Highlights d Cities possess a consistent ''core'' set of non-human microbes d Urban microbiomes echo important features of cities and city-life d Antimicrobial resistance genes are widespread in cities d Cities contain many novel bacterial and viral species

Abstract. We study the dynamical behaviour of a smooth vector field on a 3-manifold near a heteroclinic network. Under some generic assumptions on the network, we prove that every path on the network is followed by a neighbouring trajectory of the vector field -there is switching on the network. We also show that near the network there is an infinite number of hyperbolic suspended horseshoes. This leads to the existence of a horseshoe of suspended horseshoes with the shape of the network.Our results are motivated by an example constructed by Field (Lectures on Bifurcations, Dynamics, and Symmetry, Pitman Research Notes in Mathematics Series 356, Longman,1996) where we have observed, numerically, the existence of such a network.

We describe all heteroclinic networks in R 4 made of simple heteroclinic cycles of types B or C, with at least one common connecting trajectory. For networks made of cycles of type B, we study the stability of the cycles that make up the network as well as the stability of the network. We show that even when none of the cycles has strong stability properties the network as a whole may be quite stable. We prove, and provide illustrative examples of, the fact that the stability of the network does not depend a priori uniquely on the stability of the individual cycles.

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We construct examples of vector fields on a three-sphere, amenable to analytic proof of properties that guarantee the existence of complex behavior. The examples are restrictions of symmetric polynomial vector fields in R 4 and possess heteroclinic networks producing switching and nearby suspended horseshoes. The heteroclinic networks in our examples are persistent under symmetry preserving perturbations. We prove that some of the connections in the networks are the transverse intersection of invariant manifolds. The remaining connections are symmetry-induced. The networks lie in an invariant three-sphere and may involve connections exclusively between equilibria or between equilibria and periodic trajectories. The same construction technique may be applied to obtain other examples with similar features.

We provide conditions guaranteeing that certain classes of robust heteroclinic networks are asymptotically stable. We study the asymptotic stability of ac-networks-robust heteroclinic networks that exist in smooth Z n 2 -equivariant dynamical systems defined in the positive orthant of R n . Generators of the group Z n 2 are the transformations that change the sign of one of the spatial coordinates. The ac-network is a union of hyperbolic equilibria and connecting trajectories, where all equilibria belong to the coordinate axes (not more than one equilibrium per axis) with unstable manifolds of dimension one or two. The classification of ac-networks is carried out by describing all possible types of associated graphs. We prove sufficient conditions for asymptotic stability of ac-networks. The proof is given as a series of theorems and lemmas that are applicable to the ac-networks and to more general types of networks. Finally, we apply these results to discuss the asymptotic stability of several examples of heteroclinic networks.

We study the dynamics near heteroclinic networks for which all eigenvalues of the linearization at the equilibria are real. A common connection and an assumption on the geometry of its incoming and outgoing directions exclude even the weakest forms of switching (i.e. along this connection). The form of the global transition maps, and thus the type of the heteroclinic cycle, plays a crucial role in this. We look at two examples in R 5 , the House and Bowtie networks, to illustrate complex dynamics that may occur when either of these conditions is broken. For the House network, there is switching along the common connection, while for the Bowtie network we find switching along a cycle.

We determine the properties of the core-periphery model with three regions and compare our results with those of the standard 2-region model. The conditions for the stability of dispersion and concentration are established. As in the 2-region model, dispersion and concentration can be simultaneously stable. We show that the 3-region (2-region) model favours the concentration (dispersion) of economic activity. Furthermore, we provide some results for the n-region model. We show that the stability of concentration of the 2-region model implies that of any model with an even number of regions. JEL classification: R12, R23