ANDERSON: A STOCHASTIC MODEL OF FISH SCHOOL SIZE 



which show PiX) vs. X for each parameter varied 

 with relative values v' = 0.1, 1, 10, where v' = 1 is 

 the relative value corresponding with the fit to the 

 observed distribution (Figure 3). In the figures the 

 shape of PiX) varies between a well-defined sharp 

 peak, in which only a narrow range of diameters 

 are probable, and a broad low peak, in which a 

 larger range of somewhat larger diameters are 

 probable. A brief discussion of the sensitivity and 

 some possible biological implications follows. 



The density, or number offish per square meter 

 of horizontal area of a school, has a strong effect on 

 the probability distribution (Figure 4). A dense 

 grouping offish, corresponding with large values 

 of p, favors a narrow range of small school diam- 

 eters while a low density favors a wide range of 

 larger diameters. Serebrov (1976) illustrated that 

 the density offish in a school is highly correlated 

 to fish length according to the relation p = 1/iLKf 

 where p is the density in number offish per cubic 

 meter, L is fish length in centimeters, and K is a 

 constant, with average value 2.44. If we assume 

 the density per square meter is proportional to the 

 density per cubic meter, then with Serebrov's rela- 

 tion, p responds to the one-third power of fish 

 length averaged over the ensemble of schools mak- 

 ing up the observations. Fish length becomes a 

 sensitive parameter, e.g., the relative change in p 

 from 1 to 10 in Figure 4 corresponds to a relative 

 change in the average fish length from 1 to 0.46. 



a 



For small values of the entrance rate into 

 schools, small diameters are favored, and as the 

 entrance rate increases a wide range of large 

 schools is favored (Figure 5). 



The rate a has units of number of fish entering 

 the school per unit time, and if we envision the 

 entrance event as the chance encounter and join- 

 ing of two schools, then a should be proportional to 

 the average number offish in the schools divided 

 by the average time interval between encounters 

 of schools. The time interval between encounters 

 could decrease as the stock population increases if 

 the number of schools per unit area increases. 

 Thus, the entrance rate could increase with in- 

 creases in the stock population of an area. This 

 reasoning suggests that larger stocks would con- 

 tain a wide range of school sizes and small stocks 



would contain a narrow range of small school 

 sizes. 



The parameter /3 is the coefficient for the aver- 

 age exit rate offish from a school and has units of 

 t^ . If we envision the loss mechanism as a random 

 dividing of the school into two fractions, with the 

 time interval between the divisions being random, 

 then the average time interval is proportional to 

 (3~^. Thus, larger values of ^ correspond to short 

 time intervals between divisions and small values 

 correspond to large time intervals between school 

 divisions. The interval as expressed by (3 has a 

 significant effect on the probability distribution 

 PiX), with small values favoring a narrow range 

 of small diameters and large values favoring a 

 wide range of larger diameters (Figure 6). 



The randomness in the schooling process is 

 quantified in the model by the incremental stan- 

 dard deviation a, which has units of t~^^^. For 

 small levels of randomness, small a, the probabil- 

 ity distribution converges on the deterministic 

 steady-state diameter which is defined 



A'o = (oc/p^) 



1/2 



(a/6) 



1/2 



For the model fit this gives Xo = 17.3 m. The 

 convergence is evident in Figure 7 in which Xq 

 changes from 3 to 14 to 17 with a changing from 10 

 to 1 to 0.1. At larger values of a the system has 

 more random character and the probability dis- 

 tribution spreads away from the deterministic 

 value Xq. Expressed as the incremental variance 

 a^, with the dimension t~^ , the term is the diffu- 

 sion coefficient of probability in X space, since the 

 Fokker-Planck equation is in fact a diffusion equa- 

 tion of probability in X and t. 



(Tr 



The incremental standard deviation of the ex- 

 pansion rate, cr, is defined by Equation (7) and is 

 related to the probability equation through r ac- 

 cording to Equation ( 9). It has a small effect on the 

 probability distribution with larger values pro- 

 ducing a broadening of the probability distribution 

 and a shift towards larger diameters. Increasingly, 

 smaller values asymptotically approach a stable 



321 



