Theoretical studies on the necessary number of components in mixtures

Summary Theoretical studies on the necessary numbers of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to risk considerations — using the binomial distribution and the Polya-distribution. The risk r of a mixture has been defined as the probability of catastrophic losses (catastrophe = decrease of productivity of q% or more by susceptibilities of the components). Using 1) the binomial distribution and 2) its generalization, the Polya-distribution, and several simplifying assumptions, the risks r = r (x, a, q, n) have been calculated numerically (n = number of components in the mixture, a = parameter for the intensity of contagion and dispersion of susceptibilities (for example: diseases and epidemics), x = probability of susceptibility). The Polya-model reduces to the binomial case if a = 0. The main results are: 1. For each number n of components the risk r decreases markedly with decreasing x (for each q and for each a). 2. For x < = q the risk r decreases with an increasing number n of components (for each a). 3. For each number n of components and x and q with x < q the risk r increases with increasing a. 4. For given q, x and a the functions r = r(n) are asymptotic for larger numbers n of components with n > n^*. In spite of further increasing numbers of components in the mixture the risk remains almost constant. For all situations, where the risk decreases with increasing n these numbers n^*, therefore, can be considered as necessary numbers of components in mixtures, n^* depends on q, x and a. Nevertheless, a global and rough conclusion can be formulated: In many situations one obtains necessary numbers of 30–40 components for a0 and 20–30 components for a = 0.