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The ECH capacities are a sequence of real numbers associated to any symplectic four-manifold, which are monotone with respect to symplectic embeddings. It is known that for a compact star-shaped domain in R^4, the ECH capacities asymptotically recover the volume of the domain. We conjecture, with a heuristic argument, that generically the error term in this asymptotic formula converges to a constant determined by a "Ruelle invariant" which measures the average rotation of the Reeb flow on the boundary. Our main result is a proof of this conjecture for a large class of toric domains. As a corollary, we obtain a general obstruction to symplectic embeddings of open toric domains with the same volume. For more general domains in R^4, we bound the error term with an improvement on the previously known exponent from 2/5 to 1/4., v3: updated references, to appear in J. Fixed Point Theory and Applications

Additional file 2: Table S1. Environmental conditions at sites of heather collection.

Additional file 3: Table S2. Average egg hatching (%) and standard deviation (SE) for C. vulgaris extracts at decreasing concentrations (n = 3). Table S3. Average egg hatching (%) and SE for Spanish heather samples at decreasing concentrations (n = 3).

In a previous paper, we defined an "elementary" alternative to the ECH capacities of symplectic four-manifolds, using max-min energy of holomorphic curves subject to point constraints, and avoiding the use of Seiberg-Witten theory. In the present paper we use a variant of this construction to define an alternative to the ECH spectrum of a contact three-manifold. The alternative spectrum has applications to Reeb dynamics in three dimensions. In particular, we adapt ideas from a previous joint paper with Edtmair to obtain quantitative closing lemmas for Reeb vector fields in three dimensions. For the example of an irrational ellipsoid, we obtain a sharp quantitative closing lemma., Comment: 32 pages; v2 added some remarks to the introduction and updated references

Additional file 1: Text S1. RELACS: Heather sampling protocol. Figure S1. Collection of 10 heather samples using a “W” pattern over a 10-m2 area.

Further details of derivations, data analysis, and figures.

In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define "nonequivariant contact homology" and "S^1-equivariant contact homology", both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact structure. Our construction uses Morse-Bott theory and is related to the positive S^1-equivariant symplectic homology of Bourgeois-Oancea. However, instead of working with Hamiltonian Floer homology, we work directly in contact geometry, using families of almost complex structures. When cylindrical contact homology can also be defined, it agrees with the tensor product of the S^1-equivariant contact homology with Q. We also present examples showing that the S^1-equivariant contact homology contains interesting torsion information. In a subsequent paper we will use obstruction bundle gluing to extend the above story to closed three-manifolds with dynamically convex contact forms, which in particular will prove that their cylindrical contact homology has a lift to integer coefficients which depends only on the contact structure., Comment: 93 pages; v3 corrected a typo and added a reference to the introduction; to appear in Journal of Topology

Additional sensitivity analyses showing robustness of the model findings to variations in demographic and disease parameters.

Although many ecological properties of species respond to climate change, their evolutionary responses are poorly understood. Here, we use data from long-term demographic studies to predict evolutionary responses of three herbaceous perennial orchid species, Cypripedium parviflorum, C. candidum and Ophrys sphegodes, to predicted climate changes in the habitats they occupy. We focus on the evolution of sprouting probability, because all three species exhibit long-term vegetative dormancy, i.e. individual plants may not emerge above-ground, potentially for several consecutive years. The drivers of all major vital rates for populations of the species were analysed with GLMMs. High-dimensionality function-based matrix projection models were then developed to serve as core elements of deterministic and stochastic adaptive dynamics models used to analyse the adaptive context of sprouting in all populations. We then used regional climate forecasts, derived from high-resolution general atmospheric circulation models, of increased mean annual temperatures and spring precipitation at the occupied sites, to predict evolutionary trends in sprouting. The models predicted that C. parviflorum and O. sphegodes will evolve higher and lower probabilities of sprouting, respectively, by the end of the twenty-first century, whereas, after considerable variation, the probability of sprouting in C. candidum will return to its current level. These trends appear to be driven by relationships between mortality and size: in C. parviflorum and C. candidum, mortality is negatively related to size in the current year but positively related to growth since the previous year, whereas in O. sphegodes, mortality is positively related to size.

Although many ecological properties of species respond to climate change, their evolutionary responses are poorly understood. Here, we use data from long-term demographic studies to predict evolutionary responses of three herbaceous perennial orchid species, Cypripedium parviflorum, C. candidum and Ophrys sphegodes, to predicted climate changes in the habitats they occupy. We focus on the evolution of sprouting probability, because all three species exhibit long-term vegetative dormancy, i.e. individual plants may not emerge above-ground, potentially for several consecutive years. The drivers of all major vital rates for populations of the species were analysed with GLMMs. High-dimensionality function-based matrix projection models were then developed to serve as core elements of deterministic and stochastic adaptive dynamics models used to analyse the adaptive context of sprouting in all populations. We then used regional climate forecasts, derived from high-resolution general atmospheric circulation models, of increased mean annual temperatures and spring precipitation at the occupied sites, to predict evolutionary trends in sprouting. The models predicted that C. parviflorum and O. sphegodes will evolve higher and lower probabilities of sprouting, respectively, by the end of the twenty-first century, whereas, after considerable variation, the probability of sprouting in C. candidum will return to its current level. These trends appear to be driven by relationships between mortality and size: in C. parviflorum and C. candidum, mortality is negatively related to size in the current year but positively related to growth since the previous year, whereas in O. sphegodes, mortality is positively related to size.

Although many ecological properties of species respond to climate change, their evolutionary responses are poorly understood. Here, we use data from long-term demographic studies to predict evolutionary responses of three herbaceous perennial orchid species, Cypripedium parviflorum, C. candidum and Ophrys sphegodes, to predicted climate changes in the habitats they occupy. We focus on the evolution of sprouting probability, because all three species exhibit long-term vegetative dormancy, i.e. individual plants may not emerge above-ground, potentially for several consecutive years. The drivers of all major vital rates for populations of the species were analysed with GLMMs. High-dimensionality function-based matrix projection models were then developed to serve as core elements of deterministic and stochastic adaptive dynamics models used to analyse the adaptive context of sprouting in all populations. We then used regional climate forecasts, derived from high-resolution general atmospheric circulation models, of increased mean annual temperatures and spring precipitation at the occupied sites, to predict evolutionary trends in sprouting. The models predicted that C. parviflorum and O. sphegodes will evolve higher and lower probabilities of sprouting, respectively, by the end of the twenty-first century, whereas, after considerable variation, the probability of sprouting in C. candidum will return to its current level. These trends appear to be driven by relationships between mortality and size: in C. parviflorum and C. candidum, mortality is negatively related to size in the current year but positively related to growth since the previous year, whereas in O. sphegodes, mortality is positively related to size.

Although many ecological properties of species respond to climate change, their evolutionary responses are poorly understood. Here, we use data from long-term demographic studies to predict evolutionary responses of three herbaceous perennial orchid species, Cypripedium parviflorum, C. candidum and Ophrys sphegodes, to predicted climate changes in the habitats they occupy. We focus on the evolution of sprouting probability, because all three species exhibit long-term vegetative dormancy, i.e. individual plants may not emerge above-ground, potentially for several consecutive years. The drivers of all major vital rates for populations of the species were analysed with GLMMs. High-dimensionality function-based matrix projection models were then developed to serve as core elements of deterministic and stochastic adaptive dynamics models used to analyse the adaptive context of sprouting in all populations. We then used regional climate forecasts, derived from high-resolution general atmospheric circulation models, of increased mean annual temperatures and spring precipitation at the occupied sites, to predict evolutionary trends in sprouting. The models predicted that C. parviflorum and O. sphegodes will evolve higher and lower probabilities of sprouting, respectively, by the end of the twenty-first century, whereas, after considerable variation, the probability of sprouting in C. candidum will return to its current level. These trends appear to be driven by relationships between mortality and size: in C. parviflorum and C. candidum, mortality is negatively related to size in the current year but positively related to growth since the previous year, whereas in O. sphegodes, mortality is positively related to size.

Although many ecological properties of species respond to climate change, their evolutionary responses are poorly understood. Here, we use data from long-term demographic studies to predict evolutionary responses of three herbaceous perennial orchid species, Cypripedium parviflorum, C. candidum and Ophrys sphegodes, to predicted climate changes in the habitats they occupy. We focus on the evolution of sprouting probability, because all three species exhibit long-term vegetative dormancy, i.e. individual plants may not emerge above-ground, potentially for several consecutive years. The drivers of all major vital rates for populations of the species were analysed with GLMMs. High-dimensionality function-based matrix projection models were then developed to serve as core elements of deterministic and stochastic adaptive dynamics models used to analyse the adaptive context of sprouting in all populations. We then used regional climate forecasts, derived from high-resolution general atmospheric circulation models, of increased mean annual temperatures and spring precipitation at the occupied sites, to predict evolutionary trends in sprouting. The models predicted that C. parviflorum and O. sphegodes will evolve higher and lower probabilities of sprouting, respectively, by the end of the twenty-first century, whereas, after considerable variation, the probability of sprouting in C. candidum will return to its current level. These trends appear to be driven by relationships between mortality and size: in C. parviflorum and C. candidum, mortality is negatively related to size in the current year but positively related to growth since the previous year, whereas in O. sphegodes, mortality is positively related to size.

Although many ecological properties of species respond to climate change, their evolutionary responses are poorly understood. Here, we use data from long-term demographic studies to predict evolutionary responses of three herbaceous perennial orchid species, Cypripedium parviflorum, C. candidum and Ophrys sphegodes, to predicted climate changes in the habitats they occupy. We focus on the evolution of sprouting probability, because all three species exhibit long-term vegetative dormancy, i.e. individual plants may not emerge above-ground, potentially for several consecutive years. The drivers of all major vital rates for populations of the species were analysed with GLMMs. High-dimensionality function-based matrix projection models were then developed to serve as core elements of deterministic and stochastic adaptive dynamics models used to analyse the adaptive context of sprouting in all populations. We then used regional climate forecasts, derived from high-resolution general atmospheric circulation models, of increased mean annual temperatures and spring precipitation at the occupied sites, to predict evolutionary trends in sprouting. The models predicted that C. parviflorum and O. sphegodes will evolve higher and lower probabilities of sprouting, respectively, by the end of the twenty-first century, whereas, after considerable variation, the probability of sprouting in C. candidum will return to its current level. These trends appear to be driven by relationships between mortality and size: in C. parviflorum and C. candidum, mortality is negatively related to size in the current year but positively related to growth since the previous year, whereas in O. sphegodes, mortality is positively related to size.

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the relative grading on ECH; the ECH differential counts holomorphic curves in the symplectization of Y whose relative homology classes have ECH index 1. A known index inequality implies that such curves are (mostly) embedded and satisfy some additional constraints. In this paper we prove four new results about the ECH index. First, we refine the relative grading on ECH to an absolute grading, which associates to each union of Reeb orbits a homotopy class of oriented 2-plane fields on Y. Second, we extend the ECH index inequality to symplectic cobordisms between three-manifolds with Hamiltonian structures, and simplify the proof. Third, we establish general inequalities on the ECH index of unions and multiple covers of holomorphic curves in cobordisms. Finally, we define a new relative filtration on ECH, or any other kind of contact homology of a contact 3-manifold, which is similar to the ECH index and related to the Euler characteristic of holomorphic curves. This does not give new topological invariants except possibly in special situations, but it is a useful computational tool., Comment: 47 pages; added a reference and minor clarifications suggested by referee, to appear in Yashafest proceedings

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This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u_+ and u_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total multiplicity of the negative ends of u_+ at covers of gamma agrees with the total multiplicity of the positive ends of u_- at covers of gamma. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue u_+ and u_- to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. We establish a combinatorial formula for the signed count of such gluings. As an application, we deduce that the differential d in embedded contact homology satisfies d^2=0. This paper explains the more algebraic aspects of the story, and proves the above formulas using some analytical results from part II., Comment: 100 pages; slightly revised with new pictures following referee's suggestions; to appear in Journal of Symplectic Geometry

Let X be a compact oriented Riemannian manifold and let $\phi:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $\phi$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $\phi$ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $\phi$; and (c) a kind of Reidemeister torsion of X determined by the homotopy class of $\phi$. When $\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs) the Meng-Taubes relation between the Seiberg- Witten invariants and the ``Milnor torsion'' of X., Comment: 35 pages, AMS-LaTeX (one hand-drawn figure available on request)

Livestock disease controls are often linked to movements between farms, for example, via quarantine and pre- or post-movement testing. Designing effective controls, therefore, benefits from accurate assessment of herd-to-herd transmission. Household models of human infections make use of R*, the number of groups infected by an initial infected group, which is a metapopulation level analogue of the basic reproduction number R0 that provides a better characterization of disease spread in a metapopulation. However, existing approaches to calculate R* do not account for individual movements between locations which means we lack suitable tools for livestock systems. We address this gap using next-generation matrix approaches to capture movements explicitly and introduce novel tools to calculate R* in any populations coupled by individual movements. We show that depletion of infectives in the source group, which hastens its recovery, is a phenomenon with important implications for design and efficacy of movement-based controls. Underpinning our results is the observation that R* peaks at intermediate livestock movement rates. Consequently, under movement-based controls, infection could be controlled at high movement rates but persist at intermediate rates. Thus, once control schemes are present in a livestock system, a reduction in movements can counterintuitively lead to increased disease prevalence. We illustrate our results using four important livestock diseases (bovine viral diarrhoea, bovine herpes, Johne's disease and Escherichia coli O157) that each persist across different movement rate ranges with the consequence that a change in livestock movements could help control one disease, but exacerbate another.

Livestock disease controls are often linked to movements between farms, for example, via quarantine and pre- or post-movement testing. Designing effective controls, therefore, benefits from accurate assessment of herd-to-herd transmission. Household models of human infections make use of R*, the number of groups infected by an initial infected group, which is a metapopulation level analogue of the basic reproduction number R0 that provides a better characterization of disease spread in a metapopulation. However, existing approaches to calculate R* do not account for individual movements between locations which means we lack suitable tools for livestock systems. We address this gap using next-generation matrix approaches to capture movements explicitly and introduce novel tools to calculate R* in any populations coupled by individual movements. We show that depletion of infectives in the source group, which hastens its recovery, is a phenomenon with important implications for design and efficacy of movement-based controls. Underpinning our results is the observation that R* peaks at intermediate livestock movement rates. Consequently, under movement-based controls, infection could be controlled at high movement rates but persist at intermediate rates. Thus, once control schemes are present in a livestock system, a reduction in movements can counterintuitively lead to increased disease prevalence. We illustrate our results using four important livestock diseases (bovine viral diarrhoea, bovine herpes, Johne's disease and Escherichia coli O157) that each persist across different movement rate ranges with the consequence that a change in livestock movements could help control one disease, but exacerbate another.