Theoretical studies on the necessary number of components in mixtures - 1. Number of components and yield stability

Abstract

Theoretical studies on the optimal numbers of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to phenotypic yield stability (measured by the parameter 'variance'). For each component i, i = 1, 2,..., n, a parameter ui with 0 ≦ ui ≦ 1 has been introduced reflecting the different survival and yielding ability of the components. For the stochastic analysis the mean of each ui is denoted by u1 and its variance by σi2For the character 'total yield' the phenotypic variance V can be explicitly expressed dependent on 1) the number n of components in the mixture, 2) the mean {Mathematical expression} of the σi23) the variance of the σi24) the ratio {Mathematical expression} and 5) the ratio σi2/χ2 where χ denotes the mean of the ui and σu2is the variance of the uj. According to the dependence of the phenotypic stability on these factors some conclusions can be easily derived from this V-formula. Furthermore, two different approaches for a calculation of necessary or optimal numbers of components using the phenotypic variance V are discussed: A. Determination of 'optimal' numbers in the sense that a continued increase of the number of components brings about no further significant effect according to stability. B. A reduction of b % of the number of components but nevertheless an unchanged stability can be realized by an increase of the mean χ of the ui by 1% (with {Mathematical expression} and σu2assumed to be unchanged). Numerical results on n (from A) and 1 (from B) are given. Computing the coefficient of variation v for the character 'total yield' and solving for the number n of components one obtains an explicit expression for n dependent on v and the factors 2.-5. mentioned above. In the special case of equal variances, σi2= σo2for each i, the number n depends on v, x = (σ0/χ)2 and y = (σu/χ)2. Detailed numerical results for n = n (v, x, y) are given. For x ≦ 1 and y ≦ 1 one obtains n = 9, 20 and 79 for v = 0.30, 0.20 and 0.10, respectively while for x ≦ 1 and arbitrary y-values the results are n = 11, 24 and 95. © 1985 Springer-Verlag.