A Mathematical Approach to the Prediction of Differences in the Relative Ability of Races of Puccinia Graminis Tritici to Survive When Mixed

Abstract

A model describing the population trends for any number of races of rust in the urediospore stage during a single growing season is presented. The general case for the field is subsequently modified to describe trends in a population of rust races under constant conditions. A term which gives an estimate of the relative survival ability of each race in the population is finally obtained. This term allows the trends in the composition of the mixture to be predicted. The theoretical proportion of strain 21 Anz-2,3,7 in amixture of strains 21 Anz-2,7 and 21 Anz-2,3,7 of P. graminis trifici over a number of generations was compared with the proportion of strain 21 Anz-2,3,7 in an experimental population of the same two strains over the same period. The model was found to be a good first approximation to the results obtained with the experimental population.

Relative survival ability in a polycyclic system of Puccinia coronata avenae monosporic isolates in biracial mixtures on universal oat suscepts

“…After m cycles or generations, the proportion of race 1, and that of race 2,7^=1-?^, can be calculated. Ogle et al (1973) found a very good fit of the model to data obtained by Ogle and Brown (1971) on components of reproductive rates for two strains of P. graminis tritici isolated from the Australia-New…”

“…Strain interactions that change sporulation rates should effect deviations of observed survival ability from the expected. There fore, the good fit of the model (Ogle et , 1973) to data from two P. graminis tritici strains from Australia (Ogle and Brown, 1971) negates the likelihood of such interactions between the two strains.…”

“…Explanations of instability under laboratory conditions by differ ences in rates of reproduction are elegantly epitomized in a mathematical model developed by Ogle et (1973) to predict results of racial mix tures in the wheat stss rust fungus. An index representing an increment in the number of uredia during a certain period, is calculated for race i.…”

“…The predominance of one isolate in mixed populations expresses higher rates of increase of that isolate relative to other components in the pop ulation (Cournoyer, 1970;Leonard, 1969b;Ogle et al, 1973;Vanderplank, 1968). Thus, it is likely that isolates of race 290 in experiment 2, of race 264B in experiment 3, and of race 264A in experiment 5 have higher rates of increase than the isolates that declined in their mixtures.…”

“…Tliis anphasis is evident in models (Cournoyer, 1970;Leonard, 1969b;Ogle et al, 1973;Vanderplank, 1968) that describe unstable equilibria in mixed rust populations as expressions of differences in rates of growth, infection, and/or reproduction of components of populations.…”