Ecologically Sustainable Partitioning of a Metapopulations Network

A stable populations’ network is hard to interrupt without any ecological consequences. A communication blockage between patches may have a negative impact on populations’ stability or it may destabilize the populations in the ecological network. This paper deals with the construction of a safe way passing through the metapopulations habitat such that populations remain stable. To our best knowledge, this is a completely novel idea, which uses a combined approach of dynamical system stability analysis and graph partitioning algorithms. Our study finds that such safe construction is possible if algebraic connectivity of the graph components is stronger than the spatially local instabilities in respective components. Dynamical system stability analysis determines the threshold value for the stability, while a graph partitioning algorithm ensures the populations’ stability in graph components after the partitioning. The dynamics of the populations on spatially discrete patches (graph nodes) and their spatial communication with other patches is described by a set of differential equations with the general kinetics. By reversing the Turing-instability definition, the stability conditions of the considered discrete diffusive system carried out and are noted to depend upon the local dynamics of the metapopulations and the network connectivity i.e., the Fielder value of the Laplacian matrix of the graph. In the paper, necessary and sufficient conditions on the removal of some graph edges are determined to establish the stability in the remaining graph network. The exhaustive procedure and an efficient heuristic bisection graph partitioning algorithm are proposed to get the desired partition. By the exhaustive procedure, all desired partitions can be obtained and the best possible partition around a given arbitrary partition is provided by the heuristic bisection partitioning algorithm. Along the discussion of theoretical findings, detailed examples are presented that validates the claim of sustainable network cut(s).