Modelling and Simulation of Food Network

The analysis of networks of ecological throphic transfers is a useful complement to simulation modelling in the quest for understanding whole-ecosystem dynamics. Throphic networks can be studied in quantitive and systematic fashion at several levels. The purpose of this paper is to investigate a system of parabolic equations with discrete time delays describing a simple throphic food network which consists of the n + 1 type living organisms zooplankton, phytoplankton and microorganisms), the n type dissolved organic and inorganic nutrients and detritus. When the delays are relatively small, our predictions are also identical to the predictions given by the corresponding PDE. The system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations. Stability analysis of equilibria and some numerical examples are given. INTRODUCTION In ecosystem modelling, food network design is a central task. Ecological networks are quantitative, graph-based descriptions of ecosystems, consisting of compartments (trophospecies and nutrient pools) that exchange fluxes of nutrient or energy. One approach that has been used to account for the lack of empirically derived data is the development of simple algorithms to construct hypothetical, but ecologically realistic networks. However, there have been two distinct approaches marked by the initial assumptions one makes. The first group, based on population/community ecology, focuses strictly on who eats whom, producing structures involving primary producers, grazers, and predators, but explicitly lacks decomposers and detritus. As a result, these networks do not typically contain cycling. The second group, based on ecosystem ecology focuses on energy flowing the system and includes all functional groups including detritus (dead organic material) and decomposers (Fath at al., 2007). The interactions along the detritus feedback cycle are likely to occur at a different time scale than the conventional predation. This problem could be addressed by describing interactions with time-delayed differential equations (Halnes at al., 2007). The focus in most food web analyses has been on how they are arranged, affects different statistical and stability properties of model food webs. The dynamical theory of food webs has been based typically on local and global stability analysis, i.e. an existence and stability of equilibrum.