Evolution of extrafloral nectaries: adaptive process and selective regime changes from forest to savanna

Much effort has been devoted to understanding the function of extrafloral nectaries (EFNs) for antplantherbivore interactions. However, the pattern of evolution of such structures throughout the history of plant lineages remains unexplored. In this study, we used empirical knowledge on plant defences mediated by ants as a theoretical framework to test specific hypotheses about the adaptive role of EFNs during plant evolution. Emphasis was given to different processes (neutral or adaptive) and factors (habitat change and trade-offs with new trichomes) that may have affected the evolution of antplant associations. We measured seven EFN quantitative traits in all 105 species included in a well-supported phylogeny of the tribe Bignonieae (Bignoniaceae) and collected field data on antEFN interactions in 32 species. We identified a positive association between ant visitation (a surrogate of ant guarding) and the abundance of EFNs in vegetative plant parts and rejected the hypothesis of phylogenetic conservatism of EFNs, with most traits presenting K-values < 1. Modelling the evolution of EFN traits using maximum likelihood approaches further suggested adaptive evolution, with static-optimum models showing a better fit than purely drift models. In addition, the abundance of EFNs was associated with habitat shifts (with a decrease in the abundance of EFNs from forest to savannas), and a potential trade-off was detected between the abundance of EFNs and estipitate glandular trichomes (i.e. trichomes with sticky secretion). These evolutionary associations suggest divergent selection between species as well as explains K-values < 1. Experimental studies with multiple lineages of forest and savanna taxa may improve our understanding of the role of nectaries in plants. Overall, our results suggest that the evolution of EFNs was likely associated with the adaptive process which probably played an important role in the diversification of this plant group.

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.