A Class Of Reaction Diffusion Problem Description Pepa Model

The stochastic process algebra PEPA is a high-level formal description language, which has enjoyed successful applications in performance modelling of computer and communication systems. In order to model and analyse system biology and biochemical networks, the PEPA language has been further extended into Bio-PEPA. However, the capabilities of PEPA, Bio-PEPA and other discrete state-based modelling formalisms, are challenged by the size and complexity of large scale systems. This problem is usually termed as the state-space explosion problem. Recently, a novel approach—fluid approximation—has been developed to cope with this problem, which results in a set of ordinary differential equations (ODEs) to approximate the underling continuous-time Markov chains (CTMCs).We noticed that when PEPA or Bio-PEPA is being used to described the considered systems such as ecological systems, the spatial content of the environment has been ignored in the corresponding ODE models (as well as the PEPA or Bio-PEPA models), which are formulated in relation to the time evolution of uniform distributions in the habitat. However, in many ecological systems, the species under consideration may disperse spatially as well as evolve in time. In fact, due to resource limitation, individuals usually migrate to regions of lower population density. In recent years, great attention has been paid to the effect of dispersion of population in a bounded habitat, and in this situation the governing equations for the population densities are described by a system of reaction-diffusion equations. In stead of the behaviour of the populations of species considered in the context of CTMCs and ODEs, the evolution of the densities at each point in a region are dealt with in the PDEs context. The motivation of this research is trying to expand the applications of the fluid approximation approach for the PEPA or Bio-PEPA language, as well as contribute to understand the spatial information of a system which can be modelled by CTMCs or stochastic process algebras.Chapter1will present the background of PEPA, including history, development and applied range of PEPA, and the main task of this article. In Chapter2, syntax, operational semantics, and numerical representation for PEPA will be given. In addition, we will introduce two significant approximation approachs—fluid approximation and diffusion approximation.In Chapter3,an extension from the ODEs to a reaction-diffusion system for a simple PEPA model will be provided. Furthermore, some fundamental properties about the reaction-diffusion equations such as the existence and uniqueness, positivity and convergence of the solution will be established.Chapter4will present the corresponding numerical simulations by Matlab to illustrate the main result.Finially some discussions and future work will be given in Chapter5.