On Baranov's Paper

Abstract

About thirty years ago T. B_ARANOV developed a theory of exploitation of fish stock, entitled “On the question of the biological basis of fisheries”^*. Recently Prof. W. E. R_ICKER translated this paper into English. Denoting by the letter n the abundance of any group of fish and by the letter t time, he assumed that the decrease dn in the number of fish in the small time dt is proportional to the abundance of that groun and put dn/dt=-k₁n…… (1). Following to the results of Heincke he assumed that the increase of length of fish is proportional to time (t) and put t=rl…… (3). From equations (1) and (3), after integrating, we have the relation n=n₀e^-kl…… (4), where the coefficient k=k₁r is called the coefficient of decrease.
In Japan, Spring Herring are caught in the Hokkaido District from 3 age group to 8 age group. Since 1910 Hokkaido Fishery Experiment Station has published the statistcs of the catch of herring for each year and the age composition of herring which calculated from the scales. The growth of herring of each age was determined by K_AWAT using the scales of age 8 herrings, Tsuda had published the wean body weight of herring of each age. From these data I have made Table 1, from which we know that the increase of body length in each age is not constant and growth curve will be represented more correctly by the Logistic curve. From Table 1, as the number of fish is decreasing exponentially after 5 age, we can obtain the following values of constant. M=33.33, a=0.246, b=0.448, k=1.12, αk=0.307…(^*). From equation. (13) we obtain the decreasing rate ψ=1-a^k…… (16), where α=(M-l₂)/(M-l₁) and l₁ and l₂ are body length for age t₁ and t₂. Table 1 show that body length of age 3 is l₁ ?? 24cm. and that of age 4 is l₂ ?? 27cm, hence α=0.675 and ?? =0.355. Similarly from age 2 to age 3, ?? =0.33.
The number R of fish of commercial size (that is, fish whose length is greater than L ) is given by the integral (17) and (18), where L/M=β, 0<β<1. The whole weight of the fish population of commercial size is given by the integral (19), where we assume that the body weight of a fish is given by the formula p=wl³, (w=constant). Mean body weight is given by the formula (21) and q_k is given by (22). These formula are more complicated than the original ones which were obtained under simple assumptions. For nonintcgral values of k, the numerical value of q_k can not be obtained easily. For the herring the mean body weight is 79.09 monme (1 monme=3.75gr.). If we take 45 monme for 3 age or 60 monme for 4 age as L, then q=1.76-1.32. On the other hand β=0.75-0.81. If we make the table of q_k for each valne of β and k, then we shall find that for the above mentioned value of β and q, the value of k is about 1.1 and this is nearly equall to the value reported^*.