FISHERY BULLETIN: VOL. 78, NO. 4 



Risk Analysis 



For any population there is a positive probabil- 

 ity of its extinction, though for most populations 

 this is negligibly small. However, the probability 

 will increase as the population size is reduced, e.g., 

 by direct or indirect action of man. Such direct 

 action may be a harvest which is uncontrolled or 

 one which is controlled by fixed rules that do not 

 consider stochastic fluctuations in the environ- 

 ment. 



Random fluctuations in the environment or 

 population which cause increased mortality or re- 

 duced births can lead to extinction, particularly if 

 the population is at a very low level. Moreover, the 

 longer the population is maintained there, the 

 greater the probability of extinction. Such risks of 

 extinction have for a long time been the subject of 

 study in population theory, but most such models 

 are rather simple and include only statistical 

 variation within the population but not externally 

 imposed stresses. We now develop a model that 

 expresses probabilities of extinction as a function 

 of average population growth and variability of 

 environmental stresses. In this model we assume 

 that there is an average increment for each year 

 but, superimposed on this, a variability of the en- 

 vironment which may result in the actual change 

 being positive or negative. We define a stochastic 

 process which is the sum of annual increments 

 which are normally distributed with mean fx and 

 variance <j^, and with successive increments inde- 

 pendent. It is knowm from the theory of stochastic 

 processes (c.f.. Cox and Miller 1965:58) that the 

 probability of the process being absorbed by bar- 

 riers at levels "a" above the initial value or "b" 

 below the initial value are equal to 



1 - exp {2 ixbla-^) 



and 



exp(-2 fjLa/ar^) - exp{2fib/(T^) 



exp(-2 fi/a^) - 1 

 exp(-2 /Lta/o-2) - exp(2 fx b / cr^) 



respectively. If 6 is set equal to the initial value, 

 the second of these represents the probability of 

 extinction. It is difficult to specify appropriate 

 values for a. For example, a = 100 implies that 

 with 95% probability the actual increase might 

 vary from 200 above to 200 below the mean. Such 

 variations are not unreasonable in the Arctic en- 

 vironment. We do not know if they are this large or 

 not but catastrophic mortality due to ice condi- 



tions has been recorded (Sleptsov 1948, as cited by 

 Tomilin 1957, in text but no citation given). We 

 have assumed that stresses are independent 

 events from year to year. To apply this we consider 

 the female population which, in a total of 2,000, 

 will number about 1,000. Extinction clearly occurs 

 if this component falls to zero. We arbitrarily have 

 assumed that if the female population reaches 

 2,000, the population is safe from extinction. If a 

 larger "safe" upper limit is chosen, then the prob- 

 abilities of extinction will be greater. It should be 

 noted that the level of 2,000 females (or any other 

 upper limit) is not an absorbing barrier in the 

 sense that zero, the lower limit, is. However, to 

 simplify the model, we have chosen a range within 

 which it is reasonable to assume average growth is 

 approximately constant — over a wider range the 

 growth parameter must change. Because we have 

 assumed constant ^t (average increment), the 

 probabilities of a population becoming extinct 

 with 1,000 females are easily computed from the 

 given formula for various levels of average annual 

 increase and various levels of environmental 

 stress as expressed by standard deviation. 



RESULTS 



Risk Analysis 



The effect of the level of exploitation on the risk 

 of extinction is shown in Table 2. A catch of 10 

 whales shifts the average increase downward by 

 that amount; i.e., one moves one column to the left 

 in the table. A continuing kill of 30 whales shifts 

 the probabilities three columns to the left. Thus, if 

 present net recruitment were 50 whales and the 

 environmental perturbation were represented by 

 o- = 141.4, the probability of extinction according 

 to this model would be increased from 0.01 to 0.12 

 with a continuing 30 whale kill. 



Initial Stock Size 



It can be seen that the initial stock estimates are 

 little affected by the estimates or assumptions of 



Table 2. — Probabilities of extinction for stochastic process with 

 normally distributed independent additive increments. 



848 



