Abstract.- Models used by fish- 

 eries managers and population ecol- 

 ogists to analyze or predict the size- 

 frequency distribution of populations 

 usually incoiporate assumptions con- 

 cerning recruitment, mortality, and 

 individual growth that provide math- 

 ematical simplicity, but also con- 

 strain the shapes of size distribu- 

 tions. In this paper we derive a 

 predictive model for size-frequency 

 distributions which assumes constant 

 reciiiitment and mortality, and Brcniy- 

 Bertalanffy growth, then examine 

 the effects of specific violations of 

 these assumptions on the potential 

 shapes of size distributions. Although 

 bimodal and strongly unimodal size- 

 frec|uency distributions are not pos- 

 sible under these assumptions, the 

 model indicates that specific age- 

 related changes in the mortality rate 

 {Z) and the growth coefficient (A') 

 are required to obtain these distribu- 

 tions. Shifts in ZIK with age from 

 growth-dominated (Z/A'<1) to mor- 

 tality-dominated (ZIK>\) usually 

 result in strongly unimodal size-fre- 

 quency distributions. Stable bimodal 

 distributions require shifts from mor- 

 tality-dominated to growth-dominated 

 conditions via age-related changes in 

 Z. K, or both. Non-equilibrium con- 

 ditions or events such as pulses in 

 recruitment or mortality can also 

 modify size-frequency distributions, 

 but these effects are usually transi- 

 ent. These results indicate that infer- 

 ences concerning the demographic 

 dynamics of a population may be 

 derived sinijily by obsemng the shape 

 of its size-frequency distribution. 



Inferring Demographic Processes from 

 Size-Frequency Distributions: Simple 

 Models Indicate Specific Patterns 

 of Growth and Mortality 



James P. Barry 

 Mia J. Tegner 



A-OOI Scnpps Institution of Oceanography 

 La Jolla, California 92093 



Manuscript accepted 24 August 1989. 

 Fishery Bulletin, U.S. 88:13-19. 



Population ecologists and wildlife 

 managers are often interested in 

 identifying spatial and temporal pat- 

 terns of growth and mortality in 

 order to understand the dynamics of 

 populations. Because of logistic or 

 other constraints in their abilities to 

 directly quantify mortality and 

 growth, however, ecologists often 

 rely on the analysis of size distribu- 

 tions to infer these parameters 

 (Cassie 1954; Beverton and Holt 

 1956; Ricker 1958, 1975; Ebert 1973; 

 Van Sickle 1977b; Pauly and Morgan 

 1987). A population's size-frecjuency 

 distribution (hereafter referred to as 

 size distribution or size structure) 

 results from its recent history of re- 

 cruitment and mortality, integrated 

 with the growth rates of individuals. 

 Temporal or spatial changes in the 

 size structure of a population must 

 therefore reflect changes in one or 

 more of these parameters. For exam- 

 ple, red sea urchins Strongylocentro- 

 tus Jranciscfuius are long-lived and 

 have size distributions that are rela- 

 tively stable through time, but vary 

 in space due to geographic changes 

 in recruitment and mortality related 

 to the distribution of predators and 

 dispersal of larvae (Tegner and Barry 

 Unpubl.). In some locations urchins 

 have persistently bimodal size struc- 

 tures, while in others, the population 

 is unimodal or amodal. 



Techniques for size-frequency 

 analysis usually combine models of 

 the growth rates of individuals with 



mortality rates to describe observed 

 size-frequency distributions. Because 

 growth and mortality both affect the 

 shape of the distribution, knowledge 

 of one parameter may allow deduc- 

 tion of the other from the shape of 

 the size-frequency distribution. Since 

 growth, mortality, and recruitment 

 can vary considerably, as well as in- 

 dependently of one another, analyses 

 of size distributions typically utilize 

 several simplifying assumptions. 



In this paper, we show that al- 

 though simplified models commonly 

 employed for growth and mortality 

 result in a limited range of size struc- 

 tures, they can nonetheless be in- 

 dicative of the demographic patterns 

 of populations. Thus, examination of 

 size-frequency distributions may 

 allow reasonable inferences concern- 

 ing the dynamics of a population. In 

 particular, bimodal size distributions, 

 that are "impossible" under typical 

 model assumptions but are typical of 

 red urchins in the Southern Califor- 

 nia Bight, must arise from particular 

 patterns of age or size-specific changes 

 in demographic parameters. In order 

 to generate these size structures, 

 assumptions concerning constant 

 mortality and growth coefficients are 

 not tenable. 



The most basic assumption of most 

 models is that the population is stable 

 and has a stationary age structure. 

 Thus, recruitment is taken to be invar- 

 iant from year to year, or continuous, 

 and mortality rates are presumed 



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