Stochastic ontogenetic allometry: the statistical dynamics of relative growth

In the absence of stochasticity, allometric growth throughout ontogeny is axiomatically described by the logarithm-transformed power-law model, θt = log(a) b + kφ(t), where θt ≡ θ(t) and φt ≡ 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 θt and φt, the ontogenetic allometry coefficients, log(a) b 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, θt and φt. Unfortunately, there is no probabilistic model for analyzing these informative ontogenetic statistical moments.

This study treats θt and φt 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 ≡ Dt(u<Ωt>)/Dt(v<Ωt>), where <Ωt> is the expected value of the ratio of stochastic θt to stochastic φt, and u<Ωt> and v<Ωt> are the numerator and the denominator of <Ωt>, 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 θt and φt as differentiable real-valued functions of t.

Failure to account for the inherent stochasticity in both θt and φt 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 θt and the stochastic process φt are linearly related, k can vary with t.