Stochastic Versus Deterministic Description

The increase in the death rate per individual pjn with growing n could, for example, be due to limitations in food resources. An analogous effect, namely a decrease in the specific birth rate X /n for growing n will be neglected as it leads to essentially the same type of results. The birth and death rates per individual as defined in (4.64 b) are shown in Fig. 4.9. 

With these equations the condition of detailed balance is fulfilled in a trivial manner. The exact stationary solution, see (4.39), is obtained in the form 

This solution describes a population which has completely died out. The solution is stable because from nothing comes nothing . Besides this trivial solution, however, there exists a quasi-stationary solution for which the eventual extinction occurs extremely slowly. 

In the conventional approximation the quasi-stationary solution is determined by neglecting (n - l)/n in (4.66) and by inserting the approximate factors 

This conventional quasi-stationary distribution (4.68) or (4.70), however, has completely the wrong form for small n due to neglecting the factors (n - l)/n. On the other hand, it will be seen that the probability of complete extinction in the quasi-stationary solution depends decisively on the exact form for small values of n. 

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