Abstract. -The shape of a size- 

 frequency distribution is the result 

 of age- or size-specific rates of growth 

 and survival, their variability, and 

 seasonal and interannual variation 

 in recruitment. Simulation of size 

 distributions can be used to gain in- 

 sight into the underlying processes 

 that give rise to observed size struc- 

 ture of organisms in the field, but 

 the utility of this approach depends 

 critically on underlying assumptions. 

 Incorrect judgment of the signifi- 

 cance of assumptions can lead to 

 erroneous conclusions concerning 

 the causes of bi- or polymodal 

 distributions. 



Using the Brody-Bertalanffy 

 growth model and a constant sur- 

 vival rate, bi- and polymodal distri- 

 butions can be generated when re- 

 cruitment is pulsed. Even with as 

 many as 10 recruitment episodes per 

 year, size distributions show several 

 modes. A sampling of the literature 

 indicates that most fish and marine 

 invertebrates have pulsed rather 

 than continuous recruitment; thus, 

 when very little is known about a 

 species, pulsed rather than continu- 

 ous recruitment would be the better 

 assumption when interpreting the 

 shapes of size distributions. 



Our simulations differ from those 

 conducted by Barry & Tegner (1990) 

 who assumed continuous and con- 

 stant recruitment and focused on 

 changing growth and survival param- 

 eters to explain bimodal size struc- 

 ture. These authors also suggested 

 that their analysis was appropriate 

 for interpreting the dynamics of red 

 sea urchins Strongylocentrotus 

 franciscanus. We have been docu- 

 menting settlement of both red and 

 purple (S. purpuratus) sea urchins. 

 At La Jolla, California, neither 

 species showed continuous settle- 

 ment; rather, both species had pulses 

 of settlement in spring 1990 and 

 1991. 



Although age-specific variation in 

 growth or mortality parameters can 

 result in bimodal size distributions, 

 it is more likely that such distribu- 

 tions are caused by seasonal pulses 

 of recruitment. 



Inferring demographic processes 

 from size-frequency distributions: 

 Effect of pulsed recruitment on 

 simple models 



Thomas A. Ebert 

 Stephen C. Schroeter 

 John D. Dixon 



Department of Biology, San Diego State University 

 San Diego. California 92182-0057 



Manuscript accepted 22 January 1993. 

 Fishery Bulletin, U.S. 91:237-243 ( 1993). 



For many organisms, size data are 

 easy to gather and size-frequency dis- 

 tributions are common in the litera- 

 ture. In many cases, they provide the 

 only clues to the underlying dynam- 

 ics of growth, survival, and recruit- 

 ment. Thus, it is understandable that 

 an extensive literature exists con- 

 cerning their analysis. One general 

 research approach has focused on the 

 separation of size distributions into 

 components (e.g., Harding 1949, 

 Cassie 1950, Bhattacharya 1967, 

 Young & Skillman 1975, Macdonald 

 & Pitcher 1979). A second approach 

 has attempted to use size data ei- 

 ther to estimate mortality when 

 growth parameters are known (e.g., 

 Beverton & Holt 1956, Smith 1972, 

 Van Sickle 1977ab, Ebert 1981 and 

 1987, Sainsbury 1982) or to estimate 

 both growth and mortality param- 

 eters (e.g., Green 1970, Ebert 1973 

 and 1987, Saila & Lough 1981, 

 Fournier & Breen 1983, Pauly 1987). 

 A third approach has modeled size 

 distributions to gain insight into the 

 underlying processes that give rise 

 to observed distributions (e.g., Craig 

 & Oertel 1966, DeAngelis & Coutant 

 1982, Barry & Tegner 1990, Hartnoll 

 & Bryant 1990). Simulations of size 

 distributions are metaphors of the 

 dynamic processes that give rise to 

 actual size distributions. The utility 

 of simulation depends critically on 

 the underlying assumptions. If the 

 significance of any of the assumptions 



is wrongly judged, one may be led to 

 erroneous conclusions concerning un- 

 derlying dynamics. 



As an approach to explaining bi- 

 modal size distributions, Barry & 

 Tegner (1990) presented a determin- 

 istic model for the development of 

 size distributions that has seven as- 

 sumptions: (1) Brody-Bertalanffy 

 growth, (2) constant rate of mortal- 

 ity, (3) constant and continuous re- 

 cruitment, (4) strict determinism for 

 growth, so ct=0 for all sizes at an 

 age, (5) strict determinism for sur- 

 vival, so rr=0 for numbers at an age, 

 (6) population growth rate per indi- 

 vidual, r, equal to 0, and (7) a stable 

 size distribution equivalent to a 

 stable age distribution. Bimodal size 

 distributions are not possible with 

 these seven assumptions, yet bimo- 

 dality is commonly observed. Accord- 

 ingly, one or more of the assumptions 

 must be violated. Barry & Tegner 

 focused on the assumptions con- 

 cerning growth and survival and 

 concluded that "...bimodality can 

 develop only from an increase in 

 survivorship with age or an increase 

 in the growth coefficient with age, or 

 both." In particular, they argued that 

 size distributions of red sea urchins 

 Strongylocentrotus franciscanus re- 

 quired age- or size-specific changes 

 of the growth-rate constant, K, in the 

 Brody-Bertalanffy equation, the mor- 

 tality coefficient, Z, in an exponen- 

 tial survival curve, or both. 



237 



