Network ecology is an emerging field that allows researchers to conceptualize and analyse ecological networks and their dynamics. Here, we focus on the dynamics of ecological networks in response to environmental changes. Specifically, we formalize how network topologies constrain the dynamics of ecological systems into a unifying framework in network ecology that we refer to as the ‘ecological network dynamics framework’. This framework stresses that the interplay between species interaction networks and the spatial layout of habitat patches is key to identifying which network properties (number and weights of nodes and links) and trade-offs among them are needed to maintain species interactions in dynamic landscapes. We conclude that to be functional, ecological networks should be scaled according to species dispersal abilities in response to landscape heterogeneity. Determining how such effective ecological networks change through space and time can help reveal their complex dynamics in a changing world.

Despite evidence that seasonal variation may lead to the persistence of competing species, studies on the effect of seasonality on community network structures are sparse. Identifying whether seasonal network changes are the result of turnover or rewiring (i.e. rearrangement of interactions among species), also remains understudied in multi‐trophic communities. Using species abundance data for 38 species over three years (from nine sites across central/eastern United States) and a novel tree‐based inference method, we constructed seasonal networks for a multi‐trophic freshwater stream fish community. We found that seasonality influences species interactions, particularly through rewiring (81%) as compared to species turnover (19%). Moreover, the number of rewiring interactions was best explained by fish status as a piscivore/non‐piscivore and species maximum length (R2 = 0.41). Our findings suggest that rewiring may be a dominant process in stream fish communities experiencing seasonal environments and that traits linked to trophic‐level could be a good indicator of a species contribution to rewiring. As networks dominated by rewiring may be more robust, this network approach could be a valuable conservation tool for identifying which biological relationships must be retained for communities to avoid extinction.

Luna et al. (2022) concluded that the environment contributes to explaining specialisation in open plant–pollinator networks. When reproducing their study, we instead found that network size alone largely explained the variation in their specialisation metrics. Thus, we question whether empirical network specialisation is driven by the environment.

No abstract

Mathieu functions of period π or 2π, also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross-section. This paper surveys and recapitulates the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly to do with double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues, and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of anti-secularity predates that of Lindstedt. For interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: