Stochastic Ontogenetic Allometry: The Statistical Dynamics of Relative Growth

BACKGROUND: In the absence of stochasticity, allometric growth throughout ontogeny is axiomatically described by the logarithm-transformed power-law model, [Image: see text], where [Image: see text] and [Image: see text] are the logarithmic sizes of two traits at any given time t. Realistically, however, stochasticity is an inherent property of ontogenetic allometry. Due to the inherent stochasticity in both [Image: see text] and [Image: see text], the ontogenetic allometry coefficients, [Image: see text] and k, can vary with t and have intricate temporal distributions that are governed by the central and mixed moments of the random ontogenetic growth functions, [Image: see text] and [Image: see text]. Unfortunately, there is no probabilistic model for analyzing these informative ontogenetic statistical moments. METHODOLOGY/PRINCIPAL FINDINGS: This study treats [Image: see text] and [Image: see text] as correlated stochastic processes to formulate the exact probabilistic version of each of the ontogenetic allometry coefficients. In particular, the statistical dynamics of relative growth is addressed by analyzing the allometric growth factors that affect the temporal distribution of the probabilistic version of the relative growth rate, [Image: see text], where [Image: see text] is the expected value of the ratio of stochastic [Image: see text] to stochastic [Image: see text], and [Image: see text] and [Image: see text] are the numerator and the denominator of [Image: see text], respectively. These allometric growth factors, which provide important insight into ontogenetic allometry but appear only when stochasticity is introduced, describe the central and mixed moments of [Image: see text] and [Image: see text] as differentiable real-valued functions of t. CONCLUSIONS/SIGNIFICANCE: Failure to account for the inherent stochasticity in both [Image: see text] and [Image: see text] leads not only to the miscalculation of k, but also to the omission of all of the informative ontogenetic statistical moments that affect the size of traits and the timing and rate of development of traits. Furthermore, even though the stochastic process [Image: see text] and the stochastic process [Image: see text] are linearly related, k can vary with t.