Beyond R0 : demographic models for variability of lifetime reproductive output

© The Author(s), 2011. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in PLoS One 6 (2011): e20809, doi:10.1371/journal.pone.0020809.The net reproductive rate measures the expected lifetime reproductive output of an individual, and plays an important role in demography, ecology, evolution, and epidemiology. Well-established methods exist to calculate it from age- or stage-classified demographic data. As an expectation, provides no information on variability; empirical measurements of lifetime reproduction universally show high levels of variability, and often positive skewness among individuals. This is often interpreted as evidence of heterogeneity, and thus of an opportunity for natural selection. However, variability provides evidence of heterogeneity only if it exceeds the level of variability to be expected in a cohort of identical individuals all experiencing the same vital rates. Such comparisons require a way to calculate the statistics of lifetime reproduction from demographic data. Here, a new approach is presented, using the theory of Markov chains with rewards, obtaining all the moments of the distribution of lifetime reproduction. The approach applies to age- or stage-classified models, to constant, periodic, or stochastic environments, and to any kind of reproductive schedule. As examples, I analyze data from six empirical studies, of a variety of animal and plant taxa (nematodes, polychaetes, humans, and several species of perennial plants).Supported by National Science Foundation Grant DEB-0816514 and by a Research Award from the Alexander von Humboldt Foundation

A mini foxtail millet with an Arabidopsis-like life cycle as a C4 model system

Over the past few decades, several plant species, including Arabidopsis thaliana, Brachypodium distachyon and rice (Oryza sativa), have been adopted as model plants for various aspects of research. These species, especially Arabidopsis, have had vital roles in making fundamental discoveries and technological advances 1. However, all these model plants use C 3 photosynthe-sis, and discoveries made in these species are not always transferable to, or representative of, C 4 plants such as maize (Zea mays), sor-ghum (Sorghum bicolor) and millets, which are efficient fixers of atmospheric CO 2 into biomass. Thus, it is critical to develop a new model system for studies in these and many other C 4 plants 2. Foxtail millet (S. italica) is a cereal crop that was domesticated from its wild ancestor, green foxtail (Setaria viridis). These two species are evolutionarily close to several bioenergy crops, including switchgrass (Panicum virgatum), napiergrass (Pennisetum purpu-reum) and pearl millet (Pennisetum glaucum), and major cereals such as sorghum, maize and rice 3. In addition, extensive genetic diversity exists in Setaria, with approximately 30,000 accessions preserved in China, India, Japan and the United States 3 as valuable resources for gene-function dissection and elite-allele mining 4. In recent years, the whole-genome sequences of foxtail millet and green foxtail have been made available 5-9 , and both species have been proposed as C 4 model plant systems 3,6. Between these two species, foxtail millet is more suitable as a model plant due to the seed shattering and dor-mancy in green foxtail. Nevertheless, the relatively long life cycle (usually 4-5 months per generation) and large plant size (1-2 m in height) limit the use of foxtail millet as a model plant 3,10-12. To overcome such limitations, we have recently developed a large fox-tail millet ethyl methane sulfonate (EMS)-mutagenized population using Jingu21, a high-yield, high-grain-quality elite variety widely grown in north China in the past few decades. From the mutant population, we identified a miniature mutant (dubbed xiaomi) with a life cycle similar to that of Arabidopsis. Subsequently, we developed genomics and transcriptomics resources and a protocol for efficient transformation of xiaomi, as essential parts of the toolbox for the research community