We give a self-contained introduction to the theory of directed graphs, leading up to the relationship between the Perron-Frobenius eigenvectors of a graph and its autocatalytic sets. Then we discuss a particular dynamical system on a fixed but arbitrary graph, that describes the population dynamics of species whose interactions are determined by the graph. The attractors of this dynamical system are described as a function of graph topology. Finally we consider a dynamical system in which the graph of interactions of the species coevolves with the populations of the species. We show that this system exhibits complex dynamics including self-organization of the network by autocatalytic sets, growth of complexity and structure, and collapse of the network followed by recoveries. We argue that a graph theoretic classification of perturbations of the network is helpful in predicting the future impact of a perturbation over short and medium time scales.

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We consider the population dynamics of a set of species whose network of catalytic interactions is described by a directed graph. The relationship between the attractors of this dynamics and the underlying graph theoretic structures like cycles and autocatalytic sets is discussed. It is shown that when the population dynamics is suitably coupled to a slow dynamics of the graph itself, the network evolves towards increasing complexity driven by autocatalytic sets. Some quantitative measures of network complexity are described.

In this paper we study a harvesting problem in the presence of a predator and a tax. The objective is to maximize the monetary social benefit as well as prevent the predator from extinction, keeping the ecological balance.

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