Modelling And Simulation Of Filter Adaptation By Daphnia

Abstract

KEYWORDSmathematical model of ecosystem, stability analysis, feeding adaptation of Daphnia, optimal control problem, adaptive critic and recurrent neural network, numerical simulation ABSTRACT A model of feeding adaptation of a filter feeder in the framework of a pelagic ecosystem is constructed by modifying the simplified ecosystem model AQUAMOD for the presence of several different sized species of algae. Stability analysis of equilibria and some numerical simulation is given. It is shown that Hopf bifurcation may occur depending on filtration rate. Also the assumption that filtration adaptability represents optimization type process is incorporated. Two possible strategies were followed: an instantaneous optimality at each time interval and an integral formulation, maximization of the integral biomass. The optimal control problem is transcribed into nonlinear programming problem, which is implemented with adaptive critic neural network and recurrent neural network for solving nonlinear projection equations. Results show that adaptive critic based systematic approach and neural network solving of nonlinear equations hold promise for obtaining the optimal control with control and state constraints. MODEL DESCRIPTIONIn this paper we consider a simplified ecosystem model AQUAMOD for the presence of several different sized species of algae. The model consists of phosphorus (x 1 ) as a limiting nutrient for growth of four species of different sized algae (x 2 − x 5 ) and zooplankton (x 6 ). Similar models of n species of microorganisms competing exploitatively for one, two or more growth-limiting nutrients are used to study continuous culture of microorganisms in chemostat under constant condition (Smith et al., 1995) without any predators. The model is described by the following system of ordinary differential equation (1)