GENERAL DISCUSSION 3<55 



K. P. Anderson: There is a mathematical model for a population 

 assumed to show a constant mortality rate and a constant birth rate. This 

 model consists of: 



P,(N=o) -^i 



This means that under certain conditions there is a tendency to population 

 decline, and to maintain a constant mean the number of animals must be 

 very large. 



In such a system, N, the number of animals tends to o as t, time, tends to 

 infmity. The whole system can originate from one specimen. 



J. G. Skellam: This is a well-known property of Markovian stochastic 

 processes. If the Markovian framework is not assumed, this kind of extinction 

 process need not occur. In fact doubts have been raised whether biological 

 stochastic processes are necessarily Markovian. 



M. H. Williamson: Assuming that Markovian stochastic processes 

 were operating, how long would be needed for the chance extinction of a 

 species ? 



J. G. Skellam: It depends on generation time. If «, the number of 

 animals is low, the processes can be quite rapid. If very large, t — the time 

 involved — tends towards a geological time scale. In fact, Thomas Park's 

 experimental Tribolium populations, though containing very few animals 

 initially, showed extinctions in only a tiny proportion of cases. It is unlikely 

 that random extinction is of practical importance in many ecological 

 experiments. 



L. P. Slobodkin: The formula imphes that variance at a given time is 

 cumulative or multiplicative. It is important to know whether populations 

 are Markovian or not. Thus Smith, working at Harvard, took Andrewartha's 

 thrips data and related the annual censuses to mortality. The variances in the 

 same month from one year to the next should be the same as for any other 

 month, and all should increase with time if a Markov system was applicable. 

 It was found that this was not so. Variances may well be able to throw a 

 system out of a Markovian framework and make chances of stochastic 

 extinction nil. 



T. B. Reynoldson: One must remember that the population in the 

 field is far from being a simple unit. 



C. W. Hume: Surely the theory is also out in assuming that the popula- 

 tion can start from a single individual? 



J. G. Skellam: Bacteria probably can. Sex has not yet been found 

 everywhere ! 



