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We analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle’s invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid.

In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation., Revised after referee comments

Game spaces in which an organism must repeatedly compete with an opponent for mutually exclusive outcomes are critical methodologies for understanding decision-making under pressure. In the non-transitive game rock, paper, scissors (RPS), the only technique that guarantees the lack of exploitation is to perform randomly in accordance with mixed-strategy. However, such behavior is thought to be outside bounded rationality and so decision-making can become deterministic, predictable, and ultimately exploitable. This review identifies similarities across economics, neuroscience, nonlinear dynamics, human, and animal cognition literatures, and provides a taxonomy of RPS strategy. RPS strategies are discussed in terms of (a) whether the relevant computations require sensitivity to item frequency, the cyclic relationships between responses, or the outcome of the previous trial, and (b) whether the strategy is framed around the self or other. The negative implication of this taxonomy is that despite the differences in cognitive economy and recursive thought, many of the identified strategies are behaviorally isomorphic. This makes it difficult to infer strategy from behavior. The positive implication is that this isomorphism can be used as a novel design feature in furthering our understanding of the attribution, agency, and acquisition of strategy in RPS and other game spaces.

In this study, the spreading of a liquid droplet on the surface of a fibrous paper and its penetration into the paper is studied. The spreading of the droplet was visualized using confocal microscopy and the penetration depth was quantified using Automatic Scanning Absorptiometry (ASA) measurements. The three-dimensional structure of the paper was obtained through micro-tomography imaging with a resolution of 0.9 µm. The obtained images were used to reconstruct the pore space, which was in turn used in direct numerical simulations of penetration of a droplet into paper. Simulations were performed using open source code OpenFOAM, which solves equations of two-phase flow (in our case air and water) in pores based on the Volume of Fluid Method. Simulation results showed a good agreement with the experimental observations. In particular, the dimensions of spreading area of a droplet and the depth of penetration were simulated reasonably well. Then, we used the model to investigate effects of changes in various liquid properties on spreading and penetration of a droplet liquid. We made calculations for three different values of contact angle (CA): 0°, 60°, and 120°. We found the largest penetration depth for CA = 0. For CA = 60 and CA = 120, we found that the liquid droplet moved sideways from the jetted location, which is not favorable in inkjet printing. We also made simulations with larger values for viscosity and density, based on properties of an ink-based liquid used in inkjet printing. The results have shown a slower spreading and penetration compared with water. The model can be used to study effects of changes in either ink physical properties or paper layer microstructure on final spreading/penetration extent.

Theoretical and experimental research studies have shown that ecosystems governed by non-transitive competition networks tend to maintain high levels of biodiversity. The theoretical body of work, however, has mainly focused on competition networks in which the outcomes of competition events are predetermined and hence deterministic, and where all species are identical up to their competitive relationships, an assumption that may limit the applicability of theoretical results to real-life situations. In this paper, we aim to probe the robustness of the link between biodiversity and non-transitive competition by introducing a three-dimensional winning probability parameter space, making the outcomes of competition events in a three-species in silico ecosystem uncertain. While two degenerate points in this parameter space have been the subject of previous studies, we investigate the remaining settings, which equip the species with distinct competitive abilities. We find that the impact of this modification depends on the spatial dimension of the system. When the system is well mixed, it collapses to monoculture, as is also the case in the non-transitive deterministic setting. In one dimension, chaotic patterns emerge, which tend to maintain biodiversity, and a power law relates the time that species manage to coexist to the degree of uncertainty regarding competition event outcomes. In two dimensions, the formation of spiral wave patterns ensures that biodiversity is maintained for moderate degrees of uncertainty, while considerable deviations from the non-transitive deterministic setting have strong negative effects on species coexistence. It can hence be concluded that non-transitive competition can still produce coexistence when the assumption of deterministic competition is abandoned. When the system collapses to monoculture, one observes a "survival of the strongest" law, as the species that has the highest probability of defeating its competitors has the best odds to become the sole survivor.

The response of paper to humidity variations is a complex, inherently multi-scale problem. The hygroscopic swelling of individual fibres and their interactions within the fibrous network govern the macroscopic, sheet-level response. At this scale, moisture induced instabilities and out-of-plane deformations may occur, which are critical for a number of industrial applications. This work specifically focuses on several aspects of this important issue. A macroscopic phenomenological hygro-mechanical model is first proposed, which aims at predicting moisture induced out-of-plane deformations in paper sheets. The constitutive model is based on the relation between these deformations and typical irreversible phenomena associated to the history of paper manufacturing, i.e. the release of dried-in strains. The model is used to describe bending induced by moisture gradients through the thickness of the sheet as well as buckling due to moisture variation in the presence of mechanical constraints. The results of the model show that the anisotropic sheet-level hygro-expansion has a strong influence on the instability phenomena. Moreover, a comparison with experiments provides adequate semi-quantitative estimates. An additional step is made towards the multi-scale understanding of paper hygro-mechanics. The fundamental physical mechanisms governing the macroscopic moisture induced response are investigated on the basis of the underlying fibrous network. To this aim, a meso-structural model is developed which consists of a network of fibres randomly positioned in a planar region according to an orientation probability density function. A series of network simulations reveals that upon moisture content variations the expansion of the inter-fibre bonding regions essentially drives the overall deformation. Particularly in the case of anisotropic fibre orientation, this explains the origin of the macro-scale anisotropic hygro-expansion, which is essential for the observed sheet-level instability phenomena.

Cockling of paper is a common problem occurring in the production, storage and end-use of paper. It is usually induced by a moisture content change. In many cases, cockling is an irreversible phenomenon; i.e. the initial shape is not obtained although the initial moisture content is restored. This kind of moisture content change occurs in copying machines and in the printing process, for example. In this paper, we present a continuum mechanical model, which is used to study the irreversible cockling of paper. In the model, paper is treated as an orthotropic elasto-plastic material and the model takes into account the small-scale variation of fibre orientation. The model is used to show the importance of the through-thickness moisture gradient on the cockling phenomenon during a cyclic moisture content change. The results suggest that the moisture gradient is a crucial factor for the irreversible cockling.

The rheological nature of paper or board is usually treated either as elasto-plastic or as viscoelastic depending on the studied paper making process or behavior in converting and end use. In this paper we study several stress–strain curve models and the determination of material parameters from an elasto-plastic point of view. Finally, a suitable approach for all stress–strain curves measured from 180 strips is constructed using a linear function for an elastic region and a nonlinear function for a strain hardening region. This model determines a proportional limit (elastic limit) and gives fairly elegant dependencies between material/fitting parameters and two important factors of mechanical properties of paper: dry solids content and anisotropy. In this paper the dependency of a plastic strain on dry solids content and anisotropy is estimated using the introduced stress–strain curve model. Correspondingly, the model can be used to estimate many other mechanical behaviors, for example, the tension differences arising from non-uniform moisture content of the paper web profile. However, the main target of this study is to produce competent parameters based on modeled stress–strain curves for further construction of a material model. This elasto-plastic material model will be utilized in out-of-plane deformation and fracture models.

An in-plane elasto-plastic material model and a hygroexpansivity-shrinkage model for paper and board are introduced in this paper. The input parameters for both models are fiber orientation anisotropy and dry solids content. These two models, based on experimental results, could be used in an analytical approach to estimate, for example, plastic strain and shrinkage in simple one-dimensional cases, but for studies of the combined and more complicated effects of hygro-elasto-plastic behavior, a numerical finite element model was constructed. The finite element approach also offered possibilities for studying different structural variations of an orthotropic sheet as well as buckling behavior and internal stress situations under local strain differences. A few examples are presented of the effect of the anisotropy and moisture streaks under stretching and drying conditions on strain differences and buckling. The internal stresses were studied through a case in which the drying of different layers occurred at different stages. Both the anisotropy and moisture streaks were capable of rendering the buckling of the sample visible. The permanency of these defects highly depends on several process stages and tension conditions of the sheet, as demonstrated in this paper. The application possibilities of the hygro-elasto-plastic model are diverse, including investigation into several phenomena and defects appearing in drying, converting and printing process conditions peerReviewed