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This study mainly aims to explore the existence of global attractor for a modified Swift-Hohenberg equation. The method we use was the classical existence theorem of global attractors and the theory of semigroups. Use this method we prove that the equation exist a global attractor in H 1 2 space, and the global attractor attracts whatever bounded subset of H 1 2 in the H 1 2 -norm.

In this paper, we show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.

This short note presents an explicit step-by-step proof of the existence theorem of an optimal control problem applied to a deterministic model for a vector-borne disease.

Co‐existence theories fail to adequately explain observed community patterns (diversity and composition) because they mainly address local extinctions. For a more complete understanding, the regional processes responsible for species formation and geographic dispersal should also be considered. The species pool concept holds that local variation in community patterns is dependent primarily on the availability of species, which is driven by historical diversification and dispersal at continental and landscape scales. However, empirical evidence of historical effects is limited. This slow progress can be attributed to methodological difficulties in determining the characteristics of historical species pools and how they contributed to diversity patterns in contemporary landscapes. A role of landscape‐scale dispersal limitation in determining local community patterns has been demonstrated by numerous seed addition experiments. However, disentangling general patterns of dispersal limitation in communities still requires attention. Distinguishing habitat‐specific species pools can help to meet both applied and theoretical challenges. In conservation biology, the use of absolute richness may be uninformative because the size of species pools varies between habitats. For characterizing the dynamic state of individual communities, biodiversity relative to species pools provides a balanced way of assessing communities in different habitats. Information about species pools may also be useful when studying community assembly rules, because it enables a possible mechanism of trait convergence (habitat filtering) to be explicitly assessed. Empirical study of the role of historic effects and dispersal on local community patterns has often been restricted due to methodological difficulties in determining habitat‐specific species pools. However, accumulating distributional, ecological and phylogenetic information, as well as use of appropriate model systems (e.g. archipelagos with known biogeographic histories) will allow the species pool concept to be applied effectively in future research.

Question: Which functional diversity indices have the power to reveal changes in community assembly processes along abiotic stress gradients? Is their power affected by stochastic processes and variations in species richness along stress gradients? Methods: We used a simple community assembly model to explore the power of functional diversity indices across a wide range of ecological contexts. The model assumes that with declining stress the influence of niche complementarity on species fitness increases while that of environmental filtering decreases. We separately incorporated two trait-independent stochastic processes — mass and priority effects — in simulating species occurrences and abundances along a hypothetical stress gradient. We ran simulations where species richness was constant along the gradient, or increased, decreased or varied randomly with declining stress. We compared observed values for two indices of functional richness — total functional dendrogram length (FD) and convex hull volume (FRic) — with a matrix-swap null model (yielding indices SESFD and SESFRic) to remove any trivial effects of species richness. We also compared two indices that measure both functional richness and functional divergence — Rao quadratic entropy (Rao) and functional dispersion (FDis) — with a null model that randomizes abundances across species but within communities. This converts them to pure measures of functional divergence (SESRao and SESFDis). Results: When mass effects operated, only SESRao and SESFDis gave reasonable power, irrespective of how species richness varied along the stress gradient. FD, FRic, Rao and FDis had low power when species richness was constant, and variation in species richness greatly influenced their power. SESFRic and SESFD were unaffected by variation in species richness. When priority effects operated, FRic, SESFRic, Rao and FDis had good power and were unaffected by variation in species richness. Variation in species richness greatly affected FD and SESFD. SESRao and SESFDis had low power in the priority effects model but were unaffected by variation in species richness. Conclusions: Our results demonstrate that a reliable test for changes in assembly processes along stress gradients requires functional diversity indices measuring either functional richness or functional divergence. We recommend using SESFRic as a measure of functional richness and either SESRao or SESFDis (which are very closely related mathematically) as a measure of functional divergence. Used together, these indices of functional richness and functional divergence provide good power to test for increasing niche complementarity with declining stress across a broad range of ecological contexts.

We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving L² distance from a datum. Such functionals are known to attain their infima in the BV space. Under the assumption that the domain of integration is convex, we prove that if the datum is in W 1,1, then the functional has a minimizer in W 1,1. In fact, the minimizer inherits W 1,p regularity from the datum for any p ∈ [1,+∞]. We also obtain a quantitative bound on the singular part of the gradient of the minimizer in the case that the datum is in BV. We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth.

The chemotaxis-Navier-Stokes system (⋆){nt+u⋅∇namp;= Δn−∇⋅(nχ(c)∇c),ct+u⋅∇camp;= Δc−nf(c),ut+(u⋅∇)uamp;= Δu+∇P+n∇Φ,∇⋅uamp;= 0\\begin{equation*} (\\star )\\qquad \\qquad \\qquad \\quad \\begin {cases} n_t + u\\cdot \\nabla n & =\\ \\ \\Delta n - \\nabla \\cdot (n\\chi (c)\\nabla c),\\\[1mm] c_t + u\\cdot \\nabla c & =\\ \\ \\Delta c-nf(c), \\\[1mm] u_t + (u\\cdot \\nabla )u & =\\ \\ \\Delta u + \\nabla P + n \\nabla \\Phi , \\\[1mm] \\nabla \\cdot u & =\\ \\ 0 \\end{cases} \\qquad \\qquad \\qquad \\quad \\end{equation*} is considered under boundary conditions of homogeneous Neumann type for nn and cc, and Dirichlet type for uu, in a bounded convex domain Ω⊂R3\\Omega \\subset \\mathbb {R}^3 with smooth boundary, where Φ∈W1,∞(Ω)\\Phi \\in W^{1,\\infty }(\\Omega ) and χ\\chi and ff are sufficiently smooth given functions generalizing the prototypes χ≡const.\\chi \\equiv const. and f(s)=sf(s)=s for s≥0s\\ge 0. It is known that for all suitably regular initial data n0,c0n_0, c_0 and u0u_0 satisfying 0≢n0≥00\\not \\equiv n_0\\ge 0, c0≥0c_0\\ge 0 and ∇⋅u0=0\\nabla \\cdot u_0=0, a corresponding initial-boundary value problem admits at least one global weak solution which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. The present paper shows that after some relaxation time, this solution enjoys further regularity properties and thereby complies with the concept of eventual energy solutions, which is newly introduced here and which inter alia requires that two quasi-dissipative inequalities are ultimately satisfied. Moreover, it is shown that actually for any such eventual energy solution (n,c,u)(n,c,u) there exists a waiting time T0∈(0,∞)T_0\\in (0,\\infty ) with the property that (n,c,u)(n,c,u) is smooth in Ω¯×[T0,∞)\\bar \\Omega \\times [T_0,\\infty ) and that n(x,t)→n0¯,c(x,t)→0andu(x,t)→0\\begin{eqnarray*} n(x,t)\\to \\overline {n_0}, \\qquad c(x,t)\\to 0 \\qquad \\mbox {and} \\qquad u(x,t)\\to 0 \\end{eqnarray*} hold as t→∞t\\to \\infty, uniformly with respect to x∈Ωx\\in \\Omega. This resembles a classical result on the three-dimensional Navier-Stokes system, asserting eventual smoothness of arbitrary weak solutions thereof which additionally fulfill the associated natural energy inequality. In consequence, our results inter alia indicate that under the considered boundary conditions, the possibly destabilizing action of chemotactic cross-diffusion in (⋆\\star) does not substantially affect the regularity properties of the fluid flow at least on large time scales.

We recount and discuss some of the most important methods and blow-up criteria for analyzing solutions of Keller-Segel chemotaxis models. First, we discuss the results concerning the global existence, boundedness and blow-up of solutions to parabolic-elliptic type models. Thereafter we describe the global existence, boundedness and blow-up of solutions to parabolic-parabolic models. The numerical analysis of these models is still at a rather early stage only. We recollect quite a few of the known results on numerical methods also and direct the attention to a number of open problems in this domain.

Questions: Mechanisms of community assembly are increasingly explored by combining community and species trait data with null models. By investigating if the traits of existing species are more similar (trait convergence) or more dissimilar (trait divergence) than expected by chance, these tests relate observed patterns to different existence mechanisms. Do null models accurately detect trait convergence and divergence? Are different null models equally good at detecting these two opposing patterns? How important are the species pool and other constraints that are considered by different null models? Methods: We applied ten common randomizations to communities that were simulated in a process-based model. Results: Null models good at detecting biotic processes differed from those null models that revealed abiotic processes. In particular, limiting similarity (detected through divergence) was better detected by randomizations that release the link between species abundance and trait values, whereas environmental filtering (detected through convergence of an environmental response trait) was identified by randomizations that keep this link. In general, using species abundance data provided better results than using presence-absence data, particularly within given limited environmental conditions. Weaker competitor exclusion (detected through convergence of a competition-related trait) was only detected when no environmental filtering was acting on the simulated assembly, which points to difficulties in disentangling biotic and abiotic convergence in natural communities, especially when data are randomized across habitats. Conclusions: Overall the results manifest the importance of the pool of species over which randomizations are applied; in particular whether randomizations are conducted across or within given habitats. Taken together, our findings show that different null models must be combined and applied to a carefully chosen pool of species and species abundance data to ensure that co-existence mechanisms can be properly assessed. We utilize the results to (1) discuss how different constraints implied in the different null models affect the outcomes of our tests, and (2) provide some basic recommendations on how to choose null models, given the data available and questions being asked.

This work focuses on an extension to the May-Nowak model for virus dynamics, additionally accounting for diffusion in all components and chemotactically directed motion of healthy cells in response to density gradients in the population of infected cells. The first part of the paper presents a number of simulations with the aim of investigating how far the model can depict interesting patterns. A rigorous analysis of the initial-boundary value problem is presented in a second part, where a statement on global classical solvability for arbitrarily large initial data is derived under an appropriate smallness assumption on the chemotactic coefficient. Two additional results on asymptotic stabilisation indicate that the so-called basic reproduction number retains its crucial influence on the large time behavior of solutions, as is well-known from results on the May-Nowak system.