Stochastic ontogenetic allometry: The statistical dynamics of relative growth

Abstract

Background: In the absence of stochasticity, allometric growth throughout ontogeny is axiomatically described by the logarithm-transformed power-law model, θ t = log ab+kφ t, where θ t ≡ θ(t) and φ t ≡ φ(t) 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 θ t and φ t, the ontogenetic allometry coefficients, log ab 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, θ t and φ t. Unfortunately, there is no probabilistic model for analyzing these informative ontogenetic statistical moments. Methodology/Principal Findings: This study treats θ t and φ t 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, k ≡ D t(u〈Ω t〉)/D t(v〈Ω t〉), where 〈Ω t〉 is the expected value of the ratio of stochastic θ t to stochastic φ t, and u〈Ω t〉 and v〈Ω t〉 are the numerator and the denominator of 〈Ω t〉, 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 θ t and φ t as differentiable real-valued functions of t. Conclusions/Significance: Failure to account for the inherent stochasticity in both θ t and φ t 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 θ t and the stochastic process φ t are linearly related, k can vary with t. © 2011 Anthony Papadopoulos.