Fishery Bulletin 88(1). 1990 



CONSTANT MORTALITY 



A 



CC 

 O 



SIZE 



Figure 2 



Hypothetical size-frequency distributions for populations exhibiting 

 constant mortahty and von Bertalanffy growth. (A) Mortality [Z] 

 is constant for all sizes. The growth coefficient (K) is also assumed 

 to be a constant. Due to these assumptions, the shape of size- 

 frequency distributions are limited to those general forms presented 

 in b-d. according to the relative magnitudes of Z and K. (B) Z is 

 less than K over all S. (C) Z equals K. (D) Z is greater than A'. 



and high (constant) survivorship (Van Sicl<le 1977a. 

 Power 1978, DeAngeiis and Mattice 1979), is not ade- 

 quate, assuming simple Brody-Bertalanffy growth and 

 constant mortality under steady-state conditions. There 

 must be a more complicated age dependence of growth 

 rate, or an age dependence of mortality, to account for 

 a bimodal stable size distribution. 



Because the signs of A^o, Sj, S^, and A' are all 

 positive, the sign of dNIdS in equation (10) is deter- 

 mined by the first term (ZIK-1). U Z is greater than 

 K. the slope is negative and the size structure is 

 dominated by juveniles (Fig. 2). When Z is less than 

 K, the slope is positive, up to S^. and the population 

 is dominated by adults. These can be termed mortality- 

 dominated and growth-dominated populations, 

 respectively. 



Effects of age-related rates 

 of growth and mortality 



Let us now relax the assumption that A' and Z are in- 

 dependent of age. Age-specific variation in these coef- 

 ficients can result in a bimodal or unimodal size 

 distribution, depending upon the relative magnitude of 

 A' and Z. The derivations of similar models for age- 

 varying K and Z are more complicated, but we can 

 evaluate the effect of such changes simply by consider- 

 ing a combination of size distributions generated with 



VARIABLE MORTALITY 



A DECREASING 



i-' 



B INCREASING 



U 



SIZE OR AGE 



Figure 3 



1 f the assumptions of constancy for Z and K are violated, the shape 

 of a size distribution varies according to the pattern of the violation. 

 The figures on the left-hand column indicate a shift (step function) 

 in the relative values of Z and K, with age or size. The figures in 

 the three right columns show the general shape of the size-frequency 

 distribution for the specified conditions, created by combining the 

 predicted size distributions for before and after the change in K or 

 Z. (Row A) For a decrease in the ratio of ZIK with larger size, 

 biniodality is possible, depending upon the magnitude of Z and K. 

 When Z is near in value to K. and ZIK shifts from greater than to 

 less than unity, liimodality is littely. For cases where Z is always 

 greater than K or less than A', the size-frequency distribution is 

 dominated by juveniles, or adults, respectively. (Row B) For an 

 increase in ZIK, the size distribution may be unimodal with a mode 

 in any position, but cannot be bimodal. For each of the three right 

 columns, the appropriate value of ZIK for the interval S„ to S„ 

 is indicated. 



different values for one or both of these coefficients. 

 The important implications remain unchanged. If A' or 

 Z changes with size such that the ratio of ZIK shifts 

 from greater than to less than unity as size increases, 

 the slope of the size distribution will change from 

 negative to positive: conditions required for a bimodal 

 distribution (Fig. 3). In contrast, a shift from growth- 

 dominated {ZIK<\) to mortality-dominated (ZIK>\) 

 conditions produces a strongly unimodal pattern, with 

 the position of the mode determined by the size where 

 Z = K. 



Obviously, variation in the ratio of ZZ/C can arise from 

 size or age-specific changes in the value of Z, A', or t)oth. 

 Although Bertalanffy-type growth curves assume K to 

 be independent of age, the growth intervals of species 

 are frequently shown to exhibit variations in K with 

 age, with K commonly decreasing slightly with age 

 (Ricker 1975). For example, decreases in K with size 

 or age are evident from the increasing slope of Walford 

 plots for sea urchins Strongylocentrotus purpuratus 

 presented by Russell (1987). Estimates of A from Rus- 

 sell's Figure 3 decrease from approximately 0.5 to 0.05 

 from small to large urchins at most locations. Assum- 



