Based on allometric theory and scaling laws, numerous mathematical models have been proposed to study ontogenetic growth patterns of animals. Although deterministic models have provided valuable insight into growth dynamics, animal growth often deviates from strict deterministic patterns due to stochastic factors such as genetic variation and environmental fluctuations. In this study, we extend a general model for ontogenetic growth proposed by West et al. to stochastic models for ontogenetic growth by incorporating stochasticity using white noise. According to data variance fitting for stochasticity, we propose two stochastic models for ontogenetic growth, one is for determinate growth and one is for indeterminate growth. To develop a universal stochastic process for ontogenetic growth across diverse species, we approximate stochastic trajectories of two stochastic models and apply random time change and obtain a geometric Brownian motion with a multiplier of an exponential time factor. We conduct detailed mathematical analysis and numerical analysis for our stochastic models. Our stochastic models predict well not only average growth, but also variations of growth within species. This stochastic framework may be extended to studies of other growth phenomena.

Influence lines are indispensable tools for visualizing and analyzing the dynamic variations in force factors induced by external loads within structural systems. Among these methods, the energetic approach stands as a widely employed technique, rooted in the fundamental principle of work done by changing forces. It enables engineers to transform intricate structural analysis problems into manageable ones by exploring the first derivatives of the radius vector, which represent infinitesimal velocity or displacement. This methodology seamlessly interweaves concepts such as carrier motion, relative motion, and the construction of mechanisms, bringing fresh perspectives to the analysis of influence lines. In this article, we explore the nuances of these novel methods within the domain of mechanism theory. Through comprehensive elaboration and analysis, we elucidate the underlying principles and practical applications of Jacobian contours. Crucially, we introduce a straightforward, rapid, and programmable approach, promising to revolutionize influence line determination in structural engineering. This method bridges the gap between theory and practice, offering the potential to elevate the accuracy, efficiency, and adaptability of influence line analysis. As such, it represents a significant advancement in the field of structural and applied mechanics, with broad-reaching implications for engineering practice.

Brassica napus is an important vegetable and oil crop worldwide. The research is meaningful for yield and plant architecture in B. napus. In this study, one natural mutant (MT) line with determinate and capitulum-like inflorescence was chosen for further study. The paraffin sectioning detected that the inflorescence apex began to split and developed into one terminal flower, starting from five-leaf stage. Genetic analysis indicated that the segregation patterns of inflorescences in the F2 populations supported a digenic inheritance model, which was further approved by BSA-Seq technique. The BSA-Seq method detected two QTL region on C02 (14.27-18.41 Mb) and C06 (32.98-33.68 Mb) for the genetic control of determinate inflorescences in MT plants. In addition, the expression profile in MT compared with WT was analysized, and a total of 133 candidate genes for regulating the flower development (75 genes, 56.4%), shoot meristem development (29 genes, 21.8%) and inflorescence meristem development (13 genes, 9.8%) were identified. Then one joint analysis combing BSA-Seq and RNA-Seq identified two candidate gene of BncTFL1 and BncAP1 for regulating the MT phenotype. Besides, the potential utilization of the MT plants was also discussed.