Amount of Food Protein and the Accumulation of Body Protein in Fish:III. Economical Amount of Food Protein
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It has been shown that we have the following relation between the increase of body protein p and the feeding amount of protein x p=e-a-bx(x-x0), where x0 is the amount of food protein equivalent to the maintenance protein and a and b are constants.* Let the unit price of fish protein be h and that of food protein k, then the gain by culture Y will be Y=he-a-bx(x-x0)-kx. Putting h/k=m and Y/k=y, we have y=me-a-bx(x-x0)-x. (1) Let xE be the most economical amount of food protein for a given value of m, then y will take the maximum va ue when x=xE. Since, however, x lies always between x0 and x0+1/b (the amount of food protein which produces the maximum growth*) and dy/dx=me-a-bx( ?? +bx0-bx)-1 d2y/dr2=mbe-a-bx(bx-bx0-2), we have the following relation between m and xE, m=e ?? +bxE/1+bx0-bxE (2) In the equation (2), we observe that XE varies parallel with m; in other words, the higher is the price of fish protein the more profitable is it to increase the feeding amount of protein. If the price of food protein is neg igible as compared with the price of fish protein, the econom-ical amount of food protein becomes x0+1/b, agreeing with the feeding amount for the maximum growth. Since xE denotes only the most economical (inc ?? uding the case of the minimum loss) amount of food protein for a given value of m, it depends quite on the value of m whether one can make or not a profit with economical feeding amount. In order to solve this question we must find the minimum m at the point of economical balance. Since at this point y=0, we have m=xea+bx/x-x0 (3) The value of m in this equation is minimum when x=bx0+√b2x02+4bx0/2b, and takes the following value mm ?? n=(1+B)e-a+B, where B=bx0+√b2x02+4bx0/2 Thus in order that the fish culture can be in commercial profit, the price of fish protein should be at least (1+B)e-a+B times as much as the price of food protein, or the price of fish should be at least r/s(1+B)e-a+B times as much as the price of feedstuff, where r and s are the percentages of protein in fish and feedstuff respectively. The value of the minimum m can be also found by solving (2) and (3) simultaneously. The maximum profit ymax for a given value of m can be found by putting the corre-sponding value of xE into (1) ymax=xE-x0/1+bx0-bxE (4) From (4) and (2) we can observe that the relation between the maximum profit and m is expressed by a concave increasing curve, which shows that the profit by fish culture is far greater than proportional to the value of m.
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