And Bio-PEPA Model Of A Class-related Reaction Diffusion Problems

The stochastic process algebra PEPA is a formal description technique for the study of the behavior of concurrent systems by algebraic means. It has enjoyed successful applications in performance modeling of computer and communication systems. With the development of science and technology, PEPA and its semantics have recently been extended to model biological systems. So a new language Bio-PEPA is produced. It is the modification to PEPA, which can describe biochemical systems preferably. In order to cope with massive quantities of processes (as is usually the case when considering biological reactions), the model is interpreted in terms of a small set of coupled ordinary differential equations (ODES) instead of a large state space continuous time Markov chain (CTMCS).The article will introduce a special Bio-PEPA model. First five ordinary differential equations are derived from it.On this basis, we not only consider the behavior of solution with time, but also consider the impact of the changing of the spatial location on the solution of the equations. Thus the diffusion is introduced and we can derive five partial differential equations. This article mainly deals with the properties of the reaction-diffusion system.Chapter1introduces the origin of the problem, includes PEPA、Bio-PEPA、and the main task of the paper.In Chapter2, firstly, we give some instructions about the Bio-PEPA language, includes biochemical networks, the syntax and the semantics, and instructions from biochemical networks to Bio-PEPA. Then a Bio-PEPA model is given and ordinary differential equations and corresponding partial differential equations are established.Chapter3is devoted to the reaction-diffusion problems. Section1gives the existence and uniqueness of the equilibrium solution of the reaction-diffusion system. In section2we will turn the five equations into three equations, which are equivalent to the preceding problem in some certain conditions. Then we prove the local stability of the equilibrium solution of the reaction-diffusion system. Section3deals with the simplified problem, the global stability of the equilibrium solution of the simplified problem is given by using the method of upper and lower solutions. Our results show that the equilibrium solution of the simplified question tends to a constant.In Chapter4, the numerical simulations of the stability of the solutions are given by using Matlab to illustrate our theoretical results.Chapter5contains a summary of the article and the outlook for the future work.