论文标题
柔软承载能力的环境中的种群最终被灭绝
Populations in environments with a soft carrying capacity are eventually extinct
论文作者
论文摘要
考虑一个人口，其成员逐步变化或死亡（消失），但否则是相当普遍的。将初始（非随机）大小表示$ z_0 $，而$ n $ th更改的大小则为$ c_n $，$ n = 1，2，\ ldots $。人口规模因此，作为$ z_1 = z_0+c_1，\ z_2 = z_1+c_2 $等等，无限期或直到由于灭绝而没有进一步的大小更改，否。因此，灭绝是最终的，因此$ z_n = 0 $表示$ z_ {n+1} = 0 $，而没有任何其他有限吸收的人口尺寸。我们对连续变化之间的持续时间没有任何假设。在现实世界或更具体的模型中，这些模型可能会有所不同，具体取决于个人寿命分布及其相互依存关系，手头的年龄分布和干预环境。变化可能具有不同的分布。基本的假设是有一个{\ em携带能力}，即非负数$ k $，因此，鉴于整个过去的历史，对变更的有条件期望是无阳性的，只要人口超过承载能力。此外，为避免不必要的技术，我们假设更改$ c_n $等于-1（一个人死亡），条件（给定的）概率均匀地脱离了0。避免平价现象是一种简单而不是非常限制的方法，它与马尔可夫设置中的不可约性有关。直截了当的，但内容和含义深远，其结果是所有这些人群都必须消失。从数学上讲，它是基本的收敛属性以及达到吸收灭绝状态的积极概率。
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.
