Justification of the Pustovoit-Dyakov empirical formula determining the optimal sunflower phenotype by yield

The article uses the results of field experiments by A. B. Dyakov for 1960-1968, carried out at V.S. Pustovoit All-Russian Research Institute of Oil Crops (VNIIMK) under the guidance of Academician V.S. Pustovoit. Particular attention is paid to the results of the study of influence of plant population on sunflower yield and plant growth. To describe the dependence between sunflower yield and the nutrition area of the plant earlier (Academician Pustovoit V.S. and Dyakov A.B., 1967), it was proposed to use an empirical formula that takes into account the area of nutrition (x), the maximum possible yield of the plant (A), the effect of growth factors (c) and parameter (b), which determines the part of the nutritional area of the sunflower plant, which does not directly affect the seed yield. The sunflower yield curves constructed according to the proposed empirical formula for 250 experiments satisfy the condition of a 5% level of significance. The use of the empirical formula made it possible to investigate many practical issues of sunflower breeding and physiology. Taking into account the basic provisions of mathematical biophysics and the mathematical theory of the struggle for existence, an ordinary differential equation of the first order was established to describe the relative rate of change in the yield of a plant (P.F. Verhulst's equation). Based on the solution of this equation (with the corresponding initial conditions), a new rigorous analytical formula for determining the plant yield is obtained, taking into account the factors listed above. The article contains calculations showing good agreement between the data obtained by the empirical formula of Pustovoit-Dyakov and by the new formula: the difference is less than 10%. This formula can be used in the study and solution of various issues of plant physiology (not only sunflower) and in the optimization of agrotechnical measures. The analysis made it possible to strictly prove and substantiate the validity of the empirical formula of Pustovoit-Dyakov, which determines the optimal phenotype of sunflower in terms of yield.