CONFIDENCE LIMITS FOR POPULATION PROJECTIONS 

 WHEN VITAL RATES VARY RANDOMLY 



Tim Gerrodette/ Daniel Goodman,^ aito Jay Barlow^ 



ABSTRACT 



Due to unpredictable future environmental changes, population growth is more realistically viewed as a 

 stochastic than a deterministic process. Environmental variablity is modeled by allowing the population's 

 survival and fecundity rates to be correlated random variables. The expected future population vector and 

 its variance-covariance matrix are computed. The projected total future population size is approximately log- 

 normally distributed, but confidence limits for future population size can be more accurately computed from 

 the distribution of the realized factor of increase. Numerical examples illustrate how the calculation of con- 

 fidence limits for future population size and of the probability that the population will increase in size can be 

 applied to the management of living resources. 



The predicted size of an age-structured population 

 can be projected if its initial size, age distribution, 

 and vital rates are known (e.g., Leslie 1945; Keyfitz 

 1968). Such population projections are commonly 

 used in fisheries and wildlife management when age- 

 specific fecundity and mortality rates are available. 

 However, there is uncertainty in such projections. 

 First, we rarely know vital rates exactly; rather, we 

 have estimates of the true rates, and these estimates 

 are subject to sampling and other types of errors. 

 Second, the true rates themselves are not constant 

 with time. Environmental conditions are always 

 changing, and the vital rates would be expected to 

 change in response. To an extent, the changes of con- 

 ditions may themselves be forecast and incorporated 

 into a population model. Some changes, however, are 

 unpredictable, and these changes give rise to fluctua- 

 tions in the vital rates which make our estimates of 

 population size for some future time less certain. 

 Nevertheless, it may still be possible to make proba- 

 bilistic predictions about future population size given 

 some statistical knowledge about the fluctuating 

 vital rates. 



In this paper we limit ourselves to consideration of 

 the second of these problems, projecting age- 

 structured populations when mortality and fecundity 



'Scripps Institution of Oceanography, University of California at 

 San Diego, La Jolla, CA 92093; present address: Southwest 

 Fisheries Center Honolulu Laboratory, National Marine Fisheries 

 Service, NOAA, P.O. Box 3830, Honolulu, HI 96812. 



^Scripps Institution of Oceanography, University of California at 

 San Diego, La Jolla, CA 92093; present address: Department of 

 Biology, Montana State University, Bozeman, MT 59717. 



^Scripps Institution of Oceanography, University of California at 

 San Diego, La Jolla, CA 92093; present address: Southwest 

 Fisheries Center, La Jolla Laboratory, National Marine Fisheries 

 Service, NOAA, P.O. Box 271, La JoUa, CA 92038. 



Manuscript accepted May 1984. 



FISHERY BULLETIN: VOL. 83, NO. 3, 1985. 



rates vary randomly with time. Recently this topic 

 has been of interest and controversy in a more 

 theoretical context (Boyce 1977; Cohen 1979a, b; 

 Daley 1979; Tuljapurkar and Orzack 1980; Tuljapur- 

 kar 1982; Slade and Levenson 1982). In spite of 

 earlier results to the contrary (Boyce 1977), analyses 

 (Sykes 1969; Cohen 1977), and simulations (Slade 

 and Levenson 1982) have shown that when vital 

 rates fluctuate randomly with no serial correlation, 

 the expectation of population size at a future time 

 will be exactly equal to the population size projected 

 using the mean vital rates in a deterministic projec- 

 tion. For application in fisheries and wildlife manage- 

 ment, the problem is that the distribution of future 

 population sizes will often be strongly skewed. This 

 skew means that the mean and variance of future 

 population size, even if known, are not sufficient to 

 characterize the distribution and, in particular, not 

 sufficient to compute confidence hmits for total 

 population size. In this paper we examine two trans- 

 formations of this skewed distribution which approx- 

 imate a normal distribution, and evaluate the ac- 

 curacy of confidence limits computed from these 

 transformations. 



As pointed out by several of the authors cited 

 above and earlier by Lewontin and Cohen (1969) for 

 a non-age-structured population, stochastic effects 

 can cause the modal or most likely population trajec- 

 tory to decline to extinction, even though the ex- 

 pected or mean population size is growing at a 

 geometric rate. Clearly, if we are to use population 

 projections in fisheries and wildlife management, we 

 should be concerned about the effects of natural 

 variability on the results of our projections. In 

 response to this concern, we have written two com- 



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