Concentration Phenomena of a Semilinear Elliptic Equation with Large Advection in an Ecological Model

We consider a reaction-diffusion-advection equation arising from a biological model of migrating species. The qualitative properties of the globally attracting solution are studied and in some cases the limiting profile is determined. In particular, a conjecture of Cantrell, Cosner and Lou on concentration phenomena is resolved under mild conditions. Applications to a related parabolic competition system is also discussed

Identify successful marketing communication strategies that apply to a small hair salon

A selected organisation provides hair dressing services and hair products to customers. The aim of this report is to analyse how the small hair salon could improve their marketing communication strategies in order to attract more customers and enhance the relationship between customers and the organisation. The approach to collecting information was to use a questionnaire with 50 participants, to gather primary information and to conduct a secondary research study. The result of this research was to decide that the role of a successful marketing communication strategy is to attract the customer to consume. In order to make the marketing communication strategy successful, it needs to choose a suitable channel that enables it to connect with the customer. New media is an effective channel that can promote the business to the customer and interact with them. New media is also suitable for a small business to use. A recommendation for the organisation is they create their own website page, Facebook page, YouTube video and WeChat group to promote themselves and interact with customers. Those channels are popular in New Zealand, with a high number of active users. Most the organisation customers like to use those channels too, so if the organisation applies those channels to their marketing communication strategy they will be able to attract customers and persuade them to consume more products

Holomorphic vector fields and quadratic differentials on planar triangular meshes

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a M\"obius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gau{\ss} map and prescribed Hopf differential.Comment: 17 pages; final version, to appear in "Advances in Discrete Differential Geometry", ed. A. I. Bobenko, Springer, 2016; references adde

Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented