The forest of mutations associated to a multitype branching forest is obtained by merging together all vertices in each of its clusters and by preserving connections between them. (Here, by cluster, we mean a maximal connected component of the forest in which all vertices have the same type.) We first show that the forest of mutations of any multitype branching forest is itself a branching forest. Then we give its progeny distribution and we describe some of its crucial properties in terms of the initial progeny distribution. We also obtain the limiting behaviour of the number of mutations both when the total number of individuals tends to ∞ and when the number of roots tends to ∞. The continuous-time case is then investigated by considering multitype branching forests with edge lengths. When mutations are nonreversible, we give a representation of their emergence times which allows us to describe the asymptotic behaviour of the latter, under certain conditions on the mutation rates. These results have potential relevance for emergence of mutations in population cells, particularly for genetic evolution of cancer or development of infectious diseases.

Research was conducted to determine the optimum population and row spacing for corn production and for suppressing velvetleaf growth and seed production. Corn was grown in a factorial arrangement of three populations targeted at 64,000 (1 ×), 96,000 (1.5 ×), or 128,000 (2 ×) plants ha−1 and two row spacings of 38 or 76 cm. Influences on corn were determined in weed-free plots, and influences on velvetleaf were determined for target plants established at 1.5–m intervals along the center of corn interrows. Four velvetleaf plantings were made at weekly intervals beginning at corn planting. Corn row spacing had little influence on corn or velvetleaf. Corn yield exhibited a parabolic response to population with a maximum of approximately 90,000 plants ha−1 in one year, no response to population in another year, and a linear decline with increasing population in a dry year. Velvetleaf seed production was reduced 69 to 94% by the 1.5 × population and 99% by the 2 × population compared to the standard 1 × population when velvetleaf emerged with corn. Velvetleaf seed production was eliminated when velvetleaf emerged at or later than corn leaf stages 3, 5, and 6 for corn populations of 2 ×, 1.5 ×, and 1 ×, respectively. Reduced velvetleaf seed production was correlated with lower positioning of plants in the corn canopy and reduced light availability. Results suggest that higher corn populations could aid integrated weed management strategies by reducing seed production and limiting the build-up of weed populations.

Snap beans are a common processing vegetable whose yield and quality can be reduced when a few weeds emerge with or soon after the crop. The effect of redroot pigweed's emergence time and density on snap bean growth and yield was studied. Redroot pigweed, at four densities, was seeded with snap beans (early) or at the first trifoliate leaf stage (late). In 1998 the yield loss at 8 redroot pigweed plants m−1 row was 42 and 58%, whereas in 1999 it was 39 and 48% for late- and early-planted redroot pigweed, respectively. The effect of redroot pigweed density on snap bean yield loss was predicted with the hyperbolic yield equation. Coefficient A (percent yield loss as weed density approaches infinity), determined from the hyperbolic equation, varied from 47 to 63% and coefficient I (percent yield loss as weed density approaches zero) varied from 10 to 32% depending on the year and time of weed emergence, with the greater values for early-emerging redroot pigweed. Snap bean pod number and biomass were reduced as the density of early-emerging redroot pigweed increased. Regardless of the density, late-emerging redroot pigweed had less effect on snap bean growth and yield than did early-emerging redroot pigweed. The hyperbolic yield loss equation may be useful for growers to predict the effect of redroot pigweed in their fields on snap bean yields.