Theoretical studies on the necessary number of components in mixtures : 3. Number of components and risk...

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 a≠0 and 20-30 components for a = 0.