On the Solutions of Time-fractional Bacterial Chemotaxis in a Diffusion Gradient Chamber

The study of pattern formation in bacterial colonies is of particular interest both from the biological and physical points of view [1]. Growth and development of bacteria population shows a great variety of geometrical shapes. Adler [2-3] was the first to obtain bacterial population waves in the form of concentric circle. Golding et al. [4] are interested in the opposite case, where nutrient supply is limited. For a more realistic description of colonial development on a nutrient-poor surface, they take into account the interaction of bacteria with the nutrient fieldn(x, t). Chemotaxis means changes in the movement of the cell in response to a gradient of certain chemical fields [5-8]. The movement is biased along the gradient either in the gradient direction or in the opposite direction. Miyata and Sasaki [9] estimate the distance between spots generated by the bacteria colony model with chemotactic activity. A diffusion gradient chamber (DGC) is a bacterial growth device that can be used to study chemotaxis and other biological properties [10]. A DGC is square in shape and has a reservoir on each side. Our mathematical model of this device describes the motion of three chemical species: the cell population, the chemoattractant and the nutrient. It also describes the interaction between these species. Details on construction of the DGC and description of the biological experiments using DGCs are given in [10] and [11]. Our mathematical model of this device describes the motion of three chemical species: the cell population, the chemoattractant and the nutrient. It also describes the interaction between these species. Let u(x, t), g(x, t), s(x, t) and q(x, t) be the concentrations of cells, nutrient, chemoattractant substrate and oxygen. Then u, g, sand q are in general governed by the following equations: