不同的分布。基本假设是存在{ \em承载能力}，即非负数$K$，使得在给定完整的过去历史的情况下，每当人口超过承载能力时，变化的条件期望是非正的。此外，为了避免不必要的技术细节，我们假设变化$C_n$等于-1（一个个体死亡），条件（给定过去）概率均匀地远离0。这是一个简单的，不太严格的方法来避免奇偶现象，它与马尔可夫设置中的不可约性有关。直接的，但在内容和影响深远，后果是，所有这些人口必须灭绝。在数学上，它遵循下鞅收敛性质和达到吸收灭绝状态的正概率。
英文摘要：
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 $n$th 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. Changes may have quite varying distributions. The basic assumption is that there is a {\em 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 submartingale convergence property and positive probability of reaching the absorbing extinction state.
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