ANDERSON; A STOCHASTIC MODEL OF FISH SCHOOL SIZE 



chovy schools during the daylight hours is disk- 

 like. Including unknown factors that alter the 

 three-dimensional shape through a stochastic 

 term the relation between the school diameter and 

 number is expressed as 



N = X'{p + €it)) nIA 



(2) 



where X is the diameter of a school with a disklike 

 shape, p is the average density of fish per unit 

 horizontal area, and e( ^) is the stochastic variation 

 on the density and has white noise character. Re- 

 write Equation (2) as 



N 



pX^il + hit)) 



(3) 



i{t) = 6(0 + (a + 5(0) s i-hiny 



m = l 



The last term in Equation (6) is the stochastic 

 contraction rate and is related to the stochastic 

 exit rate according to the relation 



dh(t) oo 



7(0 = 7(0+ S, {-h(t)y 



dt ^" = 



We assume the components i(t) and^'iO are sta- 

 tionary random processes with zero mean values. 

 Then their statistics can be characterized by the 

 relations 



where p is a generalized density with p = p-rrlA and 

 hit) is the stochastic density normalized top giv- 

 ing /?(n = e(t)lp. 



Using Equation (3) in Equation (1) and differen- 

 tiating yields 



dX 



a 



/3X 



+ 



5(0 



dt 2pX{l+h{t)) 



y{t)X dh{t)ldt 



2{1 + h{t)) 



2pX{l + h{t)) 



(4) 



To separate the deterministic and stochastic parts 

 of Equation (4) assume p>\e{t) \ soh{t)''^<l giving 

 the convergent series 



1 



= 1-/2(0 + ^(0^ -h{tf + ... 



l + /i(0 



i + ^li i-h{t)y 



(5) 



iit)i{t + t') = o]q.{t') 



jXt)j{t+t') - O^jQjit') 



(7) 



where cr^i and cr^j are positive constants that quan- 

 tify the level of the stochastic inputs and are re- 

 ferred to as "incremental variances." The qit') 

 terms are autocorrelation functions that quantify 

 the spectrum characteristics of the stochastic in- 

 puts as a function of separation time t'. 



To investigate the probability characteristics of 

 X in Equation (6) we must combine the stochastic 

 terms into a single stochastic input, in a manner 

 that retains the statistical characteristics of the 

 terms. From Equation (7) we define the ratio 



ritr = 



Ht)i{t+t') ^ ^2^_(^>) 

 Jit)jit+t') ^J 9/(^') 



Using Equation (5) in Equation (4) and separating 

 deterministic and stochastic parts in powers of X 

 yields the stochastic dynamic equation for X as 



dX oc liX i{t) j{t)X 



= + + 



dt 2pX 2 2pX 2 



(6) 



The first term on the right side of Equation (6) is 

 the deterministic expansion rate of a school and 

 the second term is the deterministic contraction 

 rate. The third term is the stochastic expansion 

 rate and is principally related to the stochastic 

 entrance rate. The stochastic component in ex- 

 panded form is 



We assume the correlation functions are similar 

 enough to make the simplification qi( t')lqj( t') ~ 1. 

 Then we relate the two stochastic terms in Equa- 

 tion (6) as 



i{t) = rj{t) 



(8) 



where r is the ratio of the intensities of the inputs 

 and is defined as 



^,/^;. 



(9) 



In effect with Equations (8) and (9) we are taking 

 the ratio of the two stochastic terms to be equal to 



317 



