The population genetics of the potato cyst‐nematode, Heterodera rostochiensis: mathematical models to simulate the effects of growing eelworm ‐resistant potatoes bred from Solatium tuberosum ssp. andigena

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Abstract
SUMMARY: To follow population changes when potato varieties with resistance to Heterodera rostochiensis derived from Solanum tuberosum ssp. andigena were grown on infested land, a computer programme was written including three mathematical relationships: (1) a law relating multiplication to pre‐cropping density; (2) two mathematical models of inheritance of ability to overcome resistance; (3) a law relating the proportion of larvae able to become female to pre‐cropping population density. The programme also included four parameters: (1) the maximum possible reproductive rate; (2) the fraction of the population (eggs) not participating in reproduction when potatoes are grown and carried over to the following year unchanged; (3) the fraction carried over annually when other crops are grown; (4) the frequency of larvae able to become female in the population initially. Population density was measured relative to the equilibrium density and was therefore independent of the units in which density is usually measured.After supplying a range of parameters for all the above to include those likely to be encountered in practice, the changes expected (a) in the frequency of larvae able to become female in the roots of resistant varieties and (b) in population density were computed for resistant varieties grown continuously or alternately with susceptible varieties in crop rotations of different lengths. Because well established field populations are relatively dense, observed reproductive rates are small and rarely approach the maximum possible. Reproductive rate is therefore a relatively unimportant determinant of genetic change. The fraction of the population carried over to the following year is more important because it affects the length of a crop rotation necessary to make loss of potato yield acceptable, determines what the multiplication rate will be and influences the speed of genetic change by providing a reservoir of initial type males which backcross with any genetically different females that may develop on the roots of resistant plants.No experiment seems to have been done specifically to determine the parameters needed to calculate population changes. Some values can be obtained from the literature but mostly they must be guessed.When the law relating the proportion of larvae able to become female to pre‐cropping population density was included in the computations, it had little effect initially but later, after several generations, it delayed genetic change.Two field experiments, one by Huijsman (1961) another by Williams (1958) and Cole & Howard (1962 a), provide some of the variables needed to compute trends in population density. Best fitting variables were computed for the data in these experiments by the method of maximum likelihood. The computed parameters for one experiment were not very realistic but those for the other were in line with what would be expected in practice and tended to favour the hypothesis that larvae able to become female in the roots of resistant plants are double recessives (aa).The computations lead us to suggest that the best policy for potato growers who have fields suitable for resistant varieties is to alternate resistant with susceptible varieties in a crop rotation containing potatoes every 3 or 4 years.