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We study the interplay of ecological and evolutionary dynamics in communities composed of populations with contrasting time-scales. In such communities, genetic variation of individual traits can cause population transitions between stationary and cyclic ecological regimes, hence abrupt variations in fitness. Such abrupt variations raise ridges in the adaptive landscape, where the populations are poised between equilibrium and cyclic coexistence and along which evolutionary trajectories can remain sliding for long times or halt at special points called evolutionary pseudo-equilibria. These novel phenomena should be generic to all systems in which ecological interactions cause fitness to vary discontinuously. They are demonstrated by the analysis of a predator-prey community, with one adaptive trait for each population. The eco-evolutionary dynamics of the system show a number of other distinctive features, including evolutionary extinction and two forms of Red Queen dynamics. One of them is characterized by intermittent bouts of cyclic oscillations of the two populations.

Disclosing the main features of the structure of a network is crucial to understand a number of static and dynamic properties, such as robustness to failures, spreading dynamics, or collective behaviours. Among the possible characterizations, the core-periphery paradigm models the network as the union of a dense core with a sparsely connected periphery, highlighting the role of each node on the basis of its topological position. Here we show that the core-periphery structure can effectively be profiled by elaborating the behaviour of a random walker. A curve—the core-periphery profile—and a numerical indicator are derived, providing a global topological portrait. Simultaneously, a coreness value is attributed to each node, qualifying its position and role. The application to social, technological, economical, and biological networks reveals the power of this technique in disclosing the overall network structure and the peculiar role of some specific nodes.

We show in this paper how numerical bifurcation analysis can be used to study the evolution of genetically transmitted phenotypic traits. For this, we consider the standard Rosenzweig-MacArthur prey-predator model [The American Naturalist, 97 (1963), pp. 209-223] and, following the so-called adaptive dynamics approach, we derive from it a second-order evolutionary model composed of two ODEs, one for the prey trait and one for the predator trait. Then, we perform a detailed bifurcation analysis of the evolutionary model with respect to various environmental and demographic parameters. Surprisingly, the evolutionary dynamics turn out to be much richer than the population dynamics. Up to three evolutionary attractors can be present, and the bifurcation diagrams contain numerous global bifurcations and codimension-2 bifurcation points. Interesting biological properties can be extracted from these bifurcation diagrams. In particular, one can conclude that evolution of the traits can be cyclic and easily promote prey species diversity.

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