Mathematical and Stochastic Growth Models

Many statistical and mathematical models of growth are developed in the literature and effectively applied to various conditions in the existent world that involve many research problems in the different fields of applied statistics. Nevertheless, still, there is an equally large number of conditions, which have not yet been mathematically or statistically modeled, due to the complex situations or formed models are mathematically or statistically inflexible. The present study is based on mathematical and stochastic growth models. The specification of both the growth models is depicted. A detailed study of newly modified growth models is mentioned. This research will give substantial information on growth models, such as proposed modified exponential growth models and their specifications clearly motioned which gives scope for future research.

On bucket increasing trees, clustered increasing trees and increasing diamonds

In this work we analyse bucket increasing tree families. We introduce two simple stochastic growth processes, generating random bucket increasing trees of size n, complementing the earlier result of Mahmoud and Smythe (1995, Theoret. Comput. Sci.144 221–249.) for bucket recursive trees. On the combinatorial side, we define multilabelled generalisations of the tree families d-ary increasing trees and generalised plane-oriented recursive trees. Additionally, we introduce a clustering process for ordinary increasing trees and relate it to bucket increasing trees. We discuss in detail the bucket size two and present a bijection between such bucket increasing tree families and certain families of graphs called increasing diamonds, providing an explanation for phenomena observed by Bodini et al. (2016, Lect. Notes Comput. Sci.9644 207–219.). Concerning structural properties of bucket increasing trees, we analyse the tree parameter $K_n$ . It counts the initial bucket size of the node containing label n in a tree of size n and is closely related to the distribution of node types. Additionally, we analyse the parameters descendants of label j and degree of the bucket containing label j, providing distributional decompositions, complementing and extending earlier results (Kuba and Panholzer (2010), Theoret. Comput. Sci.411(34–36) 3255–3273.).

Counting individual bacteria during high-throughput growth in microfluidic droplets

Bacterial growth in microfluidic droplets is relevant in biotechnology, in microbial ecology, and in understanding stochastic population dynamics in small populations. However, it has proved challenging to automate measurement of absolute bacterial numbers within droplets, forcing the use of proxy measures for population size. Here we present a microfluidic device and imaging protocol that allows high-resolution imaging of thousands of droplets, such that individual bacteria stay in the focal plane and can be counted automatically. We demonstrate the use of this approach to track stochastic growth of multiple replicate Escherichia coli populations within droplets.

Achieving functional neuronal dendrite structure through sequential stochastic growth and retraction

Class I ventral posterior dendritic arborisation (c1vpda) proprioceptive sensory neurons respond to contractions in the Drosophila larval body wall during crawling. Their dendritic branches run along the direction of contraction, possibly a functional requirement to maximise membrane curvature during crawling contractions. Although the molecular machinery of dendritic patterning in c1vpda has been extensively studied, the process leading to the precise elaboration of their comb-like shapes remains elusive. Here, to link dendrite shape with its proprioceptive role, we performed long-term, non-invasive, in vivo time-lapse imaging of c1vpda embryonic and larval morphogenesis to reveal a sequence of differentiation stages. We combined computer models and dendritic branch dynamics tracking to propose that distinct sequential phases of stochastic growth and retraction achieve efficient dendritic trees both in terms of wire and function. Our study shows how dendrite growth balances structure–function requirements, shedding new light on general principles of self-organisation in functionally specialised dendrites.