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We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canonical equation of Adaptive Dynamics (AD). The equation deterministically describes the expected long-term evolution of inheritable traits - phenotypes or strategies - of coevolving populations, in the limit of rare and small mutations. In the vicinity of a stable equilibrium of the AD canonical equation, a mutant type can invade and coexist with the present - resident - types, whereas the fittest always win far from equilibrium. After coexistence, residents and mutants effectively diversify, according to the enlarged canonical equation, only if natural selection favors outer rather than intermediate traits - the equilibrium being evolutionarily unstable, rather than stable. Though the conditions for evolutionary branching - the joint effect of resident-mutant coexistence and evolutionary instability - have been known for long, the unfolding of the bifurcation has remained a missing tile of AD, the reason being related to the nonsmoothness of the mutant invasion fitness after branching. In this paper, we develop a methodology that allows the approximation of the invasion fitness after branching in terms of the expansion of the (smooth) fitness before branching. We then derive a canonical model for the branching bifurcation and perform its unfolding around the loss of evolutionary stability. We cast our analysis in the simplest (but classical) setting of asexual, unstructured populations living in an isolated, homogeneous, and constant abiotic environment; individual traits are one-dimensional; intra- as well as inter-specific ecological interactions are described in the vicinity of a stationary regime. [ABSTRACT FROM AUTHOR]

We present in this paper the first example of chaotic evolutionary dynamics in biology. We consider a Lotka-Volterra tritrophic food chain composed of a resource, its consumer, and a predator species, each characterized by a single adaptive phenotypic trait, and we show that for suitable modeling and parameter choices the evolutionary trajectories approach a strange attractor in the three-dimensional trait space. The study is performed through the bifurcation analysis of the so-called canonical equation of Adaptive Dynamics, the most appropriate modeling approach to long-term evolutionary dynamics. [ABSTRACT FROM AUTHOR]

Through the analysis of a mathematical model we construct in this paper the full catalog of possible love stories among standard (i.e. secure and non-synergic) individuals. For this, we first distinguish between robust and fragile couples, i.e. couples which have one or two stable equilibria. We also distinguish between high and low quality relationships by looking at the sign of the feelings among the partners. Finally, we split fragile couples into two distinct classes, namely those that start from the state of indifference to evolve toward their most desirable equilibrium, as opposed to those that have an unfavorable evolution. The boundaries separating the various classes in any two-dimensional parameter space are either bifurcation curves or special curves characterized by the annihilation of a suitably defined indicator associated to the equilibria or to their basins of attraction. The analysis is performed in part analytically and in part numerically (through continuation methods), and is mainly focused on the role of the appeals of the individuals. The results are in agreement with common wisdom and observations and show, in particular, that love stories can evolve positively even if one of the two individuals has negative feelings in the initial phase of the relationship. [ABSTRACT FROM AUTHOR]

The paper considers the evolution of a particular class of networks of identical chaotic oscillators, namely that of ecological networks. In these networks, nodes represent patches where a certain number of plant and animal populations interact on ecological timescale, arcs represent migration flows due to dispersal, and Darwinian evolution is responsible for variations, on a longer evolutionary timescale, of the demographic parameters characterizing the populations. Up to now, this problem has been mainly studied with reference to single-population patches described by one-dimensional discrete-time models and by considering only the dispersal rates of migrating populations as an evolving trait. Here, we propose a method of investigation which allows to study multipopulation patches described by continuous-time models with evolving traits influencing various demographic parameters (including or not dispersal). The method is casted within the frame of the so-called master stability function approach for the analysis of synchronization of coupled systems, and the results obtained in a first and very simple application support the conjecture that evolution drives ecological networks toward weak forms of synchronization. [ABSTRACT FROM AUTHOR]

The paper describes the slow evolution of two adaptive traits that regulate the interactions between two mutualistic populations (e.g. a flowering plant and its insect pollinator). For frozen values of the traits, the two populations can either coexist or go extinct. The values of the traits for which populations extinction is guaranteed are therefore of no interest from an evolutionary point of view. In other words, the evolutionary dynamics must be studied only in a viable subset of trait space, which is bounded due to the physiological cost of extreme trait values. Thus, evolutionary dynamics experience so-called border collision bifurcations, when a system invariant in trait space hits the border of the viable subset. The unfolding of standard and border collision bifurcations with respect to two parameters of biological interest is presented. The algebraic and boundary-value problems characterizing the border collision bifurcations are described together with some details concerning their computation. [ABSTRACT FROM AUTHOR]