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Turning attention to the topology of the fitness-landscape, the functional obligations assuring growth, survival, and reproduction must be known and quantified. For early vascular land plants, these obligations can be inferred from living tracheophytes and undoubtedly include the requirement to intercept sunlight, to mechanically sustain the weight of aerial organs, and to be able to produce and disperse diaspores some distance from parental plants. Fortunately, each of these tasks can be quantified by means of closed-form equations derived from biophysics or biomechanics. For example, the efficiency E of a computer-simulated phenotype to intercept solar irradiance (of intensity I measured perpendicular to its surfaces) is given by the formula
where Sp is the total unshaded surface area of the phenotype projected toward incident light, S is the total surface area of the phenotype, and Θ is the incident solar angle, which varies between 0° and 180° in each diurnal cycle (Niklas and Kerchner, 1984). Because the magnitude of I is independent of Θ (assuming atmospheric conditions are clear), Eq. 1a reduces to
Although the total projected area Sp of each morphology varies as a function of Θ, it also depends upon the orientation of axes in addition to the extent to which neighboring axes shade one another. All of these variables can be dealt with by even the most simple computer.
In terms of mechanical stability, the maximum bending stresses σmax that develop in a cylindrical plant axis may be computed from the formula
where M is the bending moment, which has units of force times length, and + and - denote tensile and compressive stresses, respectively (Niklas, 1992). For any value of d, the bending stresses are directly proportional to the bending moment that, when expressed in terms of Φ and γ, is given by the formula