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where ρ is the bulk density of tissues used to fabricate a plant and g is gravitational acceleration. Assuming that the bulk density varies little from one plant to another, M α γ-1 sin Φ. Thus, the bending moment and therefore the maximum bending stresses are minimized when Φ = 90° (i.e., a vertically oriented axis).
In terms of spore dispersal, an elementary ballistic model suffices:
where x is the lateral distance of transport, H is the height at which spores are released from the parent plant, UT is the average settling velocity of spores, and Uh is the horizontal wind speed averaged between H and ground level, which is assumed to parabolically diminish from the top to the base of the plant (Okubo and Levin, 1989). Assuming that UT is independent of H and that Uh is proportional to H, the maximum lateral transport distance for spores is proportional to the square of plant height, x α H2, indicating that even a small increase in plant height confers a selective advantage to spore dispersal. However, the number of spores a plant produces is as important to its reproductive success as the distance spores are transported (Niklas, 1986). Thus, the fitness contribution of the number of spores produced per plant must be considered in addition to the fitness contributed by elevating spores above ground level. Assuming that spores are produced at the tips of branches and that the number of spores per sporangium varies little among hypothetical phenotypes, reproductive fitness R is maximized by maximizing both the number n and height of branch tips: R = f(n, H).
The foregoing implicitly assumes that fitness is proportional to E and R and inversely proportional to M. Assuming that each of these three functional obligations contributes equally and independently to fitness, the most parsimonious mathematical expression for the total fitness F of a phenotype is the geometric mean of E and R divided by M—i.e., F = [(E)(R)]1/2M-1. Importantly, the topology of the fitness-landscape for each of the three functional tasks differs because different functional obligations can have different phenotypic requirements. For example, phenotypes that maximize light interception also maximize their bending moments. However, a phenotype with a high fitness in terms of its ability to garner irradiate energy for photosynthesis will have a low fitness in terms of its high probability for mechanical failure. Consequently, when both of these functions are considered simultaneously, the objective of a walk is to optimize E/M. Although E cannot be maximized without maximizing M, some functional obligations reinforce the same morphological solution because their phenotypic require-