Barry and Tegner: A predictive model for size^frequency distribution 



ing the reported instantaneous mortality rates (Z-O.l) 

 to be constant, the age-specific Z/K ratio indicates that 

 the size-specific population dynamics switch from 

 growth-dominated {Z/K<1) to mortality-dominated 

 (ZIK>1), which might account for the observed strong- 

 ly unimodal size-frequency distributions. Other species 

 of urchins (Tegner and Dayton 1977, 1981; Himmelman 

 1986; Tegner and Barry Unpubl.) as well as many 

 species of long-lived arctic fishes (DeAngelis and Mat- 

 tice 1979) exhibit bimodal size-frequency distributions 

 and must, under steady-state conditions, undergo a 

 shift from mortality-dominated to growth-dominated 

 population dynamics. 



In that the growth rates of individuals usually 

 decrease with age (Ricker 1975, but see Campbell 1979, 

 Himmelman 1986), the most likely cause of persistently 

 bimodal size distributions under steady-state conditions 

 is an even greater decrease in the mortality rate of 

 large individuals. A reduction in mortality, assuming 

 a constant or slightly decreasing growth coefficient, 

 could allow ZIK to shift from >1 to <1: conditions 

 necessary for bimodality. 



Evidence for lower mortality with larger size or age 

 is common. Many species exhibit type III survivorship 

 curves of decreasing mortality with age (Deevey 1947, 

 Odum 1971, Wilson and Bossert 1971). For example, 

 because lobsters preferentially consumed small-sized 

 red urchins Strongylocentrotus franciscanus (Tegner 

 and Levin 1983) and sheephead Semicossyphus pulcher 

 repeatedly select ?,vasL\\ev S. franciscanus when offered 

 a choice of sizes (Tegner and Dayton 1981), predation 

 mortality apparently decreases with size ( = age) in 

 urchins. In addition, geographic differences in the size 

 structure of red urchins are related to the distribution 

 of predators, with bimodal size-frequency distributions 

 found where these predators are most abundant 

 (Tegner and Barry Unpubl.). Intraspecific competition 

 for resources and high adult survivorship appear to 

 limit the growth or survival rates of juveniles or both 

 for arctic fishes (Johnson 1976), green sea urchins 

 (Himmelman 1986), as well as forest trees (Harper 

 1977), often leading to a bimodal distribution of sizes; 

 however, bimodality in these populations may arise 

 from stochastic, age-specific changes in growth (DeAn- 

 gelis and Coutant 1982). 



Unstable or non-equilibrium conditions 



There are conspicuous alternative causes of bimodal 

 size-frequency structures for populations with unstable 

 or non-stationary age compositions. In particular, 

 species that have seasonal recruitment and live for only 

 two years (e.g., blue crabs; Hines et al. 1987) have per- 

 sistent, but recruitment-controlled bimodality. For 



longer-lived species, relaxation of the assumptions of 

 a stable age structure and stationary size distribution 

 allow for transient, but perhaps persistent, variations 

 in age and size structures due to temporal variation in 

 recruitment and mortality. Interannual variation in 

 recruitment, to a population normally dominated by an 

 adult mode, can skew the age structure and occasional- 

 ly produce a mode of juveniles, thereby resulting in 

 bimodality. This feature will, however, deteriorate as 

 the juveniles grow and merge into the adult mode. 

 Similarly, a mortality-dominated population can 

 become bimodal when a large pulse of juveniles grows 

 to adult size, before being eventually depleted by mor- 

 tality. In both cases bimodality is a transient feature 

 of the size structure. How long it will persist is deter- 

 mined by the rapidity with which individuals grow to 

 asymptotic size as well as the range of variation in 

 recruitment. Recruitment pulses leading to unstable 

 size and age distributions have been reported for 

 several species (Hjort 1914, 1926; Ebert 1983; Cowen 

 1985; Johnson et al. 1986; Paine 1986; Pearse and 

 Hines 1987; Raymond and Scheibling 1987). 



Variation in growth and mortality within an age 

 cohort, due to stochastic processes and genetic vari- 

 ability, can disrupt the deterministic character of 

 growth and survivorship processes leading to a highly 

 variable age and size structure, even within a single 

 cohort. Intraspecific competition for resources can in- 

 duce bimodality within a single cohort if growth to a 

 particular size confers a great competitive advantage, 

 leading to even more rapid gi'owth (DeAngelis and Cou- 

 tant 1982). For example, if recruitment by juveniles 

 into the adult size classes is regulated by stochastic pro- 

 cesses that provide limiting resources to a few juveniles 

 upon the removal of adult individuals, the size struc- 

 ture of a cohort may become bimodal. This is appar- 

 ently ty]3ical of arctic fishes (Johnson 1976), large- 

 mouth bass (Shelton et al. 1979; Timmons et al. 1980), 

 gi'een sea urchins (Himmelman 1986), stalked barnacles 

 (Page 1986), and many species of forest trees (Harper 

 1977). 



Value of simple size-frequency 

 distribution models 



As shown in this analysis, even very simple models of 

 size-frequency distributions, with perhaps unrealistic 

 assumptions, can provide valuable information con- 

 cerning the growth and mortality schedules of popula- 

 tions. Although the range of potential size structures 

 is constrained by model assumptions such that bimodal 

 size distributions, or unimodal distributions with the 

 mode centered away from Sd or S^ , are not possible, 

 we can still utilize these models to identify likely 



