Rapid plant diversification: Planning for an evolutionary future

Systematic conservation planning is a branch of conservation biology that seeks to identify spatially explicit options for the preservation of biodiversity. Alternative systems of conservation areas are predictions about effective ways of promoting the persistence of biodiversity; therefore, they should consider not only biodiversity pattern but also the ecological and evolutionary processes that maintain and generate species. Most research and application, however, has focused on pattern representation only. This paper outlines the development of a conservation system designed to preserve biodiversity pattern and process in the context of a rapidly changing environment. The study area is the Cape Floristic Region (CFR), a biodiversity hotspot of global significance, located in southwestern Africa. This region has experienced rapid (post-Pliocene) ecological diversification of many plant lineages; there are numerous genera with large clusters of closely related species (flocks) that have subdivided habitats at a very fine scale. The challenge is to design conservation systems that will preserve both the pattern of large numbers of species and various natural processes, including the potential for lineage turnover. We outline an approach for designing a system of conservation areas to incorporate the spatial components of the evolutionary processes that maintain and generate biodiversity in the CFR. We discuss the difficulty of assessing the requirements for pattern versus process representation in the face of ongoing threats to biodiversity, the difficulty of testing the predictions of alternative conservation systems, and the widespread need in conservation planning to incorporate and set targets for the spatial components (or surrogates) of processes.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.