Homogeneous Equilibria Research Articles

Turing Instabilities are Not Enough to Ensure Pattern Formation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Counterion-controlled phase equilibria in a charge-regulated polymer solution.

We study phase equilibria in a minimal model of charge-regulated polymer solutions. Our model consists of a single polymer species whose charge state arises from protonation-deprotonation processes in the presence of a dissolved acid, whose anions serve as screening counterions. We explicitly account for variability in the polymers' charge states. Homogeneous equilibria in this model system are characterised by the total concentration of polymers, the concentration of counter-ions and the charge distributions of polymers which can be computed with the help of analytical approximations. We use these analytical results to characterise how parameter values and solution acidity influence equilibrium charge distributions and identify for which regimes uni-modal and multi-modal charge distributions arise. We then study the interplay between charge regulation, solution acidity and phase separation. We find that charge regulation has a significant impact on polymer solubility and allows for non-linear responses to the solution acidity: Re-entrant phase behaviour is possible in response to increasing solution acidity. Moreover, we show that phase separation can yield to the coexistence of local environments characterised by different charge distributions.

Existence and qualitative analysis of a fully cross-diffusive predator-prey system with nonlinear taxis sensitivity

This paper investigates a fully cross-diffusive predator-prey system with nonlinear taxis sensitivity(⋆){ut=D1uxx−χ1(uk1vx)x+μ1u(λ1−u+a1v),vt=D2vxx+χ2(vk2ux)x+μ2v(λ2−v−a2u), under homogeneous boundary conditions of Neumann type for u and v, in an open bounded interval Ω⊂R, where Di,χi,λi,ai>0 and μi≥0 for i∈{1,2}. In Tao and Winkler (2021) [1]; (2022) [2], Tao and Winkler studied the system (⋆) with k1=k2=1 and obtained the global existence and asymptotic behavior. We study the system (⋆) with k1,k2≠1 and prove that:•If k1∈(1,4031) and k2∈[14,12−9k18−5k1), then the model (⋆) possesses global weak solutions for arbitrarily large positive initial data in H1(Ω).•If k1∈(1,1110) and k2∈[23,12−9k18−5k1), then the global weak solution (u,v) of (⋆) with μ1=μ2=0 stabilizes toward homogeneous equilibria (1|Ω|∫Ωu0,1|Ω|∫Ωv0).•If k1∈(1,98) and k2∈[12,12−9k18−5k1), then the global weak solution (u,v) of (⋆) with μ1,μ2>0 converges to constant stable steady state (λ1,0) provided that λ2≤a2λ1 and the tactic sensitivities χ1 and χ2 are suitably small.

On the stability of homogeneous equilibria in the Vlasov–Poisson system on

The goal of this article is twofold. First, we investigate the linearized Vlasov–Poisson system around a family of spatially homogeneous equilibria in (the unconfined setting). Our analysis follows classical strategies from physics (Binney and Tremaine 2008 Galactic Dynamics (Princeton University Press); Landau 1946 Acad. Sci. USSR. J. Phys. 10 25–34; Penrose 1960 Phys. Fluids 3 258–65) and their subsequent mathematical extensions (Bedrossian et al 2022 SIAM J. Math. Anal. 54 4379–406; Degond 1986 Trans. Am. Math. Soc. 294 435–53; Glassey and Schaeffer 1994 Transp. Theory Stat. Phys. 23 411–53; Grenier et al 2021 Math. Res. Lett. 28 1679–702; Han-Kwan et al 2021 Commun. Math. Phys. 387 1405–40; Mouhot and Villani 2011 Acta Math. 207 29–201). The main novelties are a unified treatment of a broad class of analytic equilibria and the study of a class of generalized Poisson equilibria. For the former, this provides a detailed description of the associated Green’s functions, including in particular precise dissipation rates (which appear to be new), whereas for the latter we exhibit explicit formulas. Second, we review the main result and ideas in our recent work (Ionescu et al 2022 (arXiv:2205.04540)) on the full global nonlinear asymptotic stability of the Poisson equilibrium in .

Stability of a diffusive-delayed HCV infection model with general cell-to-cell incidence function incorporating immune response and cell proliferation.

In this work, we analyse the dynamics of a five-dimensional hepatitis C virus infection mathematical model including the spatial mobility of hepatitis C virus particles, the transmission of hepatitis C virus infection by mitosis process of infected hepatocytes with logistic growth, time delays, antibody response and cytotoxic T lymphocyte (CTL) immune response with general incidence functions for both modes of infection transmission, namely virus-to-cell as well as cell-to-cell. Firstly, we prove rigorously the existence, the uniqueness, the positivity and the boundedness of the solution of the initial value and boundary problem associated with the new constructed model. Secondly, we found that the basic reproductive number is the sum of the basic reproduction number determined by cell-free virus infection, determined by cell-to-cell infection and determined by proliferation of infected cells. It is proved the existence of five spatially homogeneous equilibria known as infection-free, immune-free, antibody response, CTL response and antibody and CTL responses. By using the linearization methods, the local stability of the latter is established under some rigorous conditions. Finally, we proved the existence of periodic solutions by highlighting the occurrence of a Hopf bifurcation for a certain threshold value of one delay.

Revisiting the Thermodynamic Characteristics of Titanium Fluorides

New values for the standard enthalpies of formation of titanium fluorides at 0 K have been obtained by revisiting data on homogeneous and heterogeneous equilibria with their participation (kJ/mol): –1403.2 ± 2.7 for TiF3(c), –1144.2 ± 3.3 for TiF3, –2349.4 ± 6.3 for Ti2F5(c); –933.7 ± 6.8 for TiF2(c), and –650.4 ± 2.4 TiF2. The thermodynamic metastability of TiF2(c) has been shown.

Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction–diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

Qualitative analysis for a diffusive predator-prey model with hunting cooperation and holling type III functional response

The Spatio-temporal pattern induced by self-diffusion of a predator-prey model with Boiling type III functional response that incorporates the hunting cooperation between predators has been investigated in this paper. For the local model without structure, the stability of non-negative equilibria with or without collaborative hunting in predators is studied. For the Spatio-temporal model, we analyze the effect of hunting cooperation term on diffusion-driven Turing instability of the homogeneous positive equilibria. To get an idea about patterns formation near the Turing bifurcation, we derive and give a detailed study of the amplitude equation using the multiple-scale analysis. Our result shows that hunting cooperation plays a crucial role in determining the stability and the Turing bifurcation of the model, which is in sharp contrast to the case without cooperation in hunting. Furthermore, some numerical simulations are illustrated to visualize the complex dynamic behavior of the model.

Depinning transition of travelling waves for particle chains

In this paper we apply Aubry–Mather theory for equilibria of 1D Hamiltonian lattice systems and the theory of invariant ordered circles to investigate the depinning transition of travelling waves for particle chains. Assume A < B are two critical values such that the particle chain has three homogeneous equilibria if the driving force . It is already known that there exist transition thresholds of the driving force such that the particle chain has stationary fronts but no travelling fronts for and travelling fronts but no stationary fronts if or . The novelty of our approach is that we prove the transition threshold () coincides with the upper (lower) limit of the upper (lower) depinning force as the rotation number tends to zero from the right. Based on this conclusion, we demonstrate that when the driving force , besides stationary fronts there are various kinds of equilibria with rotation numbers close to zero such that the spatial shift map has positive topological entropy on the set of equilibria. Furthermore, we give a necessary and sufficient condition for the absence of propagation failure, i.e. , in terms of a minimal foliation. Finally we show that are continuous with respect to potential functions in C 1 topology.

A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation

This work is concerned with a two-component parabolic system accounting for a doubly cross-diffusive interaction mechanism which was predicted in Tsyganov et al. (2003) [48] as responsible for the occurrence of certain solitary propagating waves in so-called pursuit-evasion systems. This system formally possesses two basic entropy-like structures, but especially in the presence of large data the regularity features thereby implied seem insufficient to ensure global extensibility of local-in-time classical solutions provided by known results on classical solvability in general parabolic systems of not necessarily tridiagonal type. Attempting to nevertheless develop a basic theory of existence and qualitative behavior, the manuscript firstly constructs global solutions within a natural concept of weak solvability and for arbitrarily large data, and secondly derives a result on large-time stabilization toward homogeneous equilibria. A major challenge connected with this appears to consist in designing a suitable regularization which complies with the two requirements of asserting global solvability in the corresponding approximate systems on the one hand, and of retaining consistency with essential structural properties on the other. To adequately cope with this, a fourth-order regularization is pursued which, besides essentially respecting said entropy features, conforms to the fundamental sine qua non of positivity preservation by involving thin-film type degeneracies in the associated artificial diffusion operators. Here the use of embeddings enforces a restriction to spatially one-dimensional settings, in which an apparently novel refinement of Gagliardo-Nirenberg interpolation reveals a crucial L1 compactness feature of corresponding cross-diffusive fluxes.

Asymptotic profile of a two-dimensional Chemotaxis–Navier–Stokes system with singular sensitivity and logistic source

This paper is concerned with a spatially two-dimensional version of a chemotaxis system with logistic cell proliferation and death, for a singular tactic response of standard logarithmic type, and with interaction with a surrounding incompressible fluid through transport and buoyancy. Systems of this form are of significant relevance to the understanding of chemotaxis-fluid interaction, but the rigorous knowledge of their qualitative properties is yet far from complete. In this direction, using the conditional energy functional method, the present work provides some interesting contributions by establishing results on global boundedness, and especially on large time stabilization toward homogeneous equilibria, under mild assumptions on the initial data and appropriate conditions on the strength of the damping death effects.

Landau damping in one dimensional periodic inhomogeneous collisionless plasmas

Landau damping in a collisionless plasma is a well-known example of wave particle interaction. In the past, this phenomenon was addressed for homogeneous equilibria in the linear and non-linear limit of the perturbation amplitude. However, in reality, equilibria are almost always inhomogeneous or non-uniform in space. Considering a one dimensional, collisionless, unmagnetized, electrostatic plasma with stationary ions and kinetic electrons in a periodic inhomogeneous exact equilibrium of scale k0−1 as the starting point, the fate of a small amplitude (linear) perturbation of scale k−1 is investigated using a Vlasov–Poisson solver. Three different spatial regimes, namely, k0 &amp;gt; k, k0 ∼ k, and k0 &amp;lt; k, are addressed. In the k0 &amp;gt; k regime, long wavelength perturbation k is found to generate (k ± Nk0) modes (where N is an integer), which allows damping of long wavelength perturbation in an inhomogeneous plasma and formation of phase-space vortices at phase velocities vϕ = ω/(k ± Nk0). Perhaps for the first time, novel phenomena such as “inhomogeneity induced Landau damping arrest” and “inhomogeneity induced plasma echo” are observed in k0 ∼ k and k0 &amp;lt; k regimes, respectively. New scaling laws as a function of inhomogeneity amplitude are also reported.

Mathematical models for diseases in wildlife populations with indirect transmission

In this paper, five different models for five different kinds of diseases occurring in wildlife populations are introduced. In all models, a logistic growth term is taken into account and the disease is transmitted to the susceptible population indirectly through an environment reservoir. The time evolution of these diseases is described together with its spatial propagation. The character of spatial homogeneous equilibria against the uniform and non-uniform perturbations together with the occurrence of Hopf bifurcations are discussed through a linear stability analysis. No Turing instability is observed. The partial differential field equations are also integrated numerically to validate the stability results herein obtained and to extract additional information on the temporal and spatial behavior of the different diseases.

Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium

The asymptotic behavior of the solutions of the second order linearized Vlasov-Poisson system around homogeneous equilibria is derived. It provides a fine description of some nonlinear and multidimensional phenomena such as the existence of Best frequencies. Numerical results for the 1D×1D and 2D×2D Vlasov-Poisson system illustrate the effectiveness of this approach.

Thermodynamic analysis of the copper (I) homogeneous and heterogeneous speciation in ammonium thiosulfate leaching systems

The paper presents a new thermodynamic approach to studying mixed ligand complex formation reactions in the multi-component two-phase system “cuprous oxide-ammoniacal copper (I) thiosulfate aqueous solution” under real conditions. The quintessence of the developed approach consists in the thermodynamic analysis of concurrent reactions in the heterogeneous system Cu(I) – NH3 – S2O32− – H2O on the basis of the introduced notion of generalized reaction equations. A detailed thermodynamic analysis in ammonium thiosulfate leaching systems has considered the formation of mixed ligand complex species Cu(OH)i(NH3)j(S2O3)k. The new feature is to focus on the chemical description of the overall process of complex formation consisting of a series of concurrent reactions, where the nature and ratio of the concentrations of chemical species, formed in such reactions, depend on the ratio of the concentrations of the metal Cu+ and ligands, temperature, and other thermodynamic parameters. On the basis of the selected thermodynamic data for involved species, the thermodynamic stability areas of solid phase and the repartition of soluble and insoluble chemical species towards the solution pH and a number of total reagent concentrations in heterogeneous mixtures have been investigated. The diagrams of heterogeneous and homogeneous chemical equilibria are used for the graphical representation of complex equilibria and distribution of soluble and insoluble species in examined systems. The used approach utilizes thermodynamic relationships coupled with original mass balance constraints, where the solid phases are explicitly expressed. The obtained results are useful for optimization of the overall scheme of thiosulfate leaching in gold hydrometallurgy to minimize reagent consumption and maximize gold extraction.

Multichromatic travelling waves for lattice Nagumo equations

We discuss multichromatic front solutions to the bistable Nagumo lattice differential equation. Such fronts connect the stable spatially homogeneous equilibria with spatially heterogeneous n-periodic equilibria and hence are not monotonic like the standard monochromatic fronts. In contrast to the bichromatic case, our results show that these multichromatic fronts can disappear and reappear as the diffusion coefficient is increased. In addition, these multichromatic waves can travel in parameter regimes where the monochromatic fronts are also free to travel. This leads to intricate collision processes where an incoming multichromatic wave can reverse its direction and turn into a monochromatic wave.

Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions

This work is concerned with a prototypical model for the spatio-temporal evolution of a forager–exploiter system, consisting of two species which simultaneously consume a common nutrient, and which interact through a taxis-type mechanism according to which individuals from the exploiter subpopulation move upward density gradients of the forager subgroup. Specifically, the model [Formula: see text] for the population densities [Formula: see text] and [Formula: see text] of foragers and exploiters, as well as the nutrient concentration [Formula: see text], is considered in smoothly bounded domains [Formula: see text], [Formula: see text]. It is first shown that under an explicit condition linking the sizes of the resource production rate [Formula: see text] and of the initial nutrient concentration, an associated Neumann-type initial-boundary value problem admits a global solution within an appropriate generalized concept. The second of the main results asserts stabilization of these solutions toward spatially homogeneous equilibria in the large time limit, provided that [Formula: see text] satisfies a mild assumption on temporal decay. To the best of our knowledge, these are the first rigorous analytical results addressing taxis-type cross-diffusion mechanisms coupled in a cascade-like manner as in (⋆).

Global stability of homogeneous steady states in scaling-invariant spaces for a Keller–Segel–Navier–Stokes system

In this paper, we study the global stability of homogeneous equilibria in Keller–Segel–Navier–Stokes equations in scaling-invariant spaces. We prove that for any given 0<M<1+μ1 with μ1 being the first eigenvalue of Neumann Laplacian, the initial–boundary value problem of the Keller–Segel–Navier–Stokes system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to (M,M,0) in the norm of Ld/2(Ω)×W˙1,d(Ω)×Ld(Ω) and satisfies a natural average mass condition. Our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup. Our result suggests a new observation that nontrivial classical solution for Keller–Segel–Navier–Stokes equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large, correspondingly.

Bichromatic Travelling Waves for Lattice Nagumo Equations

We discuss bichromatic (two-color) front solutions to the bistable Nagumo lattice differential equation. Such fronts connect the stable spatially homogeneous equilibria with spatially heterogeneous...

A Numerical Investigation of the Pinning Phenomenon in Quasi-Periodic Frenkel Kontrova Model Under an External Force

We perform a numerical study on Frenkel–Kontrova (FK) models in quasi-periodic media subject to a constant external force. The quasi-periodic FK models appear in several physical problems such as deposition, spin waves in 1-D, planar dislocations in 3-D either in a quasi-crystal or on a cleaved surface with irrational slopes of a periodic crystal. The phenomenon that we study is pinning. When there is no force, the very homogeneous equilibria are abundant, but when there is a constant force, the uniformly distributed equilibrium states just slide away and the physically relevant equilibria are configurations with frequencies (inverse of densities in some interpretation) that are resonant with the frequencies of the medium. We use a dynamical interpretation of the equilibrium equation and perform a systematic Monte-Carlo exploration of the equilibria. We study the geometry and quantitative properties of the equilibria after the application of a constant external force. This leads to several quantitative regularities on the abundance and geometric properties of the pinned states in the models we consider. This complements recent qualitative studies.