238 



Fishery Bulletin 91(2), 1993 



There are three issues that we would like to ex- 

 plore: (1) Possible causes of bimodal size distribu- 

 tions and, in particular, the consequences of pulsed 

 recruitment, (2) applicability of the Barry & Tegner 

 model to sea urchins in California, and (3) general 

 applicability of the Barry & Tegner model. 



Size-distribution simulation 



We simulated several size distributions to show how 

 the Barry & Tegner model works. Growth was mod- 

 eled using the Brody-Bertalanffy equation for individual 

 growth: 



S t = Sjl-be Kt ) 



where S t = size at time t after birth or settlement 

 S„ = asymptotic size 

 K = growth rate coefficient 

 , S„-S B 



(1) 



(2) 



S R = size at t=0 when organisms begin to grow 

 according to Eq. 1. 



Sometimes Eq. 1 is written 



S t = Sjl-e- Bt -V) 



where t = time at which size would be 

 b = e Kt °. 



(3) 



(4) 



Cohort survival was modeled so that the mortality rate 

 was constant: 



N t = N e" zt 



(5) 



where N t = number remaining in a cohort at time t 

 N tl = initial number in a cohort 

 Z = mortality rate coefficient. 



~ 1.0 



CO 



2> 0.8- 



5 °6 



E 



CD 



£ 0.4 



2 



o 



B 



0.0 



2 



Size at t 



Equations 1 and 5 were used to generate a number- 

 density distribution that was integrated over segments 

 of arbitrary size to produce a size-frequency distribu- 

 tion. The first step was to calculate sizes at particular 

 ages (Eq. 1) and then to estimate numbers in a cohort 

 that survived to each age (Eq. 5). The number surviv- 

 ing to a specific size was generated using a constant 

 time-interval and Z<K (Fig. 1). The size-intervals 

 shorten because growth follows Eq. 1. For Fig. 1A, the 

 time-interval between recruitment episodes is 1 unit, 

 which, for purposes of this discussion, we call lyr. 

 Changing the time-interval for reproduction, that is, 

 the number of evenly-spaced reproductive episodes/yr, 

 changes the number and spacing of lines in the graph. 

 In Fig. IB we used 10 episodes/yr; that is, an interval 

 of O.lyr. In our simulation, we assumed that the popu- 

 lation was periodically stable and stationary. 



Stable and stationary structure for a population with 

 pulsed recruitment means that demographic structure 

 and number of individuals change between recruit- 

 ment episodes but are the same across episodes. When 

 the same times after recruitment episodes are com- 

 pared, the age and size structures of the population 

 are the same (stable structure) and the numbers of 

 individuals are the same (stationary structure). The 

 relationship between density and time would be saw- 

 toothed, with an average slope of 0, but with increased 

 numbers-at-recruitment and declining numbers up to 

 the point of another recruitment episode. As the num- 

 ber of recruitment events increases, the height of each 

 tooth becomes smaller until it is smooth when recruit- 

 ment is continuous. The vertical lines in Fig. 1 repre- 

 sent not only the progression of a single cohort through 

 time, but also cohorts that make up the stable and 

 stationary population. 



A size-frequency distribution was constituted by sum- 

 ming all lines within an arbitrary interval of 1.0 size 

 units (Fig. 2). With just one reproductive episode/yr 

 (Fig. 2A), size distribution is poly- 

 modal. The overall shape of the size 

 distribution, which is formed by draw- 

 ing an envelope around the distribu- 

 tion, has a negative slope. Over an 

 increasing range of recruitment 

 frequencies, the polymodal nature is 

 preserved, so that with 10 episodes/yr 

 (Fig. 2C) modes still are evident but 

 the overall shape shows a positive 

 slope. When there are 100 episodes/yr, 

 size distribution approaches the con- 

 tinuous form used by Barry & Tegner: 



B 



i 



10 



Figure 1 



Number (N t ) vs. Size (S,) where N,=N„e z ' and S,=S.( 1-be K, l. (A) 1 recruitment 

 event/yr; (B) 10 recruitment events/yr. 



NS 



KS 



t^-tr 



M(H 



