Fishery Bulletin 88(1). 1990 



constant over time. In addition to these assumptions, 

 the population is usually required to conform to strict- 

 ly deterministic equations that describe growth and 

 abundance. Often, mortality rate is also assumed to be 

 independent of age (Green 1970, Ebert 1973), resulting 

 in a type II survivorship curve (Deevey 1947). Growth 

 rates of individuals are frequently assumed to fit a 

 Brody-Bertalanffy function (von Bertalanffy 1957, 

 Ricker 1975) in which size increases at a rate propor- 

 tional to the distance from the maximum size. For 

 models in which one or another of these assumptions 

 is relaxed, see Van Sickle (1977a, b), DeAngelis and 

 Mattice (1979), DeAngelis and Coutant (1982), and 

 several papers in Pauly and Morgan (1987). 



The applicability of these simplifying assumptions 

 varies considerably among species and populations. For 

 some species (e.g., certain long-lived fishes of arctic 

 lakes), age structures are stable and size distributions 

 are stationary, with fairly constant recruitment from 

 year to year (Johnson 1976). For many species, how- 

 ever, the assumption of a stable age structure is un- 

 realistic, due to considerable interannual variability in 

 recruitment to the zero age class (Johnson et al. 1986, 

 Pearse and Mines 1987, Raymond and Scheibling 1987, 

 Barry 1989). For red urchins, although recruitment 

 varies between years, the shape of its size distribution 

 is characteristically constant from year to year (Tegner 

 and Barry Unpubl.). The assumption of an age-invari- 

 ant mortality rate is also probably unwarranted for 

 many, if not, most species. A more common pattern 

 is high juvenile mortality, followed by high adult sur- 

 vival {ty[)e III survivorship). Models incorporating these 

 simplifying, but perhaps unrealistic, assumptions (dis- 

 cussed above) can, nevertheless, be of value in iden- 

 tifying patterns of individual growth as well as in 

 estimating recruitment and mortality for the popula- 

 tion. In many cases these properties may otherwise be 

 unobtainable. 



In this paper we are concerned with population 

 dynamics that result in a bimodal distribution of sizes. 

 Bimodal size distributions have been reported for 

 several species, and are of considerable interest (John- 

 son 1976; Tegner and Dayton 1977, 1981; Shelton et al. 

 1979; Timmons et al. 1980; Stein and Pearcy 1982; 

 Wilson 1983; Pollard 1985; Page 1986; Tegner and 

 Barry Unpubl.), especially for populations that are 

 apparently stable. Intuitively, bimodality may develop 

 and persist under equilibrium conditions by a combina- 

 tion of rapid growth of individuals to adult size and high 

 survival rates. Thus, a mode of juveniles may be distinct 

 from an adult mode comprised of several age classes 

 that overlap in size, with relatively few intermediate- 

 sized, rapidly .growing individuals. DeAngelis and Mat- 

 tice (1979) and Power (1978) suggested that bimodality 

 may arise from this sort of "pileup" of individuals at 



larger size classes due to a decrease in growth rate at 

 adult size, even with constant mortality. Mortality 

 decreases the number of older individuals, but a large 

 number are left clustered near the upper size limit. 

 Here we show that bimodal size stnicture must develop 

 from particular patterns of age-specific growth, mor- 

 tality, or both, that are not possible with commonly 

 employed models. Specifically, bimodality can develop 

 only from an increase in survivorship with age or an 

 increase in the growth coefficient with age, or both. 

 Even though simple models are limited in their range 

 of size distributions, we can use these models to iden- 

 tify deviations from them that are necessary or suffi- 

 cient to produce particular size distributions, such as 

 bimodal or strongly unimodal distributions. 



Derivation of a 



simple size-frequency model 



The change in abundance of a cohort can be repre- 

 sented as, 



dN 

 dt 



= -ZN 



(1) 



where A'' is the number of individuals alive in the cohort 

 at time /, and Z is the instantaneous mortality rate for 

 the population. Assuming that Z is constant over time 

 and independent of age and size, this equation can be 

 integrated to obtain a simple decreasing exponential 

 function for the number of individuals versus time. 



A^, 



A„c-2('-'"' 



(2) 



where f,, is the time of recruitment or birth and A,, is 

 the abundance of the population at time ^|. The equa- 

 tion can be simplified slightly l)y defining at)undance 

 in terms of age rather than time; age (t) equals t-t-Q, 

 or time since recruitment. Hence, equation (2) becomes 



A, = /V„ f 



-Zi 



(3) 



If we now assume that the population is stable, with 

 constant recruitment, this function describes both the 

 time series of abundance for a single cohort and the 

 stable age structure of the population. 



Brody-Bertalanffy growth is characterized by expo- 

 nentially decaying growth in size, with no lag during 

 early life. The general form of this deterministic equa- 

 tion is 



S(l_6e-A-„-u) 



(4) 



where S, is the size of an individual at age t (i.e., at 

 time / after /„, the time of iiirth or recruitment), S is 



