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This paper investigates the fuzzy control design problem for reaction-diffusion memristive neural networks (MNNs). Initially, by introducing the logical switched functions, the original reaction-diffusion MNN is transformed into another model form. Subsequently, a Takagi-Sugeno (T-S) fuzzy PDE model is devoted to accurately representing the reaction-diffusion MNN. Then, via the obtained T-S fuzzy model, under the hypothesis that the actuators and sensors are collocated while the spatial domain is separated into several subdomains, a Lyapunov-based membership-function-dependent fuzzy control design employing a finite number of actuators and sensors is developed in terms of linear matrix inequalities, such that the closed-loop reaction-diffusion MNN is exponentially stable. In final, numerical simulations illustrate the effectiveness of the proposed fuzzy control design method. [ABSTRACT FROM AUTHOR]

We study the network reconstruction problem for an epidemic reaction–diffusion system. These systems are an extension of deterministic, compartmental models to a graph setting, where the reactions within the nodes are coupled by a diffusion dynamics. We study the influence of the diffusion rate and the network topology, on the reconstruction and prediction problems, both from a theoretical and experimental standpoint. Results first show that for almost every network, the reconstruction problem is well posed. Then, we show that the faster the diffusion dynamics, the harder the reconstruction, but that increasing the sampling rate may help in this respect. Second, we demonstrate that it is possible to classify symmetrical networks generating the same trajectories, and that the prediction problem can still be solved satisfyingly, even when the network topology makes exact reconstruction difficult. [ABSTRACT FROM AUTHOR]

These lecture notes address the controllability under state constraints of reaction-diffusion equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities.The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle.We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics.The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome.We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems' dynamics and, in particular, the use of traveling waves and steady-states' paths, can be employed to build controls driving the system to the desired target.We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction.Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and results. [ABSTRACT FROM AUTHOR]

This article focuses on the composite $\mathcal {H}_{\infty }$ synchronization problem for jumping reaction–diffusion neural networks (NNs) with multiple kinds of disturbances. Due to the existence of disturbance effects, the performance of the aforementioned system would be degraded; therefore, improving the control performance of closed-loop NNs is the main goal of this article. Notably, for these disturbances, one of them can be described as a norm-bounded, and the other is generated by an exogenous model. In order to reject the above one kind of disturbance, a disturbance observer is developed. Furthermore, combining the disturbance observer approach and conventional state-feedback control scheme, a composite disturbance rejection controller is specifically designed to compensate for the influences of the disturbances. Then, some criteria are established based on the general Lyapunov stability theory, which can ensure that the synchronization error system is stochastically stable and satisfies a fixed $\mathcal {H}_{\infty } $ performance level. A simulation example is finally presented to verify the availability of our developed method. [ABSTRACT FROM AUTHOR]

This paper investigates the synchronization of inertial reaction-diffusion Cohen-Grossberg-type neural networks. Compared with the existing works concerning reaction-diffusion neural networks, the main innovation of this paper is that two design strategies of feedback synchronization controllers are proposed based on the types of time delays. For the systems with bounded differentiable delays, the sufficient conditions for synchronization are derived under the framework of Lyapunov method. If the time delay of the addressed system is unbounded or non-differentiable, it can also realize synchronization by employing the method of variation of parameters and some analytical techniques. Moreover, the proposed methods are applicable to various boundary conditions. The correctness of the obtained criteria is verified by three numerical examples. [ABSTRACT FROM AUTHOR]

Atom- and energy-efficient chemical separations are urgently needed to meet the surging demand for critical materials that has strained supply chains and threatened environmental damage. In this study, we used reaction-diffusion coupling to separate iron, neodymium, and dysprosium ions from model feedstocks of permanent magnets, which are typically found in electronic wastes. Feedstock solutions were placed in contact with a hydrogel loaded with potassium hydroxide and/or dibutyl phosphate, resulting in complex precipitation patterns as the various metal ions diffused into the reaction medium. Specifically, we observed the precipitation of up to 40 mM of iron from the feedstock, followed by the enrichment of 73% dysprosium, and the extraction of >95% neodymium product at a further distance from the solution-gel interface. We designed a series of experiments and simulations to determine the relevant ion diffusivities, DNd = 5.4×10-10 and DDy = 5.1×10-10 m2/s, and precipitation rates, kNd = 1.0×10-5 and kDy = = 5.0×10-3 m9mol-3s-1, which enabled a numerical model to be established for predicting the distribution of products in the reaction medium. Our proof-of-concept study validates reaction-diffusion coupling as an effective and versatile approach for critical materials separations, without relying on ligands, membranes, resins, or other specialty chemicals., (© 2025 Wiley‐VCH GmbH.)

One of the most important vector-borne disease in humans is malaria, caused by Plasmodium parasite. Seasonal temperature elements have a major effect on the life development of mosquitoes and the development of parasites. In this paper, we establish and analyze a reaction-diffusion model, which includes seasonality, vector-bias, temperature-dependent extrinsic incubation period (EIP) and maturation delay in mosquitoes. In order to get the model threshold dynamics, a threshold parameter, the basic reproduction number $ R_{0} $ is introduced, which is the spectral radius of the next generation operator. Quantitative analysis indicates that when $ R_{0} < 1 $, there is a globally attractive disease-free $ \omega $-periodic solution; disease is uniformly persistent in humans and mosquitoes if $ R_{0} > 1 $. Numerical simulations verify the results of the theoretical analysis and discuss the effects of diffusion and seasonality. We study the relationship between the parameters in the model and $ R_{0} $. More importantly, how to allocate medical resources to reduce the spread of disease is explored through numerical simulations. Last but not least, we discover that when studying malaria transmission, ignoring vector-bias or assuming that the maturity period is not affected by temperature, the risk of disease transmission will be underestimate.

This article is devoted to gain a better understanding of the synchronization control mechanism in networked, complex-valued reaction-diffusion systems via an impulsive and sampled-data controller. Different from the traditional control methods with hybrid mechanism, an impulsive and sampled-data control scheme is proposed for fractional-order complex-valued reaction-diffusion neural networks (FRDCVNNs). By utilizing Lyapunov functional and inequality techniques, finite-time Mittag-Leffler synchronization criteria via an impulsive and sampled-data controller are established and presented as linear matrix inequalities (LMIs), which can ensure the finite-time synchronization of error system containing the drive and response dynamics. In addition, synchronization control mechanism on the considered system via impulsive actuator saturation and sampled-data control are applied. Simulation examples are provided to verify the efficacy of the proposed synchronization criterion and the results of practical application to image encryption scheme based on the chaotic complex system is presented. Furthermore, the proposed cryptosystem have obvious advantages of large key space and high security.

In this paper, we develop a diffusive and delayed hepatitis B virus (HBV) infection model, which incorporates both virus‐to‐cell and cell‐to‐cell transmissions along with cytotoxic T lymphocyte (CTL) immune response. Two threshold parameters ℜ0$$ {\mathfrak{\Re}}_0 $$ (basic reproduction number) and ℜ1$$ {\mathfrak{\Re}}_1 $$ (immune response reproduction number) are defined, which determine the dynamic behavior of the model. Employing the approach of Lyapunov functionals, we show that the infection‐free steady state is globally asymptotically stable when ℜ0≤1$$ {\mathfrak{\Re}}_0\le 1 $$; the immune‐free infected steady state is globally asymptotically stable when ℜ1≤1<ℜ0$$ {\mathfrak{\Re}}_1\le 1<{\mathfrak{\Re}}_0 $$; and the immune‐present steady state is globally asymptotically stable when ℜ0>1,ℜ1>1$$ {\mathfrak{\Re}}_0>1,{\mathfrak{\Re}}_1>1 $$, and φ(I)=I$$ \varphi (I)=I $$. We perform the sensitivity and uncertainly analysis on ℜ0$$ {\mathfrak{\Re}}_0 $$ with respect to parameters and find that the most important parameter that drives the HBV infection is the mortality of HBV DNA capsids. Moreover, the theoretical results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

We consider a system of two reaction-diffusion equations coming out of reversible chemistry. When the reaction happens on the totality of the domain, it is known that exponential convergence to equilibrium holds (with explicit rate). We show in this paper that this exponential convergence also holds when the reaction happens only on a given open set of a ball, thanks to an observation estimate deduced by logarithmic convexity. [ABSTRACT FROM AUTHOR]

Lattice structures can present advantageous mechanical properties while remaining remarkably lightweight. Precise lattice design can however be tricky to set up with classical 3D modeling methods as it involves very fine details. Interestingly, natural porous structures can present such lattice-like or membrane-like features which motivates to seek for more bio-inspired approaches to microstructure design. In this paper, we present a novel method to grow lattice-like and membrane-like structures within an arbitrary shape and aligned along an oriented field. Our method relies on the use of a dedicated anisotropic reaction–diffusion system guided by an orthotropic diffusion tensor field. Assuming for instance the diffusion tensor to be related to the stress analysis of a given shape allows to generate emerging stripes patterns aligned along each one of the principal stress directions independently. A globally coherent mechanical model conforming to the initial shape boundary and infilled with oriented microstructures can therefore be synthesized. Further, we demonstrate the capability of this approach to handle other types of oriented fields such as obtained through optimization of material directions in scenarios with multiple load cases. Our approach relies on spatially and temporally local operations allowing for efficient parallelization. This permits user-interaction and automated adaptation of the design, even for fine meshes over large volumes. For instance, a designer can locally erase or “draw” over the structure and let it regrow and adapt as well as enforce regions to be deliberately full or empty. The proposed approach yields smooth and conformal oriented anisotropic geometrical patterns. This is related to recent effort in the structural optimization community on the topic of optimized oriented infills and microstructure de-homogenization. One of the resulting designs is validated by means of a full scale general non-linear analysis showcasing the advantageous properties of oriented microstructures for stability and robustness to buckling. [ABSTRACT FROM AUTHOR]

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction. [ABSTRACT FROM AUTHOR]

In this article, a new competitive neural network (CNN) with reaction-diffusion terms and mixed delays is proposed. Because this network system contains reaction-diffusion terms, it belongs to a partial differential system, which is different from the existing classic CNNs. First, taking into account the spatial diffusion effect, we introduce spatial diffusion for CNNs. Furthermore, since the time delay has an essential influence on the properties of the system, we introduce mixed delays including time-varying discrete delays and distributed delays for CNNs. By constructing suitable Lyapunov–Krasovskii functionals and virtue of the theories of delayed partial differential equations, we study the global asymptotic stability for the considered system. The effectiveness and correctness of the proposed CNN model with reaction-diffusion terms and mixed delays are verified by an example. Finally, some discussion and conclusions for recent developments of CNNs are given. [ABSTRACT FROM AUTHOR]

Exponential stability is considered for delay reaction–diffusion cellular neural networks (DRDCNNs) under two cases where the state information is fully available and not fully available. When the state information of controlled system is fully available, an aperiodically intermittent boundary controller is designed to stabilize the controlled system. When the state information is not fully available, we propose an observer based on the boundary output to estimate the system state, and an observer-based aperiodically intermittent boundary controller is designed. Employing the Lyapunov functional method and Poincaré's inequality, we obtain a criterion to ensure DRDCNNs achieve the exponential stabilization. Based on our obtained results, the influence of diffusion coefficient matrix, control gains, time-delays and control proportion on the stability are studied. To illustrate the effectiveness of our theoretical results, at last, numerical examples are given. [ABSTRACT FROM AUTHOR]

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In this paper, we investigate an SVEIR epidemic model with reaction–diffusion and nonlinear incidence. We establish the well-posedness of the solutions and the basic reproduction number R 0 . Moreover, we show that the disease-free steady state is globally asymptotically stable when R 0 < 1 , whereas the disease will be persistent when R 0 > 1 . Furthermore, using the method of Lyapunov functional, we prove the global stability of the positive steady state for the spatial homogeneous model. [ABSTRACT FROM AUTHOR]

Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, $\tau $ , between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on $\tau $. A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive $\tau $. [ABSTRACT FROM AUTHOR]

The issue of H ∞ /passive master–slave synchronization for Markov jumping neural networks with reaction–diffusion terms is investigated in this paper via an event-triggered control scheme under deception attacks. To lighten the burden of limited communication bandwidth as well as ensure the control performance, an event-triggered transmission scheme is developed. Meanwhile, the randomly occurring deception attacks, which received from the event generator are assumed to modify the sign of the control signal, are taken into account. Furthermore, sufficient conditions ensuring the prescribed H ∞ /passive performance level of the neural networks, are deduced beyond Lyapunov stability theory, and the controller gains are derived dealing with the matrix convex optimization problem. At last, the availability of the approach proposed is demonstrated via a numerical example. • Different from references (Dharani et al., 2017; Wang et al., 2021), this paper considers RDTs and parameter uncertainty. • The random deception attacks is considered. ETTS is used to save bandwidth resources. • By using LKF and inequalities, the H ∞ /passive synchronization performance is achieved. [ABSTRACT FROM AUTHOR]

The article considers a two-phase problem with a free boundary for a quasilinear parabolic equation of the reaction-diffusion type with discontinuous coefficients along a moving and previously unknown line. The problem is studied according to the following scheme. First, we establish a priori estimates for the Holder norms for the first derivatives and ensure that the domain is nondegenerate. Next, we obtain a priori estimates of higher orders. The uniqueness of the solution of the problem is obtained. The existence of a solution to the problem is proved using the Leray-Schauder principle. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-species competitive quasi-linear Lotka-Volterra system equipped with a free boundary. A priori estimates of the Hölder norms are established. The global existence and uniqueness of classical solutions are established, and then the asymptotic behavior of the solutions is described. [ABSTRACT FROM AUTHOR]

We study reaction–diffusion equations of various types in the half-space. For bistable reactions with Dirichlet boundary conditions, we prove conditional uniqueness: there is a unique nonzero bounded steady state which exceeds the bistable threshold on large balls. Moreover, solutions starting from sufficiently large initial data converge to this steady state as t → ∞. For compactly supported initial data, the asymptotic speed of this propagation agrees with the unique speed c* of the one-dimensional traveling wave. We furthermore construct a traveling wave in the halfplane of speed c*. In parallel, we show analogous results for ignition reactions under both Dirichlet and Robin boundary conditions. Using our ignition construction, we obtain stronger results for monostable reactions with the same boundary conditions. For such reactions, we show in general that there is a unique nonzero bounded steady state. Furthermore, monostable reactions exhibit the hair-trigger effect: every solution with nontrivial initial data converges to this steady state as t → ∞. Given compactly supported initial data, this disturbance propagates at a speed c* equal to the minimal speed of one-dimensional traveling waves. We also construct monostable traveling waves in the Dirichlet or Robin half-plane with any speed c ≥ c*. [ABSTRACT FROM AUTHOR]

The Gray--Scott model is a widely studied autocatalytic model that exhibits a range of interesting pattern formation behavior, as well as a rich structure of dynamics that includes many ideas from a typical undergraduate dynamical systems course, and some from beyond. Gaining an understanding of the solutions to this model is arguably most easily conducted via a bifurcation analysis of corresponding ODE problems within the software XPPAUT. In this paper, we provide an introductory XPPAUT tutorial, through which we begin to expose the range of intricate patterns that the Gray--Scott model emits. [ABSTRACT FROM AUTHOR]

The Fisher–Kolmogorov–Petrovsky–Piskunov (KPP) model, and generalizations thereof, involves simple reaction–diffusion equations for biological invasion that assume individuals in the population undergo linear diffusion with diffusivity |$D$|⁠ , and logistic proliferation with rate |$\lambda $|⁠. For the Fisher–KPP model, biologically relevant initial conditions lead to long-time travelling wave solutions that move with speed |$c=2\sqrt {\lambda D}$|⁠. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher–KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically relevant initial conditions lead to travelling waves that move with speed |$c=2\sqrt {\lambda D}> 0$|⁠. This means that, for biologically relevant initial data, the Fisher–KPP model cannot be used to study invasion with |$c \ne 2\sqrt {\lambda D}$|⁠ , or retreating travelling waves with |$c < 0$|⁠. Here, we reformulate the Fisher–KPP model as a moving boundary problem and show that this reformulated model alleviates the key limitations of the Fisher–KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front that can propagate with any wave speed, |$-\infty < c < \infty $|⁠. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub. [ABSTRACT FROM AUTHOR]

Dissipativity-based asynchronous boundary stabilization problem is addressed for stochastic Markov jump reaction-diffusion systems (SMJRDSs). In practical engineering, nonsynchronous behavior between system modes and controller modes is inevitable, and the incomplete matrix information makes the problem analysis difficult, so this work considers the asynchronous stabilization. Different from the distributed control, we apply a simple boundary control strategy, which greatly reduces the cost of the control design. Note that three issues need to be addressed: 1) how to model the asynchronous behavior? 2) how to design the asynchronous boundary controller? and 3) how to process the incomplete matrix information? We deal with these problems one by one. Based on a general hidden Markov model (HMM), an asynchronous boundary feedback controller is considered. Via the Wirtinger-type inequality, Schur complement technique, and transition matrix properties, sufficient conditions ensuring exponentially mean square stability and strictly $(W, P, R)-\alpha $ dissipativity are established, which covers several special cases. Finally, a numerical example is presented to illustrate the proposed control strategies. [ABSTRACT FROM AUTHOR]

We analyse a PDE system modelling poromechanical processes (formulated in mixed form using the solid deformation, fluid pressure, and total pressure) interacting with diffusing and reacting solutes in the medium. We investigate the well-posedness of the nonlinear set of equations using fixed-point theory, Fredholm's alternative, a priori estimates, and compactness arguments. We also propose a mixed finite element method and demonstrate the stability of the scheme. Error estimates are derived in suitable norms, and numerical experiments are conducted to illustrate the mechano-chemical coupling and to verify the theoretical rates of convergence. [ABSTRACT FROM AUTHOR]

The immobilization of enzymes in solid‐state nanochannels is a new avenue for the design of biosensors with outstanding selectivity and sensitivity. This work reports the first theoretical model of an enzymatic nanochannel biosensor. The model is applied to the system previously experimentally studied by Lin, et al. (Anal. Chem. 2014, 86, 10546): a hourglass nanochannel modified by glucose oxidase for the detection of glucose. Our predictions are in good agreement with experimental observations as a function of the applied potential, pH and glucose concentration. The sensing mechanism results from the combination of three processes: i) the establishment of a steady‐state proton concentration gradient due to a reaction‐diffusion mechanism, ii) the effect of that gradient on the charge of the adsorbed enzymes and native surface groups, and iii) the effect of the resulting surface charge on the ionic current. Strategies to improve the sensor performance based on this mechanism are identified and discussed. [ABSTRACT FROM AUTHOR]

Pattern formation arising from the reaction-diffusion epidemic model is a space-time depiction of the distribution and transmission of infectious diseases. Disease control can be achieved by controlling the associated pattern formations. For an SIR reaction-diffusion epidemic model, we review its Turing pattern formations with different transmission rates under the case of a constant recovery rate. To control pattern formations of the SIR epidemic model, we introduce a regulator as a control function of the recovery rate. For a perfect control strategy, it not only leads to a desired pattern formation, but also has a small support in space-time domains. In order to obtain such a control strategy, we propose a sparse optimal control problem governed by the SIR epidemic model. We study the existence of optimal solutions, derive the first order necessary optimality system, obtain the sparsity structure of the control function, and numerically solve the control problem. Numerical results demonstrate the feasibility and effectivity of our method. [ABSTRACT FROM AUTHOR]

It is well known that integer-order neural networks with diffusion have rich spatial and temporal dynamical behaviors, including Turing pattern and Hopf bifurcation. Recently, some studies indicate that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we mainly investigate the Turing pattern in a delayed reaction-diffusion neural network with Caputo-type fractional derivative. In particular, we find that this fractional neural network can form steadily spatial patterns even if its first-derivative counterpart cannot develop any steady pattern, which implies that temporal fractional derivative contributes to pattern formation. Numerical simulations show that both fractional derivative and time delay have influence on the shape of Turing patterns. [ABSTRACT FROM AUTHOR]

We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at x = ± ∞. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed. [ABSTRACT FROM AUTHOR]

We propose a mathematical spatiotemporal epidemic SICA model with a control strategy. The spatial behavior is modeled by adding a diffusion term with the Laplace operator, which is justified and interpreted both mathematically and physically. By applying semigroup theory on the ordinary differential equations, we prove existence and uniqueness of the global positive spatiotemporal solution for our proposed system and some of its important characteristics. Some illustrative numerical simulations are carried out that motivate us to consider optimal control theory. A suitable optimal control problem is then posed and investigated. Using an effective method based on some properties within the weak topology, we prove existence of an optimal control and develop an appropriate set of necessary optimality conditions to find the optimal control pair that minimizes the density of infected individuals and the cost of the treatment program. [ABSTRACT FROM AUTHOR]

The dynamics of a mathematical model of the Calvin cycle, which is part of photosynthesis, is analysed. Since diffusion of ATP is included in the model a system of reaction-diffusion equations is obtained. It is proved that for a suitable choice of parameters there exist spatially inhomogeneous positive steady states, in fact infinitely many of them. It is also shown that all positive steady states, homogeneous and inhomogeneous, are nonlinearly unstable. The only smooth steady state which could be stable is a trivial one, where all concentrations except that of ATP are zero. It is found that in the spatially homogeneous case there are steady states with the property that the linearization about that state has eigenvalues which are not real, indicating the presence of oscillations. Numerical simulations exhibit solutions for which the concentrations are not monotone functions of time. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel coupled poroelasticity-diffusion model for the formation of extracellular oedema and infectious myocarditis valid in large deformations, manifested as an interaction between interstitial flow and the immune-driven dynamics between leukocytes and pathogens. The governing partial differential equations are formulated in terms of skeleton displacement, fluid pressure, Lagrangian porosity, and the concentrations of pathogens and leukocytes. A five-field finite element scheme is proposed for the numerical approximation of the problem, and we provide the stability analysis for a simplified system emanating from linearisation. We also discuss the construction of an adequate, Schur complement based, nested preconditioner. The produced computational tests exemplify the properties of the new model and of the finite element schemes. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global stability of the spatially nonhomogeneous steady state solution of a reaction diffusion equation with nonlocal delay under the Dirichlet boundary condition. To achieve this, we obtain the global existence and nonnegativity of solutions and give an extensive study on the properties of omega limit sets. [ABSTRACT FROM AUTHOR]

This paper presents numerical treatments for a class of singularly perturbed parabolic partial differential equations with nonlocal boundary conditions. The problem has strong boundary layers at x = 0 and x = 1. The nonstandard finite difference method was developed to solve the considered problem in the spatial direction, and the implicit Euler method was proposed to solve the resulting system of IVPs in the temporal direction. The nonlocal boundary condition is approximated by Simpsons 13 rule. The stability and uniform convergence analysis of the scheme are studied. The developed scheme is second-order uniformly convergent in the spatial direction and first-order in the temporal direction. Two test examples are carried out to validate the applicability of the developed numerical scheme. The obtained numerical results reflect the theoretical estimate.

The vertebrate axis is segmented into repetitive structures, the vertebrae. In fish, these segmented structures are thought to form from the paraxial mesoderm and the adjacent notochord. Recent work revealed an autonomous patterning mechanism in the zebrafish notochord, with inputs from the segmented paraxial mesoderm. The notochord pattern is established in a sequential manner, progressing from anterior to posterior. Building on this previous work, here, we propose a reaction wavefront theory describing notochord patterning in zebrafish. The pattern is generated by an activator–inhibitor reaction–diffusion mechanism. Cues from the paraxial mesoderm are introduced as a profile of inhibitor sinks. Reactions are turned on by a wavefront that advances from anterior to posterior. We show that this reaction wavefront ensures that a pattern is formed sequentially, in register with the cues, despite the presence of fluctuations. We find that the velocity and shape of the reaction wavefront can modulate the prevalence of defective patterns. Normal patterning is supported in a wide range of sink profile wavelengths, while a minimum sink strength is required for the pattern to follow the cues. The theory predicts that distinct defect types occur for small or large wavelengths. Thus, the reaction wavefront theory provides a possible scenario for notochord patterning, with testable predictions that prompt future experiments.

Abstract Background Reintroducing predators is a promising conservation tool to help remedy human-caused ecosystem changes. However, the growth and spread of a reintroduced population is a spatiotemporal process that is driven by a suite of factors, such as habitat change, human activity, and prey availability. Sea otters (Enhydra lutris) are apex predators of nearshore marine ecosystems that had declined nearly to extinction across much of their range by the early 20th century. In Southeast Alaska, which is comprised of a diverse matrix of nearshore habitat and managed areas, reintroduction of 413 individuals in the late 1960s initiated the growth and spread of a population that now exceeds 25,000. Methods Periodic aerial surveys in the region provide a time series of spatially-explicit data to investigate factors influencing this successful and ongoing recovery. We integrated an ecological diffusion model that accounted for spatially-variable motility and density-dependent population growth, as well as multiple population epicenters, into a Bayesian hierarchical framework to help understand the factors influencing the success of this recovery. Results Our results indicated that sea otters exhibited higher residence time as well as greater equilibrium abundance in Glacier Bay, a protected area, and in areas where there is limited or no commercial fishing. Asymptotic spread rates suggested sea otters colonized Southeast Alaska at rates of 1–8 km/yr with lower rates occurring in areas correlated with higher residence time, which primarily included areas near shore and closed to commercial fishing. Further, we found that the intrinsic growth rate of sea otters may be higher than previous estimates suggested. Conclusions This study shows how predator recolonization can occur from multiple population epicenters. Additionally, our results suggest spatial heterogeneity in the physical environment as well as human activity and management can influence recolonization processes, both in terms of movement (or motility) and density dependence.

The current architecture practice is shifting towards Green Solutions designed, produced, and operated domestically in a self-sufficient decentralized fashion, following the UN sustainability goals. The current study proposes 3D-printed bioreceptive tiles for the passive immobilization of multi-scale-length algal strains from a mixed culture of Mougeotia sp., Oedogonium foveolatum, Zygnema sp., Microspora sp., Spirogyra sp., and Pyrocystis fusiformis. This customized passive immobilization of the chosen algal strains is designed to achieve bioremediation-integrated solutions in architectural applications. The two bioreceptive tiles following the reaction-diffusion, activator-inhibitor Grier–Meinhardt model have different patterns: P1: Polar periodic, and P2: Strip labyrinth, with niche sizes of 3000 µm and 500 µm, respectively. The results revealed that P2 has a higher immobilization capacity for the various strains, particularly Microspora sp., achieving a growth rate 1.65% higher than its activated culture density compared to a 1.08% growth rate on P1, followed by P. fusiformis with 1.53% on P2 and 1.3% on P1. These results prove the correspondence between the scale and morphology of the strip labyrinth pattern of P2 and the unbranched filamentous and fusiform large unicellular morphology of the immobilized algal strains cells, with an optimum ratio of 0.05% to 0.75% niche to the cell scale. Furthermore, The Mixed Culture method offered an intertwining net that facilitated the entrapment of the various algal strains into the bioreceptive tile.

We present a reaction-advection-diffusion (RAD) model for (low temperature) uranium oxidation in a water-vapour environment, where both OH − and H • are diffusing. In this model an intermediate UH 3 phase sits between the bulk U metal and a protective surface UO 2 layer. This surface oxide layer only remains adhered up to a maximum depth Δ adh ∗ before spallation occurs leading to significantly increased diffusive transport across the spalled layer. Under these conditions, this mechanistic model is shown to support both a parabolic (∝ t) oxide growth up to the point of spallation, before smoothly transitioning to a linear (∝ t) oxidation solution at later times. In the late-stage linear regime, a UO 2 − UH 3 interface propagates into the bulk metal at a constant velocity of D 1 3 ∗ C ∗ 2 Δ adh ∗ N 2 ∗ ; D 1 3 ∗ being the diffusion coefficient of OH − in UO 2 and C ∗ / N 2 ∗ the peak relative concentration of OH − to U. This model predicts that the intermediate hydride layer approaches a constant thickness in the linear regime, with a UH 3 − U interface propagating into the bulk metal at the same velocity. The length scale of this emergent hydride layer is shown to be most sensitive to the diffusivity of OH − in UH 3 and the corresponding reaction rate constant. Plausible parameter values are shown to lead to hydride layers < 10 nm for room temperature oxidation in a vapour pressure of 20 Torr (Δ adh ∗ = 50 nm) consistent with recent atom-probe tomography results. • A single mechanistic model that captures both parabolic and linear stages of oxide growth. • The model predicts a thin emergent hydride layer during oxidation. • The hydride layer is predicted to grow in the initial (parabolic) stage before equilibrating in the late (linear) stage. • The sensitivity of the hydride layer (size) is quantified over a range of the physical parameters. • Approximate bounds on currently unknown physical parameters are given for consistency with observation. [ABSTRACT FROM AUTHOR]