p d e 2 p a th is a free and easy to use M atlab continuation/bifurcation pack age for elliptic systems of PDEs with arbitrary many components, on general two dimensional domains, and with rather general boundary conditions. The package is based on the FEM of the M atlab p d e to o lb o x , and is explained by a number of examples, including Bratu's problem, the Schnakenberg model, Rayleigh-Benard convection, and von Karman plate equations. These serve as templates to study new problems, for which the user has to provide, via M atlab function files, a description of the geometry, the boundary conditions, the coefficients of the PDE, and a rough initial guess of a solution. The basic algorithm is a one parame ter arclength-continuation with optional bifurcation detection and branch-switching. Stability calculations, error control and mesh-handling, and some elementary timeintegration for the associated parabolic problem are also supported. The con tinuation, branch-switching, plotting etc are performed via M atlab command-line function calls guided by the AUTO style. The software can be downloaded from w w w .s ta f f .u n i-o ld e n b u r g .d e /h a n n e s .u e c k e r/p d e 2 p a th , where also an on line documentation of the software is provided such that in this paper we focus more on the mathematics and the example systems.
Personality (consistent differences between individuals in behavior) and plasticity (changes within individuals in behavior) are often viewed as separate and opposing phenomena. We tested this idea by analyzing parental care reaction norms in a bird that exhibits biparental care. Personality in provisioning behavior existed (r(ic) = 0.11) and persisted despite being reduced after accounting for individual differences in environment. Plasticity was also evident and differed between the sexes. Male visit rate was associated with changes in brood size and time of day, but female visit rate was associated with changes in nestling age and date. In both sexes changes in visit rate were positively correlated with changes in their partner's visit rate. Both sexes also exhibited multidimensional reaction norms; interaction terms revealed that within-individual visit rates increased more steeply with brood size when nestlings were older, and the effect of the partner's visit rate was sensitive to variation in date, precipitation, and the focal bird's age. Individuals also varied in how they responded (reaction norm slope) to changes in nestling age and partner visits. Moreover, parental personality was interdependent with individual plasticity in several ways. Individuals of both sexes with a high visit rate also responded more positively to changes in nestling age, and males also showed this pattern with changes in partner visit rate. Explicit use of the behavioral reaction norm integrated personality and plasticity, revealed that these are not opposing concepts, and stimulated new hypotheses about sexual conflict over care and provisioning as a life-history trait.
For a Selkov-Schnakenberg model as a prototype reaction-diffusion system on two dimensional domains we use the continuation and bifurcation software pde2path to numerically calculate branches of patterns embedded in patterns, for instance hexagons embedded in stripes and vice versa, with a planar interface between the two patterns. We use the Ginzburg-Landau reduction to approximate the locations of these branches by Maxwell points for the associated Ginzburg-Landau system. For our basic model, some but not all of these branches show a snaking behaviour in parameter space, over the given computational domains. The (numerical) non-snaking behaviour appears to be related to too narrow bistable ranges with rather small Ginzburg-Landau energy differences. This claim is illustrated by a suitable generalized model. Besides the localized patterns with planar interfaces we also give a number of examples of fully localized patterns over patterns, for instance hexagon patches embedded in radial stripes, and fully localized hexagon patches over straight stripes.
This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution of a single scalar field like a density or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn-and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dipcoating. Through the different examples we illustrate how to employ the numerical tools provided by the packages AUTO07P and PDE2PATH to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues.Published as: Engelnkemper, S., Gurevich, S.V.,
As part of its Paris Agreement commitment, China pledged to peak carbon dioxide (CO 2) emissions around 2030, striving to peak earlier, and to increase the non-fossil share of primary energy to 20% by 2030. Yet by the end of 2017, China emitted 28% of the world's energy-related CO 2 emissions, 76% of which were from coal use. How China can reinvent its energy economy cost-effectively while still achieving its commitments was the focus of a three-year joint research project completed in September 2016. Overall, this analysis found that if China follows a pathway in which it aggressively adopts all cost-effective energy efficiency and CO 2 emission reduction technologies while also aggressively moving away from fossil fuels to renewable and other non-fossil resources, it is possible to not only meet its Paris Agreement Nationally Determined Contribution (NDC) commitments, but also to reduce its 2050 CO 2 emissions to a level that is 42% below the country's 2010 CO 2 emissions. While numerous barriers exist that will need to be addressed through effective policies and programs in order to realize these potential energy use and emissions reductions, there are also significant local environmental (e.g., air quality), national and global environmental (e.g., mitigation of climate change), human health, and other unquantified benefits that will be realized if this pathway is pursued in China.
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