FISHERY BULLETIN: VOL. 70, NO. 2 



vertex except Bi, and the equations were solved 

 simultaneously. In this case there are seven 

 equations, and the work of classical solution was 

 not excessive. The result was found to be: 



B2 



\V _k.,d,obr2iXY + Z) 

 W' — k^dubniXY' + Z') 



(19) 



where: 



W = 

 W = 

 X = 



Y = 

 Y' = 

 Z = 

 Z' = 



k3k4dl3hl3d34b3ih24 (Ci2i — 624) 



k2k4di2bv2d2ib2ih34{(iu — ^34) 

 kih24ia2i — 624) + k4h34{a34 — ^34) 



W4 — <74 — A 



kshn (ai3 — bn) — (^34634 — W3 — Q3 

 k2hi2 (ai2 — 612) — 6^24624 — W2 — 92 



k4d34b34h34iCl34 &34) 



k4d24b24h24 iCl24 &24) 



h 

 h 



Questions of interest here were the effects on 

 biomass ratios of the competitors as a function 

 of various competitive coefficients and exploita- 

 tion, and the difference in these effects with and 

 without predation on the competitors. "Coef- 

 ficient" values from Menshutkin (1969) were 

 introduced for the coefficients for predation by 

 Species 4 on Species 2 and Species 3 (the same 

 coefficients for both — equal predation). Basical- 

 ly the same coefficients were used for the com- 

 petition of Species 2 and Species 3 as well. 

 Coefficients were held constant except for the 

 one whose effect was being considered. Using 

 such values, the equations were simplified, and 

 in most cases Species 3 was then given the nom- 

 inal value of the competitive variable of interest 

 while the value of that variable for Species 2 was 

 made to vary above and below the nominal. This 

 range of variation of Species 2 was expressed 

 as the ratio coefficient 2/coefficient 3. The same 

 process was performed for the earlier formula- 

 tion without predation (Equation 18) . Thus ra- 

 tios B2/B3 were obtained from both cases — with 

 and without predation. 



A brief examination was made of the effect 

 of various exploitation strategies on the relative 

 stability of two model ecosystems, one with pre- 

 dation and one without predation. These sys- 

 tems are described by Equations (18) and (19), 

 and stability was measured by the change in bio- 

 mass. Figure 6 illustrates the results of various 

 types of exploitation on the two systems. It is 



Figure 6. — Effects of predation and exploitation on mod- 

 el ecosystem stability as measured by biomass ratios. 

 Curve A illustrates a 4-species subweb in which there 

 is no exploitation of the top predator. Curve B illus- 

 trates a 3-species subw^eb with no top predator. Curve C 

 illustrates a 4-species subweb with exploitation of the 

 top predator as well as prey species 2 and 3. All nu- 

 merical values of coefficients are from Menshutkin 

 (1969). The nominal value of f^ was taken as 0.3. 



apparent from an examination of this figure that 

 the most stable conditions examined involved 

 predation as well as exploitation of the predator 

 and the prey species. However, the system in- 

 volving no top predator seemed to be more stable 

 under exploitation of both prey species than the 

 system involving predation, but with no exploi- 

 tation of the top predator. 



For different types of competitive advantage 

 of one species over the other, the effect of pre- 

 dation on biomass ratios may be very different. 

 Figure 7 demonstrates the effect of unequal com- 

 petition in the coefficient d, which relates to the 

 availability of the resource to Species 2 and Spe- 

 cies 3. Without predation, the ratio B2/B3 of 

 biomasses of the competitors is always the same 

 as their d ratio. With predation, the ratio takes 

 the much different form indicated. The values 

 used for the dn/di3 ratio ranged from 2.7 to 0.37. 

 This range of values produces a full range of 

 B2/B3 ratios, from the point where Species 3 

 becomes extinct, to the point where Species 2 be- 

 comes extinct. For the coefficient d, the results 

 are not dependent upon the absolute value of d. 



390 



