A generalised framework for building complex networks of evolution equations

We present a new approach to model partial differential equation (PDE) networks. The originality of our method lies in its ability to model various types of networks, including those whose nodes (domains) are non-identical, i.e. they can have various shapes and sizes. This approach makes it possible to model phenomena with possible connections both from within domains and from their boundaries, thus offering a general overview of transmission models. We propose a general model for the scalar case, demonstrating flow conservation in the network as well as the existence, uniqueness and positivity of the solution. In addition, we develop a model for the vector case, where we establish results similar to those of the scalar case, while adding an analysis on the global bounding of the solution in mass-conserving systems. We illustrate our approach with a concrete application to a predator-prey model in a network with two nodes, to which we apply all the obtained theoretical results. Finally, numerical simulations are provided to illustrate our theoretical findings.