Department of Biological Sciences, Texas Tech University, Lubbock, Texas, United States of America.
PLoS One. 2011;6(9):e25267. doi: 10.1371/journal.pone.0025267. Epub 2011 Sep 23.
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.
METHODOLOGY/PRINCIPAL FINDINGS: 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.
CONCLUSIONS/SIGNIFICANCE: 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.
在没有随机性的情况下，整个个体发育时期的异速生长可以用对数变换幂律模型来描述，θt=log(a)b+kφ(t)，其中θt≡θ(t)和φt≡φ(t)是任意给定时间 t 时两个特征的对数大小。然而，实际上，随机性是个体发育异速生长的固有属性。由于 θt 和 φt 都具有固有随机性，因此，个体发育异速生长系数 log(a)b 和 k 可以随 t 变化，并且具有由随机个体发育生长函数 θt 和 φt 的中心和混合矩决定的复杂时间分布。不幸的是，目前还没有用于分析这些信息丰富的个体发育统计矩的概率模型。
方法/主要发现：本研究将 θt 和 φt 视为相关的随机过程，以制定每个个体发育异速生长系数的精确概率版本。具体来说，通过分析影响概率版本相对增长率（k≡Dt(u<Ωt>)/Dt(v<Ωt>)）的时间分布的异速生长因子，来解决相对生长的统计动力学问题，其中<Ωt>是随机 θt 与随机 φt 的比值的期望，u<Ωt>和 v<Ωt>分别是<Ωt>的分子和分母。这些异速生长因子提供了对个体发育异速生长的重要见解，但只有在引入随机性时才会出现，它们将 θt 和 φt 的中心和混合矩描述为 t 的可微实值函数。
结论/意义：如果不考虑 θt 和 φt 中的固有随机性，不仅会导致 k 的计算错误，还会忽略影响特征大小以及特征发育时间和速度的所有信息丰富的个体发育统计矩。此外，即使随机过程 θt 和随机过程 φt 是线性相关的，k 也可以随 t 变化。
