Membrane computing as a modeling tool for discrete systems.(Report)

INTRODUCTION

Membrane computing (Paun, 1998) is an area of computer science that abstract computing ideas and models from the structure and the functioning of living cells. This mechanism provides a platform for modeling discrete systems in which a membrane delimits a compartment from its external environment and provides local environment that regulates specific processes.

The processes evolve in parallel and non-deterministic way in which all evolution rules are simultaneously applied to all the objects. The computation halts to produce output when no rule is applied. The discrete characteristics of membrane computing allow the dynamic systems evolve in discrete steps according to the processes.

However, some of the discrete systems have been represented in Ordinary Differential Equation (ODE) (Blanchard et al., 2006) which has continuous and deterministic evolution strategy. This approach has shown limitations when the variation of concentration of an object is modeled as continuous and deterministic manner, which ignores the behaviors of discrete systems itself (Jong, 2002). Membrane computing has been identified as an alternative to address these limitations.

Prey-Predator population (Jones et al., 2003) is a discrete system that has been modeled in ODE. The same model can be represented in membrane computing by using rewriting rules. The model is simulated with membrane computing simulation strategy based on Gillespie algorithms (Gillespie, 2001; Muniyandi and Abdullah, 2010). The properties of Prey Predator population are verified with Probabilistic Model Checker (PRISM) (Kwiatkowska et al., 2002; Muniyandi et al., 2010). The results are compared with ODE approach to certify the capability of membrane computing in modeling discrete systems.

Prey-predator population: The Prey- Predator model is a biological system that describes the dynamics of Prey-Predator population. In this model, the food supply of prey species is assumed to be abundant and no threat to its growth. Meanwhile, the only food supply of predator species is the prey to determine its growth. The interaction between prey and predator is to maintain the equilibrium of Prey- Predator population over time. The prey species could grow exponentially with its unlimited food supply but the predator species act to counterbalance the prey growth rate. Therefore, two assumptions formulated for this model to maintain the equilibrium of the population. First, the size of the prey and predator population is related to the rate at which predator encountering prey. Second, the predator has to lead to natural death which is related to a rate of fixed proportion.

Based on the assumptions, the rules in the prey predator model are interpreted. First the rule over the prey in which the change in the number of prey is specified by its own growth minus the rate at which it is preyed upon. Second, the rule over predator signifies the growth of the predator population in which its growth is not necessarily equal to the rate at which it consumes the prey but there is another rule of exponential decay to represent the natural death of the predator.

The ODE model of prey-predator population: The ODE of Prey-Predator model is represented by a pair of first order, non-linear, differential equations (Jones et al., 2003). The interactions between prey and predator determine the number of prey and predator at certain time …