QUALITATIVE ANALYSIS OF MICRO-ORGANISM POOL AND ITS INVERTEBRATE PREDATOR WITH DISCRETE TIME-DELAY IN SUNDARBAN ESTUARY, INDIA

摘要

In the present study, we consider a deterministic mathematical model of a non-spatial detritus-based micro organisms and its invertebrate predator in the estuarine environment of Sundarban. The model initiates the dynamics of interaction between the micro-organism pool feeds on detritus and the invertebrate predators. Here we consider that the micro-organism pool follows logistic type growth and the predation follows Ivlev-type response function. The main focus of our study is to understand the dynamics of the micro-organisrn pool as it plays a crucial role to shape the dynamics of the food chains/webs through transmitting the energy to the higher tropic levels. We also try to understand the effects of loss rate of invertebrate predator in the model. The mathematical analysis mcludes the existence of different, feasible equilibria and their local as well as global stability. It is shown that there exists a stable limit, cycle around the interior equilibrium point. It is also observed that there exists a critical value of the food conversion efficiency rate of micro-organism pool for which there exists Hopf-bifurcation behaviour around the equilibrium point. Later we incorporate a discrete time-delay in the functional response term of the invertebrate predator. we have also derived small amplitude periodic oscillations considering the time delay as the bifurcation parameter. All the analytical results are verified through numerical simulations with respect to different values of the system parameters. Finally we discuss the role of system parameters to understand the qualitative behaviour of the model system.